{"id":4932,"date":"2018-12-11T14:07:05","date_gmt":"2018-12-11T19:07:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-quadratic-functions-using-transformations\/"},"modified":"2018-12-11T14:09:48","modified_gmt":"2018-12-11T19:09:48","slug":"graph-quadratic-functions-using-transformations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/graph-quadratic-functions-using-transformations\/","title":{"raw":"Graph Quadratic Functions Using Transformations","rendered":"Graph Quadratic Functions Using Transformations"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Graph quadratic functions of the form \\(f\\left(x\\right)={x}^{2}+k\\)<\/li>\r\n \t<li>Graph quadratic functions of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\)<\/li>\r\n \t<li>Graph quadratic functions of the form \\(f\\left(x\\right)=a{x}^{2}\\)<\/li>\r\n \t<li>Graph quadratic functions using transformations<\/li>\r\n \t<li>Find a quadratic function from its graph<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169146948008\" class=\"be-prepared\">\r\n<p id=\"fs-id1169146616422\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol id=\"fs-id1169149034590\" type=\"1\">\r\n \t<li>Graph the function \\(f\\left(x\\right)={x}^{2}\\) by plotting points.\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/da9d6ce0-a078-4ca2-97af-8cb374f040f5#fs-id1167836683384\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Factor completely: \\({y}^{2}-14y+49.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358#fs-id1167835345249\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Factor completely: \\(2{x}^{2}-16x+32.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358#fs-id1167834396304\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149172251\">\r\n<h3 data-type=\"title\">Graph Quadratic Functions of the form \\(f\\left(x\\right)={x}^{2}+k\\)<\/h3>\r\n<p id=\"fs-id1169146658984\">In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of \\(f\\left(x\\right)={x}^{2}\\) and \u2018moving\u2019 it according to information given in the function equation. We call this graphing quadratic functions using transformations.<\/p>\r\n<p id=\"fs-id1169149155466\">In the first example, we will graph the quadratic function \\(f\\left(x\\right)={x}^{2}\\) by plotting points. Then we will see what effect adding a constant, <em data-effect=\"italics\">k<\/em>, to the equation will have on the graph of the new function \\(f\\left(x\\right)={x}^{2}+k.\\)<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149113341\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148867308\">\r\n<div data-type=\"problem\" id=\"fs-id1169148958428\">\r\n<p id=\"fs-id1169149123092\">Graph \\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={x}^{2}+2,\\) and \\(h\\left(x\\right)={x}^{2}-2\\) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148959807\">\r\n<p id=\"fs-id1169144560376\">Plotting points will help us see the effect of the constants on the basic \\(f\\left(x\\right)={x}^{2}\\) graph. We fill in the chart for all three functions.<\/p>\r\n<span data-type=\"media\" data-alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals x squared plus 2, the ordered pair (x, g of x), h of x equals x squared minus 2, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals x squared plus 2 column contains the expressions 9 plus 2, 4 plus 2, 1 plus 2, 0 plus 2, 1 plus 2, 4 plus 2, and 9 plus 2. The (x, g of x) column has the ordered pairs of (negative 3, 11), (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), (2, 6), and (3, 11). In the h of x equals x squared minus 2 column, the expressions given are 9 minus 2, 4 minus 2, 1 minus 2, 0 minus 2, 1 minus 2, 4 minus 2, and 9 minus 2. In last column, (x, h of x), contains the ordered pairs (negative 3, 7), (negative 2, 2), (negative 1, negative 1), (0, negative 2), (1, negative 1), (2, 2), and (3, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals x squared plus 2, the ordered pair (x, g of x), h of x equals x squared minus 2, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals x squared plus 2 column contains the expressions 9 plus 2, 4 plus 2, 1 plus 2, 0 plus 2, 1 plus 2, 4 plus 2, and 9 plus 2. The (x, g of x) column has the ordered pairs of (negative 3, 11), (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), (2, 6), and (3, 11). In the h of x equals x squared minus 2 column, the expressions given are 9 minus 2, 4 minus 2, 1 minus 2, 0 minus 2, 1 minus 2, 4 minus 2, and 9 minus 2. In last column, (x, h of x), contains the ordered pairs (negative 3, 7), (negative 2, 2), (negative 1, negative 1), (0, negative 2), (1, negative 1), (2, 2), and (3, 7).\" \/><\/span>\r\n<p id=\"fs-id1169149356096\">The <em data-effect=\"italics\">g<\/em>(<em data-effect=\"italics\">x<\/em>) values are two more than the <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) values. Also, the <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">x<\/em>) values are two less than the <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) values. Now we will graph all three functions on the same rectangular coordinate system.