{"id":4160,"date":"2018-12-11T13:59:17","date_gmt":"2018-12-11T18:59:17","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-quadratic-functions-using-properties\/"},"modified":"2018-12-11T13:59:17","modified_gmt":"2018-12-11T18:59:17","slug":"graph-quadratic-functions-using-properties","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-quadratic-functions-using-properties\/","title":{"raw":"Graph Quadratic Functions Using Properties","rendered":"Graph Quadratic Functions Using Properties"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Recognize the graph of a quadratic function<\/li><li>Find the axis of symmetry and vertex of a parabola<\/li><li>Find the intercepts of a parabola<\/li><li>Graph quadratic functions using properties<\/li><li>Solve maximum and minimum applications<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1169147874135\" class=\"be-prepared\"><p id=\"fs-id1169147860526\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1169147844160\" type=\"1\"><li>Graph the function \\(f\\left(x\\right)={x}^{2}\\) by plotting points.<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/da9d6ce0-a078-4ca2-97af-8cb374f040f5#fs-id1167836683384\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\(2{x}^{2}+3x-2=0.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Evaluate \\(-\\frac{b}{2a}\\) when <em data-effect=\"italics\">a<\/em> = 3 and <em data-effect=\"italics\">b<\/em> = \u22126.<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147808468\"><h3 data-type=\"title\">Recognize the Graph of a Quadratic Function<\/h3><p id=\"fs-id1169147876025\">Previously we very briefly looked at the function \\(f\\left(x\\right)={x}^{2}\\), which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form \\(f\\left(x\\right)=a{x}^{2}+bx+c\\) if \\(a\\ne 0.\\) We call this kind of function a quadratic function.<\/p><div data-type=\"note\" id=\"fs-id1169147804510\"><div data-type=\"title\">Quadratic Function<\/div><p id=\"fs-id1169147845348\">A <span data-type=\"term\">quadratic function<\/span>, where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and \\(a\\ne 0,\\) is a function of the form<\/p><div data-type=\"equation\" id=\"fs-id1169147821646\" class=\"unnumbered\" data-label=\"\">\\(f\\left(x\\right)=a{x}^{2}+bx+c\\)<\/div><\/div><p id=\"fs-id1169145731259\">We graphed the quadratic function \\(f\\left(x\\right)={x}^{2}\\) by plotting points.<\/p><span data-type=\"media\" id=\"fs-id1169147876789\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 2 to 6. The parabola has a vertex at (0, 0) and also passes through the points (-2, 4), (-1, 1), (1, 1), and (2, 4). To the right of the graph is a table of values with 3 columns. The first row is a header row and labels each column, \u201cx\u201d, \u201cf of x equals x squared\u201d, and \u201cthe order pair x, f of x.\u201d In row 2, x equals negative 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair negative 3, 9. In row 3, x equals negative 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair negative 2, 4. In row 4, x equals negative 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair negative 1, 1. In row 5, x equals 0, f of x equals x squared is 0 and the ordered pair x, f of x is the ordered pair 0, 0. In row 6, x equals 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair 1, 1. In row 7, x equals 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair 2, 4. In row 8, x equals 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair 3, 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 2 to 6. The parabola has a vertex at (0, 0) and also passes through the points (-2, 4), (-1, 1), (1, 1), and (2, 4). To the right of the graph is a table of values with 3 columns. The first row is a header row and labels each column, \u201cx\u201d, \u201cf of x equals x squared\u201d, and \u201cthe order pair x, f of x.\u201d In row 2, x equals negative 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair negative 3, 9. In row 3, x equals negative 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair negative 2, 4. In row 4, x equals negative 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair negative 1, 1. In row 5, x equals 0, f of x equals x squared is 0 and the ordered pair x, f of x is the ordered pair 0, 0. In row 6, x equals 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair 1, 1. In row 7, x equals 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair 2, 4. In row 8, x equals 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair 3, 9.\"><\/span><p id=\"fs-id1169147715853\">Every quadratic function has a graph that looks like this. We call this figure a <span data-type=\"term\">parabola<\/span>.<\/p><p id=\"fs-id1169147809055\">Let\u2019s practice graphing a parabola by plotting a few points.<\/p><div data-type=\"example\" id=\"fs-id1169147774250\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147876680\"><div data-type=\"problem\" id=\"fs-id1169147835768\"><p id=\"fs-id1169147803921\">Graph \\(f\\left(x\\right)={x}^{2}-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147820867\"><p id=\"fs-id1169147721479\">We will graph the function by plotting points.<\/p><table id=\"fs-id1169147808038\" class=\"unnumbered unstyled\" summary=\"Choose integer values for x, substitute them into the equation and simplify to find f of x. Record the values of the ordered pairs in the chart. The table of values for the function f of x equals x squared minus 1 has 2 columns. The first column is labeled x and the second column is labeled f of x. When x equals 0, f of x equals negative 1. When x equals 1, f of x equals 0. When x equals negative 1, f of x equals 0. When x equals 2, f of x equals 3. When x equals negative 2, f of x equals 3. Plot the points, and then connect them with a smooth curve. The result will be the graph of the function f of x equals negative 1. The figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points identified in the table are plotted.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Choose integer values for <em data-effect=\"italics\">x<\/em>,<div data-type=\"newline\"><br><\/div>substitute them into the equation<div data-type=\"newline\"><br><\/div>and simplify to find \\(f\\left(x\\right)\\).<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Record the values of the ordered pairs in the chart.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147856021\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Plot the points, and then connect<div data-type=\"newline\"><br><\/div>them with a smooth curve. The<div data-type=\"newline\"><br><\/div>result will be the graph of the<div data-type=\"newline\"><br><\/div>function \\(f\\left(x\\right)={x}^{2}-1\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147874347\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147906436\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147940171\"><div data-type=\"problem\" id=\"fs-id1169147770846\"><p id=\"fs-id1169147855992\">Graph \\(f\\left(x\\right)=\\text{\u2212}{x}^{2}.\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147863559\"><span data-type=\"media\" id=\"fs-id1169147959678\" data-alt=\"This figure shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 0).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148037752\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147817063\"><div data-type=\"problem\" id=\"fs-id1169147828437\"><p id=\"fs-id1169147940166\">Graph \\(f\\left(x\\right)={x}^{2}+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147855659\"><span data-type=\"media\" id=\"fs-id1169147949267\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, \u22121).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, \u22121).\"><\/span><\/div><\/div><\/div><p id=\"fs-id1169147825486\">All graphs of quadratic functions of the form <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> are parabolas that open upward or downward. See <a href=\"#CNX_IntAlg_Figure_09_06_003\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_003\"><span data-type=\"media\" id=\"fs-id1169147980073\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is x squared plus 4 x plus 3. The leading coefficient, a, is greater than 0, so this parabola opens upward.The graph on the right shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is negative x squared plus 4 x plus 3. The leading coefficient, a, is less than 0, so this parabola opens downward.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is x squared plus 4 x plus 3. The leading coefficient, a, is greater than 0, so this parabola opens upward.The graph on the right shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is negative x squared plus 4 x plus 3. The leading coefficient, a, is less than 0, so this parabola opens downward.\"><\/span><\/div><p id=\"fs-id1169145666278\">Notice that the only difference in the two functions is the negative sign before the quadratic term (<em data-effect=\"italics\">x<\/em><sup>2<\/sup> in the equation of the graph in <a href=\"#CNX_IntAlg_Figure_09_06_003\" class=\"autogenerated-content\">(Figure)<\/a>). When the quadratic term, is positive, the parabola opens upward, and when the quadratic term is negative, the parabola opens downward.<\/p><div data-type=\"note\" id=\"fs-id1169148230545\"><div data-type=\"title\">Parabola Orientation<\/div><p id=\"fs-id1169147821411\">For the graph of the quadratic function <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em>, if<\/p><span data-type=\"media\" id=\"fs-id1169147854266\" data-alt=\"This images shows a bulleted list. The first bullet notes that, if a is greater than 0, then the parabola opens upward and shows an image of an upward-opening parabola. The second bullet notes that, if a is less than 0, then the parabola opens downward and shows an image of a downward-opening parabola.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This images shows a bulleted list. The first bullet notes that, if a is greater than 0, then the parabola opens upward and shows an image of an upward-opening parabola. The second bullet notes that, if a is less than 0, then the parabola opens downward and shows an image of a downward-opening parabola.\"><\/span><\/div><div data-type=\"example\" id=\"fs-id1169147810565\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148229708\"><div data-type=\"problem\" id=\"fs-id1169147800031\"><p id=\"fs-id1169147949688\">Determine whether each parabola opens upward or downward:<\/p><p id=\"fs-id1169145664867\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-3{x}^{2}+2x-4\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=6{x}^{2}+7x-9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145666225\"><p id=\"fs-id1169148233111\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169147854650\" class=\"unnumbered unstyled\" summary=\"The standard form of a quadratic equation is f of x equals a x squared plus b x plus c. This function is f of x equals negative 3 x squared plus 2 x minus 4. Find the value of a, the coefficient of x squared. For this function, a equals negative 3. Since a is negative, the parabola will open downward.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the value of \u201c<em data-effect=\"italics\">a<\/em>\u201d.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148198055\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">Since the \u201c<em data-effect=\"italics\">a<\/em>\u201d is negative, the parabola will open downward.<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169147819480\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169145662827\" class=\"unnumbered unstyled\" summary=\"The standard form of a quadratic equation is f of x equals a x squared plus b x plus c. This function is f of x equals negative 6 x squared plus 7 x minus 9. Find the value of a, the coefficient of x squared. For this function, a equals 6. Since a is positive, the parabola will open upward.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the value of \u201c<em data-effect=\"italics\">a<\/em>\u201d.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147959131\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">Since the \u201c<em data-effect=\"italics\">a<\/em>\u201d is positive, the parabola will open upward.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147837972\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147837975\"><div data-type=\"problem\" id=\"fs-id1169147823322\"><p id=\"fs-id1169147823324\">Determine whether the graph of each function is a parabola that opens upward or downward:<\/p><p id=\"fs-id1169147854838\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=2{x}^{2}+5x-2\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=-3{x}^{2}-4x+7.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145663138\"><p id=\"fs-id1169147838982\"><span class=\"token\">\u24d0<\/span> up; <span class=\"token\">\u24d1<\/span> down<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147866845\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147836676\"><div data-type=\"problem\" id=\"fs-id1169147836678\"><p id=\"fs-id1169147980356\">Determine whether the graph of each function is a parabola that opens upward or downward:<\/p><p id=\"fs-id1169147980360\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-2{x}^{2}-2x-3\\)<span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=5{x}^{2}-2x-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148080822\"><p id=\"fs-id1169147982492\"><span class=\"token\">\u24d0<\/span> down; <span class=\"token\">\u24d1<\/span> up<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147851221\"><h3 data-type=\"title\">Find the Axis of Symmetry and Vertex of a Parabola<\/h3><p id=\"fs-id1169147906619\">Look again at <a href=\"#CNX_IntAlg_Figure_09_06_003\" class=\"autogenerated-content\">(Figure)<\/a>. Do you see that we could fold each parabola in half and then one side would lie on top of the other? The \u2018fold line\u2019 is a line of symmetry. We call it the <span data-type=\"term\">axis of symmetry<\/span> of the parabola.<\/p><p id=\"fs-id1169145661242\">We show the same two graphs again with the axis of symmetry. See <a href=\"#CNX_IntAlg_Figure_09_06_007\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_007\"><span data-type=\"media\" id=\"fs-id1169147982414\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The equation of this parabola is x squared plus 4 x plus 3. The vertical line passes through the point (negative 2, 0) and has the equation x equals negative 2. The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The equation of this parabola is negative x squared plus 4 x plus 3. The vertical line passes through the point (2, 0) and has the equation x equals 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The equation of this parabola is x squared plus 4 x plus 3. The vertical line passes through the point (negative 2, 0) and has the equation x equals negative 2. The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The equation of this parabola is negative x squared plus 4 x plus 3. The vertical line passes through the point (2, 0) and has the equation x equals 2.\"><\/span><\/div><p id=\"fs-id1169147979766\">The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> is \\(x=-\\frac{b}{2a}.\\)<\/p><p id=\"fs-id1169147982212\">So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula \\(x=-\\frac{b}{2a}.\\)<\/p><span data-type=\"media\" id=\"fs-id1169147750504\" data-alt=\"Compare the function f of x equals x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times 1. The axis of symmetry equals negative 2. Next, compare the function f of x equals negative x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times negative 1. The axis of symmetry equals 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Compare the function f of x equals x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times 1. The axis of symmetry equals negative 2. Next, compare the function f of x equals negative x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times negative 1. The axis of symmetry equals 2.\"><\/span><p id=\"fs-id1169145670075\">Notice that these are the equations of the dashed blue lines on the graphs.<\/p><p id=\"fs-id1169147819842\">The point on the parabola that is the lowest (parabola opens up), or the highest (parabola opens down), lies on the axis of symmetry. This point is called the <span data-type=\"term\">vertex<\/span> of the parabola.<\/p><p id=\"fs-id1169145664752\">We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its<\/p><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">x<\/em>-coordinate is \\(-\\frac{b}{2a}.\\) To find the <em data-effect=\"italics\">y<\/em>-coordinate of the vertex we substitute the value of the <em data-effect=\"italics\">x<\/em>-coordinate into the quadratic function.<span data-type=\"media\" id=\"fs-id1169147940220\" data-alt=\"For the function f of x equals x squared plus 4 x plus 3, the axis of symmetry is x equals negative 2. The vertex is the point on the parabola with x-coordinate negative 2. Substitute x equals negative 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals the square of negative 2 plus 4 times negative 2 plus 3, so f of x equals negative 1. The vertex is the point (negative 2, negative 1). For the function f of x equals negative x squared plus 4 x plus 3, the axis of symmetry is x equals 2. The vertex is the point on the parabola with x-coordinate 2. Substitute x equals 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals 2 squared plus 4 times 2 plus 3, so f of x equals 7. The vertex is the point (2, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"For the function f of x equals x squared plus 4 x plus 3, the axis of symmetry is x equals negative 2. The vertex is the point on the parabola with x-coordinate negative 2. Substitute x equals negative 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals the square of negative 2 plus 4 times negative 2 plus 3, so f of x equals negative 1. The vertex is the point (negative 2, negative 1). For the function f of x equals negative x squared plus 4 x plus 3, the axis of symmetry is x equals 2. The vertex is the point on the parabola with x-coordinate 2. Substitute x equals 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals 2 squared plus 4 times 2 plus 3, so f of x equals 7. The vertex is the point (2, 7).\"><\/span><div data-type=\"note\" id=\"fs-id1169147962340\"><div data-type=\"title\">Axis of Symmetry and Vertex of a Parabola<\/div><p id=\"fs-id1169148232095\">The graph of the function <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> is a parabola where:<\/p><ul id=\"fs-id1169145732569\" data-bullet-style=\"bullet\"><li>the axis of symmetry is the vertical line \\(x=-\\frac{b}{2a}.\\)<\/li><li>the vertex is a point on the axis of symmetry, so its <em data-effect=\"italics\">x<\/em>-coordinate is \\(-\\frac{b}{2a}.\\)<\/li><li>the <em data-effect=\"italics\">y<\/em>-coordinate of the vertex is found by substituting \\(x=-\\frac{b}{2a}\\) into the quadratic equation.<\/li><\/ul><\/div><div data-type=\"example\" id=\"fs-id1169147906840\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147816860\"><div data-type=\"problem\" id=\"fs-id1169147816863\"><p id=\"fs-id1169147816865\">For the graph of \\(f\\left(x\\right)=3{x}^{2}-6x+2\\) find:<\/p><p id=\"fs-id1169148054384\"><span class=\"token\">\u24d0<\/span> the axis of symmetry <span class=\"token\">\u24d1<\/span> the vertex.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145728808\"><p id=\"fs-id1169145728810\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169147960277\" class=\"unnumbered unstyled can-break\" summary=\"Compare the function f of x equals 3 x squared minus 6 x plus 2 to the standard form of a quadratic function f of x equals a x squared plus b x plus c. The axis of symmetry is the vertical line x equals negative b divided by the product 2 a. Substitute the values a equals 3 and b equals negative 6 into the equation of the line of symmetry. X equals negative 6 divided by the product 2 times 3. Simplify. The axis of symmetry is the line x equals 1.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147987890\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The axis of symmetry is the vertical line<div data-type=\"newline\"><br><\/div>\\(x=-\\frac{b}{2a}\\).<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the values of \\(a,b\\) into the<div data-type=\"newline\"><br><\/div>equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147816990\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147846800\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The axis of symmetry is the line \\(x=1\\).<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169147980794\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169147982390\" class=\"unnumbered unstyled\" summary=\"Write the function f of x equals 3 x squared minus 6 x plus 2. The vertex is a point on the line of symmetry, so its x-coordinate will be x equals 1. Find f of 1. F of 1 equals 3 times 1 squared minus 6 times 1 plus 2. Simplify f of 1 equals 3 times 1 minus 6 plus 2, which equals negative 1. This result is the y-coordinate. The vertex is the point (1, negative 1).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147981078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is a point on the line of<div data-type=\"newline\"><br><\/div>symmetry, so its <em data-effect=\"italics\">x<\/em>-coordinate will be<div data-type=\"newline\"><br><\/div>\\(x=1\\).<div data-type=\"newline\"><br><\/div>Find \\(f\\left(1\\right)\\).<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145639635\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147978649\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The result is the <em data-effect=\"italics\">y<\/em>-coordinate.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145660232\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(1,-1\\right)\\).<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147880094\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147880099\"><div data-type=\"problem\" id=\"fs-id1169148037745\"><p id=\"fs-id1169148037747\">For the graph of \\(f\\left(x\\right)=2{x}^{2}-8x+1\\) find:<\/p><p id=\"fs-id1169147950358\"><span class=\"token\">\u24d0<\/span> the axis of symmetry <span class=\"token\">\u24d1<\/span> the vertex.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147965759\"><p id=\"fs-id1169147965761\"><span class=\"token\">\u24d0<\/span>\\(x=2;\\)<span class=\"token\">\u24d1<\/span>\\(\\left(2,-7\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145640275\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147846046\"><div data-type=\"problem\" id=\"fs-id1169147846049\"><p id=\"fs-id1169147797170\">For the graph of \\(f\\left(x\\right)=2{x}^{2}-4x-3\\) find:<\/p><p id=\"fs-id1169145670510\"><span class=\"token\">\u24d0<\/span> the axis of symmetry <span class=\"token\">\u24d1<\/span> the vertex.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147841589\"><p id=\"fs-id1169147841591\"><span class=\"token\">\u24d0<\/span>\\(x=1;\\)<span class=\"token\">\u24d1<\/span>\\(\\left(1,-5\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147856138\"><h3 data-type=\"title\">Find the Intercepts of a Parabola<\/h3><p id=\"fs-id1169147807792\">When we graphed linear equations, we often used the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.<\/p><p id=\"fs-id1169147870671\">Remember, at the <em data-effect=\"italics\">y<\/em>-intercept the value of <em data-effect=\"italics\">x<\/em> is zero. So to find the <em data-effect=\"italics\">y<\/em>-intercept, we substitute <em data-effect=\"italics\">x<\/em> = 0 into the function.<\/p><p id=\"fs-id1169147962725\">Let\u2019s find the <em data-effect=\"italics\">y<\/em>-intercepts of the two parabolas shown in <a href=\"#CNX_IntAlg_Figure_09_06_012\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_012\"><span data-type=\"media\" id=\"fs-id1169147905707\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The vertical line is an axis of symmetry for the parabola, and passes through the point (negative 2, 0). It has the equation x equals negative 2. The equation of this parabola is x squared plus 4 x plus 3. When x equals 0, f of 0 equals 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3). The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The vertical line is an axis of symmetry for the parabola and passes through the point (2, 0). It has the equation x equals 2. The equation of this parabola is negative x squared plus 4 x plus 3. When x equals 0, f of 0 equals negative 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The vertical line is an axis of symmetry for the parabola, and passes through the point (negative 2, 0). It has the equation x equals negative 2. The equation of this parabola is x squared plus 4 x plus 3. When x equals 0, f of 0 equals 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3). The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The vertical line is an axis of symmetry for the parabola and passes through the point (2, 0). It has the equation x equals 2. The equation of this parabola is negative x squared plus 4 x plus 3. When x equals 0, f of 0 equals negative 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3).\"><\/span><\/div><p id=\"fs-id1169145662932\">An <em data-effect=\"italics\">x<\/em>-intercept results when the value of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) is zero. To find an <em data-effect=\"italics\">x<\/em>-intercept, we let <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 0. In other words, we will need to solve the equation 0 = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> for <em data-effect=\"italics\">x<\/em>.<\/p><div data-type=\"equation\" id=\"fs-id1169147765890\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =\\hfill &amp; a{x}^{2}+bx+c\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; a{x}^{2}+bx+c\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169148231382\">Solving quadratic equations like this is exactly what we have done earlier in this chapter!<\/p><p id=\"fs-id1169145670950\">We can now find the <em data-effect=\"italics\">x<\/em>-intercepts of the two parabolas we looked at. First we will find the <em data-effect=\"italics\">x<\/em>-intercepts of the parabola whose function is <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 3.<\/p><table id=\"fs-id1169147982318\" class=\"unnumbered unstyled\" summary=\"F of x equals x squared plus 4 x plus 3. Let f of x equal 0. 0 equals x squared plus 4 x plus 3. Factor. 0 equals the product of x plus 1 and x plus 3. Use the Zero Product Property. Then x plus 1 equals 0 or x plus 3 equals 0. Solve. X equals negative 1 or x equals negative 3. The x-intercepts are the points (negative 1, 0) and (negative 3, 0).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148233010\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147966001\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148081220\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147850531\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667133\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercepts are \\(\\left(-1,0\\right)\\) and \\(\\left(-3,0\\right)\\).<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169147982116\">Now we will find the <em data-effect=\"italics\">x<\/em>-intercepts of the parabola whose function is <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">\u2212x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 3.<\/p><table id=\"fs-id1169147869012\" class=\"unnumbered unstyled can-break\" summary=\"F of x equals negative x squared plus 4 x plus 3. Let f of x equal 0. 0 equals negative x squared plus 4 x plus 3. The quadratic does not factor, so we use the Quadratic Formula. X equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. a equals negative 1, b equals 4, and c equals 3. X equals the quotient negative 4 plus or minus the square root of the difference 4 squared minus the product 4 times negative 1 times 3 divided by the product 2 times negative 1. Simplify x equals the quotient negative 4 plus or minus square root 28 divided by negative 2. X equals the quotient negative 4 plus or minus square root 28 divided by negative 2. X equals the quotient negative 4 plus or minus 2 square root 7 divided by negative 2. X equals the quotient of the product negative 2 times the expression 2 plus or minus square root 7 divided by negative 2. X equals 2 plus or minus square root 7. The intercepts are the points (2 plus square root 7, 0) and (2 minus square root 7, 0).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666010\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147950364\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">This quadratic does not factor, so<div data-type=\"newline\"><br><\/div>we use the Quadratic Formula.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147848633\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(a=-1,b=4,c=3\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147851318\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147849869\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667570\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147962229\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145731902\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercepts are \\(\\left(2+\\sqrt{7},0\\right)\\) and<div data-type=\"newline\"><br><\/div>\\(\\left(2-\\sqrt{7},0\\right)\\).