{"id":15425,"date":"2019-09-05T12:07:25","date_gmt":"2019-09-05T16:07:25","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/parabolas-2\/"},"modified":"2019-09-05T12:07:25","modified_gmt":"2019-09-05T16:07:25","slug":"parabolas-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/parabolas-2\/","title":{"raw":"Parabolas","rendered":"Parabolas"},"content":{"raw":"[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>Graph vertical parabolas<\/li><li>Graph horizontal parabolas<\/li><li>Solve applications with parabolas<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1163872563460\" class=\"be-prepared\"><p id=\"fs-id1163872803529\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1163872549876\" type=\"1\"><li>Graph: \\(y=-3{x}^{2}+12x-12.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/3de1ae34-6225-4751-be75-a17b3e0e665b#fs-id1169147828812\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve by completing the square: \\({x}^{2}-6x+6=0.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/3e7f365d-4885-4b7a-bb2a-4358cbc00d2e#fs-id1167829894368\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Write in standard form: \\(y=3{x}^{2}-6x+5.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b#fs-id1169149374763\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872698598\"><h3 data-type=\"title\">Graph Vertical Parabolas<\/h3><p id=\"fs-id1163872705754\">The next conic section we will look at is a <span data-type=\"term\">parabola<\/span>. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the <strong data-effect=\"bold\">focus,<\/strong> and the fixed line is called the <strong data-effect=\"bold\">directrix<\/strong> of the parabola.<\/p><span data-type=\"media\" id=\"fs-id1163872576446\" data-alt=\"This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.\" \/><\/span><div data-type=\"note\" id=\"fs-id1163872392082\"><div data-type=\"title\">Parabola<\/div><p id=\"fs-id1163872467441\">A <strong data-effect=\"bold\">parabola<\/strong> is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the <strong data-effect=\"bold\">focus,<\/strong> and the fixed line is called the <strong data-effect=\"bold\">directrix<\/strong> of the parabola.<\/p><span data-type=\"media\" id=\"fs-id1163872008560\" data-alt=\"This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.\" \/><\/span><\/div><p id=\"fs-id1163871995058\">Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.<\/p><table id=\"fs-id1163871908965\" summary=\"This table, titled vertical parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is y equals ax squared plus bx plus c and standard form is y equals a open parentheses x minus h close parentheses squared plus k. Row one: orientation: general form is a greater than 0, up and a less than 0 down. Standard form is the same. Row 2: Axis of symmetry: general form is x equals minus b upon 2a and standard form is x equals h. Row 3: vertex: general form, substitute x equals minus b upon 2a and solve for y; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Vertical Parabolas<\/th><\/tr><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><\/th><th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span>\\(y=a{x}^{2}+bx+c\\)<\/th><th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span>\\(y=a{\\left(x-h\\right)}^{2}+k\\) <\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) up; \\(a&lt;0\\) down<\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) up; \\(a&lt;0\\) down<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(x=-\\frac{b}{2a}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(x=h\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Substitute \\(x=-\\frac{b}{2a}\\) and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercept<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1163872841582\">The graphs show what the parabolas look like when they open up or down. Their position in relation to the <em data-effect=\"italics\">x<\/em>- or <em data-effect=\"italics\">y<\/em>-axis is merely an example.<\/p><span data-type=\"media\" id=\"fs-id1163872407207\" data-alt=\"This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.\" \/><\/span><p id=\"fs-id1163872504389\">To graph a parabola from these forms, we used the following steps.<\/p><div data-type=\"note\" id=\"fs-id1163872743524\" class=\"howto\"><div data-type=\"title\">Graph vertical parabolas \\(\\left(y=a{x}^{2}+bx+c\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\right)\\) using properties.<\/div><ol id=\"fs-id1163872471477\" type=\"1\" class=\"stepwise\"><li>Determine whether the parabola opens upward or downward.<\/li><li>Find 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.<\/li><li>Graph the parabola.<\/li><\/ol><\/div><p id=\"fs-id1163872836555\">The next example reviews the method of graphing a parabola from the general form of its equation.<\/p><div data-type=\"example\" id=\"fs-id1163872531349\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872531116\"><div data-type=\"problem\" id=\"fs-id1163872468763\"><p id=\"fs-id1163872743031\">Graph \\(y=\\text{\u2212}{x}^{2}+6x-8\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872465154\"><table id=\"fs-id1163872557188\" class=\"unnumbered unstyled can-break\" summary=\"The equation is y equals minus x squared plus 6x minus 8. This is of the form y equals ax squared plus bx plus c. Since a is minus 1, the parabola opens downward. To find the axis of symmetry, find x equals minus b upon 2a. Substituting values of b and a, we get x equals 3. This is the axis of symmetry. The vertex is on the line x equals 3. Substituting this value in the equation, we get y equals 1. The vertex is the point 3, 1. The y intercept occurs when x equals 0. Substituting in the equation and simplifying, we get y equals minus 8. The point 0, 8 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 8. The x intercept occurs when y equals 0. We substitute this in the original equation and factor the trinomial. We get x intercepts 4, 0 and 2, 0. Graph the 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-id1163872689672\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004b_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=\"center\"><span data-type=\"media\" id=\"fs-id1163872468262\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><td data-valign=\"top\" data-align=\"center\"><\/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-id1163872574710\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004c_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-id1163872520882\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004d_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-id1163872562025\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004e_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=\"center\">The axis of symmetry is \\(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-id1163872630300\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004f_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=3.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872013436\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(x=3.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872571907\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004h_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-id1163872512977\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004i_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-id1163872568047\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004j_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=\"center\">The vertex is \\(\\left(3,1\\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-id1163872703214\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004k_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-id1163872517696\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004l_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-id1163872465389\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004m_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-id1163871975475\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004n_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=\"center\">The <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(0,-8\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The point \\(\\left(0,-8\\right)\\) is three units to the left of the<span data-type=\"newline\"><br \/><\/span>line of symmetry. The point three units to the<span data-type=\"newline\"><br \/><\/span>right of the line of symmetry is \\(\\left(6,-8\\right).\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Point symmetric to the <em data-effect=\"italics\">y<\/em>-intercept is \\(\\left(6,-8\\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-id1163872840487\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004o_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\">x<\/em>-intercept occurs when \\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872538667\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872561300\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004q_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-id1163872731143\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004r_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-id1163872571007\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004s_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-id1163872512920\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004t_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=\"center\">The <em data-effect=\"italics\">x<\/em>-intercepts are \\(\\left(4,0\\right),\\left(2,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872459332\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872423654\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872450037\"><div data-type=\"problem\" id=\"fs-id1163872419334\"><p id=\"fs-id1163872464625\">Graph \\(y=\\text{\u2212}{x}^{2}+5x-6\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871868912\"><span data-type=\"media\" id=\"fs-id1163872441832\" data-alt=\"This graph shows a parabola opening downward, with x intercepts (2, 0) and (3, 0) and y intercept (0, negative 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward, with x intercepts (2, 0) and (3, 0) and y intercept (0, negative 6).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163871994444\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872502703\"><div data-type=\"problem\" id=\"fs-id1163872505505\"><p id=\"fs-id1163871923684\">Graph \\(y=\\text{\u2212}{x}^{2}+8x-12\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872478588\"><span data-type=\"media\" id=\"fs-id1163872464117\" data-alt=\"This graph shows a parabola opening downward, with vertex (4, 4) and x intercepts (2, 0) and (6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward, with vertex (4, 4) and x intercepts (2, 0) and (6, 0).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163872464760\">The next example reviews the method of graphing a <span data-type=\"term\" class=\"no-emphasis\">parabola<\/span> from the standard form of its equation, \\(y=a{\\left(x-h\\right)}^{2}+k.\\)<\/p><div data-type=\"example\" id=\"fs-id1163872545091\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872420384\"><div data-type=\"problem\" id=\"fs-id1163872467579\"><p id=\"fs-id1163872401104\">Write\\(y=3{x}^{2}-6x+5\\) in standard form and then use properties of standard form to graph the equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872689501\"><table id=\"fs-id1163871857801\" class=\"unnumbered unstyled can-break\" summary=\"The equation is 3 x squared minus 6x plus 5. Rewrite in y equals a open parentheses x minus h close parentheses squared plus k form by completing the square. We rewrite as y equals 3 open parentheses x squared minus 2x plus 1 close parentheses plus 5 minus 3. So y is 3 open parentheses x minus 1 close parentheses squared plus 2. Here, a is 3, h is 1 and k is 2. Since a is 2, the parabola opens upward. The axis of symmetry is x equals 1. The vertex is 1, 2. Find the y intercept by substituting x equal to 0 in the original equation. We get y equal to 5. The y intercept is 0, 5. The point symmetric to it is 2, 5. For finding the x intercept, we substitute y equals 0 in the original equation. We get x equal to 1 plus square root of minus 2 upon 3. The square root of a negative number tells us the solutions are complex numbers. So there are no x intercepts. Graph the parabola.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the function in \\(y=a{\\left(x-h\\right)}^{2}+k\\) form<span data-type=\"newline\"><br \/><\/span>by completing the square.<\/td><td data-valign=\"top\" data-align=\"left\">\\(y=3{x}^{2}-6x+5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(y=3\\left({x}^{2}-2x\\phantom{\\rule{1.6em}{0ex}}\\right)+5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(y=3\\left({x}^{2}-2x+1\\right)+5-3\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(y=3{\\left(x-1\\right)}^{2}+2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\">\\(a=3\\), \\(h=1\\), \\(k=2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(a=2,\\) the parabola opens upward.<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872840988\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The axis of symmetry is \\(x=h.\\)<\/td><td data-valign=\"top\" data-align=\"left\">The axis of symmetry is \\(x=1.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(1,2\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">y<\/em>-intercept by substituting \\(x=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(y=3{\\left(x-1\\right)}^{2}+2\\)<span data-type=\"newline\"><br \/><\/span>\\(y=3\u00b7{0}^{2}-6\u00b70+5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(y=5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-intercept \\(\\left(0,5\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the point symmetric to \\(\\left(0,5\\right)\\) across the axis of symmetry.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\left(2,5\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercepts.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; 3{\\left(x-1\\right)}^{2}+2\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 3{\\left(x-1\\right)}^{2}+2\\hfill \\\\ \\hfill -2&amp; =\\hfill &amp; 3{\\left(x-1\\right)}^{2}\\hfill \\\\ \\hfill -\\frac{2}{3}&amp; =\\hfill &amp; {\\left(x-1\\right)}^{2}\\hfill \\\\ \\hfill \u00b1\\sqrt{-\\frac{2}{3}}&amp; =\\hfill &amp; x-1\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The square root of a negative number<span data-type=\"newline\"><br \/><\/span>tells us the solutions are complex<span data-type=\"newline\"><br \/><\/span>numbers. So there are no <em data-effect=\"italics\">x<\/em>-intercepts.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871865347\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872407949\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872463553\"><div data-type=\"problem\" id=\"fs-id1163872458721\"><p id=\"fs-id1163872436477\"><span class=\"token\">\u24d0<\/span> Write \\(y=2{x}^{2}+4x+5\\) in standard form and <span class=\"token\">\u24d1<\/span> use properties of standard form to graph the equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871858614\"><p id=\"fs-id1163871858499\"><span class=\"token\">\u24d0<\/span>\\(y=2{\\left(x+1\\right)}^{2}+3\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163871880010\" data-alt=\"This graph shows a parabola opening upwards, with vertex (negative 1, 3) and y intercept (0, 5). It has the point minus (2, 5) on it.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening upwards, with vertex (negative 1, 3) and y intercept (0, 5). It has the point minus (2, 5) on it.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872464291\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872607894\"><div data-type=\"problem\" id=\"fs-id1163872709856\"><p id=\"fs-id1163871858546\"><span class=\"token\">\u24d0<\/span> Write \\(y=-2{x}^{2}+8x-7\\) in standard form and <span class=\"token\">\u24d1<\/span> use properties of standard form to graph the equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872407145\"><p id=\"fs-id1163871879530\"><span class=\"token\">\u24d0<\/span>\\(y=-2{\\left(x-2\\right)}^{2}+1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872517945\" data-alt=\"This graph shows a parabola opening downwards, with vertex (2, 1) and axis of symmetry x equals 2. Its y intercept is (0, negative 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downwards, with vertex (2, 1) and axis of symmetry x equals 2. Its y intercept is (0, negative 7).\" \/><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163871876982\"><h3 data-type=\"title\">Graph Horizontal Parabolas<\/h3><p id=\"fs-id1163871783082\">Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.<\/p><table id=\"fs-id1171791485940\" summary=\"This table, titled horizontal parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is x equals ay squared plus by plus c and standard form is x equals a open parentheses y minus k close parentheses squared plus h. Row one: orientation: general form is a greater than 0, right and a less than 0 left. Standard form is the same. Row 2: Axis of symmetry: general form is y equals minus b upon 2a and standard form is y equals k. Row 3: vertex: general form, substitute y equals minus b upon 2a and solve for x; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Horizontal Parabolas<\/th><\/tr><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><\/th><th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span>\\(x=a{y}^{2}+by+c\\)<\/th><th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span>\\(x=a{\\left(y-k\\right)}^{2}+h\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) right; \\(a&lt;0\\) left<\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) right; \\(a&lt;0\\) left<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=-\\frac{b}{2a}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=k\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Substitute \\(y=-\\frac{b}{2a}\\) and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercept<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1163872015008\">The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the <em data-effect=\"italics\">x<\/em>- or <em data-effect=\"italics\">y<\/em>-axis is merely an example.<\/p><span data-type=\"media\" id=\"fs-id1163872412399\" data-alt=\"This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\" \/><\/span><p id=\"fs-id1163872455604\">Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function.<\/p><p id=\"fs-id1163872546837\">To graph a <span data-type=\"term\" class=\"no-emphasis\">parabola<\/span> that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> variables.<\/p><div data-type=\"note\" id=\"fs-id1163872530085\" class=\"howto\"><div data-type=\"title\">Graph horizontal parabolas \\(\\left(x=a{y}^{2}+by+c\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}x=a{\\left(y-k\\right)}^{2}+h\\right)\\) using properties.<\/div><ol id=\"fs-id1163872471690\" type=\"1\" class=\"stepwise\"><li>Determine whether the parabola opens to the left or to the right.<\/li><li>Find the axis of symmetry.<\/li><li>Find the vertex.<\/li><li>Find the <em data-effect=\"italics\">x<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">x<\/em>-intercept across the axis of symmetry.<\/li><li>Find the <em data-effect=\"italics\">y<\/em>-intercepts.<\/li><li>Graph the parabola.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1163872571855\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872020109\"><div data-type=\"problem\" id=\"fs-id1163872472046\"><p id=\"fs-id1163872472049\">Graph \\(x=2{y}^{2}\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872514519\"><table id=\"fs-id1163872014824\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals 2y squared. Here, a is 2 and the parabola opens to the right. To find the axis of symmetry, find y equals minus b upon 2a. Substituting values, we get y equal to 0 divided by two times two. Hence y is 0. This is the axis of symmetry. The vertex is on this line. Let y be 0. Substituting in equation, we get x equals 0. The vertex is (0, 0). Since the vertex is (0, 0) both the x- and y-intercepts are the point (0, 0). To graph the parabola we need more points. In this case it is easiest to choose values of y.\" 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-id1163872838237\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(a=2,\\) the parabola opens to the right.