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169147088946\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle is the graph of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top parabola has been moved up 2 units, and the bottom has been moved down 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle is the graph of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top parabola has been moved up 2 units, and the bottom has been moved down 2 units.\" \/><\/span>\r\n<p id=\"fs-id1169149348416\">The graph of \\(g\\left(x\\right)={x}^{2}+2\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted up 2 units.<\/p>\r\n<p id=\"fs-id1169149114126\">The graph of \\(h\\left(x\\right)={x}^{2}-2\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted down 2 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169144768997\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149017068\">\r\n<div data-type=\"problem\" id=\"fs-id1169149197830\">\r\n<p id=\"fs-id1169148984114\"><span class=\"token\">\u24d0<\/span> Graph \\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={x}^{2}+1,\\) and \\(h\\left(x\\right)={x}^{2}-1\\) on the same rectangular coordinate system.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144374509\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149306542\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_302_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> The graph of \\(g\\left(x\\right)={x}^{2}+1\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted up 1 unit. The graph of \\(h\\left(x\\right)={x}^{2}-1\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted down 1 unit.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149218298\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146639326\">\r\n<div data-type=\"problem\" id=\"fs-id1169144729632\">\r\n<p id=\"fs-id1169146632351\"><span class=\"token\">\u24d0<\/span> Graph \\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={x}^{2}+6,\\) and \\(h\\left(x\\right)={x}^{2}-6\\) on the same rectangular coordinate system.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149308582\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149367723\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_303_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> The graph of \\(h\\left(x\\right)={x}^{2}+6\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted up 6 units. The graph of \\(h\\left(x\\right)={x}^{2}-6\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted down 6 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144359256\">The last example shows us that to graph a quadratic function of the form \\(f\\left(x\\right)={x}^{2}+k,\\) we take the basic parabola graph of \\(f\\left(x\\right)={x}^{2}\\) and vertically shift it up \\(\\left(k&gt;0\\right)\\) or shift it down \\(\\left(k&lt;0\\right)\\).<\/p>\r\n<p id=\"fs-id1169149003818\"><em data-effect=\"italics\">This transformation is called a vertical shift.<\/em><\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169149027327\">\r\n<div data-type=\"title\">Graph a Quadratic Function of the form \\(f\\left(x\\right)={x}^{2}+k\\) Using a Vertical Shift<\/div>\r\n<p id=\"fs-id1169148952364\">The graph of \\(f\\left(x\\right)={x}^{2}+k\\) shifts the graph of \\(f\\left(x\\right)={x}^{2}\\) vertically <em data-effect=\"italics\">k<\/em> units.<\/p>\r\n\r\n<ul id=\"fs-id1169148960205\" data-bullet-style=\"bullet\">\r\n \t<li>If <em data-effect=\"italics\">k<\/em> &gt; 0, shift the parabola vertically up <em data-effect=\"italics\">k<\/em> units.<\/li>\r\n \t<li>If <em data-effect=\"italics\">k &lt;<\/em> 0, shift the parabola vertically down \\(|k|\\) units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1169149329648\">Now that we have seen the effect of the constant, <em data-effect=\"italics\">k<\/em>, it is easy to graph functions of the form \\(f\\left(x\\right)={x}^{2}+k.\\) We just start with the basic parabola of \\(f\\left(x\\right)={x}^{2}\\) and then shift it up or down.<\/p>\r\n<p id=\"fs-id1169149008306\">It may be helpful to practice sketching \\(f\\left(x\\right)={x}^{2}\\) quickly. We know the values and can sketch the graph from there.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169146643835\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane, with vertex (0, 0). Other points on the curve are located at (negative 4, 16), (negative 3, 9), (negative 2, 4), (negative 1, 1), (1, 1), (2, 4), (3, 9), and (4, 16).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane, with vertex (0, 0). Other points on the curve are located at (negative 4, 16), (negative 3, 9), (negative 2, 4), (negative 1, 1), (1, 1), (2, 4), (3, 9), and (4, 16).\" \/><\/span>\r\n<p id=\"fs-id1169149015752\">Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169148860112\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146631695\">\r\n<div data-type=\"problem\" id=\"fs-id1169146642824\">\r\n<p id=\"fs-id1169144377376\">Graph \\(f\\left(x\\right)={x}^{2}-3\\) using a vertical shift.