<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169145640278\">We will use the decimal approximations of the <em data-effect=\"italics\">x<\/em>-intercepts, so that we can locate these points on the graph,<\/p><div data-type=\"equation\" id=\"fs-id1169145640286\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\left(2+\\sqrt{7},0\\right)\\approx \\left(4.6,0\\right)\\hfill &amp; &amp; &amp; &amp; &amp; \\left(2-\\sqrt{7},0\\right)\\approx \\left(-0.6,0\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169147806629\">Do these results agree with our graphs? See <a href=\"#CNX_IntAlg_Figure_09_06_015\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_015\"><span data-type=\"media\" id=\"fs-id1169147828136\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows the upward-opening parabola defined by the function f of x equals x squared plus 4 x plus 3 and a dashed vertical line, x equals negative 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept is (0, 3) and the x-intercepts are (negative 1, 0) and (negative 3, 0). The graph on the right shows the downward-opening parabola defined by the function f of x equals negative x squared plus 4 x plus 3 and a dashed vertical line, x equals 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7). The y-intercept is (0, 3) and the x-intercepts are (2 plus square root 7, 0), approximately (4.6, 0) and (2 minus square root, 0), approximately (negative 0.6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows the upward-opening parabola defined by the function f of x equals x squared plus 4 x plus 3 and a dashed vertical line, x equals negative 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept is (0, 3) and the x-intercepts are (negative 1, 0) and (negative 3, 0). The graph on the right shows the downward-opening parabola defined by the function f of x equals negative x squared plus 4 x plus 3 and a dashed vertical line, x equals 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7). The y-intercept is (0, 3) and the x-intercepts are (2 plus square root 7, 0), approximately (4.6, 0) and (2 minus square root, 0), approximately (negative 0.6, 0).\"><\/span><\/div><div data-type=\"note\" id=\"fs-id1169147869876\"><div data-type=\"title\">Find the Intercepts of a Parabola<\/div><p>To find the intercepts of a parabola whose function is \\(f\\left(x\\right)=a{x}^{2}+bx+c:\\)<\/p><div data-type=\"equation\" id=\"fs-id1169147847182\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\hfill \\mathbit{\\text{y}}\\mathbf{\\text{-intercept}}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\mathbit{\\text{x}}\\mathbf{\\text{-intercepts}}\\hfill \\\\ \\hfill \\text{Let}\\phantom{\\rule{0.2em}{0ex}}x=0\\phantom{\\rule{0.2em}{0ex}}\\text{and solve for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{Let}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)=0\\phantom{\\rule{0.2em}{0ex}}\\text{and solve for}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill \\end{array}\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1169145732214\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145732216\"><div data-type=\"problem\" id=\"fs-id1169145732218\"><p id=\"fs-id1169145732220\">Find the intercepts of the parabola whose function is \\(f\\left(x\\right)={x}^{2}-2x-8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147940484\"><table id=\"fs-id1169147940487\" class=\"unnumbered unstyled can-break\" summary=\"To find the y-intercept, let x equal 0 and solve for f of x. f of x equals x squared minus 2 x minus 8. F of 0 equals 0 squared minus 2 times 0 minus 8 which simplifies to yield f of 0 equals negative 8. When x equals 0, then f of 0 equals negative 8. The y-intercept is the point (0, negative 8). To find the x-intercept, let f of x equal 0 and solve for x. f of x equals x squared minus 2 x minus 8. 0 equals x squared minus 2 x minus 8. Solve by factoring. 0 equals the product of x minus 4 and x plus 2. 0 equals x minus 4 or 0 equals x plus 2. So x equals 4 or x equals negative 2. When f of x equals 0, then x equals 4 or x equals negative 2. The x-intercepts are the points (4, 0) and (negative 2, 0).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the <strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-<\/strong>intercept, let \\(x=0\\) and<div data-type=\"newline\"><br><\/div>solve for \\(f\\left(x\\right)\\).<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666243\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147981951\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983073\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">When \\(x=0\\), then \\(f\\left(0\\right)=-8\\).<div data-type=\"newline\"><br><\/div>The <em data-effect=\"italics\">y<\/em><strong data-effect=\"bold\">-<\/strong>intercept is the point \\(\\left(0,-8\\right)\\).<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">To find the <strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-<\/strong>intercept, let \\(f\\left(x\\right)=0\\) and<div data-type=\"newline\"><br><\/div>solve for \\(x\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147835691\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145662079\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670015\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147979488\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169145639575\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">When \\(f\\left(x\\right)=0\\), then \\(x=4\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.2em}{0ex}}x=-2\\).<div data-type=\"newline\"><br><\/div>The <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">-<\/strong>intercepts are the points \\(\\left(4,0\\right)\\) and<div data-type=\"newline\"><br><\/div>\\(\\left(-2,0\\right)\\).<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147959245\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147959250\"><div data-type=\"problem\" id=\"fs-id1169147959252\"><p id=\"fs-id1169147959254\">Find the intercepts of the parabola whose function is \\(f\\left(x\\right)={x}^{2}+2x-8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147826486\"><p id=\"fs-id1169147868841\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,-8\\right)\\)<em data-effect=\"italics\">x<\/em>-intercepts \\(\\left(-4,0\\right),\\left(2,0\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145663264\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145663268\"><div data-type=\"problem\" id=\"fs-id1169145663270\"><p id=\"fs-id1169145663272\">Find the intercepts of the parabola whose function is \\(f\\left(x\\right)={x}^{2}-4x-12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145661976\"><p id=\"fs-id1169145661978\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,-12\\right)\\)<em data-effect=\"italics\">x<\/em>-intercepts \\(\\left(-2,0\\right),\\left(6,0\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169145667660\">In this chapter, we have been solving quadratic equations of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. We solved for <em data-effect=\"italics\">x<\/em> and the results were the solutions to the equation.<\/p><p id=\"fs-id1169145664380\">We are now looking at quadratic functions of the form <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em>. The graphs of these functions are parabolas. The <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">-<\/strong>intercepts of the parabolas occur where <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 0.<\/p><p id=\"fs-id1169145731793\">For example:<\/p><div data-type=\"equation\" id=\"fs-id1169145731796\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccccc}\\hfill \\mathbf{\\text{Quadratic equation}}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\mathbf{\\text{Quadratic function}}\\hfill \\\\ \\hfill \\begin{array}{}\\\\ \\hfill {x}^{2}-2x-15&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill \\left(x-5\\right)\\left(x+3\\right)&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x-5=0\\phantom{\\rule{0.8em}{0ex}}x+3&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x=5\\phantom{\\rule{2.3em}{0ex}}x&amp; =\\hfill &amp; -3\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; \\hfill \\begin{array}{c}\\hfill \\text{Let}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)=0.\\hfill \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =\\hfill &amp; {x}^{2}-2x-15\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; {x}^{2}-2x-15\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; \\left(x-5\\right)\\left(x+3\\right)\\hfill \\\\ \\hfill x-5&amp; =\\hfill &amp; 0\\phantom{\\rule{0.8em}{0ex}}x+3=0\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 5\\phantom{\\rule{2.3em}{0ex}}x=-3\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(5,0\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\left(-3,0\\right)\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}x\\text{-intercepts}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169148231607\">The solutions of the quadratic function are the <em data-effect=\"italics\">x<\/em> values of the <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">-<\/strong>intercepts.<\/p><p id=\"fs-id1169145732301\">Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the functions give the <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">-<\/strong>intercepts of the graphs, the number of <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">-<\/strong>intercepts is the same as the number of solutions.<\/p><p id=\"fs-id1169147833049\">Previously, we used the <span data-type=\"term\" class=\"no-emphasis\">discriminant<\/span> to determine the number of solutions of a quadratic function of the form \\(a{x}^{2}+bx+c=0.\\) Now we can use the discriminant to tell us how many <em data-effect=\"italics\">x<\/em>-intercepts there are on the graph.<\/p><span data-type=\"media\" id=\"fs-id1169145639964\" data-alt=\"This image shows three graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies below the x-axis and the parabola crosses the x-axis at two different points. If b squared minus 4 a c is greater than 0, then the quadratic equation a x squared plus b x plus c equals 0 has two solutions, and the graph of the parabola has 2 x-intercepts. The graph in the middle shows a downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies on the x-axis, the only point of intersection between the parabola and the x-axis. If b squared minus 4 a c equals 0, then the quadratic equation a x squared plus b x plus c equals 0 has one solution, and the graph of the parabola has 1 x-intercept. The graph on the right shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies above the x-axis and the parabola does not cross the x-axis. If b squared minus 4 a c is less than 0, then the quadratic equation a x squared plus b x plus c equals 0 has no solutions, and the graph of the parabola has no x-intercepts.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows three graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies below the x-axis and the parabola crosses the x-axis at two different points. If b squared minus 4 a c is greater than 0, then the quadratic equation a x squared plus b x plus c equals 0 has two solutions, and the graph of the parabola has 2 x-intercepts. The graph in the middle shows a downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies on the x-axis, the only point of intersection between the parabola and the x-axis. If b squared minus 4 a c equals 0, then the quadratic equation a x squared plus b x plus c equals 0 has one solution, and the graph of the parabola has 1 x-intercept. The graph on the right shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies above the x-axis and the parabola does not cross the x-axis. If b squared minus 4 a c is less than 0, then the quadratic equation a x squared plus b x plus c equals 0 has no solutions, and the graph of the parabola has no x-intercepts.\"><\/span><p id=\"fs-id1169147981318\">Before you to find the values of the <em data-effect=\"italics\">x<\/em>-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.<\/p><div data-type=\"example\" id=\"fs-id1169147981328\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147981330\"><div data-type=\"problem\" id=\"fs-id1169147981332\"><p id=\"fs-id1169147981334\">Find the intercepts of the parabola for the function \\(f\\left(x\\right)=5{x}^{2}+x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148233118\"><table id=\"fs-id1169148233121\" class=\"unnumbered unstyled\" summary=\"F of x equals 5 x squared plus x plus 4. To find the y-intercept, let x equal 0 and solve for f of x. f of 0 equals 5 times the square of 0 plus 0 plus 4, so f of 0 equals 4. When x equals 0, f of 0 equals 4. The y-intercept is the point (0, 4). To find the x-intercept, let f of x equal 0 and solve for x. f of x equals 5 x squared plus x plus 4. 0 equals 5 x squared plus x plus 4. Find the value of the discriminant to predict the number of solutions which is also the number of x-intercepts. B squared minus 4 a c equals 1 squared minus 4 times 5 times 4. This simplifies to 1 minus 80, or negative 79. Since the value of the discriminant is negative, there is no real solution to the equation. There are no x-intercepts.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147853761\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">y<\/em>-intercept, let \\(x=0\\) and<div data-type=\"newline\"><br><\/div>solve for \\(f\\left(x\\right)\\).<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148233718\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667142\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">When \\(x=0\\), then \\(f\\left(0\\right)=4\\).<div data-type=\"newline\"><br><\/div>The <em data-effect=\"italics\">y<\/em>-intercept is the point \\(\\left(0,4\\right)\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">x<\/em>-intercept, let \\(f\\left(x\\right)=0\\) and<div data-type=\"newline\"><br><\/div>solve for \\(x\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730601\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147819502\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the value of the discriminant to<div data-type=\"newline\"><br><\/div>predict the number of solutions which is<div data-type=\"newline\"><br><\/div>also the number of <em data-effect=\"italics\">x<\/em>-intercepts.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{c}\\hfill {b}^{2}-4ac\\hfill \\\\ \\hfill {1}^{2}-4\u00b75\u00b74\\hfill \\\\ \\hfill 1-80\\hfill \\\\ \\hfill -79\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">Since the value of the discriminant is<div data-type=\"newline\"><br><\/div>negative, there is no real solution to the<div data-type=\"newline\"><br><\/div>equation.<div data-type=\"newline\"><br><\/div>There are no <em data-effect=\"italics\">x<\/em>-intercepts.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147875183\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147875188\"><div data-type=\"problem\" id=\"fs-id1169147875190\"><p id=\"fs-id1169147875192\">Find the intercepts of the parabola whose function is \\(f\\left(x\\right)=3{x}^{2}+4x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145664178\"><p id=\"fs-id1169145664180\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,4\\right)\\) no <em data-effect=\"italics\">x<\/em>-intercept<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145661188\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145661192\"><div data-type=\"problem\" id=\"fs-id1169145661195\"><p id=\"fs-id1169145661197\">Find the intercepts of the parabola whose function is \\(f\\left(x\\right)={x}^{2}-4x-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147819779\"><p id=\"fs-id1169147819781\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,-5\\right)\\)<em data-effect=\"italics\">x<\/em>-intercepts \\(\\left(-1,0\\right),\\left(5,0\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147846543\"><h3 data-type=\"title\">Graph Quadratic Functions Using Properties<\/h3><p id=\"fs-id1169147846548\">Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.<\/p><div data-type=\"example\" id=\"fs-id1169147828812\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph a Quadratic Function Using Properties<\/div><div data-type=\"exercise\" id=\"fs-id1169147828817\"><div data-type=\"problem\" id=\"fs-id1169147828820\"><p id=\"fs-id1169147828822\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u22126<em data-effect=\"italics\">x<\/em> + 8 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145732909\"><span data-type=\"media\" id=\"fs-id1169145732913\" data-alt=\"Step 1 is to determine whether the parabola opens upward or downward. Loot at the leading coefficient, a, in the equation. If f of x equals x squared minus 6 x plus 8, then a equals 1. Since a is positive, the parabola opens upward.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to determine whether the parabola opens upward or downward. Loot at the leading coefficient, a, in the equation. If f of x equals x squared minus 6 x plus 8, then a equals 1. Since a is positive, the parabola opens upward.\"><\/span><span data-type=\"media\" id=\"fs-id1169147979679\" data-alt=\"Step 2 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the product 2 a. For the function f of x equals x squared minust 6 x plus 8, the axis of symmetry is negative b divided by the product 2 a. x equals the opposite of negative 6 divided by the product 2 times 1. X equals 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the product 2 a. For the function f of x equals x squared minust 6 x plus 8, the axis of symmetry is negative b divided by the product 2 a. x equals the opposite of negative 6 divided by the product 2 times 1. X equals 3.\"><\/span><span data-type=\"media\" id=\"fs-id1169147979693\" data-alt=\"In step 3, find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the function. F of x equals x squared minus 6 x plus 8. F of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1. The vertex is the point (3, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 3, find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the function. F of x equals x squared minus 6 x plus 8. F of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1. The vertex is the point (3, negative 1).\"><\/span><span data-type=\"media\" id=\"fs-id1169148081131\" data-alt=\"Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. We first find f of 0 to find the y-intercept. F of x equals x squared plus 6 x plus 8, so f of 0 equals 0 squared plus 6 times 0 plus 8. F of 0 equals 8. The y-intercept is the point (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x-value 6. The point symmetric to the y-intercept is the point (6, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. We first find f of 0 to find the y-intercept. F of x equals x squared plus 6 x plus 8, so f of 0 equals 0 squared plus 6 times 0 plus 8. F of 0 equals 8. The y-intercept is the point (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x-value 6. The point symmetric to the y-intercept is the point (6, 8).\"><\/span><span data-type=\"media\" id=\"fs-id1169148081147\" data-alt=\"Step 5 is to find the x-intercepts. Find additional points if needed. We solve f of x equals 0. We can solve this quadratic equation by factoring. To find the x-intercepts, set f of x equal to 0. F of x equals x squared minus 6 x plus 8. 0 equals x squared minus 6 x plus 8. 0 equals the product of x minus 2 and x minus 4. So x equals 2 or x equals 4. The x-intercepts are the points (2, 0) and (4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to find the x-intercepts. Find additional points if needed. We solve f of x equals 0. We can solve this quadratic equation by factoring. To find the x-intercepts, set f of x equal to 0. F of x equals x squared minus 6 x plus 8. 0 equals x squared minus 6 x plus 8. 0 equals the product of x minus 2 and x minus 4. So x equals 2 or x equals 4. The x-intercepts are the points (2, 0) and (4, 0).\"><\/span><span data-type=\"media\" id=\"fs-id1169148232620\" data-alt=\"The final step, step 6, is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola. An image shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 10. The y-axis of the plane runs from negative 3 to 7. The parabola has a vertex at (3, negative 1). Other points plotted include the x-intercepts, (2, 0) and (4, 0), the y-intercept, (0, 8), and the point (6, 8) that is symmetric to the y-intercept across the axis of symmetry.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The final step, step 6, is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola. An image shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 10. The y-axis of the plane runs from negative 3 to 7. The parabola has a vertex at (3, negative 1). Other points plotted include the x-intercepts, (2, 0) and (4, 0), the y-intercept, (0, 8), and the point (6, 8) that is symmetric to the y-intercept across the axis of symmetry.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145665615\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145665620\"><div data-type=\"problem\" id=\"fs-id1169145665622\"><p id=\"fs-id1169145665624\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 2<em data-effect=\"italics\">x<\/em> \u2212 8 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148233223\"><span data-type=\"media\" id=\"fs-id1169148233227\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 9). The y-intercept of the parabola is the point (0, negative 8). The x-intercepts of the parabola are the points (negative 4, 0) and (4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 9). The y-intercept of the parabola is the point (0, negative 8). The x-intercepts of the parabola are the points (negative 4, 0) and (4, 0).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148233030\"><div data-type=\"problem\" id=\"fs-id1169148233032\"><p id=\"fs-id1169148233034\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 8<em data-effect=\"italics\">x<\/em> + 12 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148081240\"><span data-type=\"media\" id=\"fs-id1169148081244\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 15. The axis of symmetry, x equals 4, is graphed as a dashed line. The parabola has a vertex at (4, negative 4). The y-intercept of the parabola is the point (0, 12). The x-intercepts of the parabola are the points (2, 0) and (6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 15. The axis of symmetry, x equals 4, is graphed as a dashed line. The parabola has a vertex at (4, negative 4). The y-intercept of the parabola is the point (0, 12). The x-intercepts of the parabola are the points (2, 0) and (6, 0).\"><\/span><\/div><\/div><\/div><p id=\"fs-id1169147978265\">We list the steps to take in order to graph a quadratic function here.<\/p><div data-type=\"note\" id=\"fs-id1169147978268\" class=\"howto\"><div data-type=\"title\">To graph a quadratic function using properties.<\/div><ol id=\"fs-id1169147978275\" type=\"1\" class=\"stepwise\"><li>Determine whether the parabola opens upward or downward.<\/li><li>Find the equation of the axis of symmetry.<\/li><li>Find the vertex.<\/li><li>Find the <em data-effect=\"italics\">y<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept across the axis of symmetry.<\/li><li>Find the <em data-effect=\"italics\">x<\/em>-intercepts. Find additional points if needed.<\/li><li>Graph the parabola.<\/li><\/ol><\/div><p id=\"fs-id1169147819985\">We were able to find the <em data-effect=\"italics\">x<\/em>-intercepts in the last example by factoring. We find the <em data-effect=\"italics\">x<\/em>-intercepts in the next example by factoring, too.<\/p><div data-type=\"example\" id=\"fs-id1169147819999\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145665304\"><div data-type=\"problem\" id=\"fs-id1169145665306\"><p id=\"fs-id1169145665308\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2212 9 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147965895\"><table id=\"fs-id1169147965899\" class=\"unnumbered unstyled can-break\" summary=\"Compare the equation f of x equals negative x squared plus 6 x minus 9 to the standard form of a quadratic equation f of x equals a x squared plus b x plus c. The equation is in standard form, with y on one side. Since a equals negative 1, the parabola opens downward. An image shows a downward parabola shape. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals negative 6 divded by the product 2 times negative 1. This simplifies to x equals 3. The axis of symmetry is x equals 3. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals 3 is graphed on the grid. The vertex is on the line x equals 3. To find the vertex, find f of 3. F of x equals negative x squared plus 6 x minus 9. F of 3 equals negative 3 squared plus 6 times 3 minus 9. F of 3 equals negative 9 plus 18 minus 9. F of 3 equals 0. The vertex is the point (3, 0). A new graph is shown that adds the plotted point (3, 0) to the previous image. The y-intercept occurs when x equals 0. Find f of 0 by substituting x equals 0 into the function. F of x equals negative x squared plus 6 x minus 9. F of 0 equals negative 0 squared plus 6 times 0 minus 9. F of 0 equals negative 9. The y-intercept equals the point (0, negative 9). The point (0, negative 9) is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is (6, negative 9). This point is symmetric to the y-intercept across the axis of symmetry. An updated graph is shown that adds the plotted points (0, negative 9) and (6, negative 9) to the previous image. The x-intercept occurs when f of x equals 0. Set f of x equal to 0. 0 equals negative x squared plus 6 x minus 9. Factor the GCF. 0 equals the opposite of the expression x squared minus 6 x plus 9. Factor the trinomial. 0 equals the opposite of the square of the difference x minus 3. Solve for x. x equals 3. Connect the points to graph the parabola. A final graph is displayed. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals 3 is graphed on the grid. The points (3, 0), (0, negative 9) and (6, negative 9) are plotted and connected to show a downward-opening parabola.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145663980\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is \\(-1\\), the parabola opens downward.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1169145666734\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the equation of the axis of symmetry, use<div data-type=\"newline\"><br><\/div>\\(x=-\\frac{b}{2a}\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667055\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666748\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147979079\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The axis of symmetry is \\(x=3\\).<div data-type=\"newline\"><br><\/div>The vertex is on the line \\(x=3\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148080844\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(3\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670858\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670883\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147949859\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145729478\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(3,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670430\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept occurs when \\(x=0\\). Find \\(f\\left(0\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145664697\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute \\(x=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148234150\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147950479\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(0,-9\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The point \\(\\left(0,-9\\right)\\) is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is \\(\\left(6,-9\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148232516\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">Point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(6,-9\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept occurs when \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147978665\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145732623\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the GCF.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148232333\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983375\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145662857\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Connect the points to graph the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147960326\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145731494\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145731498\"><div data-type=\"problem\" id=\"fs-id1169145731500\"><p id=\"fs-id1169145731503\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 3<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 12<em data-effect=\"italics\">x<\/em> \u2212 12 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147870442\"><span data-type=\"media\" id=\"fs-id1169147870447\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (2, 0). The y-intercept (0, negative 12) is plotted as well as the axis of symmetry, x equals 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (2, 0). The y-intercept (0, negative 12) is plotted as well as the axis of symmetry, x equals 2.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147987696\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147987700\"><div data-type=\"problem\" id=\"fs-id1169147987702\"><p>Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 24<em data-effect=\"italics\">x<\/em> + 36 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147816801\"><span data-type=\"media\" id=\"fs-id1169147816805\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 30 to 20. The y-axis of the plane runs from negative 10 to 40. The parabola has a vertex at (negative 3, 0). The y-intercept (0, 36) is plotted as well as the axis of symmetry, x equals negative 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 30 to 20. The y-axis of the plane runs from negative 10 to 40. The parabola has a vertex at (negative 3, 0). The y-intercept (0, 36) is plotted as well as the axis of symmetry, x equals negative 3.\"><\/span><\/div><\/div><\/div><p id=\"fs-id1169145729161\">For the graph of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2212 9, the vertex and the <em data-effect=\"italics\">x<\/em>-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation 0 = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2212 9 is 0, so there is only one solution. That means there is only one <em data-effect=\"italics\">x<\/em>-intercept, and it is the vertex of the parabola.<\/p><p id=\"fs-id1169148233514\">How many <em data-effect=\"italics\">x<\/em>-intercepts would you expect to see on the graph of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 5?