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872550744\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007a_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 axis of symmetry, find \\(y=-\\frac{b}{2a}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871930867\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007c_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-id1163872511323\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007d_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-id1163872617865\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007e_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 \\(y=0.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is on the line\\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872646179\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872697362\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007g_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-id1163872555371\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007h_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(0,0\\right).\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1171792502294\">Since the vertex is \\(\\left(0,0\\right),\\) both the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts are the point \\(\\left(0,0\\right).\\) To graph the parabola we need more points. In this case it is easiest to choose values of <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/><\/span> <\/p><span data-type=\"media\" id=\"fs-id1163872666028\" data-alt=\"In the equation x equals 2 y squared, when y is 1, x is 2 and when y is 2, x is 8. The points are (2, 1) and (8, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the equation x equals 2 y squared, when y is 1, x is 2 and when y is 2, x is 8. The points are (2, 1) and (8, 2).\" \/><\/span><p><span data-type=\"newline\"><br \/><\/span> We also plot the points symmetric to \\(\\left(2,1\\right)\\) and \\(\\left(8,2\\right)\\) across the <em data-effect=\"italics\">y<\/em>-axis, the points \\(\\left(2,-1\\right),\\)\\(\\left(8,-2\\right).\\)<\/p><p id=\"fs-id1171791085709\">Graph the parabola.<\/p><span data-type=\"media\" id=\"fs-id1163872506672\" data-alt=\"This graph shows right opening parabola with vertex (0, 0). Four points are marked on it: point (2, 1), point (2, negative 1), point (8, 2) and point (8 minus 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows right opening parabola with vertex (0, 0). Four points are marked on it: point (2, 1), point (2, negative 1), point (8, 2) and point (8 minus 2).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872518547\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163871859575\"><div data-type=\"problem\" id=\"fs-id1163872645830\"><p id=\"fs-id1163872645832\">Graph \\(x={y}^{2}\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872839751\"><span data-type=\"media\" id=\"fs-id1163871979402\" data-alt=\"This graph shows right opening parabola with vertex at origin. Two points on it are (4, 2) and (4, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows right opening parabola with vertex at origin. Two points on it are (4, 2) and (4, negative 2).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163871880202\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872667845\"><div data-type=\"problem\" id=\"fs-id1163871925446\"><p id=\"fs-id1163871925448\">Graph \\(x=\\text{\u2212}{y}^{2}\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872457850\"><span data-type=\"media\" id=\"fs-id1163871858943\" data-alt=\"This graph shows left opening parabola with vertex at origin. Two points on it are (negative 4, 2) and (negative 4, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows left opening parabola with vertex at origin. Two points on it are (negative 4, 2) and (negative 4, negative 2).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163871924267\">In the next example, the vertex is not the origin.<\/p><div data-type=\"example\" id=\"fs-id1163871924271\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872840036\"><div data-type=\"problem\" id=\"fs-id1163872840038\"><p id=\"fs-id1163872564141\">Graph \\(x=\\text{\u2212}{y}^{2}+2y+8\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872511188\"><table id=\"fs-id1163872743219\" class=\"unnumbered unstyled can-break\" summary=\"The equation is minus y squared plus 2y plus 8. Since a is minus 1, the parabola opens to the left. To find the axis of symmetry, find y equals minus b upon 2a. The axis is y equals 1. The vertex is on this line. Substituting y equals 1 in the equation, we get x equal to 9. The vertex is (9, 1). Substituting y equal to 0 in the original equation, we get x intercept (8, 0). The symmetric point is (8, 2). Substituting x equal to 0 in the original equation, we get y intercepts (0, 4) and (0, negative 2). Connect the points to graph the 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-id1163872504881\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(a=-1,\\) the parabola opens to the left.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872502350\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010a_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 axis of symmetry, find \\(y=-\\frac{b}{2a}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872555506\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\"><span data-type=\"media\" id=\"fs-id1163872555636\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010d_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-id1163872713792\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010e_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=\"center\">The axis of symmetry is \\(y=1.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is on the line\\(y=1.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872617374\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(y=1.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872532027\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010g_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-id1163872645556\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010h_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=\"center\">The vertex is \\(\\left(9,1\\right).\\)<\/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-id1163871868721\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010i_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-id1163872697152\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010j_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-id1163872536891\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010k_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=\"center\">The <em data-effect=\"italics\">x<\/em>-intercept is \\(\\left(8,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The point \\(\\left(8,0\\right)\\) is one unit below the line of<span data-type=\"newline\"><br \/><\/span>symmetry. The symmetric point one unit<span data-type=\"newline\"><br \/><\/span>above the line of symmetry is \\(\\left(8,2\\right)\\)<\/td><td data-valign=\"top\" data-align=\"center\">Symmetric point is \\(\\left(8,2\\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-id1163872540069\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010l_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-id1163872515142\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010m_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-id1163872422916\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010n_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-id1163872686399\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010o_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-id1163872546302\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010p_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=\"center\">The <em data-effect=\"italics\">y<\/em>-intercepts are \\(\\left(0,4\\right)\\) and \\(\\left(0,-2\\right).\\)<\/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-id1163872626896\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872545783\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872520607\"><div data-type=\"problem\" id=\"fs-id1163872520609\"><p id=\"fs-id1163872548750\">Graph \\(x=\\text{\u2212}{y}^{2}-4y+12\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872570688\"><span data-type=\"media\" id=\"fs-id1163872520735\" data-alt=\"This graph shows left opening parabola with vertex (16, negative 2) and x intercept (12, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows left opening parabola with vertex (16, negative 2) and x intercept (12, 0).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872561489\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872565294\"><div data-type=\"problem\" id=\"fs-id1163872565296\"><p id=\"fs-id1163872544224\">Graph \\(x=\\text{\u2212}{y}^{2}+2y-3\\) by using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872555620\"><span data-type=\"media\" id=\"fs-id1163872019443\" data-alt=\"This graph shows left opening parabola with vertex (negative 2, 1) and x intercept minus (3, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows left opening parabola with vertex (negative 2, 1) and x intercept minus (3, 0).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163871868755\">In <a href=\"#fs-id1171791485940\" class=\"autogenerated-content\">(Figure)<\/a>, we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form \\(x=a{\\left(y-k\\right)}^{2}+h.\\) We will use this procedure in the next example.<\/p><div data-type=\"example\" id=\"fs-id1163872548704\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872570665\"><div data-type=\"problem\" id=\"fs-id1163872570667\"><p id=\"fs-id1163872549013\">Graph \\(x=2{\\left(y-2\\right)}^{2}+1\\) using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872543817\"><table id=\"fs-id1163872565265\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals 2 open parentheses y minus 2 close parentheses squared plus 1. Here, a is 2, h is 1 and k is 2. Since a is 2, the parabola opens to the right. The axis of symmetry is y equals k or y equals 2) and vertex is (h, k) or (1, 2). By substituting y equals 0 in the equation, we find x intercept (9, 0). The point symmetric to this across the axis is (9, 4). By substituting x equals 0 in the equation and simplifying, we arrive at minus 1 equals 2 open parentheses y minus 2 close parentheses squared. A square cannot be negative, so there is no real solution. So there are no y-intercepts. Graph the 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-id1163872574283\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(a=2,\\)\\(h=1,\\)\\(k=2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(a=2,\\) the parabola opens to the right.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163871783227\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_011a_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\">The axis of symmetry is \\(y=k.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{7.5em}{0ex}}\\)The axis of symmetry is \\(y=2.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{7.5em}{0ex}}\\)The vertex is \\(\\left(1,2\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercept by substituting \\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccc}\\hfill x&amp; =\\hfill &amp; 2{\\left(y-2\\right)}^{2}+1\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 2{\\left(0-2\\right)}^{2}+1\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 9\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{7.5em}{0ex}}\\)The <em data-effect=\"italics\">x<\/em>-intercept is \\(\\left(9,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the point symmetric to \\(\\left(9,0\\right)\\) across the<span data-type=\"newline\"><br \/><\/span>axis of symmetry.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{7.5em}{0ex}}\\left(9,4\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">y<\/em>-intercepts. Let \\(x=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccc}\\hfill x&amp; =\\hfill &amp; 2{\\left(y-2\\right)}^{2}+1\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 2{\\left(y-2\\right)}^{2}+1\\hfill \\\\ \\hfill -1&amp; =\\hfill &amp; 2{\\left(y-2\\right)}^{2}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">A square cannot be negative, so there is no real<span data-type=\"newline\"><br \/><\/span>solution. So there are no <em data-effect=\"italics\">y<\/em>-intercepts.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872023683\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872013292\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872519614\"><div data-type=\"problem\" id=\"fs-id1163872519616\"><p id=\"fs-id1163872555545\">Graph \\(x=3{\\left(y-1\\right)}^{2}+2\\) using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872664549\"><span data-type=\"media\" id=\"fs-id1163872009558\" data-alt=\"This graph shows a parabola opening right with vertex (2, 1) and x intercept (5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening right with vertex (2, 1) and x intercept (5, 0).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872011035\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872630754\"><div data-type=\"problem\" id=\"fs-id1163872630756\"><p id=\"fs-id1163872643897\">Graph \\(x=2{\\left(y-3\\right)}^{2}+2\\) using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872697286\"><span data-type=\"media\" id=\"fs-id1163872505694\" data-alt=\"This graph shows a parabola opening right with vertex (2, 3) and symmetric points (4, 2) and (4, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening right with vertex (2, 3) and symmetric points (4, 2) and (4, 4).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1165926769473\">In the next example, we notice the a is negative and so the parabola opens to the left.<\/p><div data-type=\"example\" id=\"fs-id1163872743277\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872543604\"><div data-type=\"problem\" id=\"fs-id1163872543606\"><p id=\"fs-id1163872543608\">Graph \\(x=-4{\\left(y+1\\right)}^{2}+4\\) using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872478313\"><table id=\"fs-id1163872464385\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals negative 4 open parentheses y plus 1 close parentheses squared plus 4. Here a is negative 4, h is 4 and k is negative 1. Since a is negative 4, the parabola opens to the left. The axis of symmetry is y equals negative 1 and vertex is (4, negative 1). Substituting y equals 0, we get x intercept (0, 0). The symmetric point across the axis is (0, negative 2). Substituting x equals 0, we get y intercepts (0, 0) and (0, negative 2). Graph the 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-id1163872506744\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(a=-4,\\)\\(h=4,\\)\\(k=-1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(a=-4,\\) the parabola opens to the left.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872840579\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_012a_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\">The axis of symmetry is \\(y=k.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{8em}{0ex}}\\)The axis of symmetry is \\(y=-1.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{8em}{0ex}}\\)The vertex is \\(\\left(4,-1\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercept by substituting \\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1.7em}{0ex}}\\begin{array}{ccc}\\hfill x&amp; =\\hfill &amp; -4{\\left(y+1\\right)}^{2}+4\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; -4{\\left(0+1\\right)}^{2}+4\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 0\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{8em}{0ex}}\\)The <em data-effect=\"italics\">x<\/em>-intercept is \\(\\left(0,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the point symmetric to \\(\\left(0,0\\right)\\) across the<span data-type=\"newline\"><br \/><\/span>axis of symmetry.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{8em}{0ex}}\\left(0,-2\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">y<\/em>-intercepts.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{8em}{0ex}}x=-4{\\left(y+1\\right)}^{2}+4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let \\(x=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccc}\\hfill 0&amp; =\\hfill &amp; -4{\\left(y+1\\right)}^{2}+4\\hfill \\\\ \\hfill -4&amp; =\\hfill &amp; -4{\\left(y+1\\right)}^{2}\\hfill \\\\ \\hfill 1&amp; =\\hfill &amp; {\\left(y+1\\right)}^{2}\\hfill \\\\ \\hfill y+1&amp; =\\hfill &amp; \u00b11\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(y=-1+1\\phantom{\\rule{1.5em}{0ex}}y=-1-1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(y=0\\phantom{\\rule{4em}{0ex}}y=-2\\)<\/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>-intercepts are \\(\\left(0,0\\right)\\) and \\(\\left(0,-2\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872434858\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872446043\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872502554\"><div data-type=\"problem\" id=\"fs-id1163872502556\"><p id=\"fs-id1163872472786\">Graph \\(x=-4{\\left(y+2\\right)}^{2}+4\\) using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872568837\"><span data-type=\"media\" id=\"fs-id1163872561835\" data-alt=\"This figure shows a parabola opening to the left with vertex (4, negative 2) and y intercepts (0, negative 1) and (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening to the left with vertex (4, negative 2) and y intercepts (0, negative 1) and (0, negative 3).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872568831\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872536885\"><div data-type=\"problem\" id=\"fs-id1163872536887\"><p id=\"fs-id1163872560738\">Graph \\(x=-2{\\left(y+3\\right)}^{2}+2\\) using properties.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872437230\"><span data-type=\"media\" id=\"fs-id1163872436951\" data-alt=\"This figure shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_313_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1165926770425\">The next example requires that we first put the equation in standard form and then use the properties.<\/p><div data-type=\"example\" id=\"fs-id1163872467394\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872467396\"><div data-type=\"problem\" id=\"fs-id1163872467398\"><p id=\"fs-id1163872502738\">Write \\(x=2{y}^{2}+12y+17\\) in standard form and then use the properties of the standard form to graph the equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872441461\"><table id=\"fs-id1163872422949\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals 2 y squared plus 12y plus 17. Rewrite in standard form by completing the square x equals 2 open parentheses y squared plus 6y plus 9 close parentheses plus 17 minus 18. This is x equals 2 open parentheses y plus 3 close parentheses squared minus 1. Here a is 2, h is negative 1 and k is negative 3. The axis is y equals negative 3. The vertex is (negative 1, negative 3). By substituting y equals 0, we get the x intercept (17, 0). Its symmetric point across the axis of symmetry is (17, negative 6). By substituting x equals 0 in the equation, we get approximate y values equal to minus 2.3 and minus 3.7. Graph the 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-id1163872470366\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the function in<span data-type=\"newline\"><br \/><\/span>\\(x=a{\\left(y-k\\right)}^{2}+h\\) form by completing<span data-type=\"newline\"><br \/><\/span>the square.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872485064\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013c_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-id1163872616679\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013d_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-id1163872515567\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\"><span data-type=\"media\" id=\"fs-id1163872458823\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(a=2,\\phantom{\\rule{1em}{0ex}}h=-1,\\phantom{\\rule{1em}{0ex}}k=-3\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(a=2,\\) the parabola opens to<span data-type=\"newline\"><br \/><\/span>the right.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872470774\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013a_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\">The axis of symmetry is \\(y=k.\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{10.4em}{0ex}}\\)The axis of symmetry is \\(y=-3.