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149025527\">\r\n<table id=\"fs-id1169149149956\" class=\"unnumbered unstyled can-break\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">We first draw the graph of \\(f\\left(x\\right)={x}^{2}\\) on\r\n<div data-type=\"newline\"><\/div>\r\nthe grid.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148828285\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane with a vertex of (0, 0) with other points on the curve located at (negative 1, 1) and (1, 1). It is the graph of f of x equals x squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane with a vertex of (0, 0) with other points on the curve located at (negative 1, 1) and (1, 1). It is the graph of f of x equals x squared.\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Determine \\(k\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149040105\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149173152\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Shift the graph \\(f\\left(x\\right)={x}^{2}\\) down 3.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148952589\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 3 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 3 units.\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169146655572\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148952706\">\r\n<div data-type=\"problem\" id=\"fs-id1169149016289\">\r\n<p id=\"fs-id1169148861142\">Graph \\(f\\left(x\\right)={x}^{2}-5\\) using a vertical shift.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144604928\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149086768\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 5 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 5 units.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149178424\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149336332\">\r\n<div data-type=\"problem\" id=\"fs-id1169148837434\">\r\n<p id=\"fs-id1169149296930\">Graph \\(f\\left(x\\right)={x}^{2}+7\\) using a vertical shift.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149156241\"><span data-type=\"media\" id=\"fs-id1169148964091\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Graph Quadratic Functions of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\)<\/h3>\r\n<p id=\"fs-id1169148861412\">In the first example, we graphed the quadratic function \\(f\\left(x\\right)={x}^{2}\\) by plotting points and then saw the effect of adding a constant <em data-effect=\"italics\">k<\/em> to the function had on the resulting graph of the new function \\(f\\left(x\\right)={x}^{2}+k.\\)<\/p>\r\n<p id=\"fs-id1169144565259\">We will now explore the effect of subtracting a constant, <em data-effect=\"italics\">h<\/em>, from <em data-effect=\"italics\">x<\/em> has on the resulting graph of the new function \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}.\\)<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149218779\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149194685\">\r\n<div data-type=\"problem\" id=\"fs-id1169149306119\">\r\n<p id=\"fs-id1169146593912\">Graph \\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.5em}{0ex}}g\\left(x\\right)={\\left(x-1\\right)}^{2},\\) and \\(h\\left(x\\right)={\\left(x+1\\right)}^{2}\\) on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149016255\">\r\n<p id=\"fs-id1169148860835\">Plotting points will help us see the effect of the constants on the basic \\(f\\left(x\\right)={x}^{2}\\) graph. We fill in the chart for all three functions.<\/p>\r\n<span data-type=\"media\" data-alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals the quantity of x minus 1 squared, the ordered pair (x, g of x), h of x equals the quantity of x plus 1 squared, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals the quantity of x minus 1 squared column contains the values of 16, 9, 4, 1, 0, 1, and 4. The (x, g of x) column has the ordered pairs of (negative 3, 1), (negative 2, 9), (negative 1, 4), (0, 1), (1, 0), (2, 1), and (3, 4). In the h of x equals the quantity of x plus 1 squared, the values given are 4, 1, 0, 1, 4, 9, and 16. In last column, (x, h of x), contains the ordered pairs (negative 3, 4), (negative 2, 1), (negative 1, 0), (0, 4), (1, negative 1), (2, 9), and (3, 16).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals the quantity of x minus 1 squared, the ordered pair (x, g of x), h of x equals the quantity of x plus 1 squared, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals the quantity of x minus 1 squared column contains the values of 16, 9, 4, 1, 0, 1, and 4. The (x, g of x) column has the ordered pairs of (negative 3, 1), (negative 2, 9), (negative 1, 4), (0, 1), (1, 0), (2, 1), and (3, 4). In the h of x equals the quantity of x plus 1 squared, the values given are 4, 1, 0, 1, 4, 9, and 16. In last column, (x, h of x), contains the ordered pairs (negative 3, 4), (negative 2, 1), (negative 1, 0), (0, 4), (1, negative 1), (2, 9), and (3, 16).\" \/><\/span>\r\n<p id=\"fs-id1169144769382\">The <em data-effect=\"italics\">g<\/em>(<em data-effect=\"italics\">x<\/em>) values and the <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">x<\/em>) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169146630286\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 1 unit, and the right curve has been moved to the right 1 unit.