<\/p><div data-type=\"example\" id=\"fs-id1169147981414\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147981416\"><div data-type=\"problem\" id=\"fs-id1169147981419\"><p id=\"fs-id1169147981421\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 5 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145729572\"><table id=\"fs-id1169145729575\" class=\"unnumbered unstyled can-break\" summary=\"Compare the equation f of x equals x squared plus 4 x plus 5 to the standard form of a quadratic equation f of x equals a x squared plus b x plus c. Since a is 1, the parabola opens upward. An image shows an upward parabola shape. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals negative 4 divded by the product 2 times 1. This simplifies to x equals negative 2. The axis of symmetry is x equals negative 2. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals negative 2 is graphed on the grid. The vertex is on the line x equals negative 2. To find the vertex, find f of negative 2. F of x equals x squared plus 4 x plus 5. F of negative 2 equals negative 2 squared plus 4 times negative 2 plus 5. F of negative 2 equals 4 minus 8 plus 5. F of negative 2 equals 1. The vertex is the point (negative 2, 1). A new graph is shown that adds the plotted point (negative 2, 1) to the previous image. The y-intercept occurs when x equals 0. Find f of 0 by substituting x equals 0 into the function. F of 0 equals 5. The y-intercept equals the point (0, 5). The point (negative 4, 5) is 2 units to the left of the line of symmetry. The point 2 units to the right of the line of symmetry is (0, 5). The point (negative 4, 5) is symmetric to the y-intercept across the axis of symmetry. An updated graph is shown that adds the plotted points (0, 5) and (negative 4, 5) to the previous image. The x-intercept occurs when f of x equals 0. Set f of x equal to 0. 0 equals x squared plus 4 x plus 5. Test the discriminant. B squared minus 4 a c equals 4 squared minus the product 4 times1 times 5. This expression simplifies to 16 minus 20, or negative 4. Since the value of the discriminant is negative, there is no real-number solution and so no x-intercept. Connect the points to graph the parabola. You may want to choose two more points for greater accuracy.A final graph is displayed. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals negative 2 is graphed on the grid. The points (negative 2, 1), (negative 4, 5) and (0, 5) are plotted and connected to show an upward-opening parabola.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145640378\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is 1, the parabola opens upward.<\/td><td data-valign=\"top\" data-align=\"right\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1169145731299\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"right\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the axis of symmetry, find \\(x=-\\frac{b}{2a}\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148234641\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230670\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230695\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The equation of the axis of symmetry is \\(x=-2\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147980412\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is on the line \\(x=-2.\\)<\/td><td data-valign=\"top\" data-align=\"right\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(x\\right)\\) when \\(x=-2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147833138\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147950571\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147857771\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983252\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(-2,1\\right)\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983154\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept occurs when \\(x=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148225974\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(0\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171791382117\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145733101\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(0,5\\right)\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The point \\(\\left(-4,5\\right)\\) is two units to the left of the line of<div data-type=\"newline\"><br><\/div>symmetry.<div data-type=\"newline\"><br><\/div>The point two units to the right of the line of<div data-type=\"newline\"><br><\/div>symmetry is \\(\\left(0,5\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147987816\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">Point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(-4,5\\right)\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept occurs when \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145639853\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730498\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Test the discriminant.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730382\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730408\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148081049\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147988243\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since the value of the discriminant is negative, there is<div data-type=\"newline\"><br><\/div>no real solution and so no <em data-effect=\"italics\">x<\/em>-intercept.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Connect the points to graph the parabola. You may<div data-type=\"newline\"><br><\/div>want to choose two more points for greater accuracy.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147982542\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145664584\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145664588\"><div data-type=\"problem\" id=\"fs-id1169145664590\"><p id=\"fs-id1169145664592\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 2<em data-effect=\"italics\">x<\/em> + 3 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145639654\"><p id=\"fs-id1169145639656\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1169145639658\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 4. The y-axis of the plane runs from negative 1 to 5. The parabola has a vertex at (1, 2). The y-intercept (0, 3) is plotted as is the line of symmetry, x equals 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 4. The y-axis of the plane runs from negative 1 to 5. The parabola has a vertex at (1, 2). The y-intercept (0, 3) is plotted as is the line of symmetry, x equals 1.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147960415\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147960419\"><div data-type=\"problem\" id=\"fs-id1169147960422\"><p id=\"fs-id1169147960424\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = \u22123<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 6<em data-effect=\"italics\">x<\/em> \u2212 4 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145730795\"><p id=\"fs-id1169145730797\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1169145730799\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 2. The y-axis of the plane runs from negative 5 to 1. The parabola has a vertex at (negative 1, negative 2). The y-intercept (0, negative 4) is plotted as is the line of symmetry, x equals negative 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 2. The y-axis of the plane runs from negative 5 to 1. The parabola has a vertex at (negative 1, negative 2). The y-intercept (0, negative 4) is plotted as is the line of symmetry, x equals negative 1.\"><\/span><\/div><\/div><\/div><p id=\"fs-id1169148229746\">Finding the <em data-effect=\"italics\">y<\/em>-intercept by finding <em data-effect=\"italics\">f<\/em> (0) is easy, isn\u2019t it? Sometimes we need to use the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span> to find the <em data-effect=\"italics\">x<\/em>-intercepts.<\/p><div data-type=\"example\" id=\"fs-id1169148229537\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148229540\"><div data-type=\"problem\" id=\"fs-id1169148229542\"><p id=\"fs-id1169148229544\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 2<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">x<\/em> \u2212 3 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147949422\"><table id=\"fs-id1169147949425\" class=\"unnumbered unstyled can-break\" summary=\"Compare the equation f of x equals 2 x squared minus 4 x minus 3 to the standard form of a quadratic equation f of x equals a x squared plus b x plus c. The equation has y on one side. Since a is 2, the parabola opens upward. An image shows an upward parabola shape. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals the opposite of negative 4 divded by the product 2 times 2. This simplifies to x equals 1. The axis of symmetry is x equals 1. The vertex is on the line x equals 1. To find the vertex, find f of 1. F of x equals 2 x squared minus 4 x minus 3. F of 1 equals 2 times 1squared minus 4 times 1 minus 3. F of 1 equals 2 minus 4 minus 3. F of 1 equals negative 5. The vertex is the point (1, negative 5). The y-intercept occurs when x equals 0. Find f of 0 by substituting x equals 0 into the function. F of 0 equals 2 times 0 squared minus 4 times 0 minus 3. F of 0 equals negative 3. The y-intercept is the point (0, negative 3). The point (0, negative 3) is 1 unit to the left of the line of symmetry. The point 1 unit to the right of the line of symmetry is (2, negative 3). The point (2, negative 3) is symmetric to the y-intercept across the axis of symmetry. The x-intercept occurs when y equals 0. Set f of x equal to 0. 0 equals 2 x squared minus 4 x minus 3. Use the Quadratic Formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Substitute the values for a, b, and c. X equals the quotient of the expression the opposite of negative 4 plus or minus the square root of the difference negative 4 squared minus the product 4 times 2 times negative 3 divided by the product 2 times 2. Simplify. X equals the quotient of the expression 4 plus or minus the square root of the sum 16 plus 24 divided by 4. Simplify inside the radical to get the quotient of 4 plus or minus square root 40 and 4. Simplify the radical. X equals the quotient of the expression 4 plus or minus 2 square root 10 and 4. Factor the GCF. X equalasa the quotient of the product 2 times the expression 2 plus or minus square root 10 divided by 4. Remove common factors. x equals the quotient of 2 plus or minus square root 10 and 2. Write as two equations. The first is x equals the quotient 2 plus square root 10 divided by 2, approximately 2.5. The second solution is x equals the quotient 2 minus square root 10 divided by 2, approximately negative 0.6. The approximate values of the x-intercepts are (2.5, 0) and (negative 0.6, 0). Graph the parabola using the points found. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals 1is graphed on the grid. The points (0, negative 3), (1, negative 5) and (2, negative 3) are plotted and connected to show an upward-opening parabola.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145731003\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is 2, the parabola opens upward. <span data-type=\"media\" id=\"fs-id1169147960646\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"right\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To find the equation of the axis of symmetry, use<div data-type=\"newline\"><br><\/div>\\(x=-\\frac{b}{2a}\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145662541\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148234442\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230977\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The equation of the axis of<\/strong><div data-type=\"newline\"><br><\/div><strong data-effect=\"bold\">symmetry is \\(x=1.\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is on the line \\(x=1\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145728428\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(1\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148226086\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148233332\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147878532\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The vertex is<\/strong>\\(\\left(1,-5\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept occurs when \\(x=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147831886\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(0\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730199\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147906862\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The <em data-effect=\"italics\">y<\/em>-intercept is<\/strong>\\(\\left(0,-3\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The point \\(\\left(0,-3\\right)\\) is one unit to the left of the line of<div data-type=\"newline\"><br><\/div>symmetry.<\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Point symmetric to the<\/strong><div data-type=\"newline\"><br><\/div><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercept is<\/strong>\\(\\left(2,-3\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The point one unit to the right of the line of<div data-type=\"newline\"><br><\/div>symmetry is \\(\\left(2,-3\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept occurs when \\(y=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147845616\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(x\\right)=0\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145731996\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Quadratic Formula.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147817004\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the values of \\(a,b,\\) and \\(c.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145728073\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145639761\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify inside the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145664488\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666940\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the GCF.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670110\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147816256\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write as two equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147854659\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022v_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Approximate the values.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230462\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022w_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The approximate values of the<\/strong><div data-type=\"newline\"><br><\/div><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts are<\/strong>\\(\\left(2.5,0\\right)\\)<strong data-effect=\"bold\">and<\/strong><div data-type=\"newline\"><br><\/div>\\(\\left(-0.6,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the parabola using the points found.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148231101\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022x_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148234549\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148234553\"><div data-type=\"problem\" id=\"fs-id1169148234555\"><p id=\"fs-id1169148234558\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 5<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 10<em data-effect=\"italics\">x<\/em> + 3 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145666849\"><span data-type=\"media\" id=\"fs-id1169145666854\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 2). The y-intercept of the parabola is the point (0, 3). The x-intercepts of the parabola are approximately (negative 1.6, 0) and (negative 0.4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 2). The y-intercept of the parabola is the point (0, 3). The x-intercepts of the parabola are approximately (negative 1.6, 0) and (negative 0.4, 0).\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147950673\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147950678\"><div data-type=\"problem\" id=\"fs-id1169147950680\"><p id=\"fs-id1169147950682\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = \u22123<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 6<em data-effect=\"italics\">x<\/em> + 5 by using its properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145663686\"><span data-type=\"media\" id=\"fs-id1169145663690\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, 8). The y-intercept of the parabola is the point (0, 5). The x-intercepts of the parabola are approximately (negative 2.6, 0) and (0.6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, 8). The y-intercept of the parabola is the point (0, 5). The x-intercepts of the parabola are approximately (negative 2.6, 0) and (0.6, 0).\"><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147979390\"><h3 data-type=\"title\">Solve Maximum and Minimum Applications<\/h3><p id=\"fs-id1169147979395\">Knowing that the <span data-type=\"term\" class=\"no-emphasis\">vertex<\/span> of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic function. The <em data-effect=\"italics\">y<\/em>-coordinate of the vertex is the <span data-type=\"term\" class=\"no-emphasis\">minimum<\/span> value of a parabola that opens upward. It is the <span data-type=\"term\" class=\"no-emphasis\">maximum<\/span> value of a parabola that opens downward. See <a href=\"#CNX_IntAlg_Figure_09_06_023\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_023\"><span data-type=\"media\" id=\"fs-id1169145664103\" data-alt=\"This figure shows 2 graphs side-by-side. The left graph shows a downward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label maximum. The right graph shows an upward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label minimum.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 graphs side-by-side. The left graph shows a downward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label maximum. The right graph shows an upward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label minimum.\"><\/span><\/div><div data-type=\"note\" id=\"fs-id1169145663361\"><div data-type=\"title\">Minimum or Maximum Values of a Quadratic Function<\/div><p id=\"fs-id1169145663367\">The <strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-coordinate of the vertex<\/strong> of the graph of a quadratic function is the<\/p><ul id=\"fs-id1169145667351\" data-bullet-style=\"bullet\"><li><em data-effect=\"italics\">minimum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">upward<\/em>.<\/li><li><em data-effect=\"italics\">maximum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">downward<\/em>.<\/li><\/ul><\/div><div data-type=\"example\" id=\"fs-id1169148234337\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148234339\"><div data-type=\"problem\" id=\"fs-id1169148234341\"><p>Find the minimum or maximum value of the quadratic function \\(f\\left(x\\right)={x}^{2}+2x-8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148232836\"><table id=\"fs-id1169148232839\" class=\"unnumbered unstyled can-break\" summary=\"Start with the function f of x equals x squared plus 2 x minus 8. Since a is positive, the parabola opens upward. The quadratic equation has a minimum. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals negative 2 divded by the product 2 times 1. This simplifies to x equals negative 1. The axis of symmetry is x equals negative 1. The vertex is on the line x equals negative 1. To find the vertex, find f of negative 1. F of x equals x squared plus 2 x minus 8. F of negative 1 equals negative 1squared plus 2 times negative 1 minus 8. F of negative 1 equals 1 minus 2 minus 8. F of negative 1 equals negative 9. The vertex is the point (negative 1, negative 9). Since the parabola has a minimum, the y-coordinate of the vertex is the minimum y-value of the quadratic equation. The minimum value of the quadratic is negative 9 and it occurs when x = negative 1. Show the graph to verify the result. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals negative 1 is graphed on the grid. The graph shows an upward-opening parabola with vertex (negative 1, negative 9).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145733011\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is positive, the parabola opens upward.<div data-type=\"newline\"><br><\/div>The quadratic equation has a minimum.<\/td><td data-valign=\"top\" data-align=\"right\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the equation of the axis of symmetry.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148231798\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148054218\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145729978\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The equation of the axis of<div data-type=\"newline\"><br><\/div>symmetry is \\(x=-1\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is on the line \\(x=-1\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147981736\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find \\(f\\left(-1\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148232232\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145733322\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147905335\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(-1,-9\\right)\\).<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since the parabola has a minimum, the <em data-effect=\"italics\">y<\/em>-coordinate of<div data-type=\"newline\"><br><\/div>the vertex is the minimum <em data-effect=\"italics\">y<\/em>-value of the quadratic<div data-type=\"newline\"><br><\/div>equation.<div data-type=\"newline\"><br><\/div>The minimum value of the quadratic is \\(-9\\) and it<div data-type=\"newline\"><br><\/div>occurs when \\(x=-1\\).<\/td><td data-valign=\"top\" data-align=\"right\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145729683\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Show the graph to verify the result.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145729053\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145729057\"><div data-type=\"problem\" id=\"fs-id1169145729059\"><p id=\"fs-id1169145729062\">Find the maximum or minimum value of the quadratic function \\(f\\left(x\\right)={x}^{2}-8x+12.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145667472\"><p id=\"fs-id1169145667474\">The minimum value of the quadratic function is \u22124 and it occurs when <em data-effect=\"italics\">x<\/em> = 4.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147979779\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147979783\"><div data-type=\"problem\" id=\"fs-id1169147979785\"><p id=\"fs-id1169147979788\">Find the maximum or minimum value of the quadratic function \\(f\\left(x\\right)=-4{x}^{2}+16x-11.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147960121\"><p id=\"fs-id1169147960123\">The maximum value of the quadratic function is 5 and it occurs when <em data-effect=\"italics\">x<\/em> = 2.<\/p><\/div><\/div><\/div><p id=\"fs-id1169147950049\">We have used the formula<\/p><div data-type=\"equation\" id=\"fs-id1169147950052\" class=\"unnumbered\" data-label=\"\">\\(h\\left(t\\right)=-16{t}^{2}+{v}_{0}t+{h}_{0}\\)<\/div><p id=\"fs-id1169145728967\">to calculate the height in feet, <em data-effect=\"italics\">h<\/em> , of an object shot upwards into the air with initial velocity, <em data-effect=\"italics\">v<\/em><sub>0<\/sub>, after <em data-effect=\"italics\">t<\/em> seconds .<\/p><p id=\"fs-id1169145730295\">This formula is a quadratic function, so its graph is a parabola. By solving for the coordinates of the vertex (<em data-effect=\"italics\">t, h<\/em>), we can find how long it will take the object to reach its maximum height. Then we can calculate the maximum height.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145730308\"><div data-type=\"problem\" id=\"fs-id1169147949312\"><p id=\"fs-id1169147949314\">The quadratic equation <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 176<em data-effect=\"italics\">t<\/em> + 4 models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet.<\/p><p id=\"fs-id1169145665091\"><span class=\"token\">\u24d0<\/span> How many seconds will it take the volleyball to reach its maximum height? <span class=\"token\">\u24d1<\/span> Find the maximum height of the volleyball.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145665105\"><p id=\"fs-id1169145665107\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}h\\left(t\\right)=-16{t}^{2}+176t+4\\hfill \\\\ \\begin{array}{c}\\text{Since}\\phantom{\\rule{0.2em}{0ex}}a\\phantom{\\rule{0.2em}{0ex}}\\text{is negative, the parabola opens}\\hfill \\\\ \\text{downward.}\\hfill \\\\ \\text{The quadratic function has a maximum.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><p id=\"fs-id1169147959899\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Find the equation of the axis of symmetry.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}t=-\\frac{b}{2a}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}t=-\\frac{176}{2\\left(-16\\right)}\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}t=5.5\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\text{The equation of the axis of symmetry is}\\hfill \\\\ t=5.5.\\hfill \\end{array}\\hfill \\\\ \\text{The vertex is on the line}\\phantom{\\rule{0.2em}{0ex}}t=5.5.\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\text{The maximum occurs when}\\phantom{\\rule{0.2em}{0ex}}t=5.5\\hfill \\\\ \\text{seconds.}\\hfill \\end{array}\\hfill \\end{array}\\)<p id=\"fs-id1169147960752\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}\\text{Find}\\phantom{\\rule{0.2em}{0ex}}h\\left(5.5\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{9em}{0ex}}h\\left(t\\right)=-16{t}^{2}+176t+4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{9em}{0ex}}h\\left(t\\right)=-16{\\left(5.5\\right)}^{2}+176\\left(5.5\\right)+4\\hfill \\\\ \\text{Use a calculator to simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{9em}{0ex}}h\\left(t\\right)=488\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{9em}{0ex}}\\text{The vertex is}\\phantom{\\rule{0.2em}{0ex}}\\left(5.5,488\\right).\\hfill \\end{array}\\)<p id=\"fs-id1169145661103\">Since the parabola has a maximum, the <em data-effect=\"italics\">h<\/em>-coordinate of the vertex is the maximum value of the quadratic function.<\/p><p id=\"fs-id1169145733203\">The maximum value of the quadratic is 488 feet and it occurs when <em data-effect=\"italics\">t<\/em> = 5.5 seconds.<\/p><p id=\"fs-id1169145733211\">After 5.5 seconds, the volleyball will reach its maximum height of 488 feet.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145733218\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145733222\"><div data-type=\"problem\" id=\"fs-id1169145733224\"><p id=\"fs-id1169147981616\">Solve, rounding answers to the nearest tenth.<\/p><p id=\"fs-id1169147981619\">The quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 128<em data-effect=\"italics\">t<\/em> + 32 is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft\/sec. How long will it take for the stone to reach its maximum height? What is the maximum height?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147980700\"><p id=\"fs-id1169147980702\">It will take 4 seconds for the stone to reach its maximum height of 288 feet.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147980709\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147980713\"><div data-type=\"problem\" id=\"fs-id1169147980715\"><p id=\"fs-id1169147980717\">A path of a toy rocket thrown upward from the ground at a rate of 208 ft\/sec is modeled by the quadratic function of <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 208<em data-effect=\"italics\">t<\/em>. When will the rocket reach its maximum height? What will be the maximum height?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148233832\"><p id=\"fs-id1169148233834\">It will 6.5 seconds for the rocket to reach its maximum height of 676 feet.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148233840\" class=\"media-2\"><p id=\"fs-id1169145640053\">Access these online resources for additional instruction and practice with graphing quadratic functions using properties.<\/p><ul id=\"fs-id1163872563321\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct1\">Quadratic Functions: Axis of Symmetry and Vertex<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct2\">Finding x- and y-intercepts of a Quadratic Function<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct3\">Graphing Quadratic Functions<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct4\">Solve Maxiumum or Minimum Applications<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct5\">Quadratic Applications: Minimum and Maximum<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169147982432\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1169147982440\" data-bullet-style=\"bullet\"><li>Parabola Orientation <ul id=\"fs-id1169147982448\" data-bullet-style=\"bullet\"><li>For the graph of the quadratic function \\(f\\left(x\\right)=a{x}^{2}+bx+c,\\) if <ul id=\"fs-id1169148229448\" data-bullet-style=\"bullet\"><li><em data-effect=\"italics\">a<\/em> &gt; 0, the parabola opens upward.<\/li><li><em data-effect=\"italics\">a<\/em> &lt; 0, the parabola opens downward.<\/li><\/ul><\/li><\/ul><\/li><li>Axis of Symmetry and Vertex of a Parabola The graph of the function \\(f\\left(x\\right)=a{x}^{2}+bx+c\\) is a parabola where: <ul id=\"fs-id1169145730086\" data-bullet-style=\"bullet\"><li>the axis of symmetry is the vertical line \\(x=-\\frac{b}{2a}.\\)<\/li><li>the vertex is a point on the axis of symmetry, so its <em data-effect=\"italics\">x<\/em>-coordinate is \\(-\\frac{b}{2a}.\\)<\/li><li>the <em data-effect=\"italics\">y<\/em>-coordinate of the vertex is found by substituting \\(x=-\\frac{b}{2a}\\) into the quadratic equation.