\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{10.4em}{0ex}}\\)The vertex is \\(\\left(-1,-3\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercept by substituting<span data-type=\"newline\"><br \/><\/span>\\(y=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1.8em}{0ex}}\\begin{array}{ccc}\\hfill x&amp; =\\hfill &amp; 2{\\left(y+3\\right)}^{2}-1\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 2{\\left(0+3\\right)}^{2}-1\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 17\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{10.4em}{0ex}}\\)The <em data-effect=\"italics\">x<\/em>-intercept is \\(\\left(17,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the point symmetric to \\(\\left(17,0\\right)\\)<span data-type=\"newline\"><br \/><\/span>across the axis of symmetry.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{10.4em}{0ex}}\\left(17,-6\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">y<\/em>-intercepts.<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span>Let \\(x=0.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccc}\\hfill x&amp; =\\hfill &amp; 2{\\left(y+3\\right)}^{2}-1\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 2{\\left(y+3\\right)}^{2}-1\\hfill \\\\ \\hfill 1&amp; =\\hfill &amp; 2{\\left(y+3\\right)}^{2}\\hfill \\\\ \\hfill \\frac{1}{2}&amp; =\\hfill &amp; {\\left(y+3\\right)}^{2}\\hfill \\\\ \\hfill y+3&amp; =\\hfill &amp; \u00b1\\sqrt{\\frac{1}{2}}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; -3\u00b1\\frac{\\sqrt{2}}{2}\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(y=-3+\\frac{\\sqrt{2}}{2}\\phantom{\\rule{1em}{0ex}}y=-3-\\frac{\\sqrt{2}}{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(y\\approx -2.3\\phantom{\\rule{2em}{0ex}}y\\approx -3.7\\)<\/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>-intercepts are \\(\\left(0,-3+\\frac{\\sqrt{2}}{2}\\right),\\left(0,-3-\\frac{\\sqrt{2}}{2}\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872570813\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872471299\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872515426\"><div data-type=\"problem\" id=\"fs-id1163872515428\"><p id=\"fs-id1163872537712\"><span class=\"token\">\u24d0<\/span> Write \\(x=3{y}^{2}+6y+7\\) in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872556946\"><p id=\"fs-id1163872556948\"><span class=\"token\">\u24d0<\/span>\\(x=3{\\left(y+1\\right)}^{2}+4\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872517880\" data-alt=\"This graph shows a parabola opening to the right with vertex (4, negative 1) and x intercept (7, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (4, negative 1) and x intercept (7, 0).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872520960\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872530074\"><div data-type=\"problem\" id=\"fs-id1163872530076\"><p id=\"fs-id1163872503156\"><span class=\"token\">\u24d0<\/span> Write \\(x=-4{y}^{2}-16y-12\\) in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872515248\"><p id=\"fs-id1163872515250\"><span class=\"token\">\u24d0<\/span>\\(x=-4{\\left(y+2\\right)}^{2}+4\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872571430\" data-alt=\"This graph shows a parabola opening to the left with vertex (4, negative 2) and x intercept minus (12, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_315_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (4, negative 2) and x intercept minus (12, 0).\" \/><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872566004\"><h3 data-type=\"title\">Solve Applications with Parabolas<\/h3><p id=\"fs-id1163872511907\">Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.<\/p><div data-type=\"example\" id=\"fs-id1163872566243\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872566245\"><div data-type=\"problem\" id=\"fs-id1163872630376\"><p id=\"fs-id1163872630378\">Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.<\/p><span data-type=\"media\" id=\"fs-id1163872566959\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 10 feet high and 20 feet wide at the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_014_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 10 feet high and 20 feet wide at the base.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872408903\"><p id=\"fs-id1163872408906\">We will first set up a coordinate system and draw the parabola. The graph will give us the information we need to write the equation of the graph in the standard form\\(y=a{\\left(x-h\\right)}^{2}+k.\\)<span data-type=\"newline\"><br \/><\/span> <\/p><table id=\"fs-id1163872571206\" class=\"unnumbered unstyled can-break\" summary=\"Let the lower left side of the bridge be the origin of the coordinate grid at the point (0, 0). Since the base is 20 feet wide the point (20, 0) represents the lower right side. The bridge is 10 feet high at the highest point. The highest point is the vertex of the parabola so the y coordinate of the vertex will be 10. Since the bridge is symmetric, the vertex must fall halfway between the left most point, (0, 0) and the rightmost point (20, 0). From this we know that the x coordinate of the vertex will also be 10. The vertex is 10, 10. So h is 10 and k is 10. Substitute the values into the standard form y equals a open parentheses x minus h close parentheses squared plus k. The value of a is still unknown. To find the value of a use one of the other points on the parabola, point (0, 0). Substituting the values into the equation, we get a equal to minus 1 by 10. Substitute the value for a into the equation. We get y equals minus 1 upon 10 open parentheses x minus 10 close parentheses squared plus 10.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Let the lower left side of the bridge be the<span data-type=\"newline\"><br \/><\/span>origin of the coordinate grid at the point \\(\\left(0,0\\right).\\)<span data-type=\"newline\"><br \/><\/span>Since the base is 20 feet wide the point<span data-type=\"newline\"><br \/><\/span>\\(\\left(20,0\\right)\\) represents the lower right side.<span data-type=\"newline\"><br \/><\/span>The bridge is 10 feet high at the highest<span data-type=\"newline\"><br \/><\/span>point. The highest point is the vertex of<span data-type=\"newline\"><br \/><\/span>the parabola so the <em data-effect=\"italics\">y<\/em>-coordinate of the<span data-type=\"newline\"><br \/><\/span>vertex will be 10.<span data-type=\"newline\"><br \/><\/span>Since the bridge is symmetric, the vertex<span data-type=\"newline\"><br \/><\/span>must fall halfway between the left most<span data-type=\"newline\"><br \/><\/span>point, \\(\\left(0,0\\right),\\) and the rightmost point<span data-type=\"newline\"><br \/><\/span>\\(\\left(20,0\\right).\\) From this we know that the<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">x<\/em>-coordinate of the vertex will also be 10.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872572181\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Identify the vertex, \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(h,k\\right)=\\left(10,10\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(h=10,\\text{\u2003}k=10\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the values into the standard form.<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span>The value of <em data-effect=\"italics\">a<\/em> is still unknown. To find<span data-type=\"newline\"><br \/><\/span>the value of <em data-effect=\"italics\">a<\/em> use one of the other points<span data-type=\"newline\"><br \/><\/span>on the parabola.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1.3em}{0ex}}\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; a{\\left(x-h\\right)}^{2}+k\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; a{\\left(x-10\\right)}^{2}+10\\hfill \\\\ \\hfill \\left(x,y\\right)&amp; =\\hfill &amp; \\left(0,0\\right)\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the values of the other point<span data-type=\"newline\"><br \/><\/span>into the equation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{3.35em}{0ex}}\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; a{\\left(x-10\\right)}^{2}+10\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; a{\\left(0-10\\right)}^{2}+10\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">a<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{2em}{0ex}}\\begin{array}{ccc}\\hfill 0&amp; =\\hfill &amp; a{\\left(0-10\\right)}^{2}+10\\hfill \\\\ \\hfill -10&amp; =\\hfill &amp; a{\\left(-10\\right)}^{2}\\hfill \\\\ \\hfill -10&amp; =\\hfill &amp; 100a\\hfill \\\\ \\hfill \\frac{-10}{100}&amp; =\\hfill &amp; a\\hfill \\\\ \\hfill a&amp; =\\hfill &amp; -\\frac{1}{10}\\hfill \\end{array}\\)<\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\">\\(y=a{\\left(x-10\\right)}^{2}+10\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the value for <em data-effect=\"italics\">a<\/em> into the<span data-type=\"newline\"><br \/><\/span>equation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1.8em}{0ex}}y=-\\frac{1}{10}{\\left(x-10\\right)}^{2}+10\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872718951\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872804356\"><div data-type=\"problem\" id=\"fs-id1163872804358\"><p id=\"fs-id1163872804373\">Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.<\/p><span data-type=\"media\" id=\"fs-id1163872804378\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 20 feet high and 40 feet wide at the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_016_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 20 feet high and 40 feet wide at the base.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163871996401\"><p id=\"fs-id1163871996403\">\\(y=-\\frac{1}{20}{\\left(x-20\\right)}^{2}+20\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872713382\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872713400\"><div data-type=\"problem\" id=\"fs-id1163872713402\"><p id=\"fs-id1163872559727\">Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.<\/p><span data-type=\"media\" id=\"fs-id1163872559731\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 5 feet high and 10 feet wide at the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 5 feet high and 10 feet wide at the base.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872803531\"><p id=\"fs-id1163872803534\">\\(y=-\\frac{1}{5}{x}^{2}+2x\\)\\(y=-\\frac{1}{5}{\\left(x-5\\right)}^{2}+5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163871999504\" class=\"media-2\"><p id=\"fs-id1163871996260\">Access these online resources for additional instructions and practice with quadratic functions and parabolas.<\/p><ul id=\"fs-id1163871996293\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37quadfunc\">Quadratic Functions<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37conhorizpbola\">Introduction to Conics and Graphing Horizontal Parabolas<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163871997815\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1163872662987\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Parabola:<\/strong> A <strong data-effect=\"bold\">parabola<\/strong> is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the <strong data-effect=\"bold\">focus,<\/strong> and the fixed line is called the <strong data-effect=\"bold\">directrix<\/strong> of the parabola.<span data-type=\"newline\"><br \/><\/span> <table id=\"fs-id1163872713464\" summary=\"This table, titled vertical parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is y equals ax squared plus bx plus c and standard form is y equals a open parentheses x minus h close parentheses squared plus k. Row one: orientation: general form is a greater than 0, up and a less than 0 down. Standard form is the same. Row 2: Axis of symmetry: general form is x equals minus b upon 2a and standard form is x equals h. Row 3: vertex: general form, substitute x equals minus b upon 2a and solve for y; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Vertical Parabolas<\/th><\/tr><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><\/th><th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span>\\(y=a{x}^{2}+bx+c\\)<\/th><th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span>\\(y=a{\\left(x-h\\right)}^{2}+k\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) up; \\(a&lt;0\\) down<\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) up; \\(a&lt;0\\) down<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(x=-\\frac{b}{2a}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(x=h\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Substitute \\(x=-\\frac{b}{2a}\\) and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>- intercept<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><\/tr><\/tbody><\/table><span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163872607467\" data-alt=\"This figure shows two parabolas with axis x equals h and vertex (h, k). The one on the left opens up and a is greater than 0. The one on the right opens down. Here a is less than 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_018_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis x equals h and vertex (h, k). The one on the left opens up and a is greater than 0. The one on the right opens down. Here a is less than 0.\" \/><\/span> <\/li><li><strong data-effect=\"bold\">How to graph vertical parabolas \\(\\left(y=a{x}^{2}+bx+c\\) or \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\right)\\) using properties.<\/strong><ol id=\"fs-id1163871863277\" type=\"1\" class=\"stepwise\"><li>Determine whether the parabola opens upward or downward.<\/li><li>Find 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.<\/li><li>Graph the parabola.<\/li><\/ol><span data-type=\"newline\"><br \/><\/span><table id=\"fs-id1163872728536\" summary=\"This table, titled horizontal parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is x equals ay squared plus by plus c and standard form is x equals a open parentheses y minus k close parentheses squared plus h. Row one: orientation: general form is a greater than 0, right and a less than 0 left. Standard form is the same. Row 2: Axis of symmetry: general form is y equals minus b upon 2a and standard form is y equals k. Row 3: vertex: general form, substitute y equals minus b upon 2a and solve for x; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Horizontal Parabolas<\/th><\/tr><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><\/th><th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span>\\(x=a{y}^{2}+by+c\\)<\/th><th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span>\\(x=a{\\left(y-k\\right)}^{2}+h\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) right; \\(a&lt;0\\) left<\/td><td data-valign=\"middle\" data-align=\"center\">\\(a&gt;0\\) right; \\(a&lt;0\\) left<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=-\\frac{b}{2a}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=k\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Substitute \\(y=-\\frac{b}{2a}\\) and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(x=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercept<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Let \\(y=0\\)<\/td><\/tr><\/tbody><\/table><span data-type=\"newline\"><br \/><\/span><span data-type=\"media\" id=\"fs-id1163872023565\" data-alt=\"This figure shows two parabolas with axis of symmetry y equals k, and vertex (h, k). The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_019_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis of symmetry y equals k, and vertex (h, k). The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\" \/><\/span><\/li><li><strong data-effect=\"bold\">How to graph horizontal parabolas \\(\\left(x=a{y}^{2}+by+c\\) or \\(x=a{\\left(y-k\\right)}^{2}+h\\right)\\) using properties.<\/strong><ol id=\"fs-id1163872706418\" type=\"1\" class=\"stepwise\"><li>Determine whether the parabola opens to the left or to the right.<\/li><li>Find the axis of symmetry.<\/li><li>Find the vertex.<\/li><li>Find the <em data-effect=\"italics\">x<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">x<\/em>-intercept across the axis of symmetry.<\/li><li>Find the <em data-effect=\"italics\">y<\/em>-intercepts.<\/li><li>Graph the parabola.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163872686249\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163872014554\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1163872622892\"><strong data-effect=\"bold\">Graph Vertical Parabolas<\/strong><\/p><p id=\"fs-id1163873805265\">In the following exercises, graph each equation by using properties.<\/p><div data-type=\"exercise\" id=\"fs-id1163872643998\"><div data-type=\"problem\" id=\"fs-id1163872573165\"><p id=\"fs-id1163872573167\">\\(y=\\text{\u2212}{x}^{2}+4x-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872444219\"><span data-type=\"media\" id=\"fs-id1163872572880\" data-alt=\"This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872572850\"><div data-type=\"problem\" id=\"fs-id1163872015072\"><p id=\"fs-id1163872015074\">\\(y=\\text{\u2212}{x}^{2}+8x-15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872510316\"><div data-type=\"problem\" id=\"fs-id1163872510318\"><p id=\"fs-id1163872510320\">\\(y=6{x}^{2}+2x-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871879845\"><span data-type=\"media\" id=\"fs-id1163871879849\" data-alt=\"This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872541582\"><div data-type=\"problem\" id=\"fs-id1163872541584\"><p id=\"fs-id1163872627634\">\\(y=8{x}^{2}-10x+3\\)<\/p><\/div><\/div><p id=\"fs-id1163872435842\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163872644217\"><div data-type=\"problem\" id=\"fs-id1163872506006\"><p id=\"fs-id1163872506008\">\\(y=\\text{\u2212}{x}^{2}+2x-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872730548\"><p id=\"fs-id1163872613320\"><span class=\"token\">\u24d0<\/span>\\(y=\\text{\u2212}{\\left(x-1\\right)}^{2}-3\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872661395\" data-alt=\"This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872628550\"><div data-type=\"problem\" id=\"fs-id1163871942141\"><p id=\"fs-id1163871942143\">\\(y=2{x}^{2}+4x+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872087819\"><div data-type=\"problem\" id=\"fs-id1163871998458\"><p id=\"fs-id1163871998460\">\\(y=-2{x}^{2}-4x-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871999528\"><p id=\"fs-id1163871998170\"><span class=\"token\">\u24d0<\/span>\\(y=-2{\\left(x+1\\right)}^{2}-3\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163871859944\" data-alt=\"This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872505265\"><div data-type=\"problem\" id=\"fs-id1163872505268\"><p id=\"fs-id1163872505270\">\\(y=3{x}^{2}-12x+7\\)<\/p><\/div><\/div><p id=\"fs-id1163872838584\"><strong data-effect=\"bold\">Graph Horizontal Parabolas<\/strong><\/p><p id=\"fs-id1163872721660\">In the following exercises, graph each equation by using properties.