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 1 unit, and the right curve has been moved to the right 1 unit.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148874734\" data-alt=\"The figure says on the first line that the graph of g of x equals the quantity x minus 1 square is the same as the graph of f of x equals x squared but shifted right 1 unit. The second line states that the graph of h of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The third line of the figure says g of x equals the quantity x minus 1 squared with an arrow underneath it pointing to the right with 1 unit written beside it. Finally, it gives h of x equals the quantity of x plus 1 squared with an arrow underneath it pointing to the left with 1 unit written beside it.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure says on the first line that the graph of g of x equals the quantity x minus 1 square is the same as the graph of f of x equals x squared but shifted right 1 unit. The second line states that the graph of h of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The third line of the figure says g of x equals the quantity x minus 1 squared with an arrow underneath it pointing to the right with 1 unit written beside it. Finally, it gives h of x equals the quantity of x plus 1 squared with an arrow underneath it pointing to the left with 1 unit written beside it.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169139944015\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148991807\">\r\n<div data-type=\"problem\" id=\"fs-id1169149345242\">\r\n<p id=\"fs-id1169148869913\"><span class=\"token\">\u24d0<\/span> Graph \\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.5em}{0ex}}g\\left(x\\right)={\\left(x+2\\right)}^{2},\\) and \\(h\\left(x\\right)={\\left(x-2\\right)}^{2}\\) on the same rectangular coordinate system.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149112894\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146646903\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 2 units, and the right curve has been moved to the right 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 2 units, and the right curve has been moved to the right 2 units.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> The graph of \\(g\\left(x\\right)={\\left(x+2\\right)}^{2}\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted left 2 units. The graph of \\(h\\left(x\\right)={\\left(x-2\\right)}^{2}\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shift right 2 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169146648281\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146741856\">\r\n<div data-type=\"problem\" id=\"fs-id1169148985707\">\r\n\r\n<span class=\"token\">\u24d0<\/span> Graph \\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.5em}{0ex}}g\\left(x\\right)={x}^{2}+5,\\) and \\(h\\left(x\\right)={x}^{2}-5\\) on the same rectangular coordinate system.\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146644234\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149298000\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 5 units, and the right curve has been moved to the right 5 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 5 units, and the right curve has been moved to the right 5 units.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> The graph of \\(g\\left(x\\right)={\\left(x+5\\right)}^{2}\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted left 5 units. The graph of \\(h\\left(x\\right)={\\left(x-5\\right)}^{2}\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted right 5 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149026032\">The last example shows us that to graph a quadratic function of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2},\\) we take the basic parabola graph of \\(f\\left(x\\right)={x}^{2}\\) and shift it left (<em data-effect=\"italics\">h<\/em> &gt; 0) or shift it right (<em data-effect=\"italics\">h<\/em> &lt; 0).<\/p>\r\n<p id=\"fs-id1169148992304\"><em data-effect=\"italics\">This transformation is called a horizontal shift<\/em>.<\/p>\r\n\r\n<div data-type=\"note\">\r\n<div data-type=\"title\">Graph a Quadratic Function of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\) Using a Horizontal Shift<\/div>\r\n<p id=\"fs-id1169148926201\">The graph of \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\) shifts the graph of \\(f\\left(x\\right)={x}^{2}\\) horizontally \\(h\\) units.<\/p>\r\n\r\n<ul id=\"fs-id1169149033754\" data-bullet-style=\"bullet\">\r\n \t<li>If <em data-effect=\"italics\">h<\/em> &gt; 0, shift the parabola horizontally left <em data-effect=\"italics\">h<\/em> units.<\/li>\r\n \t<li>If <em data-effect=\"italics\">h<\/em> &lt; 0, shift the parabola horizontally right \\(|h|\\) units.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1169148909281\">Now that we have seen the effect of the constant, <em data-effect=\"italics\">h<\/em>, it is easy to graph functions of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}.