<\/li><\/ul><\/li><li>Find the Intercepts of a Parabola <ul id=\"fs-id1169145665517\" data-bullet-style=\"bullet\"><li>To find the intercepts of a parabola whose function is \\(f\\left(x\\right)=a{x}^{2}+bx+c:\\)<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1171791466640\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\hfill \\mathbit{\\text{y}}\\mathbf{\\text{-intercept}}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\mathbit{\\text{x}}\\mathbf{\\text{-intercepts}}\\hfill \\\\ \\hfill \\text{Let}\\phantom{\\rule{0.2em}{0ex}}x=0\\phantom{\\rule{0.2em}{0ex}}\\text{and solve for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{Let}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)=0\\phantom{\\rule{0.2em}{0ex}}\\text{and solve for}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill \\end{array}\\)<\/div><\/li><\/ul><\/li><li>How to graph a quadratic function using properties. <ol id=\"fs-id1169145663066\" type=\"1\" class=\"stepwise\"><li>Determine whether the parabola opens upward or downward.<\/li><li>Find the equation of the axis of symmetry.<\/li><li>Find the vertex.<\/li><li>Find the <em data-effect=\"italics\">y<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept across the axis of symmetry.<\/li><li>Find the <em data-effect=\"italics\">x<\/em>-intercepts. Find additional points if needed.<\/li><li>Graph the parabola.<\/li><\/ol><\/li><li>Minimum or Maximum Values of a Quadratic Equation <ul id=\"fs-id1169145663471\" data-bullet-style=\"bullet\"><li>The <em data-effect=\"italics\">y<\/em>-coordinate of the vertex of the graph of a quadratic equation is the<\/li><li><em data-effect=\"italics\">minimum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">upward<\/em>.<\/li><li><em data-effect=\"italics\">maximum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">downward<\/em>.<\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148054108\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169148054112\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1169148054119\"><strong data-effect=\"bold\">Recognize the Graph of a Quadratic Function<\/strong><\/p><p id=\"fs-id1169148054126\">In the following exercises, graph the functions by plotting points.<\/p><div data-type=\"exercise\" id=\"fs-id1169148054129\"><div data-type=\"problem\" id=\"fs-id1169148054131\"><p id=\"fs-id1169148054133\">\\(f\\left(x\\right)={x}^{2}+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147978477\"><span data-type=\"media\" id=\"fs-id1169148235046\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 3).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148235063\"><div data-type=\"problem\" id=\"fs-id1169148235065\"><p id=\"fs-id1169148235067\">\\(f\\left(x\\right)={x}^{2}-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147982855\"><div data-type=\"problem\" id=\"fs-id1169147982858\"><p id=\"fs-id1169147982860\">\\(y=\\text{\u2212}{x}^{2}+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145666124\"><span data-type=\"media\" id=\"fs-id1169145666129\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 1).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145664887\"><div data-type=\"problem\" id=\"fs-id1169145664889\"><p id=\"fs-id1169145664891\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}-1\\)<\/p><\/div><\/div><p id=\"fs-id1169147982242\">For each of the following exercises, determine if the parabola opens up or down.<\/p><div data-type=\"exercise\" id=\"fs-id1169147982245\"><div data-type=\"problem\" id=\"fs-id1169145663558\"><p id=\"fs-id1169145663560\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-2{x}^{2}-6x-7\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=6{x}^{2}+2x+3\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145660490\"><p id=\"fs-id1169147949006\"><span class=\"token\">\u24d0<\/span> down <span class=\"token\">\u24d1<\/span> up<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147949020\"><div data-type=\"problem\" id=\"fs-id1169147949023\"><p id=\"fs-id1169147949025\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=4{x}^{2}+x-4\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=-9{x}^{2}-24x-16\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147816892\"><div data-type=\"problem\" id=\"fs-id1169147816894\"><p id=\"fs-id1169147816896\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-3{x}^{2}+5x-1\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=2{x}^{2}-4x+5\\)<\/div><div data-type=\"solution\" id=\"fs-id1169147987922\"><p id=\"fs-id1169147987925\"><span class=\"token\">\u24d0<\/span> down <span class=\"token\">\u24d1<\/span> up<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147987594\"><div data-type=\"problem\" id=\"fs-id1169147987596\"><p id=\"fs-id1169147987598\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)={x}^{2}+3x-4\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(f\\left(x\\right)=-4{x}^{2}-12x-9\\)<\/div><\/div><p id=\"fs-id1169147961912\"><strong data-effect=\"bold\">Find the Axis of Symmetry and Vertex of a Parabola<\/strong><\/p><p id=\"fs-id1169147961918\">In the following functions, find <span class=\"token\">\u24d0<\/span> the equation of the axis of symmetry and <span class=\"token\">\u24d1<\/span> the vertex of its graph.<\/p><div data-type=\"exercise\" id=\"fs-id1169145730691\"><div data-type=\"problem\" id=\"fs-id1169145730693\"><p id=\"fs-id1169145730695\">\\(f\\left(x\\right)={x}^{2}+8x-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147905125\"><p id=\"fs-id1169147905127\"><span class=\"token\">\u24d0<\/span>\\(x=-4\\); <span class=\"token\">\u24d1<\/span>\\(\\left(-4,-17\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147980905\"><div data-type=\"problem\" id=\"fs-id1169147980908\"><p id=\"fs-id1169147980910\">\\(f\\left(x\\right)={x}^{2}+10x+25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145732722\"><div data-type=\"problem\" id=\"fs-id1169145732724\"><p id=\"fs-id1169145732726\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+2x+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148231894\"><p id=\"fs-id1169148231896\"><span class=\"token\">\u24d0<\/span>\\(x=1\\); <span class=\"token\">\u24d1<\/span>\\(\\left(1,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147949126\"><div data-type=\"problem\" id=\"fs-id1169147949128\"><p id=\"fs-id1169147949130\">\\(f\\left(x\\right)=-2{x}^{2}-8x-3\\)<\/p><\/div><\/div><p id=\"fs-id1169148234953\"><strong data-effect=\"bold\">Find the Intercepts of a Parabola<\/strong><\/p><p id=\"fs-id1169148234959\">In the following exercises, find the intercepts of the parabola whose function is given.<\/p><div data-type=\"exercise\" id=\"fs-id1169148234963\"><div data-type=\"problem\" id=\"fs-id1169148234966\"><p id=\"fs-id1169148234968\">\\(f\\left(x\\right)={x}^{2}+7x+6\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147978967\"><p id=\"fs-id1169147978969\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,6\\right);\\)<em data-effect=\"italics\">x<\/em>-intercept \\(\\left(-1,0\\right),\\left(-6,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197891\"><div data-type=\"problem\" id=\"fs-id1169148197893\"><p>\\(f\\left(x\\right)={x}^{2}+10x-11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147845711\"><div data-type=\"problem\" id=\"fs-id1169147845714\"><p id=\"fs-id1169147845716\">\\(f\\left(x\\right)={x}^{2}+8x+12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145660279\"><p id=\"fs-id1169145660281\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,12\\right);\\)<em data-effect=\"italics\">x<\/em>-intercept \\(\\left(-2,0\\right),\\left(-6,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148231489\"><div data-type=\"problem\" id=\"fs-id1169148231491\"><p id=\"fs-id1169148231493\">\\(f\\left(x\\right)={x}^{2}+5x+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145730909\"><div data-type=\"problem\" id=\"fs-id1169145730912\"><p id=\"fs-id1169145730914\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+8x-19\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145670318\"><p><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,-19\\right);\\)<em data-effect=\"italics\">x<\/em>-intercept: none<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147988342\"><div data-type=\"problem\" id=\"fs-id1169147988344\"><p id=\"fs-id1169147988346\">\\(f\\left(x\\right)=-3{x}^{2}+x-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145665816\"><div data-type=\"problem\" id=\"fs-id1169145665819\"><p id=\"fs-id1169145665821\">\\(f\\left(x\\right)={x}^{2}+6x+13\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145659962\"><p id=\"fs-id1169145659964\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,13\\right);\\)<em data-effect=\"italics\">x<\/em>-intercept: none<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145670773\"><div data-type=\"problem\" id=\"fs-id1169145670775\"><p id=\"fs-id1169145670778\">\\(f\\left(x\\right)={x}^{2}+8x+12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147799883\"><div data-type=\"problem\" id=\"fs-id1169147799885\"><p id=\"fs-id1169147799887\">\\(f\\left(x\\right)=4{x}^{2}-20x+25\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147866283\"><p id=\"fs-id1169147866285\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,-16\\right);\\)<em data-effect=\"italics\">x<\/em>-intercept \\(\\left(\\frac{5}{2},0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147851717\"><div data-type=\"problem\" id=\"fs-id1169147851719\"><p id=\"fs-id1169147851721\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}-14x-49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147834444\"><div data-type=\"problem\" id=\"fs-id1169148053906\"><p id=\"fs-id1169148053908\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}-6x-9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148053944\"><p id=\"fs-id1169148053946\"><em data-effect=\"italics\">y<\/em>-intercept: \\(\\left(0,9\\right);\\)<em data-effect=\"italics\">x<\/em>-intercept \\(\\left(-3,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145640476\"><div data-type=\"problem\" id=\"fs-id1169145640478\"><p id=\"fs-id1169147959363\">\\(f\\left(x\\right)=4{x}^{2}+4x+1\\)<\/p><\/div><\/div><p id=\"fs-id1169147865984\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Properties<\/strong><\/p><p id=\"fs-id1169147865990\">In the following exercises, graph the function by using its properties.<\/p><div data-type=\"exercise\" id=\"fs-id1169147865994\"><div data-type=\"problem\" id=\"fs-id1169147865996\"><p id=\"fs-id1169147865998\">\\(f\\left(x\\right)={x}^{2}+6x+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147848694\"><span data-type=\"media\" id=\"fs-id1169147848698\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147848716\"><div data-type=\"problem\" id=\"fs-id1169147848718\"><p id=\"fs-id1169147848720\">\\(f\\left(x\\right)={x}^{2}+4x-12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147848778\"><div data-type=\"problem\" id=\"fs-id1169147848780\"><p id=\"fs-id1169147848782\">\\(f\\left(x\\right)={x}^{2}+4x+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147848818\"><span data-type=\"media\" id=\"fs-id1169147848822\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147848840\"><div data-type=\"problem\"><p>\\(f\\left(x\\right)={x}^{2}-6x+8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148080663\"><div data-type=\"problem\" id=\"fs-id1169148080665\"><p id=\"fs-id1169148080667\">\\(f\\left(x\\right)=9{x}^{2}+12x+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148080705\"><span data-type=\"media\" id=\"fs-id1169148080710\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148080728\"><div data-type=\"problem\" id=\"fs-id1169148080730\"><p id=\"fs-id1169148080732\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+8x-16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197394\"><div data-type=\"problem\" id=\"fs-id1169148197397\"><p id=\"fs-id1169148197399\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+2x-7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148197437\"><span data-type=\"media\" id=\"fs-id1169148197441\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_321_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197459\"><div data-type=\"problem\" id=\"fs-id1169148197461\"><p id=\"fs-id1169148197463\">\\(f\\left(x\\right)=5{x}^{2}+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197517\"><div data-type=\"problem\" id=\"fs-id1169148197519\"><p id=\"fs-id1169148197521\">\\(f\\left(x\\right)=2{x}^{2}-4x+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148197559\"><span data-type=\"media\" id=\"fs-id1169148197563\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_323_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197582\"><div data-type=\"problem\" id=\"fs-id1169148197584\"><p id=\"fs-id1169148197586\">\\(f\\left(x\\right)=3{x}^{2}-6x-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197647\"><div data-type=\"problem\" id=\"fs-id1169148197649\"><p id=\"fs-id1169148197651\">\\(f\\left(x\\right)=2{x}^{2}-4x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148197690\"><span data-type=\"media\" id=\"fs-id1169148197694\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_325_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197712\"><div data-type=\"problem\" id=\"fs-id1169148197714\"><p id=\"fs-id1169148197716\">\\(f\\left(x\\right)=-4{x}^{2}-6x-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148197778\"><div data-type=\"problem\" id=\"fs-id1169148197780\"><p id=\"fs-id1169148197782\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}-4x+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145667779\"><span data-type=\"media\" id=\"fs-id1169145667783\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_327_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145667802\"><div data-type=\"problem\" id=\"fs-id1169145667804\"><p id=\"fs-id1169145667806\">\\(f\\left(x\\right)={x}^{2}+6x+8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145667865\"><div data-type=\"problem\" id=\"fs-id1169145667867\"><p id=\"fs-id1169145667869\">\\(f\\left(x\\right)=5{x}^{2}-10x+8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145667907\"><span data-type=\"media\" id=\"fs-id1169145667911\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_329_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145667930\"><div data-type=\"problem\" id=\"fs-id1169145667932\"><p id=\"fs-id1169145667934\">\\(f\\left(x\\right)=-16{x}^{2}+24x-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145667995\"><div data-type=\"problem\" id=\"fs-id1169145667997\"><p id=\"fs-id1169145667999\">\\(f\\left(x\\right)=3{x}^{2}+18x+20\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668037\"><span data-type=\"media\" id=\"fs-id1169145668041\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_331_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668060\"><div data-type=\"problem\" id=\"fs-id1169145668062\"><p id=\"fs-id1169145668064\">\\(f\\left(x\\right)=-2{x}^{2}+8x-10\\)<\/p><\/div><\/div><p id=\"fs-id1169145668125\"><strong data-effect=\"bold\">Solve Maximum and Minimum Applications<\/strong><\/p><p id=\"fs-id1169145668132\">In the following exercises, find the maximum or minimum value of each function.<\/p><div data-type=\"exercise\" id=\"fs-id1169145668135\"><div data-type=\"problem\" id=\"fs-id1169145668137\"><p id=\"fs-id1169145668139\">\\(f\\left(x\\right)=2{x}^{2}+x-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668172\"><p id=\"fs-id1169145668174\">The minimum value is \\(-\\frac{9}{8}\\) when \\(x=-\\frac{1}{4}.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668206\"><div data-type=\"problem\" id=\"fs-id1169145668208\"><p id=\"fs-id1169145668211\">\\(y=-4{x}^{2}+12x-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668260\"><div data-type=\"problem\" id=\"fs-id1169145668262\"><p id=\"fs-id1169145668264\">\\(y={x}^{2}-6x+15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668290\"><p id=\"fs-id1169145668292\">The maximum value is 6 when <em data-effect=\"italics\">x<\/em> = 3.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668303\"><div data-type=\"problem\" id=\"fs-id1169145668305\"><p id=\"fs-id1169145668307\">\\(y=\\text{\u2212}{x}^{2}+4x-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668348\"><div data-type=\"problem\" id=\"fs-id1169145668350\"><p id=\"fs-id1169145668352\">\\(y=-9{x}^{2}+16\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668373\"><p id=\"fs-id1169145668375\">The maximum value is 16 when <em data-effect=\"italics\">x<\/em> = 0.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668386\"><div data-type=\"problem\" id=\"fs-id1169145668388\"><p id=\"fs-id1169145668391\">\\(y=4{x}^{2}-49\\)<\/p><\/div><\/div><p id=\"fs-id1169145668425\">In the following exercises, solve. Round answers to the nearest tenth.<\/p><div data-type=\"exercise\" id=\"fs-id1169145668428\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668430\"><p id=\"fs-id1169145668432\">An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 168<em data-effect=\"italics\">t<\/em> + 45 find how long it will take the arrow to reach its maximum height, and then find the maximum height.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668462\"><p id=\"fs-id1169145668464\">In 5.3 sec the arrow will reach maximum height of 486 ft.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668469\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668471\"><p id=\"fs-id1169145668474\">A stone is thrown vertically upward from a platform that is 20 feet height at a rate of 160 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 160<em data-effect=\"italics\">t<\/em> + 20 to find how long it will take the stone to reach its maximum height, and then find the maximum height.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668511\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668513\"><p id=\"fs-id1169145668515\">A ball is thrown vertically upward from the ground with an initial velocity of 109 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 109<em data-effect=\"italics\">t<\/em> + 0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668545\"><p id=\"fs-id1169145668547\">In 3.4 seconds the ball will reach its maximum height of 185.6 feet.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668552\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668554\"><p id=\"fs-id1169145668556\">A ball is thrown vertically upward from the ground with an initial velocity of 122 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 122<em data-effect=\"italics\">t<\/em> + 0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668593\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668595\"><p id=\"fs-id1169145668597\">A computer store owner estimates that by charging <em data-effect=\"italics\">x<\/em> dollars each for a certain computer, he can sell 40 \u2212 <em data-effect=\"italics\">x<\/em> computers each week. The quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +40<em data-effect=\"italics\">x<\/em> is used to find the revenue, <em data-effect=\"italics\">R<\/em>, received when the selling price of a computer is <em data-effect=\"italics\">x<\/em>, Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145668650\"><p id=\"fs-id1169145668652\">20 computers will give the maximum of ?400 in receipts.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668658\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668660\"><p id=\"fs-id1169145668662\">A retailer who sells backpacks estimates that by selling them for <em data-effect=\"italics\">x<\/em> dollars each, he will be able to sell 100 \u2212 <em data-effect=\"italics\">x<\/em> backpacks a month. The quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +100<em data-effect=\"italics\">x<\/em> is used to find the <em data-effect=\"italics\">R<\/em>, received when the selling price of a backpack is <em data-effect=\"italics\">x<\/em>. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145668721\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169145668723\"><p id=\"fs-id1169145668725\">A retailer who sells fashion boots estimates that by selling them for <em data-effect=\"italics\">x<\/em> dollars each, he will be able to sell 70 \u2212 <em data-effect=\"italics\">x<\/em> boots a week. Use the quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +70<em data-effect=\"italics\">x<\/em> to find the revenue received when the average selling price of a pair of fashion boots is <em data-effect=\"italics\">x<\/em>. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue per day.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147946274\"><p id=\"fs-id1169147946276\">He will be able to sell 35 pairs of boots at the maximum revenue of ?1,225.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946281\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147946283\"><p id=\"fs-id1169147946285\">A cell phone company estimates that by charging <em data-effect=\"italics\">x<\/em> dollars each for a certain cell phone, they can sell 8 \u2212 <em data-effect=\"italics\">x<\/em> cell phones per day. Use the quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +8<em data-effect=\"italics\">x<\/em> to find the revenue received per day when the selling price of a cell phone is <em data-effect=\"italics\">x<\/em>. Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946338\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147946340\"><p id=\"fs-id1169147946342\">A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(240 \u2212 2<em data-effect=\"italics\">x<\/em>) gives the area of the corral, <em data-effect=\"italics\">A<\/em>, for the length, <em data-effect=\"italics\">x<\/em>, of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147946382\"><p id=\"fs-id1169147946384\">The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 square feet.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946390\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147946392\"><p id=\"fs-id1169147946394\">A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic function <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(100 \u2212 2<em data-effect=\"italics\">x<\/em>) gives the area, <em data-effect=\"italics\">A<\/em>, of the dog run for the length, <em data-effect=\"italics\">x<\/em>, of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946442\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147946445\"><p id=\"fs-id1169147946447\">A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the \u201cwalls.\u201d He wants to maximize the area using 80 feet of fencing. The quadratic function <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(80 \u2212 2<em data-effect=\"italics\">x<\/em>) gives the area of the patio, where <em data-effect=\"italics\">x<\/em> is the width of one side. Find the maximum area of the patio.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147946480\"><p id=\"fs-id1169147946482\">The maximum area of the patio is 800 feet.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946488\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169147946490\"><p id=\"fs-id1169147946492\">A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them 300 feet of fencing to use to enclose part of their backyard. Use the quadratic function <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(300 \u2212 2<em data-effect=\"italics\">x<\/em>) to determine the maximum area of the fenced in yard.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1169147946529\"><h4 data-type=\"title\">Writing Exercise<\/h4><div data-type=\"exercise\" id=\"fs-id1169147946536\"><div data-type=\"problem\" id=\"fs-id1169147946539\"><p id=\"fs-id1169147946541\">How do the graphs of the functions \\(f\\left(x\\right)={x}^{2}\\) and \\(f\\left(x\\right)={x}^{2}-1\\) differ? We graphed them at the start of this section. What is the difference between their graphs? How are their graphs the same?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147946591\"><p id=\"fs-id1169147946594\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946599\"><div data-type=\"problem\" id=\"fs-id1169147946601\"><p id=\"fs-id1169147946603\">Explain the process of finding the vertex of a parabola.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946615\"><div data-type=\"problem\" id=\"fs-id1169147946618\"><p id=\"fs-id1169147946620\">Explain how to find the intercepts of a parabola.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147946624\"><p id=\"fs-id1169147946627\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147946632\"><div data-type=\"problem\" id=\"fs-id1169147946634\"><p id=\"fs-id1169147946636\">How can you use the discriminant when you are graphing a quadratic function?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169147946650\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1169147946655\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1169147946678\" data-alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can recognize the graph of a quadratic equation.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the axis of symmetry and vertex of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the intercepts of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can graph quadratic equations in two variables.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve maximum and minimum applications.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can recognize the graph of a quadratic equation.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the axis of symmetry and vertex of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the intercepts of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can graph quadratic equations in two variables.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve maximum and minimum applications.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><\/span><p id=\"fs-id1169147946672\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1169147946698\"><dt>quadratic function<\/dt><dd id=\"fs-id1169147946703\">A quadratic function, where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and \\(a\\ne 0,\\) is a function of the form \\(f\\left(x\\right)=a{x}^{2}+bx+c.\\)<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Recognize the graph of a quadratic function<\/li>\n<li>Find the axis of symmetry and vertex of a parabola<\/li>\n<li>Find the intercepts of a parabola<\/li>\n<li>Graph quadratic functions using properties<\/li>\n<li>Solve maximum and minimum applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147874135\" class=\"be-prepared\">\n<p id=\"fs-id1169147860526\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1169147844160\" type=\"1\">\n<li>Graph the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by plotting points.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/da9d6ce0-a078-4ca2-97af-8cb374f040f5#fs-id1167836683384\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58c3f50b3ce4cb8b28e68873d001d620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"135\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7137224bdc6600511c4c03a0a63f370c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/> when <em data-effect=\"italics\">a<\/em> = 3 and <em data-effect=\"italics\">b<\/em> = \u22126.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147808468\">\n<h3 data-type=\"title\">Recognize the Graph of a Quadratic Function<\/h3>\n<p id=\"fs-id1169147876025\">Previously we very briefly looked at the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/>, which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ff39a37a230408f7c9a6410a33dbe1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce78e2da43dbf8e758d3c5c14d7f44ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> We call this kind of function a quadratic function.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147804510\">\n<div data-type=\"title\">Quadratic Function<\/div>\n<p id=\"fs-id1169147845348\">A <span data-type=\"term\">quadratic function<\/span>, where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d28cb478bd8b6abe7d9573551313d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> is a function of the form<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147821646\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ff39a37a230408f7c9a6410a33dbe1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169145731259\">We graphed the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by plotting points.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147876789\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 2 to 6. The parabola has a vertex at (0, 0) and also passes through the points (-2, 4), (-1, 1), (1, 1), and (2, 4). To the right of the graph is a table of values with 3 columns. The first row is a header row and labels each column, \u201cx\u201d, \u201cf of x equals x squared\u201d, and \u201cthe order pair x, f of x.\u201d In row 2, x equals negative 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair negative 3, 9. In row 3, x equals negative 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair negative 2, 4. In row 4, x equals negative 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair negative 1, 1. In row 5, x equals 0, f of x equals x squared is 0 and the ordered pair x, f of x is the ordered pair 0, 0. In row 6, x equals 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair 1, 1. In row 7, x equals 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair 2, 4. In row 8, x equals 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair 3, 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 2 to 6. The parabola has a vertex at (0, 0) and also passes through the points (-2, 4), (-1, 1), (1, 1), and (2, 4). To the right of the graph is a table of values with 3 columns. The first row is a header row and labels each column, \u201cx\u201d, \u201cf of x equals x squared\u201d, and \u201cthe order pair x, f of x.\u201d In row 2, x equals negative 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair negative 3, 9. In row 3, x equals negative 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair negative 2, 4. In row 4, x equals negative 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair negative 1, 1. In row 5, x equals 0, f of x equals x squared is 0 and the ordered pair x, f of x is the ordered pair 0, 0. In row 6, x equals 1, f of x equals x squared is 1 and the ordered pair x, f of x is the ordered pair 1, 1. In row 7, x equals 2, f of x equals x squared is 4 and the ordered pair x, f of x is the ordered pair 2, 4. In row 8, x equals 3, f of x equals x squared is 9 and the ordered pair x, f of x is the ordered pair 3, 9.