<\/p><div data-type=\"exercise\" id=\"fs-id1163872390039\"><div data-type=\"problem\" id=\"fs-id1163872390041\"><p id=\"fs-id1163872390043\">\\(x=-2{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872838682\"><span data-type=\"media\" id=\"fs-id1163872468380\" data-alt=\"This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872512005\"><div data-type=\"problem\" id=\"fs-id1163872512008\"><p id=\"fs-id1163872511006\">\\(x=3{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872463058\"><div data-type=\"problem\" id=\"fs-id1163872463060\"><p id=\"fs-id1163872659707\">\\(x=4{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872471858\"><span data-type=\"media\" id=\"fs-id1163872601352\" data-alt=\"This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872574426\"><div data-type=\"problem\" id=\"fs-id1163872557861\"><p id=\"fs-id1163872557864\">\\(x=-4{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872557264\"><div data-type=\"problem\" id=\"fs-id1163872557266\"><p id=\"fs-id1163872557268\">\\(x=\\text{\u2212}{y}^{2}-2y+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872664303\"><span data-type=\"media\" id=\"fs-id1163872664307\" data-alt=\"This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872663524\"><div data-type=\"problem\" id=\"fs-id1163872000432\"><p id=\"fs-id1163872000434\">\\(x=\\text{\u2212}{y}^{2}-4y+5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872095357\"><div data-type=\"problem\" id=\"fs-id1163872573368\"><p id=\"fs-id1163872573370\">\\(x={y}^{2}+6y+8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872096646\"><span data-type=\"media\" id=\"fs-id1163872516356\" data-alt=\"This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872610552\"><div data-type=\"problem\" id=\"fs-id1163872610554\"><p id=\"fs-id1163872610556\">\\(x={y}^{2}-4y-12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872550366\"><div data-type=\"problem\" id=\"fs-id1163872019422\"><p id=\"fs-id1163872019424\">\\(x={\\left(y-2\\right)}^{2}+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872426618\"><span data-type=\"media\" id=\"fs-id1163872516187\" data-alt=\"This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872529947\"><div data-type=\"problem\" id=\"fs-id1163872529950\"><p id=\"fs-id1163872529952\">\\(x={\\left(y-1\\right)}^{2}+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872548729\"><div data-type=\"problem\" id=\"fs-id1163872544198\"><p id=\"fs-id1163872544200\">\\(x=\\text{\u2212}{\\left(y-1\\right)}^{2}+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871858570\"><span data-type=\"media\" id=\"fs-id1163872750269\" data-alt=\"This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872837439\"><div data-type=\"problem\" id=\"fs-id1163872837441\"><p id=\"fs-id1163872560004\">\\(x=\\text{\u2212}{\\left(y-4\\right)}^{2}+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872562094\"><div data-type=\"problem\" id=\"fs-id1163872562096\"><p id=\"fs-id1163872562098\">\\(x={\\left(y+2\\right)}^{2}+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872642821\"><span data-type=\"media\" id=\"fs-id1163871912071\" data-alt=\"This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872520663\"><div data-type=\"problem\" id=\"fs-id1163872520665\"><p id=\"fs-id1163872520667\">\\(x={\\left(y+1\\right)}^{2}+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872391569\"><div data-type=\"problem\" id=\"fs-id1163872391571\"><p id=\"fs-id1163872699904\">\\(x=\\text{\u2212}{\\left(y+3\\right)}^{2}+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871873361\"><span data-type=\"media\" id=\"fs-id1163872520699\" data-alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872505590\"><div data-type=\"problem\" id=\"fs-id1163872505593\"><p id=\"fs-id1163872565104\">\\(x=\\text{\u2212}{\\left(y+4\\right)}^{2}+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872572862\"><div data-type=\"problem\" id=\"fs-id1163872572864\"><p id=\"fs-id1163872572866\">\\(x=-3{\\left(y-2\\right)}^{2}+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872020157\"><span data-type=\"media\" id=\"fs-id1163871923731\" data-alt=\"This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871942317\"><div data-type=\"problem\" id=\"fs-id1163872730401\"><p id=\"fs-id1163872730404\">\\(x=-2{\\left(y-1\\right)}^{2}+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872505979\"><div data-type=\"problem\" id=\"fs-id1163872505981\"><p id=\"fs-id1163872628073\">\\(x=4{\\left(y+1\\right)}^{2}-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872448629\"><span data-type=\"media\" id=\"fs-id1163872448633\" data-alt=\"This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872426680\"><div data-type=\"problem\" id=\"fs-id1163872426682\"><p id=\"fs-id1163872426684\">\\(x=2{\\left(y+4\\right)}^{2}-2\\)<\/p><\/div><\/div><p id=\"fs-id1163872665896\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163872665235\"><div data-type=\"problem\" id=\"fs-id1163872572389\"><p id=\"fs-id1163872572391\">\\(x={y}^{2}+4y-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872506209\"><p id=\"fs-id1163872506211\"><span class=\"token\">\u24d0<\/span>\\(x={\\left(y+2\\right)}^{2}-9\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872542914\" data-alt=\"This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872663620\"><div data-type=\"problem\" id=\"fs-id1163872663622\"><p id=\"fs-id1163872663625\">\\(x={y}^{2}+2y-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872558332\"><div data-type=\"problem\" id=\"fs-id1163872558334\"><p id=\"fs-id1163872558336\">\\(x=-2{y}^{2}-12y-16\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871980002\"><p id=\"fs-id1163872554258\"><span class=\"token\">\u24d0<\/span>\\(x=-2{\\left(y+3\\right)}^{2}+2\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872665146\" data-alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872665071\"><div data-type=\"problem\" id=\"fs-id1163872665073\"><p id=\"fs-id1163872665075\">\\(x=-3{y}^{2}-6y-5\\)<\/p><\/div><\/div><p id=\"fs-id1163872647117\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p><p id=\"fs-id1163872647000\">In the following exercises, match each graph to one of the following equations: <span class=\"token\">\u24d0<\/span> <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">y<\/em><sup>2<\/sup> = 64 <span class=\"token\">\u24d1<\/span> <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">y<\/em><sup>2<\/sup> = 49<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span> (<em data-effect=\"italics\">x<\/em> + 5)<sup>2<\/sup> + (<em data-effect=\"italics\">y<\/em> + 2)<sup>2<\/sup> = 4 <span class=\"token\">\u24d3<\/span> (<em data-effect=\"italics\">x<\/em> \u2212 2)<sup>2<\/sup> + (<em data-effect=\"italics\">y<\/em> \u2212 3)<sup>2<\/sup> = 9 <span class=\"token\">\u24d4<\/span> <em data-effect=\"italics\">y<\/em> = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 8<em data-effect=\"italics\">x<\/em> \u2212 15 <span class=\"token\">\u24d5<\/span> <em data-effect=\"italics\">y<\/em> = 6<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 2<em data-effect=\"italics\">x<\/em> \u2212 1<\/p><div data-type=\"exercise\" id=\"fs-id1163872627995\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872627997\"><span data-type=\"media\" id=\"fs-id1163872627999\" data-alt=\"This graph shows circle with center (0, 0) and radius 8 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (0, 0) and radius 8 units.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872505906\"><p id=\"fs-id1163872505908\"><span class=\"token\">\u24d0<\/span><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872505929\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872505931\"><span data-type=\"media\" id=\"fs-id1163872505933\" data-alt=\"This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872628330\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872628332\"><span data-type=\"media\" id=\"fs-id1163872628220\" data-alt=\"This graph shows circle with center (0, 0) and radius 7 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (0, 0) and radius 7 units.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872663640\"><p id=\"fs-id1163872663642\"><span class=\"token\">\u24d1<\/span><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871946568\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872643681\"><span data-type=\"media\" id=\"fs-id1163872643683\" data-alt=\"This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872465736\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872465739\"><span data-type=\"media\" id=\"fs-id1163872465741\" data-alt=\"This graph shows circle with center (2, 3) and radius 3 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (2, 3) and radius 3 units.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872548133\"><p id=\"fs-id1163872548135\"><span class=\"token\">\u24d3<\/span><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872647979\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872647982\"><span data-type=\"media\" id=\"fs-id1163872647937\" data-alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\" \/><\/span><\/div><\/div><p id=\"fs-id1163872648028\"><strong data-effect=\"bold\">Solve Applications with Parabolas<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1163872543093\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872543096\"><p id=\"fs-id1163872648078\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p><span data-type=\"media\" id=\"fs-id1163872648082\" data-alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872648001\"><p id=\"fs-id1163872648004\">\\(y=-\\frac{1}{15}{\\left(x-15\\right)}^{2}+15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872564608\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872564610\"><p id=\"fs-id1163872564612\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p><span data-type=\"media\" id=\"fs-id1163872558048\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872554290\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872554292\"><p id=\"fs-id1163872554467\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p><span data-type=\"media\" id=\"fs-id1163872554472\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872554384\"><p id=\"fs-id1163872554386\">\\(y=-\\frac{1}{10}{\\left(x-30\\right)}^{2}+90\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872661810\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872661813\"><p id=\"fs-id1163872661815\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p><span data-type=\"media\" id=\"fs-id1163872662438\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_210_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.\" \/><\/span><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1163872664992\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1163872611307\"><div data-type=\"problem\" id=\"fs-id1163872665874\"><p id=\"fs-id1163872665876\">In your own words, define a parabola.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872665880\"><p id=\"fs-id1163872642717\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872642723\"><div data-type=\"problem\" id=\"fs-id1163872642725\"><p id=\"fs-id1163872661848\">Is the parabola \\(y={x}^{2}\\) a function? Is the parabola \\(x={y}^{2}\\) a function? Explain why or why not.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872617605\"><div data-type=\"problem\" id=\"fs-id1163872617607\"><p id=\"fs-id1163872545250\">Write the equation of a parabola that opens up or down in standard form and the equation of a parabola that opens left or right in standard form. Provide a sketch of the parabola for each one, label the vertex and axis of symmetry.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872545256\"><p id=\"fs-id1163872648060\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872648065\"><div data-type=\"problem\" id=\"fs-id1163872648068\"><p id=\"fs-id1163872517075\">Explain in your own words, how you can tell from its equation whether a parabola opens up, down, left or right.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163872554329\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1163872554423\"><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-id1163872546272\" data-alt=\"This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_211_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.\" \/><\/span><p id=\"fs-id1163872665913\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1163872665756\"><dt>parabola<\/dt><dd id=\"fs-id1163872558173\">A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.<\/dd><\/dl><\/div>","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>Graph vertical parabolas<\/li>\n<li>Graph horizontal parabolas<\/li>\n<li>Solve applications with parabolas<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872563460\" class=\"be-prepared\">\n<p id=\"fs-id1163872803529\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1163872549876\" type=\"1\">\n<li>Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb275baba90ff1754c31441f7c969833_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/3de1ae34-6225-4751-be75-a17b3e0e665b#fs-id1169147828812\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc40031db6c8bd26025485aa686b9377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#54;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/3e7f365d-4885-4b7a-bb2a-4358cbc00d2e#fs-id1167829894368\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Write in standard form: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-959d614bd6bcdc7d3cd63964e5d213f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b#fs-id1169149374763\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872698598\">\n<h3 data-type=\"title\">Graph Vertical Parabolas<\/h3>\n<p id=\"fs-id1163872705754\">The next conic section we will look at is a <span data-type=\"term\">parabola<\/span>. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the <strong data-effect=\"bold\">focus,<\/strong> and the fixed line is called the <strong data-effect=\"bold\">directrix<\/strong> of the parabola.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872576446\" data-alt=\"This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1163872392082\">\n<div data-type=\"title\">Parabola<\/div>\n<p id=\"fs-id1163872467441\">A <strong data-effect=\"bold\">parabola<\/strong> is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the <strong data-effect=\"bold\">focus,<\/strong> and the fixed line is called the <strong data-effect=\"bold\">directrix<\/strong> of the parabola.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872008560\" data-alt=\"This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.\" \/><\/span><\/div>\n<p id=\"fs-id1163871995058\">Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.<\/p>\n<table id=\"fs-id1163871908965\" summary=\"This table, titled vertical parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is y equals ax squared plus bx plus c and standard form is y equals a open parentheses x minus h close parentheses squared plus k. Row one: orientation: general form is a greater than 0, up and a less than 0 down. Standard form is the same. Row 2: Axis of symmetry: general form is x equals minus b upon 2a and standard form is x equals h. Row 3: vertex: general form, substitute x equals minus b upon 2a and solve for y; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Vertical Parabolas<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><\/th>\n<th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af4106dc6abca5e766ae7ae01481411c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66d6e02b37a1552327ac6337041618b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"136\" style=\"vertical-align: -4px;\" \/> <\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> up; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> down<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> up; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f6486da4c3e33f0fbdf54846ff952eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Substitute <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;\" \/> and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercept<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Let <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=\"middle\" data-align=\"center\">Let <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<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1163872841582\">The graphs show what the parabolas look like when they open up or down. Their position in relation to the <em data-effect=\"italics\">x<\/em>&#8211; or <em data-effect=\"italics\">y<\/em>-axis is merely an example.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872407207\" data-alt=\"This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.\" \/><\/span><\/p>\n<p id=\"fs-id1163872504389\">To graph a parabola from these forms, we used the following steps.<\/p>\n<div data-type=\"note\" id=\"fs-id1163872743524\" class=\"howto\">\n<div data-type=\"title\">Graph vertical parabolas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b58224fe43d6517a8b50a92f95dfd35f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#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;&#53;&#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;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"343\" style=\"vertical-align: -12px;\" \/> using properties.<\/div>\n<ol id=\"fs-id1163872471477\" type=\"1\" class=\"stepwise\">\n<li>Determine whether the parabola opens upward or downward.<\/li>\n<li>Find 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.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1163872836555\">The next example reviews the method of graphing a parabola from the general form of its equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872531349\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872531116\">\n<div data-type=\"problem\" id=\"fs-id1163872468763\">\n<p id=\"fs-id1163872743031\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed1de99520d686222bf5d18cb9c89258_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;&#54;&#120;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872465154\">\n<table id=\"fs-id1163872557188\" class=\"unnumbered unstyled can-break\" summary=\"The equation is y equals minus x squared plus 6x minus 8. This is of the form y equals ax squared plus bx plus c. Since a is minus 1, the parabola opens downward. To find the axis of symmetry, find x equals minus b upon 2a. Substituting values of b and a, we get x equals 3. This is the axis of symmetry. The vertex is on the line x equals 3. Substituting this value in the equation, we get y equals 1. The vertex is the point 3, 1. The y intercept occurs when x equals 0. Substituting in the equation and simplifying, we get y equals minus 8. The point 0, 8 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 8. The x intercept occurs when y equals 0. We substitute this in the original equation and factor the trinomial. We get x intercepts 4, 0 and 2, 0. Graph the 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-id1163872689672\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004b_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-df802eaa8e08eb73e8cd4d30d23d4b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> 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=\"center\"><span data-type=\"media\" id=\"fs-id1163872468262\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/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-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;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872574710\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004c_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-id1163872520882\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004d_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-id1163872562025\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004e_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=\"center\">The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf60bc9fcf312a246a055c15ee98033c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" 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-id1163872630300\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004f_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-bf60bc9fcf312a246a055c15ee98033c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872013436\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004g_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-bf60bc9fcf312a246a055c15ee98033c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872571907\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004h_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-id1163872512977\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004i_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-id1163872568047\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004j_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=\"center\">The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c867e408365ba7dbd665695abcf20b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#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-id1163872703214\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004k_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-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872517696\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004l_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-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872465389\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004m_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-id1163871975475\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004n_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=\"center\">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-e6e1b2267c0518c3de1c638018c1958f_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;&#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-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;\" \/> is three units to the left of the<span data-type=\"newline\"><br \/><\/span>line of symmetry. The point three units to the<span data-type=\"newline\"><br \/><\/span>right of the line of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39cc5c9cbbe8373ef4cb04cb8741b684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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-39cc5c9cbbe8373ef4cb04cb8741b684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#56;&#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\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872840487\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004o_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\">x<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872538667\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004p_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-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872561300\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004q_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-id1163872731143\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004r_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-id1163872571007\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004s_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-id1163872512920\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004t_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=\"center\">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-5ef81ced6525ddec145e71bc9d4dfe57_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;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872459332\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_004u_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-id1163872423654\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872450037\">\n<div data-type=\"problem\" id=\"fs-id1163872419334\">\n<p id=\"fs-id1163872464625\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-029f05a6fc320cd8439bf02845d8486b_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;&#53;&#120;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871868912\"><span data-type=\"media\" id=\"fs-id1163872441832\" data-alt=\"This graph shows a parabola opening downward, with x intercepts (2, 0) and (3, 0) and y intercept (0, negative 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward, with x intercepts (2, 0) and (3, 0) and y intercept (0, negative 6).