\\) We just start with the basic parabola of \\(f\\left(x\\right)={x}^{2}\\) and then shift it left or right.<\/p>\r\n<p id=\"fs-id1169149296601\">The next example will require a horizontal shift.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149144219\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1169148849971\">\r\n<p id=\"fs-id1169149214640\">Graph \\(f\\left(x\\right)={\\left(x-6\\right)}^{2}\\) using a horizontal shift.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146612670\">\r\n<table id=\"fs-id1169148910136\" class=\"unnumbered unstyled can-break\" summary=\"Graph f of x equals the quantity x minus 6 squared by using a horizontal shift. First draw the graph f of x equals x squared on a grid. This graph shows an upward-opening parabola on the x y-coordinate plane with a vertex of (0, 0) with other points on the curve located at (negative 1, 1) and (1, 1). It is the graph of f of x equals x squared. Determine h. F of x equals the quantity x minus h squared. F of x equals the quantity x minus 6 squared, so h is equal to 6. Shift the graph f of x equals x squared to the right by 6 units. This graph shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved to the right 6 units.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">We first draw the graph of \\(f\\left(x\\right)={x}^{2}\\) on\r\n<div data-type=\"newline\"><\/div>\r\nthe grid.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149173075\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Determine <em data-effect=\"italics\">h<\/em>.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149285273\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146665752\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Shift the graph \\(f\\left(x\\right)={x}^{2}\\) to the right 6 units.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148924608\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149023374\" class=\"try\">\r\n<div data-type=\"exercise\">\r\n<div data-type=\"problem\" id=\"fs-id1169148859376\">\r\n<p id=\"fs-id1169147089300\">Graph \\(f\\left(x\\right)={\\left(x-4\\right)}^{2}\\) using a horizontal shift.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1169149007400\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved right 4 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved right 4 units.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148843373\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148821362\">\r\n<div data-type=\"problem\" id=\"fs-id1169148838444\">\r\n<p id=\"fs-id1169149002072\">Graph \\(f\\left(x\\right)={\\left(x+6\\right)}^{2}\\) using a horizontal shift.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144376177\"><span data-type=\"media\" id=\"fs-id1169148934004\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The right curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 6 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The right curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 6 units.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148878550\">Now that we know the effect of the constants <em data-effect=\"italics\">h<\/em> and <em data-effect=\"italics\">k<\/em>, we will graph a quadratic function of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}+k\\) by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169148912189\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149344893\">\r\n<div data-type=\"problem\" id=\"fs-id1169149196976\">\r\n<p id=\"fs-id1169146814186\">Graph \\(f\\left(x\\right)={\\left(x+1\\right)}^{2}-2\\) using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149344188\">\r\n<p id=\"fs-id1169146744941\">This function will involve two transformations and we need a plan.<\/p>\r\n<p id=\"fs-id1169149000682\">Let\u2019s first identify the constants <em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148929946\" data-alt=\"F of x equals the quantity x plush 1 squared minus 2 is given on the top line with f of x equals the quanitity x minus h squared minis k on the second line. The given equation was changed to f of x equals the quantity of x minus negative 1 squared plush negative 2 on the third line. The final line says h equals negative 1 and k equals negative 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals the quantity x plush 1 squared minus 2 is given on the top line with f of x equals the quanitity x minus h squared minis k on the second line. The given equation was changed to f of x equals the quantity of x minus negative 1 squared plush negative 2 on the third line. The final line says h equals negative 1 and k equals negative 2.\" \/><\/span>\r\n<p id=\"fs-id1169149114301\">The <em data-effect=\"italics\">h<\/em> constant gives us a horizontal shift and the <em data-effect=\"italics\">k<\/em> gives us a vertical shift.<\/p>\r\n<span data-type=\"media\" data-alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared minus 2. The next lines say h equals negative 1 which means shift left 1 unit and k equals negative 2 which means shift down 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared minus 2. The next lines say h equals negative 1 which means shift left 1 unit and k equals negative 2 which means shift down 2 units.