\" \/><\/span><\/p>\n<p id=\"fs-id1169147715853\">Every quadratic function has a graph that looks like this. We call this figure a <span data-type=\"term\">parabola<\/span>.<\/p>\n<p id=\"fs-id1169147809055\">Let\u2019s practice graphing a parabola by plotting a few points.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147774250\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147876680\">\n<div data-type=\"problem\" id=\"fs-id1169147835768\">\n<p id=\"fs-id1169147803921\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dab2bfe13b69cc4906b85a0706a7cf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147820867\">\n<p id=\"fs-id1169147721479\">We will graph the function by plotting points.<\/p>\n<table id=\"fs-id1169147808038\" class=\"unnumbered unstyled\" summary=\"Choose integer values for x, substitute them into the equation and simplify to find f of x. Record the values of the ordered pairs in the chart. The table of values for the function f of x equals x squared minus 1 has 2 columns. The first column is labeled x and the second column is labeled f of x. When x equals 0, f of x equals negative 1. When x equals 1, f of x equals 0. When x equals negative 1, f of x equals 0. When x equals 2, f of x equals 3. When x equals negative 2, f of x equals 3. Plot the points, and then connect them with a smooth curve. The result will be the graph of the function f of x equals negative 1. The figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The points identified in the table are plotted.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Choose integer values for <em data-effect=\"italics\">x<\/em>,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>substitute them into the equation<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and simplify to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Record the values of the ordered pairs in the chart.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147856021\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Plot the points, and then connect<\/p>\n<div data-type=\"newline\"><\/div>\n<p>them with a smooth curve. The<\/p>\n<div data-type=\"newline\"><\/div>\n<p>result will be the graph of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a294c15a253d64d5494761235c0cd07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147874347\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147906436\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147940171\">\n<div data-type=\"problem\" id=\"fs-id1169147770846\">\n<p id=\"fs-id1169147855992\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33d48a8fdf338e2ebc700d415cafcfc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147863559\"><span data-type=\"media\" id=\"fs-id1169147959678\" data-alt=\"This figure shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148037752\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147817063\">\n<div data-type=\"problem\" id=\"fs-id1169147828437\">\n<p id=\"fs-id1169147940166\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34a76a50f760a5cb9fb8274915d63b91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147855659\"><span data-type=\"media\" id=\"fs-id1169147949267\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, \u22121).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, \u22121).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147825486\">All graphs of quadratic functions of the form <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> are parabolas that open upward or downward. See <a href=\"#CNX_IntAlg_Figure_09_06_003\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_003\"><span data-type=\"media\" id=\"fs-id1169147980073\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is x squared plus 4 x plus 3. The leading coefficient, a, is greater than 0, so this parabola opens upward.The graph on the right shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is negative x squared plus 4 x plus 3. The leading coefficient, a, is less than 0, so this parabola opens downward.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is x squared plus 4 x plus 3. The leading coefficient, a, is greater than 0, so this parabola opens upward.The graph on the right shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The general form for the equation of this graph is f of x equals a x squared plus b x plus c. The equation of this parabola is negative x squared plus 4 x plus 3. The leading coefficient, a, is less than 0, so this parabola opens downward.\" \/><\/span><\/div>\n<p id=\"fs-id1169145666278\">Notice that the only difference in the two functions is the negative sign before the quadratic term (<em data-effect=\"italics\">x<\/em><sup>2<\/sup> in the equation of the graph in <a href=\"#CNX_IntAlg_Figure_09_06_003\" class=\"autogenerated-content\">(Figure)<\/a>). When the quadratic term, is positive, the parabola opens upward, and when the quadratic term is negative, the parabola opens downward.<\/p>\n<div data-type=\"note\" id=\"fs-id1169148230545\">\n<div data-type=\"title\">Parabola Orientation<\/div>\n<p id=\"fs-id1169147821411\">For the graph of the quadratic function <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em>, if<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147854266\" data-alt=\"This images shows a bulleted list. The first bullet notes that, if a is greater than 0, then the parabola opens upward and shows an image of an upward-opening parabola. The second bullet notes that, if a is less than 0, then the parabola opens downward and shows an image of a downward-opening parabola.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This images shows a bulleted list. The first bullet notes that, if a is greater than 0, then the parabola opens upward and shows an image of an upward-opening parabola. The second bullet notes that, if a is less than 0, then the parabola opens downward and shows an image of a downward-opening parabola.\" \/><\/span><\/div>\n<div data-type=\"example\" id=\"fs-id1169147810565\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148229708\">\n<div data-type=\"problem\" id=\"fs-id1169147800031\">\n<p id=\"fs-id1169147949688\">Determine whether each parabola opens upward or downward:<\/p>\n<p id=\"fs-id1169145664867\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86fb25f93b96d1afd435c95021d81fdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c06ec5c63b147c121150519414d80c98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145666225\">\n<p id=\"fs-id1169148233111\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169147854650\" class=\"unnumbered unstyled\" summary=\"The standard form of a quadratic equation is f of x equals a x squared plus b x plus c. This function is f of x equals negative 3 x squared plus 2 x minus 4. Find the value of a, the coefficient of x squared. For this function, a equals negative 3. Since a is negative, the parabola will open downward.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the value of \u201c<em data-effect=\"italics\">a<\/em>\u201d.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148198055\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">Since the \u201c<em data-effect=\"italics\">a<\/em>\u201d is negative, the parabola will open downward.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169147819480\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169145662827\" class=\"unnumbered unstyled\" summary=\"The standard form of a quadratic equation is f of x equals a x squared plus b x plus c. This function is f of x equals negative 6 x squared plus 7 x minus 9. Find the value of a, the coefficient of x squared. For this function, a equals 6. Since a is positive, the parabola will open upward.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the value of \u201c<em data-effect=\"italics\">a<\/em>\u201d.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147959131\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">Since the \u201c<em data-effect=\"italics\">a<\/em>\u201d is positive, the parabola will open upward.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147837972\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147837975\">\n<div data-type=\"problem\" id=\"fs-id1169147823322\">\n<p id=\"fs-id1169147823324\">Determine whether the graph of each function is a parabola that opens upward or downward:<\/p>\n<p id=\"fs-id1169147854838\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6432c46c5ad3525eabc7f5f59c6a5e72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9166a9351d7245cac60adfcebed194ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145663138\">\n<p id=\"fs-id1169147838982\"><span class=\"token\">\u24d0<\/span> up; <span class=\"token\">\u24d1<\/span> down<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147866845\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147836676\">\n<div data-type=\"problem\" id=\"fs-id1169147836678\">\n<p id=\"fs-id1169147980356\">Determine whether the graph of each function is a parabola that opens upward or downward:<\/p>\n<p id=\"fs-id1169147980360\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c38d0525418dacdf1f3e18dfee1d645b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37935264b84fbb33533ecb64d6a1a4d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148080822\">\n<p id=\"fs-id1169147982492\"><span class=\"token\">\u24d0<\/span> down; <span class=\"token\">\u24d1<\/span> up<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147851221\">\n<h3 data-type=\"title\">Find the Axis of Symmetry and Vertex of a Parabola<\/h3>\n<p id=\"fs-id1169147906619\">Look again at <a href=\"#CNX_IntAlg_Figure_09_06_003\" class=\"autogenerated-content\">(Figure)<\/a>. Do you see that we could fold each parabola in half and then one side would lie on top of the other? The \u2018fold line\u2019 is a line of symmetry. We call it the <span data-type=\"term\">axis of symmetry<\/span> of the parabola.<\/p>\n<p id=\"fs-id1169145661242\">We show the same two graphs again with the axis of symmetry. See <a href=\"#CNX_IntAlg_Figure_09_06_007\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_007\"><span data-type=\"media\" id=\"fs-id1169147982414\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The equation of this parabola is x squared plus 4 x plus 3. The vertical line passes through the point (negative 2, 0) and has the equation x equals negative 2. The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The equation of this parabola is negative x squared plus 4 x plus 3. The vertical line passes through the point (2, 0) and has the equation x equals 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The equation of this parabola is x squared plus 4 x plus 3. The vertical line passes through the point (negative 2, 0) and has the equation x equals negative 2. The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The equation of this parabola is negative x squared plus 4 x plus 3. The vertical line passes through the point (2, 0) and has the equation x equals 2.\" \/><\/span><\/div>\n<p id=\"fs-id1169147979766\">The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfab3cdadd0b3fd4994e4a6616037c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1169147982212\">So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfab3cdadd0b3fd4994e4a6616037c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147750504\" data-alt=\"Compare the function f of x equals x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times 1. The axis of symmetry equals negative 2. Next, compare the function f of x equals negative x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times negative 1. The axis of symmetry equals 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Compare the function f of x equals x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times 1. The axis of symmetry equals negative 2. Next, compare the function f of x equals negative x squared plus 4 x plus 3 to the standard form of a quadratic function, f of x equals a x squared plus b x plus c. The axis of symmetry is the line x equals negative b divided by the product 2 a. Substituting for b and a yields x equals negative 4 divided by the product 2 times negative 1. The axis of symmetry equals 2.\" \/><\/span><\/p>\n<p id=\"fs-id1169145670075\">Notice that these are the equations of the dashed blue lines on the graphs.<\/p>\n<p id=\"fs-id1169147819842\">The point on the parabola that is the lowest (parabola opens up), or the highest (parabola opens down), lies on the axis of symmetry. This point is called the <span data-type=\"term\">vertex<\/span> of the parabola.<\/p>\n<p id=\"fs-id1169145664752\">We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its<\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">x<\/em>-coordinate is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ab25f94cf91879c16c63a8acc445d48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/> To find the <em data-effect=\"italics\">y<\/em>-coordinate of the vertex we substitute the value of the <em data-effect=\"italics\">x<\/em>-coordinate into the quadratic function.<span data-type=\"media\" id=\"fs-id1169147940220\" data-alt=\"For the function f of x equals x squared plus 4 x plus 3, the axis of symmetry is x equals negative 2. The vertex is the point on the parabola with x-coordinate negative 2. Substitute x equals negative 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals the square of negative 2 plus 4 times negative 2 plus 3, so f of x equals negative 1. The vertex is the point (negative 2, negative 1). For the function f of x equals negative x squared plus 4 x plus 3, the axis of symmetry is x equals 2. The vertex is the point on the parabola with x-coordinate 2. Substitute x equals 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals 2 squared plus 4 times 2 plus 3, so f of x equals 7. The vertex is the point (2, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"For the function f of x equals x squared plus 4 x plus 3, the axis of symmetry is x equals negative 2. The vertex is the point on the parabola with x-coordinate negative 2. Substitute x equals negative 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals the square of negative 2 plus 4 times negative 2 plus 3, so f of x equals negative 1. The vertex is the point (negative 2, negative 1). For the function f of x equals negative x squared plus 4 x plus 3, the axis of symmetry is x equals 2. The vertex is the point on the parabola with x-coordinate 2. Substitute x equals 2 into the function f of x equals x squared plus 4 x plus 3. F of x equals 2 squared plus 4 times 2 plus 3, so f of x equals 7. The vertex is the point (2, 7).\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1169147962340\">\n<div data-type=\"title\">Axis of Symmetry and Vertex of a Parabola<\/div>\n<p id=\"fs-id1169148232095\">The graph of the function <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> is a parabola where:<\/p>\n<ul id=\"fs-id1169145732569\" data-bullet-style=\"bullet\">\n<li>the axis of symmetry is the vertical line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfab3cdadd0b3fd4994e4a6616037c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>the vertex is a point on the axis of symmetry, so its <em data-effect=\"italics\">x<\/em>-coordinate is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ab25f94cf91879c16c63a8acc445d48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>the <em data-effect=\"italics\">y<\/em>-coordinate of the vertex is found by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d0499e5d51ece29865924215982302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/> into the quadratic equation.<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169147906840\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147816860\">\n<div data-type=\"problem\" id=\"fs-id1169147816863\">\n<p id=\"fs-id1169147816865\">For the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf86a6f18ffc805ebbfd35e17ca7d0e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> find:<\/p>\n<p id=\"fs-id1169148054384\"><span class=\"token\">\u24d0<\/span> the axis of symmetry <span class=\"token\">\u24d1<\/span> the vertex.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145728808\">\n<p id=\"fs-id1169145728810\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169147960277\" class=\"unnumbered unstyled can-break\" summary=\"Compare the function f of x equals 3 x squared minus 6 x plus 2 to the standard form of a quadratic function f of x equals a x squared plus b x plus c. The axis of symmetry is the vertical line x equals negative b divided by the product 2 a. Substitute the values a equals 3 and b equals negative 6 into the equation of the line of symmetry. X equals negative 6 divided by the product 2 times 3. Simplify. The axis of symmetry is the line x equals 1.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147987890\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The axis of symmetry is the vertical line<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d0499e5d51ece29865924215982302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfaed44949cf9cfbeb3445de33aabd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"25\" style=\"vertical-align: -4px;\" \/> into the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147816990\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147846800\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The axis of symmetry is the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169147980794\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169147982390\" class=\"unnumbered unstyled\" summary=\"Write the function f of x equals 3 x squared minus 6 x plus 2. The vertex is a point on the line of symmetry, so its x-coordinate will be x equals 1. Find f of 1. F of 1 equals 3 times 1 squared minus 6 times 1 plus 2. Simplify f of 1 equals 3 times 1 minus 6 plus 2, which equals negative 1. This result is the y-coordinate. The vertex is the point (1, negative 1).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147981078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The vertex is a point on the line of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>symmetry, so its <em data-effect=\"italics\">x<\/em>-coordinate will be<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91ba6dcc1b657dfb8a4f15162ee96b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145639635\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147978649\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The result is the <em data-effect=\"italics\">y<\/em>-coordinate.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145660232\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3ac38dbb39343c28a60a287dfb114b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147880094\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147880099\">\n<div data-type=\"problem\" id=\"fs-id1169148037745\">\n<p id=\"fs-id1169148037747\">For the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1839940310d7083e5f900d18e8569028_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> find:<\/p>\n<p id=\"fs-id1169147950358\"><span class=\"token\">\u24d0<\/span> the axis of symmetry <span class=\"token\">\u24d1<\/span> the vertex.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147965759\">\n<p id=\"fs-id1169147965761\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba5fa2318b2b97b57b5af83c04b54507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73884e0ac8ed4b11b427b3c4ca0557b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145640275\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147846046\">\n<div data-type=\"problem\" id=\"fs-id1169147846049\">\n<p id=\"fs-id1169147797170\">For the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d21dc34104e0a9bb9fa6d628c2efd1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> find:<\/p>\n<p id=\"fs-id1169145670510\"><span class=\"token\">\u24d0<\/span> the axis of symmetry <span class=\"token\">\u24d1<\/span> the vertex.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147841589\">\n<p id=\"fs-id1169147841591\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13d2b2daeddd3687c1678b253e4a2508_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91e09d029e9b798ea0b0617171c88cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147856138\">\n<h3 data-type=\"title\">Find the Intercepts of a Parabola<\/h3>\n<p id=\"fs-id1169147807792\">When we graphed linear equations, we often used the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.<\/p>\n<p id=\"fs-id1169147870671\">Remember, at the <em data-effect=\"italics\">y<\/em>-intercept the value of <em data-effect=\"italics\">x<\/em> is zero. So to find the <em data-effect=\"italics\">y<\/em>-intercept, we substitute <em data-effect=\"italics\">x<\/em> = 0 into the function.<\/p>\n<p id=\"fs-id1169147962725\">Let\u2019s find the <em data-effect=\"italics\">y<\/em>-intercepts of the two parabolas shown in <a href=\"#CNX_IntAlg_Figure_09_06_012\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_012\"><span data-type=\"media\" id=\"fs-id1169147905707\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The vertical line is an axis of symmetry for the parabola, and passes through the point (negative 2, 0). It has the equation x equals negative 2. The equation of this parabola is x squared plus 4 x plus 3. When x equals 0, f of 0 equals 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3). The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The vertical line is an axis of symmetry for the parabola and passes through the point (2, 0). It has the equation x equals 2. The equation of this parabola is negative x squared plus 4 x plus 3. When x equals 0, f of 0 equals negative 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows an upward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1) and passes through the points (negative 4, 3) and (0, 3). The vertical line is an axis of symmetry for the parabola, and passes through the point (negative 2, 0). It has the equation x equals negative 2. The equation of this parabola is x squared plus 4 x plus 3. When x equals 0, f of 0 equals 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3). The graph on the right shows an downward-opening parabola and a dashed vertical line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7) and passes through the points (0, 3) and (4, 3). The vertical line is an axis of symmetry for the parabola and passes through the point (2, 0). It has the equation x equals 2. The equation of this parabola is negative x squared plus 4 x plus 3. When x equals 0, f of 0 equals negative 0 squared plus 4 times 0 plus 3. F of 0 equals 3. The y-intercept of the graph is the point (0, 3).\" \/><\/span><\/div>\n<p id=\"fs-id1169145662932\">An <em data-effect=\"italics\">x<\/em>-intercept results when the value of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) is zero. To find an <em data-effect=\"italics\">x<\/em>-intercept, we let <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 0. In other words, we will need to solve the equation 0 = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> for <em data-effect=\"italics\">x<\/em>.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147765890\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afd3534772970ed7ce5be14018246285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"180\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"fs-id1169148231382\">Solving quadratic equations like this is exactly what we have done earlier in this chapter!<\/p>\n<p id=\"fs-id1169145670950\">We can now find the <em data-effect=\"italics\">x<\/em>-intercepts of the two parabolas we looked at. First we will find the <em data-effect=\"italics\">x<\/em>-intercepts of the parabola whose function is <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 3.<\/p>\n<table id=\"fs-id1169147982318\" class=\"unnumbered unstyled\" summary=\"F of x equals x squared plus 4 x plus 3. Let f of x equal 0. 0 equals x squared plus 4 x plus 3. Factor. 0 equals the product of x plus 1 and x plus 3. Use the Zero Product Property. Then x plus 1 equals 0 or x plus 3 equals 0. Solve. X equals negative 1 or x equals negative 3. The x-intercepts are the points (negative 1, 0) and (negative 3, 0).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148233010\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147966001\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148081220\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147850531\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667133\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebe7ea89f522e94da67a0a0622127bab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ba9cc2a7f12e65a6b3de8f34bcc16e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169147982116\">Now we will find the <em data-effect=\"italics\">x<\/em>-intercepts of the parabola whose function is <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">\u2212x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 3.<\/p>\n<table id=\"fs-id1169147869012\" class=\"unnumbered unstyled can-break\" summary=\"F of x equals negative x squared plus 4 x plus 3. Let f of x equal 0. 0 equals negative x squared plus 4 x plus 3. The quadratic does not factor, so we use the Quadratic Formula. X equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. a equals negative 1, b equals 4, and c equals 3. X equals the quotient negative 4 plus or minus the square root of the difference 4 squared minus the product 4 times negative 1 times 3 divided by the product 2 times negative 1. Simplify x equals the quotient negative 4 plus or minus square root 28 divided by negative 2. X equals the quotient negative 4 plus or minus square root 28 divided by negative 2. X equals the quotient negative 4 plus or minus 2 square root 7 divided by negative 2. X equals the quotient of the product negative 2 times the expression 2 plus or minus square root 7 divided by negative 2. X equals 2 plus or minus square root 7. The intercepts are the points (2 plus square root 7, 0) and (2 minus square root 7, 0).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666010\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147950364\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">This quadratic does not factor, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>we use the Quadratic Formula.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147848633\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdd62b8e26127ad0aaa5ff6b0e81f33b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#49;&#44;&#98;&#61;&#52;&#44;&#99;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147851318\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147849869\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667570\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147962229\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145731902\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_014h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cf9b34caa4f0a82991d5af7399608d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"83\" style=\"vertical-align: -7px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f52342293871e7f9f7f7c9f901af3b7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"83\" style=\"vertical-align: -7px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169145640278\">We will use the decimal approximations of the <em data-effect=\"italics\">x<\/em>-intercepts, so that we can locate these points on the graph,<\/p>\n<div data-type=\"equation\" id=\"fs-id1169145640286\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2d9519a838bd0cbdb0aaed6fd8d045d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"420\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1169147806629\">Do these results agree with our graphs? See <a href=\"#CNX_IntAlg_Figure_09_06_015\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_015\"><span data-type=\"media\" id=\"fs-id1169147828136\" data-alt=\"This image shows 2 graphs side-by-side. The graph on the left shows the upward-opening parabola defined by the function f of x equals x squared plus 4 x plus 3 and a dashed vertical line, x equals negative 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept is (0, 3) and the x-intercepts are (negative 1, 0) and (negative 3, 0). The graph on the right shows the downward-opening parabola defined by the function f of x equals negative x squared plus 4 x plus 3 and a dashed vertical line, x equals 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7). The y-intercept is (0, 3) and the x-intercepts are (2 plus square root 7, 0), approximately (4.6, 0) and (2 minus square root, 0), approximately (negative 0.6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows 2 graphs side-by-side. The graph on the left shows the upward-opening parabola defined by the function f of x equals x squared plus 4 x plus 3 and a dashed vertical line, x equals negative 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept is (0, 3) and the x-intercepts are (negative 1, 0) and (negative 3, 0). The graph on the right shows the downward-opening parabola defined by the function f of x equals negative x squared plus 4 x plus 3 and a dashed vertical line, x equals 2, graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (2, 7). The y-intercept is (0, 3) and the x-intercepts are (2 plus square root 7, 0), approximately (4.6, 0) and (2 minus square root, 0), approximately (negative 0.6, 0).