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163871994444\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872502703\">\n<div data-type=\"problem\" id=\"fs-id1163872505505\">\n<p id=\"fs-id1163871923684\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b5ae7e794e416e80f0754a9144f33e0_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;&#56;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872478588\"><span data-type=\"media\" id=\"fs-id1163872464117\" data-alt=\"This graph shows a parabola opening downward, with vertex (4, 4) and x intercepts (2, 0) and (6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward, with vertex (4, 4) and x intercepts (2, 0) and (6, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872464760\">The next example reviews the method of graphing a <span data-type=\"term\" class=\"no-emphasis\">parabola<\/span> from the standard form of its equation, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0551f43b12a39be94457982f93cb2b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1163872545091\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872420384\">\n<div data-type=\"problem\" id=\"fs-id1163872467579\">\n<p id=\"fs-id1163872401104\">Write<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-feade4f7aea444dadac6ddb5ae470ef4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/> in standard form and then use properties of standard form to graph the equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872689501\">\n<table id=\"fs-id1163871857801\" class=\"unnumbered unstyled can-break\" summary=\"The equation is 3 x squared minus 6x plus 5. Rewrite in y equals a open parentheses x minus h close parentheses squared plus k form by completing the square. We rewrite as y equals 3 open parentheses x squared minus 2x plus 1 close parentheses plus 5 minus 3. So y is 3 open parentheses x minus 1 close parentheses squared plus 2. Here, a is 3, h is 1 and k is 2. Since a is 2, the parabola opens upward. The axis of symmetry is x equals 1. The vertex is 1, 2. Find the y intercept by substituting x equal to 0 in the original equation. We get y equal to 5. The y intercept is 0, 5. The point symmetric to it is 2, 5. For finding the x intercept, we substitute y equals 0 in the original equation. We get x equal to 1 plus square root of minus 2 upon 3. The square root of a negative number tells us the solutions are complex numbers. So there are no x intercepts. Graph the parabola.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66d6e02b37a1552327ac6337041618b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"136\" style=\"vertical-align: -4px;\" \/> form<span data-type=\"newline\"><br \/><\/span>by completing the square.<\/td>\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-feade4f7aea444dadac6ddb5ae470ef4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" 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\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab5d7eda2e816d4ab068098cf3e15d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"178\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\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-b18dc4f94990faf8961dc3b6b0a0444b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"211\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\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-806ddff54986671c4a723bc8c4eaf360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td>\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-9418f76b7fb8efbd61d4b14b3df06bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f6645b4c9f700d9074aa66ed05eeb6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"42\" style=\"vertical-align: -1px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-633d0c333092d820a4ccc4ff860bc5da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eebb7c55daa365334583aac412e471ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> the parabola opens upward.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872840988\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\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-96cd3cfed7e131a541dfc60ec480496d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#104;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: 0px;\" \/><\/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-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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\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-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/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-4b55ec00d0efa63f10ac93a2bba77a12_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;&#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\">Find the <em data-effect=\"italics\">y<\/em>-intercept by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\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-806ddff54986671c4a723bc8c4eaf360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6be93a343914c0a0f987d5b9720359ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&middot;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&middot;&#48;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" 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\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39af43abd99adaf051fde7775af522c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" 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\"><em data-effect=\"italics\">y<\/em>-intercept <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\">Find the point symmetric to <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;\" \/> across the axis of symmetry.<\/td>\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-2da1eb750fc283f55cb9396d5536b47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#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\">Find the <em data-effect=\"italics\">x<\/em>-intercepts.<\/td>\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-ae2e2be4d02483cd11c0b450fdc79f4e_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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#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;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"127\" width=\"188\" style=\"vertical-align: -59px;\" \/><\/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 square root of a negative number<span data-type=\"newline\"><br \/><\/span>tells us the solutions are complex<span data-type=\"newline\"><br \/><\/span>numbers. So there are no <em data-effect=\"italics\">x<\/em>-intercepts.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871865347\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_005b_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-id1163872407949\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872463553\">\n<div data-type=\"problem\" id=\"fs-id1163872458721\">\n<p id=\"fs-id1163872436477\"><span class=\"token\">\u24d0<\/span> Write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3450cf8f97e85c18042485c74a7fab98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/> in standard form and <span class=\"token\">\u24d1<\/span> use properties of standard form to graph the equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871858614\">\n<p id=\"fs-id1163871858499\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66ee8f7aa405914db0a7b977aa0b92c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"135\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871880010\" data-alt=\"This graph shows a parabola opening upwards, with vertex (negative 1, 3) and y intercept (0, 5). It has the point minus (2, 5) on it.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening upwards, with vertex (negative 1, 3) and y intercept (0, 5). It has the point minus (2, 5) on it.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872464291\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872607894\">\n<div data-type=\"problem\" id=\"fs-id1163872709856\">\n<p id=\"fs-id1163871858546\"><span class=\"token\">\u24d0<\/span> Write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d84dffe2bbc1b6979339174559a7d77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/> in standard form and <span class=\"token\">\u24d1<\/span> use properties of standard form to graph the equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872407145\">\n<p id=\"fs-id1163871879530\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f26dfd7b002685e6064db59b82aa5def_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872517945\" data-alt=\"This graph shows a parabola opening downwards, with vertex (2, 1) and axis of symmetry x equals 2. Its y intercept is (0, negative 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downwards, with vertex (2, 1) and axis of symmetry x equals 2. Its y intercept is (0, negative 7).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163871876982\">\n<h3 data-type=\"title\">Graph Horizontal Parabolas<\/h3>\n<p id=\"fs-id1163871783082\">Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.<\/p>\n<table id=\"fs-id1171791485940\" summary=\"This table, titled horizontal parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is x equals ay squared plus by plus c and standard form is x equals a open parentheses y minus k close parentheses squared plus h. Row one: orientation: general form is a greater than 0, right and a less than 0 left. Standard form is the same. Row 2: Axis of symmetry: general form is y equals minus b upon 2a and standard form is y equals k. Row 3: vertex: general form, substitute y equals minus b upon 2a and solve for x; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Horizontal Parabolas<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><\/th>\n<th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a214c67af8566d1bde959ee0b988ea7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#121;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-768e98072205b6f32ac40430fa75c9dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> right; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> left<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> right; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> left<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-308a0550b99fd1d3473a29d50fed9537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#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=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-523f1a3d67678826bd298f5f6fd3916d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-308a0550b99fd1d3473a29d50fed9537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#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;\" \/> and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercept<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Let <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=\"middle\" data-align=\"center\">Let <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<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1163872015008\">The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the <em data-effect=\"italics\">x<\/em>&#8211; or <em data-effect=\"italics\">y<\/em>-axis is merely an example.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872412399\" data-alt=\"This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\" \/><\/span><\/p>\n<p id=\"fs-id1163872455604\">Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function.<\/p>\n<p id=\"fs-id1163872546837\">To graph a <span data-type=\"term\" class=\"no-emphasis\">parabola<\/span> that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> variables.<\/p>\n<div data-type=\"note\" id=\"fs-id1163872530085\" class=\"howto\">\n<div data-type=\"title\">Graph horizontal parabolas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5248eab42a069066a5a134abdf20de2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#61;&#97;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#121;&#43;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"314\" style=\"vertical-align: -12px;\" \/> using properties.<\/div>\n<ol id=\"fs-id1163872471690\" type=\"1\" class=\"stepwise\">\n<li>Determine whether the parabola opens to the left or to the right.<\/li>\n<li>Find the axis of symmetry.<\/li>\n<li>Find the vertex.<\/li>\n<li>Find the <em data-effect=\"italics\">x<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">x<\/em>-intercept across the axis of symmetry.<\/li>\n<li>Find the <em data-effect=\"italics\">y<\/em>-intercepts.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1163872571855\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872020109\">\n<div data-type=\"problem\" id=\"fs-id1163872472046\">\n<p id=\"fs-id1163872472049\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-131a874aaa5ab5d0fa123256a235a21a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872514519\">\n<table id=\"fs-id1163872014824\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals 2y squared. Here, a is 2 and the parabola opens to the right. To find the axis of symmetry, find y equals minus b upon 2a. Substituting values, we get y equal to 0 divided by two times two. Hence y is 0. This is the axis of symmetry. The vertex is on this line. Let y be 0. Substituting in equation, we get x equals 0. The vertex is (0, 0). Since the vertex is (0, 0) both the x- and y-intercepts are the point (0, 0). To graph the parabola we need more points. In this case it is easiest to choose values of y.\" 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-id1163872838237\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007b_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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eebb7c55daa365334583aac412e471ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> the parabola opens to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872550744\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007a_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 axis of symmetry, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-131ba7ee106653b845c3138bea4cd832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871930867\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007c_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-id1163872511323\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007d_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-id1163872617865\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007e_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-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/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-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872646179\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007f_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-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872697362\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007g_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-id1163872555371\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_007h_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-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#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<\/tbody>\n<\/table>\n<p id=\"fs-id1171792502294\">Since the vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> both the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts are the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> To graph the parabola we need more points. In this case it is easiest to choose values of <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872666028\" data-alt=\"In the equation x equals 2 y squared, when y is 1, x is 2 and when y is 2, x is 8. The points are (2, 1) and (8, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the equation x equals 2 y squared, when y is 1, x is 2 and when y is 2, x is 8. The points are (2, 1) and (8, 2).\" \/><\/span><\/p>\n<p><span data-type=\"newline\"><br \/><\/span> We also plot the points symmetric to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9b49b201a3efa56726407933e068d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> across the <em data-effect=\"italics\">y<\/em>-axis, the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d12b9cd0b9abd94fdac4f1989dcc0baa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76273d11a41fc3c733f6aa36a6a19817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1171791085709\">Graph the parabola.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872506672\" data-alt=\"This graph shows right opening parabola with vertex (0, 0). Four points are marked on it: point (2, 1), point (2, negative 1), point (8, 2) and point (8 minus 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows right opening parabola with vertex (0, 0). Four points are marked on it: point (2, 1), point (2, negative 1), point (8, 2) and point (8 minus 2).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872518547\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163871859575\">\n<div data-type=\"problem\" id=\"fs-id1163872645830\">\n<p id=\"fs-id1163872645832\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a06f7ac6810a816ad784b7a49b7126a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872839751\"><span data-type=\"media\" id=\"fs-id1163871979402\" data-alt=\"This graph shows right opening parabola with vertex at origin. Two points on it are (4, 2) and (4, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows right opening parabola with vertex at origin. Two points on it are (4, 2) and (4, negative 2).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163871880202\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872667845\">\n<div data-type=\"problem\" id=\"fs-id1163871925446\">\n<p id=\"fs-id1163871925448\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73f0b84c3c3f2e44f716c656bdb60a70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872457850\"><span data-type=\"media\" id=\"fs-id1163871858943\" data-alt=\"This graph shows left opening parabola with vertex at origin. Two points on it are (negative 4, 2) and (negative 4, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows left opening parabola with vertex at origin. Two points on it are (negative 4, 2) and (negative 4, negative 2).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163871924267\">In the next example, the vertex is not the origin.