\" \/><\/span>\r\n<p id=\"fs-id1169148959791\">We first draw the graph of \\(f\\left(x\\right)={x}^{2}\\) on the grid.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149013677\" data-alt=\"The figure says on the first line that the graph of f of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The second line states that the graph of f of x equals the quantity x plus 1 squared minus 2 is the same as the graph of f of x equals the quantity x plus 1 squared but shifted down 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure says on the first line that the graph of f of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The second line states that the graph of f of x equals the quantity x plus 1 squared minus 2 is the same as the graph of f of x equals the quantity x plus 1 squared but shifted down 2 units.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149349732\" data-alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 1, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left one unit to produce f of x equals the quantity of x plus 1 squared. By moving f of x equals the quantity of x plus 1 squared down 1, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 1, then another curve moved down 1 to produce f of x equals the quantity of x plus 1 squared minus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 1, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left one unit to produce f of x equals the quantity of x plus 1 squared. By moving f of x equals the quantity of x plus 1 squared down 1, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 1, then another curve moved down 1 to produce f of x equals the quantity of x plus 1 squared minus 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149309956\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149089502\">\r\n<div data-type=\"problem\" id=\"fs-id1169148924313\">\r\n<p id=\"fs-id1169148909943\">Graph \\(f\\left(x\\right)={\\left(x+2\\right)}^{2}-3\\) using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149161091\"><span data-type=\"media\" id=\"fs-id1169149009344\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 2 units to the left to produce f of x equals the quantity of x plus 2 squared. The final curve is produced by moving down 3 units to produce f of x equals the quantity of x plus 2 squared minus 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 2 units to the left to produce f of x equals the quantity of x plus 2 squared. The final curve is produced by moving down 3 units to produce f of x equals the quantity of x plus 2 squared minus 3.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149349373\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148957608\">\r\n<div data-type=\"problem\" id=\"fs-id1169148933961\">\r\n<p id=\"fs-id1169149223960\">Graph \\(f\\left(x\\right)={\\left(x-3\\right)}^{2}+1\\) using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148870526\"><span data-type=\"media\" id=\"fs-id1169149117732\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 3 units to the right to produce f of x equals the quantity of x minus 3 squared. The final curve is produced by moving up 1 unit to produce f of x equals the quantity of x minus 3squared plus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 3 units to the right to produce f of x equals the quantity of x minus 3 squared. The final curve is produced by moving up 1 unit to produce f of x equals the quantity of x minus 3squared plus 1.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148968555\">\r\n<h3 data-type=\"title\">Graph Quadratic Functions of the Form \\(f\\left(x\\right)=a{x}^{2}\\)<\/h3>\r\n<p id=\"fs-id1169146652112\">So far we graphed the quadratic function \\(f\\left(x\\right)={x}^{2}\\) and then saw the effect of including a constant <em data-effect=\"italics\">h<\/em> or <em data-effect=\"italics\">k<\/em> in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient <em data-effect=\"italics\">a<\/em> on the resulting graph of the new function \\(f\\left(x\\right)=a{x}^{2}.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148924245\" data-alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals 2 times x squared, the ordered pair (x, g of x), h of x equals one-half times x squared, and the ordered pair (x, h of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared column, the values are 4, 1, 0, 1, and 4. In the (x, f of x) column, the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 2 times x squared column contains the expressions 2 times 4, 2 times 1, 2 times 0, 2 times 1, and 2 times 4. The (x, g of x) column has the ordered pairs of (negative 2, 8), (negative 1, 2), (0, 0), (1, 2), and (2,8). In the h of x equals one-half times x squared, the expressions given are one-half times 4, one-half times 1, one-half times 0, one-half times 1, and one-half times 4. In last column, (x, h of x), contains the ordered pairs (negative 2, 2), (negative 