\" \/><\/span><\/div>\n<div data-type=\"note\" id=\"fs-id1169147869876\">\n<div data-type=\"title\">Find the Intercepts of a Parabola<\/div>\n<p>To find the intercepts of a parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6064af0045e2b14aad7aa1166556bad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1169147847182\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ac14170e97c1ca8f66a5947b1316438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"524\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169145732214\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145732216\">\n<div data-type=\"problem\" id=\"fs-id1169145732218\">\n<p id=\"fs-id1169145732220\">Find the intercepts of the parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e3d50080ee22561755f62bdb6d03679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147940484\">\n<table id=\"fs-id1169147940487\" class=\"unnumbered unstyled can-break\" summary=\"To find the y-intercept, let x equal 0 and solve for f of x. f of x equals x squared minus 2 x minus 8. F of 0 equals 0 squared minus 2 times 0 minus 8 which simplifies to yield f of 0 equals negative 8. When x equals 0, then f of 0 equals negative 8. The y-intercept is the point (0, negative 8). To find the x-intercept, let f of x equal 0 and solve for x. f of x equals x squared minus 2 x minus 8. 0 equals x squared minus 2 x minus 8. Solve by factoring. 0 equals the product of x minus 4 and x plus 2. 0 equals x minus 4 or 0 equals x plus 2. So x equals 4 or x equals negative 2. When f of x equals 0, then x equals 4 or x equals negative 2. The x-intercepts are the points (4, 0) and (negative 2, 0).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the <strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>&#8211;<\/strong>intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666243\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147981951\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983073\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e52788eaab6fb40ec47aa68b33b91e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The <em data-effect=\"italics\">y<\/em><strong data-effect=\"bold\">&#8211;<\/strong>intercept is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de801b109a641a695bdc89761979b816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">To find the <strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>&#8211;<\/strong>intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147835691\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145662079\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670015\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147979488\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169145639575\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_016h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c0e5ba2303417b15d69af18f15f2998_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"121\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">&#8211;<\/strong>intercepts are the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2809647061e2f05aa3080110836f8805_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147959245\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147959250\">\n<div data-type=\"problem\" id=\"fs-id1169147959252\">\n<p id=\"fs-id1169147959254\">Find the intercepts of the parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65e9a49844f1e45dd76c96487513af4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147826486\">\n<p id=\"fs-id1169147868841\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de801b109a641a695bdc89761979b816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercepts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f03ebc99ec1de086fdae81fe222fa95e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145663264\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145663268\">\n<div data-type=\"problem\" id=\"fs-id1169145663270\">\n<p id=\"fs-id1169145663272\">Find the intercepts of the parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0513f42b41cb75ab6441edc92216fc8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145661976\">\n<p id=\"fs-id1169145661978\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9efa561ba754c4e3dc405dfa212bdab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercepts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80e256e7f0367d33a6f1ada4f37e23e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145667660\">In this chapter, we have been solving quadratic equations of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. We solved for <em data-effect=\"italics\">x<\/em> and the results were the solutions to the equation.<\/p>\n<p id=\"fs-id1169145664380\">We are now looking at quadratic functions of the form <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em>. The graphs of these functions are parabolas. The <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">&#8211;<\/strong>intercepts of the parabolas occur where <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 0.<\/p>\n<p id=\"fs-id1169145731793\">For example:<\/p>\n<div data-type=\"equation\" id=\"fs-id1169145731796\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-836458bc8a6618216220930b38703a41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#53;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#61;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#51;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"346\" width=\"677\" style=\"vertical-align: -168px;\" \/><\/div>\n<p id=\"fs-id1169148231607\">The solutions of the quadratic function are the <em data-effect=\"italics\">x<\/em> values of the <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">&#8211;<\/strong>intercepts.<\/p>\n<p id=\"fs-id1169145732301\">Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the functions give the <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">&#8211;<\/strong>intercepts of the graphs, the number of <em data-effect=\"italics\">x<\/em><strong data-effect=\"bold\">&#8211;<\/strong>intercepts is the same as the number of solutions.<\/p>\n<p id=\"fs-id1169147833049\">Previously, we used the <span data-type=\"term\" class=\"no-emphasis\">discriminant<\/span> to determine the number of solutions of a quadratic function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff55a4ef4f0f0f08cb24335fede5832e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/> Now we can use the discriminant to tell us how many <em data-effect=\"italics\">x<\/em>-intercepts there are on the graph.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169145639964\" data-alt=\"This image shows three graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies below the x-axis and the parabola crosses the x-axis at two different points. If b squared minus 4 a c is greater than 0, then the quadratic equation a x squared plus b x plus c equals 0 has two solutions, and the graph of the parabola has 2 x-intercepts. The graph in the middle shows a downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies on the x-axis, the only point of intersection between the parabola and the x-axis. If b squared minus 4 a c equals 0, then the quadratic equation a x squared plus b x plus c equals 0 has one solution, and the graph of the parabola has 1 x-intercept. The graph on the right shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies above the x-axis and the parabola does not cross the x-axis. If b squared minus 4 a c is less than 0, then the quadratic equation a x squared plus b x plus c equals 0 has no solutions, and the graph of the parabola has no x-intercepts.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This image shows three graphs side-by-side. The graph on the left shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies below the x-axis and the parabola crosses the x-axis at two different points. If b squared minus 4 a c is greater than 0, then the quadratic equation a x squared plus b x plus c equals 0 has two solutions, and the graph of the parabola has 2 x-intercepts. The graph in the middle shows a downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies on the x-axis, the only point of intersection between the parabola and the x-axis. If b squared minus 4 a c equals 0, then the quadratic equation a x squared plus b x plus c equals 0 has one solution, and the graph of the parabola has 1 x-intercept. The graph on the right shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola lies above the x-axis and the parabola does not cross the x-axis. If b squared minus 4 a c is less than 0, then the quadratic equation a x squared plus b x plus c equals 0 has no solutions, and the graph of the parabola has no x-intercepts.\" \/><\/span><\/p>\n<p id=\"fs-id1169147981318\">Before you to find the values of the <em data-effect=\"italics\">x<\/em>-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147981328\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147981330\">\n<div data-type=\"problem\" id=\"fs-id1169147981332\">\n<p id=\"fs-id1169147981334\">Find the intercepts of the parabola for the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aac9d07dfc7995212b69322f3fb77eec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148233118\">\n<table id=\"fs-id1169148233121\" class=\"unnumbered unstyled\" summary=\"F of x equals 5 x squared plus x plus 4. To find the y-intercept, let x equal 0 and solve for f of x. f of 0 equals 5 times the square of 0 plus 0 plus 4, so f of 0 equals 4. When x equals 0, f of 0 equals 4. The y-intercept is the point (0, 4). To find the x-intercept, let f of x equal 0 and solve for x. f of x equals 5 x squared plus x plus 4. 0 equals 5 x squared plus x plus 4. Find the value of the discriminant to predict the number of solutions which is also the number of x-intercepts. B squared minus 4 a c equals 1 squared minus 4 times 5 times 4. This simplifies to 1 minus 80, or negative 79. Since the value of the discriminant is negative, there is no real solution to the equation. There are no x-intercepts.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147853761\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">y<\/em>-intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148233718\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667142\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45b43bf9d65f347836295596178da8ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The <em data-effect=\"italics\">y<\/em>-intercept is the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c3a69d33f9737210f9c4f1551f4b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the <em data-effect=\"italics\">x<\/em>-intercept, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730601\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147819502\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_018f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the value of the discriminant to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>predict the number of solutions which is<\/p>\n<div data-type=\"newline\"><\/div>\n<p>also the number of <em data-effect=\"italics\">x<\/em>-intercepts.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1001214f9e6e4c653d00ce95444bd69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#49;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#53;&middot;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#45;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"64\" style=\"vertical-align: -33px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">Since the value of the discriminant is<\/p>\n<div data-type=\"newline\"><\/div>\n<p>negative, there is no real solution to the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>There are no <em data-effect=\"italics\">x<\/em>-intercepts.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147875183\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147875188\">\n<div data-type=\"problem\" id=\"fs-id1169147875190\">\n<p id=\"fs-id1169147875192\">Find the intercepts of the parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efcc896d58c60a745b1ee77aa8851acf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145664178\">\n<p id=\"fs-id1169145664180\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c3a69d33f9737210f9c4f1551f4b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> no <em data-effect=\"italics\">x<\/em>-intercept<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145661188\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145661192\">\n<div data-type=\"problem\" id=\"fs-id1169145661195\">\n<p id=\"fs-id1169145661197\">Find the intercepts of the parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cff92733285c2ef67e68b438d221126b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147819779\">\n<p id=\"fs-id1169147819781\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f6ab1a0ed415088c10eaaa3977a4992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercepts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-341afeede1a21cdeaa6e8cf64bc6bdd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147846543\">\n<h3 data-type=\"title\">Graph Quadratic Functions Using Properties<\/h3>\n<p id=\"fs-id1169147846548\">Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147828812\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph a Quadratic Function Using Properties<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147828817\">\n<div data-type=\"problem\" id=\"fs-id1169147828820\">\n<p id=\"fs-id1169147828822\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u22126<em data-effect=\"italics\">x<\/em> + 8 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145732909\"><span data-type=\"media\" id=\"fs-id1169145732913\" data-alt=\"Step 1 is to determine whether the parabola opens upward or downward. Loot at the leading coefficient, a, in the equation. If f of x equals x squared minus 6 x plus 8, then a equals 1. Since a is positive, the parabola opens upward.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to determine whether the parabola opens upward or downward. Loot at the leading coefficient, a, in the equation. If f of x equals x squared minus 6 x plus 8, then a equals 1. Since a is positive, the parabola opens upward.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169147979679\" data-alt=\"Step 2 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the product 2 a. For the function f of x equals x squared minust 6 x plus 8, the axis of symmetry is negative b divided by the product 2 a. x equals the opposite of negative 6 divided by the product 2 times 1. X equals 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the product 2 a. For the function f of x equals x squared minust 6 x plus 8, the axis of symmetry is negative b divided by the product 2 a. x equals the opposite of negative 6 divided by the product 2 times 1. X equals 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169147979693\" data-alt=\"In step 3, find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the function. F of x equals x squared minus 6 x plus 8. F of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1. The vertex is the point (3, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 3, find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the function. F of x equals x squared minus 6 x plus 8. F of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1. The vertex is the point (3, negative 1).\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148081131\" data-alt=\"Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. We first find f of 0 to find the y-intercept. F of x equals x squared plus 6 x plus 8, so f of 0 equals 0 squared plus 6 times 0 plus 8. F of 0 equals 8. The y-intercept is the point (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x-value 6. The point symmetric to the y-intercept is the point (6, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. We first find f of 0 to find the y-intercept. F of x equals x squared plus 6 x plus 8, so f of 0 equals 0 squared plus 6 times 0 plus 8. F of 0 equals 8. The y-intercept is the point (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x-value 6. The point symmetric to the y-intercept is the point (6, 8).\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148081147\" data-alt=\"Step 5 is to find the x-intercepts. Find additional points if needed. We solve f of x equals 0. We can solve this quadratic equation by factoring. To find the x-intercepts, set f of x equal to 0. F of x equals x squared minus 6 x plus 8. 0 equals x squared minus 6 x plus 8. 0 equals the product of x minus 2 and x minus 4. So x equals 2 or x equals 4. The x-intercepts are the points (2, 0) and (4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to find the x-intercepts. Find additional points if needed. We solve f of x equals 0. We can solve this quadratic equation by factoring. To find the x-intercepts, set f of x equal to 0. F of x equals x squared minus 6 x plus 8. 0 equals x squared minus 6 x plus 8. 0 equals the product of x minus 2 and x minus 4. So x equals 2 or x equals 4. The x-intercepts are the points (2, 0) and (4, 0).\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148232620\" data-alt=\"The final step, step 6, is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola. An image shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 10. The y-axis of the plane runs from negative 3 to 7. The parabola has a vertex at (3, negative 1). Other points plotted include the x-intercepts, (2, 0) and (4, 0), the y-intercept, (0, 8), and the point (6, 8) that is symmetric to the y-intercept across the axis of symmetry.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_019f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The final step, step 6, is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola. An image shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 10. The y-axis of the plane runs from negative 3 to 7. The parabola has a vertex at (3, negative 1). Other points plotted include the x-intercepts, (2, 0) and (4, 0), the y-intercept, (0, 8), and the point (6, 8) that is symmetric to the y-intercept across the axis of symmetry.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145665615\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145665620\">\n<div data-type=\"problem\" id=\"fs-id1169145665622\">\n<p id=\"fs-id1169145665624\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 2<em data-effect=\"italics\">x<\/em> \u2212 8 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148233223\"><span data-type=\"media\" id=\"fs-id1169148233227\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 9). The y-intercept of the parabola is the point (0, negative 8). The x-intercepts of the parabola are the points (negative 4, 0) and (4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 9). The y-intercept of the parabola is the point (0, negative 8). The x-intercepts of the parabola are the points (negative 4, 0) and (4, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148233030\">\n<div data-type=\"problem\" id=\"fs-id1169148233032\">\n<p id=\"fs-id1169148233034\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 8<em data-effect=\"italics\">x<\/em> + 12 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148081240\"><span data-type=\"media\" id=\"fs-id1169148081244\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 15. The axis of symmetry, x equals 4, is graphed as a dashed line. The parabola has a vertex at (4, negative 4). The y-intercept of the parabola is the point (0, 12). The x-intercepts of the parabola are the points (2, 0) and (6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 15. The axis of symmetry, x equals 4, is graphed as a dashed line. The parabola has a vertex at (4, negative 4). The y-intercept of the parabola is the point (0, 12). The x-intercepts of the parabola are the points (2, 0) and (6, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147978265\">We list the steps to take in order to graph a quadratic function here.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147978268\" class=\"howto\">\n<div data-type=\"title\">To graph a quadratic function using properties.<\/div>\n<ol id=\"fs-id1169147978275\" type=\"1\" class=\"stepwise\">\n<li>Determine whether the parabola opens upward or downward.<\/li>\n<li>Find the equation of the axis of symmetry.<\/li>\n<li>Find the vertex.<\/li>\n<li>Find the <em data-effect=\"italics\">y<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept across the axis of symmetry.<\/li>\n<li>Find the <em data-effect=\"italics\">x<\/em>-intercepts. Find additional points if needed.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1169147819985\">We were able to find the <em data-effect=\"italics\">x<\/em>-intercepts in the last example by factoring. We find the <em data-effect=\"italics\">x<\/em>-intercepts in the next example by factoring, too.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147819999\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145665304\">\n<div data-type=\"problem\" id=\"fs-id1169145665306\">\n<p id=\"fs-id1169145665308\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2212 9 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147965895\">\n<table id=\"fs-id1169147965899\" class=\"unnumbered unstyled can-break\" summary=\"Compare the equation f of x equals negative x squared plus 6 x minus 9 to the standard form of a quadratic equation f of x equals a x squared plus b x plus c. The equation is in standard form, with y on one side. Since a equals negative 1, the parabola opens downward. An image shows a downward parabola shape. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals negative 6 divded by the product 2 times negative 1. This simplifies to x equals 3. The axis of symmetry is x equals 3. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals 3 is graphed on the grid. The vertex is on the line x equals 3. To find the vertex, find f of 3. F of x equals negative x squared plus 6 x minus 9. F of 3 equals negative 3 squared plus 6 times 3 minus 9. F of 3 equals negative 9 plus 18 minus 9. F of 3 equals 0. The vertex is the point (3, 0). A new graph is shown that adds the plotted point (3, 0) to the previous image. The y-intercept occurs when x equals 0. Find f of 0 by substituting x equals 0 into the function. F of x equals negative x squared plus 6 x minus 9. F of 0 equals negative 0 squared plus 6 times 0 minus 9. F of 0 equals negative 9. The y-intercept equals the point (0, negative 9). The point (0, negative 9) is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is (6, negative 9). This point is symmetric to the y-intercept across the axis of symmetry. An updated graph is shown that adds the plotted points (0, negative 9) and (6, negative 9) to the previous image. The x-intercept occurs when f of x equals 0. Set f of x equal to 0. 0 equals negative x squared plus 6 x minus 9. Factor the GCF. 0 equals the opposite of the expression x squared minus 6 x plus 9. Factor the trinomial. 0 equals the opposite of the square of the difference x minus 3. Solve for x. x equals 3. Connect the points to graph the parabola. A final graph is displayed. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals 3 is graphed on the grid. The points (3, 0), (0, negative 9) and (6, negative 9) are plotted and connected to show a downward-opening parabola.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145663980\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>, the parabola opens downward.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1169145666734\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the equation of the axis of symmetry, use<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d0499e5d51ece29865924215982302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145667055\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666748\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147979079\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The vertex is on the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148080844\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7454c1320ac593411a470ad12380405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670858\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670883\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147949859\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145729478\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc6a40acab1fcbe9adecd900d5d2a756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670430\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>. Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d73c012bc8b5a6f560bea30840502b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145664697\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148234150\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147950479\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52593685d80b1f67d02908c97187edb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de98e117a50e6be85d85016c3daa191d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> is three units to the left of the line of symmetry. The point three units to the right of the line of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3976ef1ff68d71667ca3289946f8d0c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148232516\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">Point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3976ef1ff68d71667ca3289946f8d0c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147978665\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145732623\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the GCF.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148232333\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983375\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145662857\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Connect the points to graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147960326\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_020u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145731494\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145731498\">\n<div data-type=\"problem\" id=\"fs-id1169145731500\">\n<p id=\"fs-id1169145731503\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 3<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 12<em data-effect=\"italics\">x<\/em> \u2212 12 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147870442\"><span data-type=\"media\" id=\"fs-id1169147870447\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (2, 0). The y-intercept (0, negative 12) is plotted as well as the axis of symmetry, x equals 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (2, 0). The y-intercept (0, negative 12) is plotted as well as the axis of symmetry, x equals 2.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147987696\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147987700\">\n<div data-type=\"problem\" id=\"fs-id1169147987702\">\n<p>Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 24<em data-effect=\"italics\">x<\/em> + 36 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147816801\"><span data-type=\"media\" id=\"fs-id1169147816805\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 30 to 20. The y-axis of the plane runs from negative 10 to 40. The parabola has a vertex at (negative 3, 0). The y-intercept (0, 36) is plotted as well as the axis of symmetry, x equals negative 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 30 to 20. The y-axis of the plane runs from negative 10 to 40. The parabola has a vertex at (negative 3, 0). The y-intercept (0, 36) is plotted as well as the axis of symmetry, x equals negative 3.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145729161\">For the graph of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2212 9, the vertex and the <em data-effect=\"italics\">x<\/em>-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation 0 = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2212 9 is 0, so there is only one solution. That means there is only one <em data-effect=\"italics\">x<\/em>-intercept, and it is the vertex of the parabola.<\/p>\n<p id=\"fs-id1169148233514\">How many <em data-effect=\"italics\">x<\/em>-intercepts would you expect to see on the graph of <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 5?<\/p>\n<div data-type=\"example\" id=\"fs-id1169147981414\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147981416\">\n<div data-type=\"problem\" id=\"fs-id1169147981419\">\n<p id=\"fs-id1169147981421\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 4<em data-effect=\"italics\">x<\/em> + 5 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145729572\">\n<table id=\"fs-id1169145729575\" class=\"unnumbered unstyled can-break\" summary=\"Compare the equation f of x equals x squared plus 4 x plus 5 to the standard form of a quadratic equation f of x equals a x squared plus b x plus c. Since a is 1, the parabola opens upward. An image shows an upward parabola shape. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals negative 4 divded by the product 2 times 1. This simplifies to x equals negative 2. The axis of symmetry is x equals negative 2. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals negative 2 is graphed on the grid. The vertex is on the line x equals negative 2. To find the vertex, find f of negative 2. F of x equals x squared plus 4 x plus 5. F of negative 2 equals negative 2 squared plus 4 times negative 2 plus 5. F of negative 2 equals 4 minus 8 plus 5. F of negative 2 equals 1. The vertex is the point (negative 2, 1). A new graph is shown that adds the plotted point (negative 2, 1) to the previous image. The y-intercept occurs when x equals 0. Find f of 0 by substituting x equals 0 into the function. F of 0 equals 5. The y-intercept equals the point (0, 5). The point (negative 4, 5) is 2 units to the left of the line of symmetry. The point 2 units to the right of the line of symmetry is (0, 5). The point (negative 4, 5) is symmetric to the y-intercept across the axis of symmetry. An updated graph is shown that adds the plotted points (0, 5) and (negative 4, 5) to the previous image. The x-intercept occurs when f of x equals 0. Set f of x equal to 0. 0 equals x squared plus 4 x plus 5. Test the discriminant. B squared minus 4 a c equals 4 squared minus the product 4 times1 times 5. This expression simplifies to 16 minus 20, or negative 4. Since the value of the discriminant is negative, there is no real-number solution and so no x-intercept. Connect the points to graph the parabola. You may want to choose two more points for greater accuracy.A final graph is displayed. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals negative 2 is graphed on the grid. The points (negative 2, 1), (negative 4, 5) and (0, 5) are plotted and connected to show an upward-opening parabola.