<\/p>\n<div data-type=\"example\" id=\"fs-id1163871924271\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872840036\">\n<div data-type=\"problem\" id=\"fs-id1163872840038\">\n<p id=\"fs-id1163872564141\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd39eda974c27bdc8726bfb89badb1ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872511188\">\n<table id=\"fs-id1163872743219\" class=\"unnumbered unstyled can-break\" summary=\"The equation is minus y squared plus 2y plus 8. Since a is minus 1, the parabola opens to the left. To find the axis of symmetry, find y equals minus b upon 2a. The axis is y equals 1. The vertex is on this line. Substituting y equals 1 in the equation, we get x equal to 9. The vertex is (9, 1). Substituting y equal to 0 in the original equation, we get x intercept (8, 0). The symmetric point is (8, 2). Substituting x equal to 0 in the original equation, we get y intercepts (0, 4) and (0, negative 2). Connect the points to graph the 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-id1163872504881\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c865fb1787a14d8e5ea32aa0dcc7529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> the parabola opens to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872502350\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010a_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 axis of symmetry, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-131ba7ee106653b845c3138bea4cd832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872555506\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\"><span data-type=\"media\" id=\"fs-id1163872555636\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010d_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-id1163872713792\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010e_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=\"center\">The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4acfdc8013815a0ea0c7059786c6402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/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-f4acfdc8013815a0ea0c7059786c6402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872617374\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010f_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-f4acfdc8013815a0ea0c7059786c6402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872532027\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010g_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-id1163872645556\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010h_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=\"center\">The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98a3b86886ca4d1e8ba9f27c02774619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#49;&#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\">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-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871868721\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010i_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-id1163872697152\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010j_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-id1163872536891\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010k_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=\"center\">The <em data-effect=\"italics\">x<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81fc9fcab5d76c4841e410e8f1a11832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#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\">The point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52a836a9aba870ffb4036b68b4cb0c99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> is one unit below the line of<span data-type=\"newline\"><br \/><\/span>symmetry. The symmetric point one unit<span data-type=\"newline\"><br \/><\/span>above the line of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9b49b201a3efa56726407933e068d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#50;&#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=\"center\">Symmetric point is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a189c7c9e7c26f0f299cea31ee065cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#50;&#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\">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-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872540069\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010l_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-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872515142\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010m_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-id1163872422916\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010n_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-id1163872686399\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010o_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-id1163872546302\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010p_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=\"center\">The <em data-effect=\"italics\">y<\/em>-intercepts are <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;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f644d689ddf634b480d4ff320ff89c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#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\">Connect the points to graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872626896\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_010q_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-id1163872545783\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872520607\">\n<div data-type=\"problem\" id=\"fs-id1163872520609\">\n<p id=\"fs-id1163872548750\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07930b48ce241b2cf609079ea3af7a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872570688\"><span data-type=\"media\" id=\"fs-id1163872520735\" data-alt=\"This graph shows left opening parabola with vertex (16, negative 2) and x intercept (12, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows left opening parabola with vertex (16, negative 2) and x intercept (12, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872561489\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872565294\">\n<div data-type=\"problem\" id=\"fs-id1163872565296\">\n<p id=\"fs-id1163872544224\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f1937b6ceac36bbd758d5f7e181210a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/> by using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872555620\"><span data-type=\"media\" id=\"fs-id1163872019443\" data-alt=\"This graph shows left opening parabola with vertex (negative 2, 1) and x intercept minus (3, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows left opening parabola with vertex (negative 2, 1) and x intercept minus (3, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163871868755\">In <a href=\"#fs-id1171791485940\" class=\"autogenerated-content\">(Figure)<\/a>, we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61a0adef03f36027f84fe7445f34374e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#104;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/> We will use this procedure in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872548704\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872570665\">\n<div data-type=\"problem\" id=\"fs-id1163872570667\">\n<p id=\"fs-id1163872549013\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3fc7a8bd2b3ba95dfce61c3e3f5b9a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/> using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872543817\">\n<table id=\"fs-id1163872565265\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals 2 open parentheses y minus 2 close parentheses squared plus 1. Here, a is 2, h is 1 and k is 2. Since a is 2, the parabola opens to the right. The axis of symmetry is y equals k or y equals 2) and vertex is (h, k) or (1, 2). By substituting y equals 0 in the equation, we find x intercept (9, 0). The point symmetric to this across the axis is (9, 4). By substituting x equals 0 in the equation and simplifying, we arrive at minus 1 equals 2 open parentheses y minus 2 close parentheses squared. A square cannot be negative, so there is no real solution. So there are no y-intercepts. Graph the 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-id1163872574283\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eebb7c55daa365334583aac412e471ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19f5eec9e2096ffb4bfe597b31e07d42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-633d0c333092d820a4ccc4ff860bc5da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eebb7c55daa365334583aac412e471ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> the parabola opens to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163871783227\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_011a_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\">The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afa765d3f555400c4f5b69f3a3d3d4a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\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-7b6bd83670c6c128f55bcd7d2ee07880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fda069db202961fbab9483df8729b66d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\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-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\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-7b6bd83670c6c128f55bcd7d2ee07880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b55ec00d0efa63f10ac93a2bba77a12_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;&#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\">Find the <em data-effect=\"italics\">x<\/em>-intercept by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3e7ab3bf632cddfb5e5c4939569ffa5_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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#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;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#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;&#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=\"62\" width=\"156\" style=\"vertical-align: -23px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\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-7b6bd83670c6c128f55bcd7d2ee07880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The <em data-effect=\"italics\">x<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-754a083691ae02221203e3591b7f4f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#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\">Find the point symmetric to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e80d3ec8d2361b248d4337b97bcc361_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> across the<span data-type=\"newline\"><br \/><\/span>axis of symmetry.<\/td>\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-7d388fc1cc9837f5af8121c0c09c2233_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#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\">Find the <em data-effect=\"italics\">y<\/em>-intercepts. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9d368ce08b3fa065adea49de696a225_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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#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;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#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=\"68\" width=\"168\" style=\"vertical-align: -28px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">A square cannot be negative, so there is no real<span data-type=\"newline\"><br \/><\/span>solution. So there are no <em data-effect=\"italics\">y<\/em>-intercepts.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872023683\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_011c_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-id1163872013292\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872519614\">\n<div data-type=\"problem\" id=\"fs-id1163872519616\">\n<p id=\"fs-id1163872555545\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f953f832da7792687e029ea556ed4a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/> using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872664549\"><span data-type=\"media\" id=\"fs-id1163872009558\" data-alt=\"This graph shows a parabola opening right with vertex (2, 1) and x intercept (5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening right with vertex (2, 1) and x intercept (5, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872011035\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872630754\">\n<div data-type=\"problem\" id=\"fs-id1163872630756\">\n<p id=\"fs-id1163872643897\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26541e4c97cffc258437152d0d6adc32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/> using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872697286\"><span data-type=\"media\" id=\"fs-id1163872505694\" data-alt=\"This graph shows a parabola opening right with vertex (2, 3) and symmetric points (4, 2) and (4, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening right with vertex (2, 3) and symmetric points (4, 2) and (4, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926769473\">In the next example, we notice the a is negative and so the parabola opens to the left.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872743277\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872543604\">\n<div data-type=\"problem\" id=\"fs-id1163872543606\">\n<p id=\"fs-id1163872543608\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9620736823bbe90a96c8062a651327f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/> using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872478313\">\n<table id=\"fs-id1163872464385\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals negative 4 open parentheses y plus 1 close parentheses squared plus 4. Here a is negative 4, h is 4 and k is negative 1. Since a is negative 4, the parabola opens to the left. The axis of symmetry is y equals negative 1 and vertex is (4, negative 1). Substituting y equals 0, we get x intercept (0, 0). The symmetric point across the axis is (0, negative 2). Substituting x equals 0, we get y intercepts (0, 0) and (0, negative 2). Graph the 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-id1163872506744\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9991c38e9674f694d4f10de5596656e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-012e3507a76cfdeee5a7f53eb47b04df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fe385ef91df1be21fe481876057bcb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"55\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9991c38e9674f694d4f10de5596656e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/> the parabola opens to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872840579\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_012a_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\">The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afa765d3f555400c4f5b69f3a3d3d4a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\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-2c539148683ba11410ba46f819d41cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1144e182f26c1fd5166b5411a3ed3cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\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-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\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-2c539148683ba11410ba46f819d41cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65fe2760e5f7600eb05caea522df3c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#49;&#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\">Find the <em data-effect=\"italics\">x<\/em>-intercept by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa6aa0d368c922cc0e0e76a207f50949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#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;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#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;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"171\" style=\"vertical-align: -23px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\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-2c539148683ba11410ba46f819d41cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The <em data-effect=\"italics\">x<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#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\">Find the point symmetric to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> across the<span data-type=\"newline\"><br \/><\/span>axis of symmetry.<\/td>\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-5f714d378145eef279ce082072649219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#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\">Find the <em data-effect=\"italics\">y<\/em>-intercepts.<\/td>\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-2aefb481980e8eb0af586191cb8e3d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/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-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa9fc06763c10ceb6cf49a1ceada9dd3_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;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#43;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"90\" width=\"201\" style=\"vertical-align: -39px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14e81955e7bd19b425351bdc352ea6e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#43;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#45;&#49;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"198\" 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=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69a858424b85df233c6f9983b4997346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#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;&#121;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"168\" 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\">The <em data-effect=\"italics\">y<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f644d689ddf634b480d4ff320ff89c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#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\">Graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872434858\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_012c_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-id1163872446043\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872502554\">\n<div data-type=\"problem\" id=\"fs-id1163872502556\">\n<p id=\"fs-id1163872472786\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79b02820f2a064d60f3c677673bffab7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/> using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872568837\"><span data-type=\"media\" id=\"fs-id1163872561835\" data-alt=\"This figure shows a parabola opening to the left with vertex (4, negative 2) and y intercepts (0, negative 1) and (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening to the left with vertex (4, negative 2) and y intercepts (0, negative 1) and (0, negative 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872568831\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872536885\">\n<div data-type=\"problem\" id=\"fs-id1163872536887\">\n<p id=\"fs-id1163872560738\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-621ce7a12965a9df91ca7b79414c0a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/> using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872437230\"><span data-type=\"media\" id=\"fs-id1163872436951\" data-alt=\"This figure shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_313_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926770425\">The next example requires that we first put the equation in standard form and then use the properties.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872467394\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872467396\">\n<div data-type=\"problem\" id=\"fs-id1163872467398\">\n<p id=\"fs-id1163872502738\">Write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d45397298101bbdf75d5e42651118475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#43;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/> in standard form and then use the properties of the standard form to graph the equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872441461\">\n<table id=\"fs-id1163872422949\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x equals 2 y squared plus 12y plus 17. Rewrite in standard form by completing the square x equals 2 open parentheses y squared plus 6y plus 9 close parentheses plus 17 minus 18. This is x equals 2 open parentheses y plus 3 close parentheses squared minus 1. Here a is 2, h is negative 1 and k is negative 3. The axis is y equals negative 3. The vertex is (negative 1, negative 3). By substituting y equals 0, we get the x intercept (17, 0). Its symmetric point across the axis of symmetry is (17, negative 6). By substituting x equals 0 in the equation, we get approximate y values equal to minus 2.3 and minus 3.7. Graph the 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-id1163872470366\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\">Rewrite the function in<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-768e98072205b6f32ac40430fa75c9dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"137\" style=\"vertical-align: -4px;\" \/> form by completing<span data-type=\"newline\"><br \/><\/span>the square.