1, one-half), (0, 0), (1, one-half), and (2, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals 2 times x squared, the ordered pair (x, g of x), h of x equals one-half times x squared, and the ordered pair (x, h of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared column, the values are 4, 1, 0, 1, and 4. In the (x, f of x) column, the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 2 times x squared column contains the expressions 2 times 4, 2 times 1, 2 times 0, 2 times 1, and 2 times 4. The (x, g of x) column has the ordered pairs of (negative 2, 8), (negative 1, 2), (0, 0), (1, 2), and (2,8). In the h of x equals one-half times x squared, the expressions given are one-half times 4, one-half times 1, one-half times 0, one-half times 1, and one-half times 4. In last column, (x, h of x), contains the ordered pairs (negative 2, 2), (negative 1, one-half), (0, 0), (1, one-half), and (2, 2).\" \/><\/span>\r\n<p id=\"fs-id1169149279870\">If we graph these functions, we can see the effect of the constant <em data-effect=\"italics\">a<\/em>, assuming <em data-effect=\"italics\">a<\/em> &gt; 0.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149113873\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of g of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half). The wider curve, h of x equals one-half x squared, has a vertex at (0,0) and other points of (negative 2, 2) and (2,2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of g of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half). The wider curve, h of x equals one-half x squared, has a vertex at (0,0) and other points of (negative 2, 2) and (2,2).\" \/><\/span>\r\n<p id=\"fs-id1169149042294\">To graph a function with constant <em data-effect=\"italics\">a<\/em> it is easiest to choose a few points on \\(f\\left(x\\right)={x}^{2}\\) and multiply the <em data-effect=\"italics\">y<\/em>-values by <em data-effect=\"italics\">a<\/em>.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169149003706\">\r\n<div data-type=\"title\">Graph of a Quadratic Function of the form \\(f\\left(x\\right)=a{x}^{2}\\)<\/div>\r\n<p id=\"fs-id1169149157942\">The coefficient <em data-effect=\"italics\">a<\/em> in the function \\(f\\left(x\\right)=a{x}^{2}\\) affects the graph of \\(f\\left(x\\right)={x}^{2}\\) by stretching or compressing it.<\/p>\r\n\r\n<ul id=\"fs-id1169144874738\" data-bullet-style=\"bullet\">\r\n \t<li>If \\(0&lt;|a|&lt;1,\\) the graph of \\(f\\left(x\\right)=a{x}^{2}\\) will be \u201cwider\u201d than the graph of \\(f\\left(x\\right)={x}^{2}.\\)<\/li>\r\n \t<li>If \\(|a|&gt;1\\), the graph of \\(f\\left(x\\right)=a{x}^{2}\\) will be \u201cskinnier\u201d than the graph of \\(f\\left(x\\right)={x}^{2}.\\)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1169138882032\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149027796\">\r\n<div data-type=\"problem\" id=\"fs-id1169148881265\">\r\n<p id=\"fs-id1169148969134\">Graph \\(f\\left(x\\right)=3{x}^{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148872077\">\r\n<p id=\"fs-id1169149004284\">We will graph the functions \\(f\\left(x\\right)={x}^{2}\\) and \\(g\\left(x\\right)=3{x}^{2}\\) on the same grid. We will choose a few points on \\(f\\left(x\\right)={x}^{2}\\) and then multiply the <em data-effect=\"italics\">y<\/em>-values by 3 to get the points for \\(g\\left(x\\right)=3{x}^{2}.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149041586\" data-alt=\"The table depicts the effect of constants on the basic function of x squared. The table has 3 columns labeled x, f of x equals x squared with the ordered pair (x, f of x), and g of x equals 3 times x squared with the ordered pair (x, g of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared with the ordered pair (x, f of x), the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 3 times x squared with the ordered pair (x, g of x) column has the ordered pairs of (negative 2, 12) because 3 times 4 equals 12, (negative 1, 3) because 3 times 1 equals 3, (0, 0) because 3 times 0 equals 0, (1, 3) because 3 times 1 equals 3, and (2,12) because 3 times 4 equals 12. The graph beside the table shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). The slimmer curve of g of x equals 3 times x squared has a vertex at (0,0) and other points given of (negative 2, 12), (negative 1, 3), (1, 3), and (2,12).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table depicts the effect of constants on the basic function of x squared. The table has 3 columns labeled x, f of x equals x squared with the ordered pair (x, f of x), and g of x equals 3 times x squared with the ordered pair (x, g of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared with the ordered pair (x, f of x), the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 3 times x squared with the ordered pair (x, g of x) column has the ordered pairs of (negative 2, 12) because 3 times 4 equals 12, (negative 1, 3) because 3 times 1 equals 3, (0, 0) because 3 times 0 equals 0, (1, 3) because 3 times 1 equals 3, and (2,12) because 3 times 4 equals 12. The graph beside the table shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). The slimmer curve of g of x equals 3 times x squared has a vertex at (0,0) and other points given of (negative 2, 12), (negative 1, 3), (1, 3), and (2,12).