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145640378\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is 1, the parabola opens upward.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1169145731299\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"right\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the axis of symmetry, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d0499e5d51ece29865924215982302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148234641\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230670\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230695\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The equation of the axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147980412\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The vertex is on the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86f872935a384592f05d5fdc077a0a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"right\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86f872935a384592f05d5fdc077a0a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147833138\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147950571\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147857771\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983252\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e86393e45c0f6cf9bb7fcf130d3db9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147983154\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148225974\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db4f958b4c4deacd90047ad0c954283a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171791382117\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145733101\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1111596661e846aebf3d4f873566ff24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> is two units to the left of the line of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>symmetry.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The point two units to the right of the line of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147987816\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">Point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1111596661e846aebf3d4f873566ff24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145639853\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730498\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Test the discriminant.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730382\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730408\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148081049\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147988243\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since the value of the discriminant is negative, there is<\/p>\n<div data-type=\"newline\"><\/div>\n<p>no real solution and so no <em data-effect=\"italics\">x<\/em>-intercept.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Connect the points to graph the parabola. You may<\/p>\n<div data-type=\"newline\"><\/div>\n<p>want to choose two more points for greater accuracy.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147982542\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_021u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145664584\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145664588\">\n<div data-type=\"problem\" id=\"fs-id1169145664590\">\n<p id=\"fs-id1169145664592\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 2<em data-effect=\"italics\">x<\/em> + 3 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145639654\">\n<p id=\"fs-id1169145639656\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169145639658\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 4. The y-axis of the plane runs from negative 1 to 5. The parabola has a vertex at (1, 2). The y-intercept (0, 3) is plotted as is the line of symmetry, x equals 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 4. The y-axis of the plane runs from negative 1 to 5. The parabola has a vertex at (1, 2). The y-intercept (0, 3) is plotted as is the line of symmetry, x equals 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147960415\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147960419\">\n<div data-type=\"problem\" id=\"fs-id1169147960422\">\n<p id=\"fs-id1169147960424\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = \u22123<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 6<em data-effect=\"italics\">x<\/em> \u2212 4 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145730795\">\n<p id=\"fs-id1169145730797\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169145730799\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 2. The y-axis of the plane runs from negative 5 to 1. The parabola has a vertex at (negative 1, negative 2). The y-intercept (0, negative 4) is plotted as is the line of symmetry, x equals negative 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 2. The y-axis of the plane runs from negative 5 to 1. The parabola has a vertex at (negative 1, negative 2). The y-intercept (0, negative 4) is plotted as is the line of symmetry, x equals negative 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148229746\">Finding the <em data-effect=\"italics\">y<\/em>-intercept by finding <em data-effect=\"italics\">f<\/em> (0) is easy, isn\u2019t it? Sometimes we need to use the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span> to find the <em data-effect=\"italics\">x<\/em>-intercepts.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148229537\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148229540\">\n<div data-type=\"problem\" id=\"fs-id1169148229542\">\n<p id=\"fs-id1169148229544\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 2<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">x<\/em> \u2212 3 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147949422\">\n<table id=\"fs-id1169147949425\" class=\"unnumbered unstyled can-break\" summary=\"Compare the equation f of x equals 2 x squared minus 4 x minus 3 to the standard form of a quadratic equation f of x equals a x squared plus b x plus c. The equation has y on one side. Since a is 2, the parabola opens upward. An image shows an upward parabola shape. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals the opposite of negative 4 divded by the product 2 times 2. This simplifies to x equals 1. The axis of symmetry is x equals 1. The vertex is on the line x equals 1. To find the vertex, find f of 1. F of x equals 2 x squared minus 4 x minus 3. F of 1 equals 2 times 1squared minus 4 times 1 minus 3. F of 1 equals 2 minus 4 minus 3. F of 1 equals negative 5. The vertex is the point (1, negative 5). The y-intercept occurs when x equals 0. Find f of 0 by substituting x equals 0 into the function. F of 0 equals 2 times 0 squared minus 4 times 0 minus 3. F of 0 equals negative 3. The y-intercept is the point (0, negative 3). The point (0, negative 3) is 1 unit to the left of the line of symmetry. The point 1 unit to the right of the line of symmetry is (2, negative 3). The point (2, negative 3) is symmetric to the y-intercept across the axis of symmetry. The x-intercept occurs when y equals 0. Set f of x equal to 0. 0 equals 2 x squared minus 4 x minus 3. Use the Quadratic Formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Substitute the values for a, b, and c. X equals the quotient of the expression the opposite of negative 4 plus or minus the square root of the difference negative 4 squared minus the product 4 times 2 times negative 3 divided by the product 2 times 2. Simplify. X equals the quotient of the expression 4 plus or minus the square root of the sum 16 plus 24 divided by 4. Simplify inside the radical to get the quotient of 4 plus or minus square root 40 and 4. Simplify the radical. X equals the quotient of the expression 4 plus or minus 2 square root 10 and 4. Factor the GCF. X equalasa the quotient of the product 2 times the expression 2 plus or minus square root 10 divided by 4. Remove common factors. x equals the quotient of 2 plus or minus square root 10 and 2. Write as two equations. The first is x equals the quotient 2 plus square root 10 divided by 2, approximately 2.5. The second solution is x equals the quotient 2 minus square root 10 divided by 2, approximately negative 0.6. The approximate values of the x-intercepts are (2.5, 0) and (negative 0.6, 0). Graph the parabola using the points found. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals 1is graphed on the grid. The points (0, negative 3), (1, negative 5) and (2, negative 3) are plotted and connected to show an upward-opening parabola.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145731003\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is 2, the parabola opens upward. <span data-type=\"media\" id=\"fs-id1169147960646\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"right\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To find the equation of the axis of symmetry, use<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d0499e5d51ece29865924215982302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145662541\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148234442\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230977\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The equation of the axis of<\/strong><\/p>\n<div data-type=\"newline\"><\/div>\n<p><strong data-effect=\"bold\">symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29163feacef7bfd88b9b5d136f8fef91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The vertex is on the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145728428\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91ba6dcc1b657dfb8a4f15162ee96b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148226086\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148233332\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147878532\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The vertex is<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65261358a1796b5039acb865fda2665a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">y<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147831886\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d73c012bc8b5a6f560bea30840502b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145730199\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147906862\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The <em data-effect=\"italics\">y<\/em>-intercept is<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a02f2acb07c1c1b796ba6e9846dec18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-017f20ea0c6fda3470cedb20ea0b5537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> is one unit to the left of the line of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>symmetry.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Point symmetric to the<\/strong><\/p>\n<div data-type=\"newline\"><\/div>\n<p><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercept is<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1564d1d2328bb6bd9e7b30e6d573d2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The point one unit to the right of the line of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1564d1d2328bb6bd9e7b30e6d573d2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The <em data-effect=\"italics\">x<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e8ef70615fdaee8588017ac1fdd2da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147845616\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145731996\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Quadratic Formula.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147817004\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56bcbb839cd565be266206e54cab5663_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"29\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9642cdcdda4b3ba04e90b50a66ecb92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145728073\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145639761\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify inside the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145664488\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145666940\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the GCF.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145670110\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147816256\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write as two equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147854659\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022v_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Approximate the values.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148230462\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022w_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">The approximate values of the<\/strong><\/p>\n<div data-type=\"newline\"><\/div>\n<p><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts are<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e668f238be9e938c40e815c8daafd20f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#46;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><strong data-effect=\"bold\">and<\/strong><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c7d7e7951438bf11451cbe631400c50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola using the points found.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148231101\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_022x_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148234549\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148234553\">\n<div data-type=\"problem\" id=\"fs-id1169148234555\">\n<p id=\"fs-id1169148234558\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = 5<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 10<em data-effect=\"italics\">x<\/em> + 3 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145666849\"><span data-type=\"media\" id=\"fs-id1169145666854\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 2). The y-intercept of the parabola is the point (0, 3). The x-intercepts of the parabola are approximately (negative 1.6, 0) and (negative 0.4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 2). The y-intercept of the parabola is the point (0, 3). The x-intercepts of the parabola are approximately (negative 1.6, 0) and (negative 0.4, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147950673\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147950678\">\n<div data-type=\"problem\" id=\"fs-id1169147950680\">\n<p id=\"fs-id1169147950682\">Graph <em data-effect=\"italics\">f<\/em> (<em data-effect=\"italics\">x<\/em>) = \u22123<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 6<em data-effect=\"italics\">x<\/em> + 5 by using its properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145663686\"><span data-type=\"media\" id=\"fs-id1169145663690\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, 8). The y-intercept of the parabola is the point (0, 5). The x-intercepts of the parabola are approximately (negative 2.6, 0) and (0.6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, 8). The y-intercept of the parabola is the point (0, 5). The x-intercepts of the parabola are approximately (negative 2.6, 0) and (0.6, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147979390\">\n<h3 data-type=\"title\">Solve Maximum and Minimum Applications<\/h3>\n<p id=\"fs-id1169147979395\">Knowing that the <span data-type=\"term\" class=\"no-emphasis\">vertex<\/span> of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic function. The <em data-effect=\"italics\">y<\/em>-coordinate of the vertex is the <span data-type=\"term\" class=\"no-emphasis\">minimum<\/span> value of a parabola that opens upward. It is the <span data-type=\"term\" class=\"no-emphasis\">maximum<\/span> value of a parabola that opens downward. See <a href=\"#CNX_IntAlg_Figure_09_06_023\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_09_06_023\"><span data-type=\"media\" id=\"fs-id1169145664103\" data-alt=\"This figure shows 2 graphs side-by-side. The left graph shows a downward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label maximum. The right graph shows an upward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label minimum.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 graphs side-by-side. The left graph shows a downward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label maximum. The right graph shows an upward opening parabola plotted in the x y-plane. An arrow points to the vertex with the label minimum.\" \/><\/span><\/div>\n<div data-type=\"note\" id=\"fs-id1169145663361\">\n<div data-type=\"title\">Minimum or Maximum Values of a Quadratic Function<\/div>\n<p id=\"fs-id1169145663367\">The <strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-coordinate of the vertex<\/strong> of the graph of a quadratic function is the<\/p>\n<ul id=\"fs-id1169145667351\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">minimum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">upward<\/em>.<\/li>\n<li><em data-effect=\"italics\">maximum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">downward<\/em>.<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169148234337\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148234339\">\n<div data-type=\"problem\" id=\"fs-id1169148234341\">\n<p>Find the minimum or maximum value of the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65e9a49844f1e45dd76c96487513af4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148232836\">\n<table id=\"fs-id1169148232839\" class=\"unnumbered unstyled can-break\" summary=\"Start with the function f of x equals x squared plus 2 x minus 8. Since a is positive, the parabola opens upward. The quadratic equation has a minimum. To find the axis of symmetry, find x equals negative b divided by the product 2 a. Substitute values to yield x equals negative 2 divded by the product 2 times 1. This simplifies to x equals negative 1. The axis of symmetry is x equals negative 1. The vertex is on the line x equals negative 1. To find the vertex, find f of negative 1. F of x equals x squared plus 2 x minus 8. F of negative 1 equals negative 1squared plus 2 times negative 1 minus 8. F of negative 1 equals 1 minus 2 minus 8. F of negative 1 equals negative 9. The vertex is the point (negative 1, negative 9). Since the parabola has a minimum, the y-coordinate of the vertex is the minimum y-value of the quadratic equation. The minimum value of the quadratic is negative 9 and it occurs when x = negative 1. Show the graph to verify the result. A graph shows an x y-coordinate grid. X values range from negative 10 to 10 and y values ranges from negative 10 to 10. The vertical line x equals negative 1 is graphed on the grid. The graph shows an upward-opening parabola with vertex (negative 1, negative 9).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145733011\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">a<\/em> is positive, the parabola opens upward.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The quadratic equation has a minimum.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the equation of the axis of symmetry.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148231798\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148054218\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145729978\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The equation of the axis of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The vertex is on the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147981736\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-134041031ac2f255a6139c40c1ff81ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148232232\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145733322\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147905335\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d713689ec966ac6e37c89486312ba573_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since the parabola has a minimum, the <em data-effect=\"italics\">y<\/em>-coordinate of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the vertex is the minimum <em data-effect=\"italics\">y<\/em>-value of the quadratic<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The minimum value of the quadratic is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and it<\/p>\n<div data-type=\"newline\"><\/div>\n<p>occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169145729683\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_024i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Show the graph to verify the result.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145729053\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145729057\">\n<div data-type=\"problem\" id=\"fs-id1169145729059\">\n<p id=\"fs-id1169145729062\">Find the maximum or minimum value of the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ea2527682d6b860ec5f558c50fdd977_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145667472\">\n<p id=\"fs-id1169145667474\">The minimum value of the quadratic function is \u22124 and it occurs when <em data-effect=\"italics\">x<\/em> = 4.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147979779\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147979783\">\n<div data-type=\"problem\" id=\"fs-id1169147979785\">\n<p id=\"fs-id1169147979788\">Find the maximum or minimum value of the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b2a8beb8b420720a35231eddac66377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;&#45;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"195\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147960121\">\n<p id=\"fs-id1169147960123\">The maximum value of the quadratic function is 5 and it occurs when <em data-effect=\"italics\">x<\/em> = 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147950049\">We have used the formula<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147950052\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36a64205daa0fa08e2d4e1423a7a3537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#116;&#43;&#123;&#104;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"186\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1169145728967\">to calculate the height in feet, <em data-effect=\"italics\">h<\/em> , of an object shot upwards into the air with initial velocity, <em data-effect=\"italics\">v<\/em><sub>0<\/sub>, after <em data-effect=\"italics\">t<\/em> seconds .<\/p>\n<p id=\"fs-id1169145730295\">This formula is a quadratic function, so its graph is a parabola. By solving for the coordinates of the vertex (<em data-effect=\"italics\">t, h<\/em>), we can find how long it will take the object to reach its maximum height. Then we can calculate the maximum height.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145730308\">\n<div data-type=\"problem\" id=\"fs-id1169147949312\">\n<p id=\"fs-id1169147949314\">The quadratic equation <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 176<em data-effect=\"italics\">t<\/em> + 4 models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet.<\/p>\n<p id=\"fs-id1169145665091\"><span class=\"token\">\u24d0<\/span> How many seconds will it take the volleyball to reach its maximum height? <span class=\"token\">\u24d1<\/span> Find the maximum height of the volleyball.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145665105\">\n<p id=\"fs-id1169145665107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b502acdf0d9775f033fc3c3096fe4069_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#54;&#116;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#44;&#32;&#116;&#104;&#101;&#32;&#112;&#97;&#114;&#97;&#98;&#111;&#108;&#97;&#32;&#111;&#112;&#101;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#111;&#119;&#110;&#119;&#97;&#114;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#32;&#104;&#97;&#115;&#32;&#97;&#32;&#109;&#97;&#120;&#105;&#109;&#117;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"629\" style=\"vertical-align: -36px;\" \/><\/p>\n<p id=\"fs-id1169147959899\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee83cf9b304f9bf7f3f716b778e1a04b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#97;&#120;&#105;&#115;&#32;&#111;&#102;&#32;&#115;&#121;&#109;&#109;&#101;&#116;&#114;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#54;&#125;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#61;&#53;&#46;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#97;&#120;&#105;&#115;&#32;&#111;&#102;&#32;&#115;&#121;&#109;&#109;&#101;&#116;&#114;&#121;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#116;&#61;&#53;&#46;&#53;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#118;&#101;&#114;&#116;&#101;&#120;&#32;&#105;&#115;&#32;&#111;&#110;&#32;&#116;&#104;&#101;&#32;&#108;&#105;&#110;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#61;&#53;&#46;&#53;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#109;&#97;&#120;&#105;&#109;&#117;&#109;&#32;&#111;&#99;&#99;&#117;&#114;&#115;&#32;&#119;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#61;&#53;&#46;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#101;&#99;&#111;&#110;&#100;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"771\" style=\"vertical-align: -68px;\" \/><\/p>\n<p id=\"fs-id1169147960752\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-665b9eaedc07e86e1abf209973376b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#54;&#116;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#97;&#32;&#99;&#97;&#108;&#99;&#117;&#108;&#97;&#116;&#111;&#114;&#32;&#116;&#111;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#56;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#118;&#101;&#114;&#116;&#101;&#120;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#53;&#44;&#52;&#56;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"708\" style=\"vertical-align: -38px;\" \/><\/p>\n<p id=\"fs-id1169145661103\">Since the parabola has a maximum, the <em data-effect=\"italics\">h<\/em>-coordinate of the vertex is the maximum value of the quadratic function.<\/p>\n<p id=\"fs-id1169145733203\">The maximum value of the quadratic is 488 feet and it occurs when <em data-effect=\"italics\">t<\/em> = 5.5 seconds.<\/p>\n<p id=\"fs-id1169145733211\">After 5.5 seconds, the volleyball will reach its maximum height of 488 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145733218\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145733222\">\n<div data-type=\"problem\" id=\"fs-id1169145733224\">\n<p id=\"fs-id1169147981616\">Solve, rounding answers to the nearest tenth.<\/p>\n<p id=\"fs-id1169147981619\">The quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 128<em data-effect=\"italics\">t<\/em> + 32 is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft\/sec. How long will it take for the stone to reach its maximum height? What is the maximum height?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147980700\">\n<p id=\"fs-id1169147980702\">It will take 4 seconds for the stone to reach its maximum height of 288 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147980709\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147980713\">\n<div data-type=\"problem\" id=\"fs-id1169147980715\">\n<p id=\"fs-id1169147980717\">A path of a toy rocket thrown upward from the ground at a rate of 208 ft\/sec is modeled by the quadratic function of <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 208<em data-effect=\"italics\">t<\/em>. When will the rocket reach its maximum height? What will be the maximum height?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148233832\">\n<p id=\"fs-id1169148233834\">It will 6.5 seconds for the rocket to reach its maximum height of 676 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148233840\" class=\"media-2\">\n<p id=\"fs-id1169145640053\">Access these online resources for additional instruction and practice with graphing quadratic functions using properties.<\/p>\n<ul id=\"fs-id1163872563321\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct1\">Quadratic Functions: Axis of Symmetry and Vertex<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct2\">Finding x- and y-intercepts of a Quadratic Function<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct3\">Graphing Quadratic Functions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct4\">Solve Maxiumum or Minimum Applications<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFunct5\">Quadratic Applications: Minimum and Maximum<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169147982432\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169147982440\" data-bullet-style=\"bullet\">\n<li>Parabola Orientation\n<ul id=\"fs-id1169147982448\" data-bullet-style=\"bullet\">\n<li>For the graph of the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74e95bcdad9817005268179a7abe0817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"161\" style=\"vertical-align: -4px;\" \/> if\n<ul id=\"fs-id1169148229448\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">a<\/em> &gt; 0, the parabola opens upward.<\/li>\n<li><em data-effect=\"italics\">a<\/em> &lt; 0, the parabola opens downward.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Axis of Symmetry and Vertex of a Parabola The graph of the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ff39a37a230408f7c9a6410a33dbe1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/> is a parabola where:\n<ul id=\"fs-id1169145730086\" data-bullet-style=\"bullet\">\n<li>the axis of symmetry is the vertical line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfab3cdadd0b3fd4994e4a6616037c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>the vertex is a point on the axis of symmetry, so its <em data-effect=\"italics\">x<\/em>-coordinate is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ab25f94cf91879c16c63a8acc445d48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>the <em data-effect=\"italics\">y<\/em>-coordinate of the vertex is found by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d0499e5d51ece29865924215982302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/> into the quadratic equation.