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872485064\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872616679\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872515567\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_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\"><span data-type=\"media\" id=\"fs-id1163872458823\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the constants <em data-effect=\"italics\">a, h, k<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd71b90e10769de11021207042fe9fb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#61;&#45;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"206\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eebb7c55daa365334583aac412e471ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> the parabola opens to<span data-type=\"newline\"><br \/><\/span>the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1163872470774\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013a_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\">The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afa765d3f555400c4f5b69f3a3d3d4a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\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-86d8afce888b42528ef64b096768deff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The axis of symmetry is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c969882b6218a3357058a4f00796c77f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\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-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\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-86d8afce888b42528ef64b096768deff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The vertex is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59171119206d5c457b9713bf74f27414_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#51;&#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\">Find the <em data-effect=\"italics\">x<\/em>-intercept by substituting<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75302a626d14800af3f2a671e9c7e23e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#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;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#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;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"156\" style=\"vertical-align: -24px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\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-86d8afce888b42528ef64b096768deff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>The <em data-effect=\"italics\">x<\/em>-intercept is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-270354cf44c3f4086570af8a17d0d7ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#55;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the point symmetric to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb361ecb1a2975abb43ced821f9dd3af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#55;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>across the axis of symmetry.<\/td>\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-6d16244ca5b9db2ca0b7f3ee7b5544be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#55;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">y<\/em>-intercepts.<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20f87f0bfd29e3ac1fc6d2217e27eb55_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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#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;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#43;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#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=\"150\" width=\"186\" style=\"vertical-align: -70px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9541fd232e66d9aee534f379f1916711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#45;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"215\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9c808591bd2e63413b056f0d2e2aff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#45;&#50;&#46;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#45;&#51;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"175\" 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\">The <em data-effect=\"italics\">y<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18dbea833531ef39f23c5401dcbe7e69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"224\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872570813\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_013g_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-id1163872471299\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872515426\">\n<div data-type=\"problem\" id=\"fs-id1163872515428\">\n<p id=\"fs-id1163872537712\"><span class=\"token\">\u24d0<\/span> Write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42c8d00daea4b6a0c9b9871b86133acb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#121;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/> in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872556946\">\n<p id=\"fs-id1163872556948\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a166bec1a950317d9a5fd2781019f76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"135\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872517880\" data-alt=\"This graph shows a parabola opening to the right with vertex (4, negative 1) and x intercept (7, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (4, negative 1) and x intercept (7, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872520960\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872530074\">\n<div data-type=\"problem\" id=\"fs-id1163872530076\">\n<p id=\"fs-id1163872503156\"><span class=\"token\">\u24d0<\/span> Write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a28c7672554ad2861b107a633f630772_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#121;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"161\" style=\"vertical-align: -4px;\" \/> in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872515248\">\n<p id=\"fs-id1163872515250\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79b02820f2a064d60f3c677673bffab7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872571430\" data-alt=\"This graph shows a parabola opening to the left with vertex (4, negative 2) and x intercept minus (12, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_315_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (4, negative 2) and x intercept minus (12, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872566004\">\n<h3 data-type=\"title\">Solve Applications with Parabolas<\/h3>\n<p id=\"fs-id1163872511907\">Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872566243\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872566245\">\n<div data-type=\"problem\" id=\"fs-id1163872630376\">\n<p id=\"fs-id1163872630378\">Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872566959\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 10 feet high and 20 feet wide at the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_014_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 10 feet high and 20 feet wide at the base.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872408903\">\n<p id=\"fs-id1163872408906\">We will first set up a coordinate system and draw the parabola. The graph will give us the information we need to write the equation of the graph in the standard form<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0551f43b12a39be94457982f93cb2b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> <\/p>\n<table id=\"fs-id1163872571206\" class=\"unnumbered unstyled can-break\" summary=\"Let the lower left side of the bridge be the origin of the coordinate grid at the point (0, 0). Since the base is 20 feet wide the point (20, 0) represents the lower right side. The bridge is 10 feet high at the highest point. The highest point is the vertex of the parabola so the y coordinate of the vertex will be 10. Since the bridge is symmetric, the vertex must fall halfway between the left most point, (0, 0) and the rightmost point (20, 0). From this we know that the x coordinate of the vertex will also be 10. The vertex is 10, 10. So h is 10 and k is 10. Substitute the values into the standard form y equals a open parentheses x minus h close parentheses squared plus k. The value of a is still unknown. To find the value of a use one of the other points on the parabola, point (0, 0). Substituting the values into the equation, we get a equal to minus 1 by 10. Substitute the value for a into the equation. We get y equals minus 1 upon 10 open parentheses x minus 10 close parentheses squared plus 10.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Let the lower left side of the bridge be the<span data-type=\"newline\"><br \/><\/span>origin of the coordinate grid at the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>Since the base is 20 feet wide the point<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f446073f44ccd35139a43184833b7b80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> represents the lower right side.<span data-type=\"newline\"><br \/><\/span>The bridge is 10 feet high at the highest<span data-type=\"newline\"><br \/><\/span>point. The highest point is the vertex of<span data-type=\"newline\"><br \/><\/span>the parabola so the <em data-effect=\"italics\">y<\/em>-coordinate of the<span data-type=\"newline\"><br \/><\/span>vertex will be 10.<span data-type=\"newline\"><br \/><\/span>Since the bridge is symmetric, the vertex<span data-type=\"newline\"><br \/><\/span>must fall halfway between the left most<span data-type=\"newline\"><br \/><\/span>point, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> and the rightmost point<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc52ea2966282c48fc80b349ef388859_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> From this we know that the<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">x<\/em>-coordinate of the vertex will also be 10.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872572181\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the vertex, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d093087ae083fe467ca87a94f875ba8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#44;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" 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=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3faacdf409092961cd5162bf17a99e9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#49;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8195;&#125;&#107;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values into the standard form.<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span>The value of <em data-effect=\"italics\">a<\/em> is still unknown. To find<span data-type=\"newline\"><br \/><\/span>the value of <em data-effect=\"italics\">a<\/em> use one of the other points<span data-type=\"newline\"><br \/><\/span>on the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-095c5d807e446b0187482d45a6dd0166_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#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;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#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=\"66\" width=\"206\" style=\"vertical-align: -27px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values of the other point<span data-type=\"newline\"><br \/><\/span>into the equation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-625b34da98663cab5c76b2bf6eac750b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#51;&#53;&#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;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#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;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"176\" style=\"vertical-align: -16px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d099fd0d25030d44c501bcea11e5ca6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#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;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#48;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#48;&#125;&#123;&#49;&#48;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#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=\"113\" width=\"195\" style=\"vertical-align: -52px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff0cd2150be37b8a3d33905da710a708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the value for <em data-effect=\"italics\">a<\/em> into the<span data-type=\"newline\"><br \/><\/span>equation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c54b923f86c8619c7a52a1edf77422_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"175\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872718951\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872804356\">\n<div data-type=\"problem\" id=\"fs-id1163872804358\">\n<p id=\"fs-id1163872804373\">Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872804378\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 20 feet high and 40 feet wide at the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_016_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 20 feet high and 40 feet wide at the base.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163871996401\">\n<p id=\"fs-id1163871996403\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32e9a296815a1b2f6d2370fbc3e53f2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#48;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"175\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872713382\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872713400\">\n<div data-type=\"problem\" id=\"fs-id1163872713402\">\n<p id=\"fs-id1163872559727\">Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872559731\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 5 feet high and 10 feet wide at the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 5 feet high and 10 feet wide at the base.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872803531\">\n<p id=\"fs-id1163872803534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27cbf789147685cf5b024e5682a9ccbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"116\" style=\"vertical-align: -6px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-583000fb64168f5e47589944faf22a2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"149\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163871999504\" class=\"media-2\">\n<p id=\"fs-id1163871996260\">Access these online resources for additional instructions and practice with quadratic functions and parabolas.<\/p>\n<ul id=\"fs-id1163871996293\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37quadfunc\">Quadratic Functions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37conhorizpbola\">Introduction to Conics and Graphing Horizontal Parabolas<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163871997815\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1163872662987\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Parabola:<\/strong> A <strong data-effect=\"bold\">parabola<\/strong> is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the <strong data-effect=\"bold\">focus,<\/strong> and the fixed line is called the <strong data-effect=\"bold\">directrix<\/strong> of the parabola.<span data-type=\"newline\"><br \/><\/span><br \/>\n<table id=\"fs-id1163872713464\" summary=\"This table, titled vertical parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is y equals ax squared plus bx plus c and standard form is y equals a open parentheses x minus h close parentheses squared plus k. Row one: orientation: general form is a greater than 0, up and a less than 0 down. Standard form is the same. Row 2: Axis of symmetry: general form is x equals minus b upon 2a and standard form is x equals h. Row 3: vertex: general form, substitute x equals minus b upon 2a and solve for y; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Vertical Parabolas<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><\/th>\n<th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af4106dc6abca5e766ae7ae01481411c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66d6e02b37a1552327ac6337041618b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> up; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> down<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> up; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><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=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f6486da4c3e33f0fbdf54846ff952eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Substitute <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;\" \/> and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>&#8211; intercept<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Let <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=\"middle\" data-align=\"center\">Let <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<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163872607467\" data-alt=\"This figure shows two parabolas with axis x equals h and vertex (h, k). The one on the left opens up and a is greater than 0. The one on the right opens down. Here a is less than 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_018_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis x equals h and vertex (h, k). The one on the left opens up and a is greater than 0. The one on the right opens down. Here a is less than 0.\" \/><\/span> <\/li>\n<li><strong data-effect=\"bold\">How to graph vertical parabolas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecc01ead2bd0409f76a22c1c0c2604aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -7px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-927787f213e43ccfb6b01346ed30a7de_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;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> using properties.<\/strong>\n<ol id=\"fs-id1163871863277\" type=\"1\" class=\"stepwise\">\n<li>Determine whether the parabola opens upward or downward.<\/li>\n<li>Find 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.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<p><span data-type=\"newline\"><br \/><\/span><\/p>\n<table id=\"fs-id1163872728536\" summary=\"This table, titled horizontal parabolas, has 3 columns, 5 rows and a header row. The header row labeled the second and third column general form and standard form respectively. General form is x equals ay squared plus by plus c and standard form is x equals a open parentheses y minus k close parentheses squared plus h. Row one: orientation: general form is a greater than 0, right and a less than 0 left. Standard form is the same. Row 2: Axis of symmetry: general form is y equals minus b upon 2a and standard form is y equals k. Row 3: vertex: general form, substitute y equals minus b upon 2a and solve for x; standard form is point h, k. Row 4: y intercept: general and standard forms, let x be 0. Row 5: x intercept: general and standard forms, let y be 0.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Horizontal Parabolas<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><\/th>\n<th data-valign=\"top\" data-align=\"left\">General form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a214c67af8566d1bde959ee0b988ea7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#121;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/th>\n<th data-valign=\"top\" data-align=\"left\">Standard form<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-768e98072205b6f32ac40430fa75c9dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> right; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> left<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> right; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c2f9a17945ee163942aad7a36538cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> left<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Axis of symmetry<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-308a0550b99fd1d3473a29d50fed9537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#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=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-523f1a3d67678826bd298f5f6fd3916d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Vertex<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Substitute <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-308a0550b99fd1d3473a29d50fed9537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#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;\" \/> and<span data-type=\"newline\"><br \/><\/span>solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">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;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercept<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Let <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=\"middle\" data-align=\"center\">Let <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<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/><\/span><span data-type=\"media\" id=\"fs-id1163872023565\" data-alt=\"This figure shows two parabolas with axis of symmetry y equals k, and vertex (h, k). The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_019_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two parabolas with axis of symmetry y equals k, and vertex (h, k). The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">How to graph horizontal parabolas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e59aab23f4d8b09d844cf2c79b74c51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#61;&#97;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#121;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -7px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5341bdea5fcecb987c5421efaf7b224f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"137\" style=\"vertical-align: -4px;\" \/> using properties.