<\/li>\n<\/ul>\n<\/li>\n<li>Find the Intercepts of a Parabola\n<ul id=\"fs-id1169145665517\" data-bullet-style=\"bullet\">\n<li>To find the intercepts of a parabola whose function is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6064af0045e2b14aad7aa1166556bad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"166\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1171791466640\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ac14170e97c1ca8f66a5947b1316438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"524\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>How to graph a quadratic function using properties.\n<ol id=\"fs-id1169145663066\" type=\"1\" class=\"stepwise\">\n<li>Determine whether the parabola opens upward or downward.<\/li>\n<li>Find the equation of the axis of symmetry.<\/li>\n<li>Find the vertex.<\/li>\n<li>Find the <em data-effect=\"italics\">y<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept across the axis of symmetry.<\/li>\n<li>Find the <em data-effect=\"italics\">x<\/em>-intercepts. Find additional points if needed.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<\/li>\n<li>Minimum or Maximum Values of a Quadratic Equation\n<ul id=\"fs-id1169145663471\" data-bullet-style=\"bullet\">\n<li>The <em data-effect=\"italics\">y<\/em>-coordinate of the vertex of the graph of a quadratic equation is the<\/li>\n<li><em data-effect=\"italics\">minimum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">upward<\/em>.<\/li>\n<li><em data-effect=\"italics\">maximum<\/em> value of the quadratic equation if the parabola opens <em data-effect=\"italics\">downward<\/em>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148054108\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169148054112\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169148054119\"><strong data-effect=\"bold\">Recognize the Graph of a Quadratic Function<\/strong><\/p>\n<p id=\"fs-id1169148054126\">In the following exercises, graph the functions by plotting points.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148054129\">\n<div data-type=\"problem\" id=\"fs-id1169148054131\">\n<p id=\"fs-id1169148054133\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79d33ba440ae182f1cdc0f6b2ee89f49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147978477\"><span data-type=\"media\" id=\"fs-id1169148235046\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148235063\">\n<div data-type=\"problem\" id=\"fs-id1169148235065\">\n<p id=\"fs-id1169148235067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27d02dbda05c8bfd937503f2f2145f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147982855\">\n<div data-type=\"problem\" id=\"fs-id1169147982858\">\n<p id=\"fs-id1169147982860\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4867154db2d25e5817c7f7f1a30c34d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145666124\"><span data-type=\"media\" id=\"fs-id1169145666129\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145664887\">\n<div data-type=\"problem\" id=\"fs-id1169145664889\">\n<p id=\"fs-id1169145664891\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b888ebb855c0a08eed63ec6e59a38cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147982242\">For each of the following exercises, determine if the parabola opens up or down.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147982245\">\n<div data-type=\"problem\" id=\"fs-id1169145663558\">\n<p id=\"fs-id1169145663560\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5582760a212eb6e115493034278a777_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-171973da5391040e321a978609e04797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145660490\">\n<p id=\"fs-id1169147949006\"><span class=\"token\">\u24d0<\/span> down <span class=\"token\">\u24d1<\/span> up<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147949020\">\n<div data-type=\"problem\" id=\"fs-id1169147949023\">\n<p id=\"fs-id1169147949025\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dda06c11af3576c71f6cbc369be5643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63a7d5462158d89857d0be00a3fc345a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#120;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"191\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147816892\">\n<div data-type=\"problem\" id=\"fs-id1169147816894\">\n<p id=\"fs-id1169147816896\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8376aa9e1333e16574ad18548321c3b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d054833f4dc42b7c6515fb978ccb335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169147987922\">\n<p id=\"fs-id1169147987925\"><span class=\"token\">\u24d0<\/span> down <span class=\"token\">\u24d1<\/span> up<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147987594\">\n<div data-type=\"problem\" id=\"fs-id1169147987596\">\n<p id=\"fs-id1169147987598\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4fd3c1c44aebe8962e1e7aea9a560294_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20e064fdb405dacdea995ba4ac6cd118_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169147961912\"><strong data-effect=\"bold\">Find the Axis of Symmetry and Vertex of a Parabola<\/strong><\/p>\n<p id=\"fs-id1169147961918\">In the following functions, find <span class=\"token\">\u24d0<\/span> the equation of the axis of symmetry and <span class=\"token\">\u24d1<\/span> the vertex of its graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145730691\">\n<div data-type=\"problem\" id=\"fs-id1169145730693\">\n<p id=\"fs-id1169145730695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fac59ae2b37fe5c6c8003d0c7f36652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147905125\">\n<p id=\"fs-id1169147905127\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/>; <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9380a0b1127e1e014e21066645956ad7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#49;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147980905\">\n<div data-type=\"problem\" id=\"fs-id1169147980908\">\n<p id=\"fs-id1169147980910\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2aae04f75b376bb53f88f9bd8f9b124a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145732722\">\n<div data-type=\"problem\" id=\"fs-id1169145732724\">\n<p id=\"fs-id1169145732726\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c01b88b5d2af9952ee24dde34f86bdf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148231894\">\n<p id=\"fs-id1169148231896\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/>; <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce830c2dfa3b70e2906cf4d1b7248973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147949126\">\n<div data-type=\"problem\" id=\"fs-id1169147949128\">\n<p id=\"fs-id1169147949130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e68de16b16b4d106682d39eb18a2484_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148234953\"><strong data-effect=\"bold\">Find the Intercepts of a Parabola<\/strong><\/p>\n<p id=\"fs-id1169148234959\">In the following exercises, find the intercepts of the parabola whose function is given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148234963\">\n<div data-type=\"problem\" id=\"fs-id1169148234966\">\n<p id=\"fs-id1169148234968\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c2d7cdc9f8b28160e3eb046e1814163_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147978967\">\n<p id=\"fs-id1169147978969\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66b975e76001a761f8bccafa711f6482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercept <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1acdf5b555233d4aa34d71f59987689_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197891\">\n<div data-type=\"problem\" id=\"fs-id1169148197893\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb4dd2426b3971c1cb56311e5f09c860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147845711\">\n<div data-type=\"problem\" id=\"fs-id1169147845714\">\n<p id=\"fs-id1169147845716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b6050240289bd8b39da37af7ab06f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145660279\">\n<p id=\"fs-id1169145660281\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7f5a90906e45727cea1a10c6ed318b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercept <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e4dfe2737ca7caa6a62163add3bb301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148231489\">\n<div data-type=\"problem\" id=\"fs-id1169148231491\">\n<p id=\"fs-id1169148231493\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e31b988934069c51bdc795e191df093_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145730909\">\n<div data-type=\"problem\" id=\"fs-id1169145730912\">\n<p id=\"fs-id1169145730914\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d355f60c1a91965787add43f41d315a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145670318\">\n<p><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbb69aceba0b65bc4f4681e78e375084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercept: none<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147988342\">\n<div data-type=\"problem\" id=\"fs-id1169147988344\">\n<p id=\"fs-id1169147988346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-023836cef878776750bf145fc81192e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145665816\">\n<div data-type=\"problem\" id=\"fs-id1169145665819\">\n<p id=\"fs-id1169145665821\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fc6d45e8f6619a7ed8f16630fa41098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145659962\">\n<p id=\"fs-id1169145659964\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-414c1c1fa1ac37ce78d44ba32b1bf133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercept: none<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145670773\">\n<div data-type=\"problem\" id=\"fs-id1169145670775\">\n<p id=\"fs-id1169145670778\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b6050240289bd8b39da37af7ab06f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147799883\">\n<div data-type=\"problem\" id=\"fs-id1169147799885\">\n<p id=\"fs-id1169147799887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b88285026d1f5ea8a385df084edd818b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#43;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147866283\">\n<p id=\"fs-id1169147866285\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a7adf920ab9cf2623d641ae07501fe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercept <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d03f220a14669e7447fa24af6cc39b37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147851717\">\n<div data-type=\"problem\" id=\"fs-id1169147851719\">\n<p id=\"fs-id1169147851721\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d25b92f4b9be4736687860866ebfc15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147834444\">\n<div data-type=\"problem\" id=\"fs-id1169148053906\">\n<p id=\"fs-id1169148053908\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d70578d7d4c821e294c046d5a39a184d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148053944\">\n<p id=\"fs-id1169148053946\"><em data-effect=\"italics\">y<\/em>-intercept: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb6b502574c6ba2923930bfe098f8c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">x<\/em>-intercept <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ba9cc2a7f12e65a6b3de8f34bcc16e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145640476\">\n<div data-type=\"problem\" id=\"fs-id1169145640478\">\n<p id=\"fs-id1169147959363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad7ca4be480e75bd778106f884b7c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147865984\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Properties<\/strong><\/p>\n<p id=\"fs-id1169147865990\">In the following exercises, graph the function by using its properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147865994\">\n<div data-type=\"problem\" id=\"fs-id1169147865996\">\n<p id=\"fs-id1169147865998\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7764e8629197413d70a63c624bc5a664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147848694\"><span data-type=\"media\" id=\"fs-id1169147848698\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147848716\">\n<div data-type=\"problem\" id=\"fs-id1169147848718\">\n<p id=\"fs-id1169147848720\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1677c89a2469ba2687ac8f1ec0d9a646_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147848778\">\n<div data-type=\"problem\" id=\"fs-id1169147848780\">\n<p id=\"fs-id1169147848782\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b5881b5e6ee44e3322983780b195d9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147848818\"><span data-type=\"media\" id=\"fs-id1169147848822\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147848840\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f1b9b782091679ba0aa268bfd172d41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148080663\">\n<div data-type=\"problem\" id=\"fs-id1169148080665\">\n<p id=\"fs-id1169148080667\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97f3e1342b01db9122b77790d2fd6d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148080705\"><span data-type=\"media\" id=\"fs-id1169148080710\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148080728\">\n<div data-type=\"problem\" id=\"fs-id1169148080730\">\n<p id=\"fs-id1169148080732\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4af2e80e5072314de828d998cb00c262_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197394\">\n<div data-type=\"problem\" id=\"fs-id1169148197397\">\n<p id=\"fs-id1169148197399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e4d5ad6a1c51e6225ca932ad7df9244_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148197437\"><span data-type=\"media\" id=\"fs-id1169148197441\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_321_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197459\">\n<div data-type=\"problem\" id=\"fs-id1169148197461\">\n<p id=\"fs-id1169148197463\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97e1a71f50cf65288fc469fc0d3309d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197517\">\n<div data-type=\"problem\" id=\"fs-id1169148197519\">\n<p id=\"fs-id1169148197521\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24f3f708ca8a1fa02f814955fba31c4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148197559\"><span data-type=\"media\" id=\"fs-id1169148197563\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_323_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197582\">\n<div data-type=\"problem\" id=\"fs-id1169148197584\">\n<p id=\"fs-id1169148197586\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f15a9be6bdd54556c4507265027a79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197647\">\n<div data-type=\"problem\" id=\"fs-id1169148197649\">\n<p id=\"fs-id1169148197651\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bd6b81f6a945255621e2d9669189963_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148197690\"><span data-type=\"media\" id=\"fs-id1169148197694\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_325_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197712\">\n<div data-type=\"problem\" id=\"fs-id1169148197714\">\n<p id=\"fs-id1169148197716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0aa8f7d39715a0a134ef6d8664ef362b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148197778\">\n<div data-type=\"problem\" id=\"fs-id1169148197780\">\n<p id=\"fs-id1169148197782\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6565b7b214cd33873fcbea9885d72441_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145667779\"><span data-type=\"media\" id=\"fs-id1169145667783\" data-alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_327_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145667802\">\n<div data-type=\"problem\" id=\"fs-id1169145667804\">\n<p id=\"fs-id1169145667806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4a00ebc4eea9ee54b5fdeff90efcb25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145667865\">\n<div data-type=\"problem\" id=\"fs-id1169145667867\">\n<p id=\"fs-id1169145667869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d387e1230499bacffd130fa18193c82f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145667907\"><span data-type=\"media\" id=\"fs-id1169145667911\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_329_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145667930\">\n<div data-type=\"problem\" id=\"fs-id1169145667932\">\n<p id=\"fs-id1169145667934\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9a42477f6ecf5f58d5ccb3d1e60f82f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"191\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145667995\">\n<div data-type=\"problem\" id=\"fs-id1169145667997\">\n<p id=\"fs-id1169145667999\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26b4e0379d82ae427ec54a998e6595fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668037\"><span data-type=\"media\" id=\"fs-id1169145668041\" data-alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_331_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668060\">\n<div data-type=\"problem\" id=\"fs-id1169145668062\">\n<p id=\"fs-id1169145668064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6878d6a93409d773441810417e63e8f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145668125\"><strong data-effect=\"bold\">Solve Maximum and Minimum Applications<\/strong><\/p>\n<p id=\"fs-id1169145668132\">In the following exercises, find the maximum or minimum value of each function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145668135\">\n<div data-type=\"problem\" id=\"fs-id1169145668137\">\n<p id=\"fs-id1169145668139\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c93f0200bd4a49c7d6856083166ef1d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668172\">\n<p id=\"fs-id1169145668174\">The minimum value is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a12e48d665ce6c2155b798ff369fd5d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc1f9d48d68a03fa60a0dd5306bfcb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668206\">\n<div data-type=\"problem\" id=\"fs-id1169145668208\">\n<p id=\"fs-id1169145668211\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5970150a99a2f667657d4d6e6858d2fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668260\">\n<div data-type=\"problem\" id=\"fs-id1169145668262\">\n<p id=\"fs-id1169145668264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-287a52a8bcf70c9a9c4dff851eda48c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668290\">\n<p id=\"fs-id1169145668292\">The maximum value is 6 when <em data-effect=\"italics\">x<\/em> = 3.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668303\">\n<div data-type=\"problem\" id=\"fs-id1169145668305\">\n<p id=\"fs-id1169145668307\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a54c30be3f364a6b550aeeea978baf85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668348\">\n<div data-type=\"problem\" id=\"fs-id1169145668350\">\n<p id=\"fs-id1169145668352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a2771993c9b77578b97d49f392116cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668373\">\n<p id=\"fs-id1169145668375\">The maximum value is 16 when <em data-effect=\"italics\">x<\/em> = 0.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668386\">\n<div data-type=\"problem\" id=\"fs-id1169145668388\">\n<p id=\"fs-id1169145668391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3f14a42ff33253750f1ab9713704169_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145668425\">In the following exercises, solve. Round answers to the nearest tenth.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145668428\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668430\">\n<p id=\"fs-id1169145668432\">An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 168<em data-effect=\"italics\">t<\/em> + 45 find how long it will take the arrow to reach its maximum height, and then find the maximum height.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668462\">\n<p id=\"fs-id1169145668464\">In 5.3 sec the arrow will reach maximum height of 486 ft.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668469\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668471\">\n<p id=\"fs-id1169145668474\">A stone is thrown vertically upward from a platform that is 20 feet height at a rate of 160 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 160<em data-effect=\"italics\">t<\/em> + 20 to find how long it will take the stone to reach its maximum height, and then find the maximum height.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668511\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668513\">\n<p id=\"fs-id1169145668515\">A ball is thrown vertically upward from the ground with an initial velocity of 109 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 109<em data-effect=\"italics\">t<\/em> + 0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668545\">\n<p id=\"fs-id1169145668547\">In 3.4 seconds the ball will reach its maximum height of 185.6 feet.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668552\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668554\">\n<p id=\"fs-id1169145668556\">A ball is thrown vertically upward from the ground with an initial velocity of 122 ft\/sec. Use the quadratic function <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">t<\/em>) = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 122<em data-effect=\"italics\">t<\/em> + 0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668593\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668595\">\n<p id=\"fs-id1169145668597\">A computer store owner estimates that by charging <em data-effect=\"italics\">x<\/em> dollars each for a certain computer, he can sell 40 \u2212 <em data-effect=\"italics\">x<\/em> computers each week. The quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +40<em data-effect=\"italics\">x<\/em> is used to find the revenue, <em data-effect=\"italics\">R<\/em>, received when the selling price of a computer is <em data-effect=\"italics\">x<\/em>, Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145668650\">\n<p id=\"fs-id1169145668652\">20 computers will give the maximum of ?400 in receipts.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668658\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668660\">\n<p id=\"fs-id1169145668662\">A retailer who sells backpacks estimates that by selling them for <em data-effect=\"italics\">x<\/em> dollars each, he will be able to sell 100 \u2212 <em data-effect=\"italics\">x<\/em> backpacks a month. The quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +100<em data-effect=\"italics\">x<\/em> is used to find the <em data-effect=\"italics\">R<\/em>, received when the selling price of a backpack is <em data-effect=\"italics\">x<\/em>. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145668721\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169145668723\">\n<p id=\"fs-id1169145668725\">A retailer who sells fashion boots estimates that by selling them for <em data-effect=\"italics\">x<\/em> dollars each, he will be able to sell 70 \u2212 <em data-effect=\"italics\">x<\/em> boots a week. Use the quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +70<em data-effect=\"italics\">x<\/em> to find the revenue received when the average selling price of a pair of fashion boots is <em data-effect=\"italics\">x<\/em>. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue per day.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147946274\">\n<p id=\"fs-id1169147946276\">He will be able to sell 35 pairs of boots at the maximum revenue of ?1,225.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946281\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147946283\">\n<p id=\"fs-id1169147946285\">A cell phone company estimates that by charging <em data-effect=\"italics\">x<\/em> dollars each for a certain cell phone, they can sell 8 \u2212 <em data-effect=\"italics\">x<\/em> cell phones per day. Use the quadratic function <em data-effect=\"italics\">R<\/em>(<em data-effect=\"italics\">x<\/em>) = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> +8<em data-effect=\"italics\">x<\/em> to find the revenue received per day when the selling price of a cell phone is <em data-effect=\"italics\">x<\/em>. Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946338\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147946340\">\n<p id=\"fs-id1169147946342\">A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(240 \u2212 2<em data-effect=\"italics\">x<\/em>) gives the area of the corral, <em data-effect=\"italics\">A<\/em>, for the length, <em data-effect=\"italics\">x<\/em>, of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147946382\">\n<p id=\"fs-id1169147946384\">The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 square feet.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946390\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147946392\">\n<p id=\"fs-id1169147946394\">A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic function <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(100 \u2212 2<em data-effect=\"italics\">x<\/em>) gives the area, <em data-effect=\"italics\">A<\/em>, of the dog run for the length, <em data-effect=\"italics\">x<\/em>, of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946442\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147946445\">\n<p id=\"fs-id1169147946447\">A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the \u201cwalls.\u201d He wants to maximize the area using 80 feet of fencing. The quadratic function <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(80 \u2212 2<em data-effect=\"italics\">x<\/em>) gives the area of the patio, where <em data-effect=\"italics\">x<\/em> is the width of one side. Find the maximum area of the patio.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147946480\">\n<p id=\"fs-id1169147946482\">The maximum area of the patio is 800 feet.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946488\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169147946490\">\n<p id=\"fs-id1169147946492\">A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them 300 feet of fencing to use to enclose part of their backyard. Use the quadratic function <em data-effect=\"italics\">A<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">x<\/em>(300 \u2212 2<em data-effect=\"italics\">x<\/em>) to determine the maximum area of the fenced in yard.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169147946529\">\n<h4 data-type=\"title\">Writing Exercise<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169147946536\">\n<div data-type=\"problem\" id=\"fs-id1169147946539\">\n<p id=\"fs-id1169147946541\">How do the graphs of the functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a294c15a253d64d5494761235c0cd07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/> differ? We graphed them at the start of this section. What is the difference between their graphs? How are their graphs the same?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147946591\">\n<p id=\"fs-id1169147946594\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946599\">\n<div data-type=\"problem\" id=\"fs-id1169147946601\">\n<p id=\"fs-id1169147946603\">Explain the process of finding the vertex of a parabola.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946615\">\n<div data-type=\"problem\" id=\"fs-id1169147946618\">\n<p id=\"fs-id1169147946620\">Explain how to find the intercepts of a parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147946624\">\n<p id=\"fs-id1169147946627\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147946632\">\n<div data-type=\"problem\" id=\"fs-id1169147946634\">\n<p id=\"fs-id1169147946636\">How can you use the discriminant when you are graphing a quadratic function?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169147946650\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1169147946655\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147946678\" data-alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can recognize the graph of a quadratic equation.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the axis of symmetry and vertex of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the intercepts of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can graph quadratic equations in two variables.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve maximum and minimum applications.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can recognize the graph of a quadratic equation.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the axis of symmetry and vertex of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can find the intercepts of a parabola.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can graph quadratic equations in two variables.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve maximum and minimum applications.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\" \/><\/span><\/p>\n<p id=\"fs-id1169147946672\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1169147946698\">\n<dt>quadratic function<\/dt>\n<dd id=\"fs-id1169147946703\">A quadratic function, where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d28cb478bd8b6abe7d9573551313d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> is a function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a0b6fa82f59c470088d6e34f484552d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4160","chapter","type-chapter","status-publish","hentry"],"part":3677,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4160\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3677"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=4160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=4160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=4160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=4160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}