<\/strong>\n<ol id=\"fs-id1163872706418\" type=\"1\" class=\"stepwise\">\n<li>Determine whether the parabola opens to the left or to the right.<\/li>\n<li>Find the axis of symmetry.<\/li>\n<li>Find the vertex.<\/li>\n<li>Find the <em data-effect=\"italics\">x<\/em>-intercept. Find the point symmetric to the <em data-effect=\"italics\">x<\/em>-intercept across the axis of symmetry.<\/li>\n<li>Find the <em data-effect=\"italics\">y<\/em>-intercepts.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163872686249\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163872014554\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1163872622892\"><strong data-effect=\"bold\">Graph Vertical Parabolas<\/strong><\/p>\n<p id=\"fs-id1163873805265\">In the following exercises, graph each equation by using properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872643998\">\n<div data-type=\"problem\" id=\"fs-id1163872573165\">\n<p id=\"fs-id1163872573167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-060b44ead1d070782961944e01065ffb_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;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872444219\"><span data-type=\"media\" id=\"fs-id1163872572880\" data-alt=\"This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872572850\">\n<div data-type=\"problem\" id=\"fs-id1163872015072\">\n<p id=\"fs-id1163872015074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efc32fd0dcb0d7732f6656b8509d192a_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;&#56;&#120;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872510316\">\n<div data-type=\"problem\" id=\"fs-id1163872510318\">\n<p id=\"fs-id1163872510320\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad1ac3f62be96cfdc25626aeb4c3625d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871879845\"><span data-type=\"media\" id=\"fs-id1163871879849\" data-alt=\"This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872541582\">\n<div data-type=\"problem\" id=\"fs-id1163872541584\">\n<p id=\"fs-id1163872627634\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcc65aea8a0bb3739817ff27d46a11ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872435842\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872644217\">\n<div data-type=\"problem\" id=\"fs-id1163872506006\">\n<p id=\"fs-id1163872506008\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-542b562a02f9188de24159ad7a6d355c_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;&#50;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872730548\">\n<p id=\"fs-id1163872613320\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-822568aba42181e5118cf2ad13d8d69e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"126\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872661395\" data-alt=\"This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872628550\">\n<div data-type=\"problem\" id=\"fs-id1163871942141\">\n<p id=\"fs-id1163871942143\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac45acf7f38a404c45ce6901a30c05d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872087819\">\n<div data-type=\"problem\" id=\"fs-id1163871998458\">\n<p id=\"fs-id1163871998460\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7c37bf860211fa12559f95c82e2698a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871999528\">\n<p id=\"fs-id1163871998170\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9347152fc270d93d554c7b193c19e36f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871859944\" data-alt=\"This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872505265\">\n<div data-type=\"problem\" id=\"fs-id1163872505268\">\n<p id=\"fs-id1163872505270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e14996073ea5ab2eaa846d09c0b7d40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872838584\"><strong data-effect=\"bold\">Graph Horizontal Parabolas<\/strong><\/p>\n<p id=\"fs-id1163872721660\">In the following exercises, graph each equation by using properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872390039\">\n<div data-type=\"problem\" id=\"fs-id1163872390041\">\n<p id=\"fs-id1163872390043\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed2dc971a6ee266c63a2ac4f4c636c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872838682\"><span data-type=\"media\" id=\"fs-id1163872468380\" data-alt=\"This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872512005\">\n<div data-type=\"problem\" id=\"fs-id1163872512008\">\n<p id=\"fs-id1163872511006\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-995d115b5fbf2340a9c4e3aae844f855_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872463058\">\n<div data-type=\"problem\" id=\"fs-id1163872463060\">\n<p id=\"fs-id1163872659707\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e913c3a3c6cdf3000753a63ad0da3369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872471858\"><span data-type=\"media\" id=\"fs-id1163872601352\" data-alt=\"This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872574426\">\n<div data-type=\"problem\" id=\"fs-id1163872557861\">\n<p id=\"fs-id1163872557864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3edbe4cd197fae4ef47d6ca12990b5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872557264\">\n<div data-type=\"problem\" id=\"fs-id1163872557266\">\n<p id=\"fs-id1163872557268\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d431e52390c32e1c274db2382f7d57dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872664303\"><span data-type=\"media\" id=\"fs-id1163872664307\" data-alt=\"This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872663524\">\n<div data-type=\"problem\" id=\"fs-id1163872000432\">\n<p id=\"fs-id1163872000434\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a86bcdeac5e06df07514b46b2ab0ed92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872095357\">\n<div data-type=\"problem\" id=\"fs-id1163872573368\">\n<p id=\"fs-id1163872573370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9407f5d32483182fc728a9ad4bed8e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#121;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872096646\"><span data-type=\"media\" id=\"fs-id1163872516356\" data-alt=\"This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872610552\">\n<div data-type=\"problem\" id=\"fs-id1163872610554\">\n<p id=\"fs-id1163872610556\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ef8082a40bea66918c5a48cd1278f0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872550366\">\n<div data-type=\"problem\" id=\"fs-id1163872019422\">\n<p id=\"fs-id1163872019424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e3ee325d84a76fedefd5d6c5624e781_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872426618\"><span data-type=\"media\" id=\"fs-id1163872516187\" data-alt=\"This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872529947\">\n<div data-type=\"problem\" id=\"fs-id1163872529950\">\n<p id=\"fs-id1163872529952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f04af8b7cecb91cc6d2952cd26ff035c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872548729\">\n<div data-type=\"problem\" id=\"fs-id1163872544198\">\n<p id=\"fs-id1163872544200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a71a0b4b297df7025beb86c51bd6946a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871858570\"><span data-type=\"media\" id=\"fs-id1163872750269\" data-alt=\"This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872837439\">\n<div data-type=\"problem\" id=\"fs-id1163872837441\">\n<p id=\"fs-id1163872560004\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5286abcea93823333c364975a571c69f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872562094\">\n<div data-type=\"problem\" id=\"fs-id1163872562096\">\n<p id=\"fs-id1163872562098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-345c91eceb9293356aaa29ed19ba3563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872642821\"><span data-type=\"media\" id=\"fs-id1163871912071\" data-alt=\"This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872520663\">\n<div data-type=\"problem\" id=\"fs-id1163872520665\">\n<p id=\"fs-id1163872520667\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b658f47901125d612461df8b15c0d5b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872391569\">\n<div data-type=\"problem\" id=\"fs-id1163872391571\">\n<p id=\"fs-id1163872699904\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77f20aff95e9fb3b00160eebaefb4fe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871873361\"><span data-type=\"media\" id=\"fs-id1163872520699\" data-alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872505590\">\n<div data-type=\"problem\" id=\"fs-id1163872505593\">\n<p id=\"fs-id1163872565104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d54d842f12207e2c9bcc892f902bd473_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872572862\">\n<div data-type=\"problem\" id=\"fs-id1163872572864\">\n<p id=\"fs-id1163872572866\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c377e9593924e9bb28200a2fdb45f5c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872020157\"><span data-type=\"media\" id=\"fs-id1163871923731\" data-alt=\"This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871942317\">\n<div data-type=\"problem\" id=\"fs-id1163872730401\">\n<p id=\"fs-id1163872730404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f12e068b6e2d9d80f4f7ea02b83c67e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872505979\">\n<div data-type=\"problem\" id=\"fs-id1163872505981\">\n<p id=\"fs-id1163872628073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d752759191a32a666cf0385e286a62f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872448629\"><span data-type=\"media\" id=\"fs-id1163872448633\" data-alt=\"This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872426680\">\n<div data-type=\"problem\" id=\"fs-id1163872426682\">\n<p id=\"fs-id1163872426684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3044aaa9c6937fa8229b472a71988d41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872665896\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> use properties of the standard form to graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872665235\">\n<div data-type=\"problem\" id=\"fs-id1163872572389\">\n<p id=\"fs-id1163872572391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22ee08f385bedc09988ef63e6b849c7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872506209\">\n<p id=\"fs-id1163872506211\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24dd47c88a542ce43bd31c3308274275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"126\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872542914\" data-alt=\"This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872663620\">\n<div data-type=\"problem\" id=\"fs-id1163872663622\">\n<p id=\"fs-id1163872663625\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e26a640c37e3539f9d081ce04c73e38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#51;\" 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-id1163872558332\">\n<div data-type=\"problem\" id=\"fs-id1163872558334\">\n<p id=\"fs-id1163872558336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07126c4db1199212ce3f552f3b5a7295_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871980002\">\n<p id=\"fs-id1163872554258\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-621ce7a12965a9df91ca7b79414c0a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872665146\" data-alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872665071\">\n<div data-type=\"problem\" id=\"fs-id1163872665073\">\n<p id=\"fs-id1163872665075\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71000db5762bbdc98098e605cb7b50c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872647117\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p>\n<p id=\"fs-id1163872647000\">In the following exercises, match each graph to one of the following equations: <span class=\"token\">\u24d0<\/span> <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">y<\/em><sup>2<\/sup> = 64 <span class=\"token\">\u24d1<\/span> <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">y<\/em><sup>2<\/sup> = 49<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span> (<em data-effect=\"italics\">x<\/em> + 5)<sup>2<\/sup> + (<em data-effect=\"italics\">y<\/em> + 2)<sup>2<\/sup> = 4 <span class=\"token\">\u24d3<\/span> (<em data-effect=\"italics\">x<\/em> \u2212 2)<sup>2<\/sup> + (<em data-effect=\"italics\">y<\/em> \u2212 3)<sup>2<\/sup> = 9 <span class=\"token\">\u24d4<\/span> <em data-effect=\"italics\">y<\/em> = \u2212<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 8<em data-effect=\"italics\">x<\/em> \u2212 15 <span class=\"token\">\u24d5<\/span> <em data-effect=\"italics\">y<\/em> = 6<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 2<em data-effect=\"italics\">x<\/em> \u2212 1<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872627995\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872627997\"><span data-type=\"media\" id=\"fs-id1163872627999\" data-alt=\"This graph shows circle with center (0, 0) and radius 8 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (0, 0) and radius 8 units.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872505906\">\n<p id=\"fs-id1163872505908\"><span class=\"token\">\u24d0<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872505929\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872505931\"><span data-type=\"media\" id=\"fs-id1163872505933\" data-alt=\"This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872628330\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872628332\"><span data-type=\"media\" id=\"fs-id1163872628220\" data-alt=\"This graph shows circle with center (0, 0) and radius 7 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (0, 0) and radius 7 units.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872663640\">\n<p id=\"fs-id1163872663642\"><span class=\"token\">\u24d1<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871946568\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872643681\"><span data-type=\"media\" id=\"fs-id1163872643683\" data-alt=\"This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872465736\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872465739\"><span data-type=\"media\" id=\"fs-id1163872465741\" data-alt=\"This graph shows circle with center (2, 3) and radius 3 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (2, 3) and radius 3 units.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872548133\">\n<p id=\"fs-id1163872548135\"><span class=\"token\">\u24d3<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872647979\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872647982\"><span data-type=\"media\" id=\"fs-id1163872647937\" data-alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163872648028\"><strong data-effect=\"bold\">Solve Applications with Parabolas<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872543093\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872543096\">\n<p id=\"fs-id1163872648078\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872648082\" data-alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (negative 5, negative 2) and radius 2 units.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872648001\">\n<p id=\"fs-id1163872648004\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-322ee3ebe72b4c5c1dcfcc59aeceba9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"174\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872564608\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872564610\">\n<p id=\"fs-id1163872564612\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872558048\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872554290\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872554292\">\n<p id=\"fs-id1163872554467\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872554472\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872554384\">\n<p id=\"fs-id1163872554386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80b5168b546902afc427d6efea895010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"175\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872661810\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872661813\">\n<p id=\"fs-id1163872661815\">Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin (0, 0).<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872662438\" data-alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_210_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1163872664992\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1163872611307\">\n<div data-type=\"problem\" id=\"fs-id1163872665874\">\n<p id=\"fs-id1163872665876\">In your own words, define a parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872665880\">\n<p id=\"fs-id1163872642717\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872642723\">\n<div data-type=\"problem\" id=\"fs-id1163872642725\">\n<p id=\"fs-id1163872661848\">Is the parabola <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0e2ebf2a1b63bd75c0c19696a700e4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -4px;\" \/> a function? Is the parabola <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a06f7ac6810a816ad784b7a49b7126a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -4px;\" \/> a function? Explain why or why not.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872617605\">\n<div data-type=\"problem\" id=\"fs-id1163872617607\">\n<p id=\"fs-id1163872545250\">Write the equation of a parabola that opens up or down in standard form and the equation of a parabola that opens left or right in standard form. Provide a sketch of the parabola for each one, label the vertex and axis of symmetry.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872545256\">\n<p id=\"fs-id1163872648060\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872648065\">\n<div data-type=\"problem\" id=\"fs-id1163872648068\">\n<p id=\"fs-id1163872517075\">Explain in your own words, how you can tell from its equation whether a parabola opens up, down, left or right.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163872554329\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1163872554423\"><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-id1163872546272\" data-alt=\"This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_02_211_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1163872665913\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/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-id1163872665756\">\n<dt>parabola<\/dt>\n<dd id=\"fs-id1163872558173\">A parabola is all points in a plane that are the same distance from a fixed point and a fixed line.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15425","chapter","type-chapter","status-publish","hentry"],"part":15253,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15425","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\/15425\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/15253"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15425\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15425"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15425"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15425"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}