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149188800\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149150153\">\r\n<div data-type=\"problem\" id=\"fs-id1169148952025\">\r\n<p id=\"fs-id1169144645882\">Graph \\(f\\left(x\\right)=-3{x}^{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146652009\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149026718\" data-alt=\"The graph shows the upward-opening parabola on the x y-coordinate plane of f of x equals x squared that has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). Also shown is a downward-opening parabola of f of x equals negative 3 times x squared. It has a vertex of (0,0) with other points at (negative 1, negative 3) and (1, negative 3)\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the upward-opening parabola on the x y-coordinate plane of f of x equals x squared that has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). Also shown is a downward-opening parabola of f of x equals negative 3 times x squared. It has a vertex of (0,0) with other points at (negative 1, negative 3) and (1, negative 3)\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148988111\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148943450\">\r\n<div data-type=\"problem\" id=\"fs-id1169148917724\">\r\n<p id=\"fs-id1169148964704\">Graph \\(f\\left(x\\right)=2{x}^{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149223770\"><span data-type=\"media\" id=\"fs-id1169144382865\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149065202\">\r\n<h3 data-type=\"title\">Graph Quadratic Functions Using Transformations<\/h3>\r\n<p id=\"fs-id1169149095024\">We have learned how the constants <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">h<\/em>, and <em data-effect=\"italics\">k<\/em> in the functions, \\(f\\left(x\\right)={x}^{2}+k,\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right)={\\left(x-h\\right)}^{2},\\) and \\(f\\left(x\\right)=a{x}^{2}\\) affect their graphs. We can now put this together and graph quadratic functions \\(f\\left(x\\right)=a{x}^{2}+bx+c\\) by first putting them into the form \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) by completing the square. This form is sometimes known as the vertex form or standard form.<\/p>\r\n<p id=\"fs-id1169149087320\">We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148938795\" data-alt=\"This figure shows the difference when completing the square with a quadratic equation and a quadratic function. For the quadratic equation, start with x squared plus 8 times x plus 6 equals zero. Subtract 6 from both sides to get x squared plus 8 times x equals negative 6 while leaving space to complete the square. Then, complete the square by adding 16 to both sides to get x squared plush 8 times x plush 16 equals negative 6 plush 16. Factor to get the quantity x plus 4 squared equals 10. For the quadratic function, start with f of x equals x squared plus 8 times x plus 6. The second line shows to leave space between the 8 times x and the 6 in order to complete the square. Complete the square by adding 16 and subtracting 16 on the same side to get f of x equals x squared plus 8 times x plush 16 plus 6 minus 16. Factor to get f of x equals the quantity of x plush 4 squared minus 10.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the difference when completing the square with a quadratic equation and a quadratic function. For the quadratic equation, start with x squared plus 8 times x plus 6 equals zero. Subtract 6 from both sides to get x squared plus 8 times x equals negative 6 while leaving space to complete the square. Then, complete the square by adding 16 to both sides to get x squared plush 8 times x plush 16 equals negative 6 plush 16. Factor to get the quantity x plus 4 squared equals 10. For the quadratic function, start with f of x equals x squared plus 8 times x plus 6. The second line shows to leave space between the 8 times x and the 6 in order to complete the square. Complete the square by adding 16 and subtracting 16 on the same side to get f of x equals x squared plus 8 times x plush 16 plus 6 minus 16. Factor to get f of x equals the quantity of x plush 4 squared minus 10.\" \/><\/span>\r\n<p id=\"fs-id1169148956985\">When we complete the square in a function with a coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> that is not one, we have to factor that coefficient from just the <em data-effect=\"italics\">x<\/em>-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the <em data-effect=\"italics\">x<\/em>-terms.<\/p>\r\n<p id=\"fs-id1169149156980\">Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149374763\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169144563134\">\r\n<div data-type=\"problem\" id=\"fs-id1169149370896\">\r\n<p id=\"fs-id1169148889707\">Rewrite \\(f\\left(x\\right)=-3{x}^{2}-6x-1\\) in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148968618\">\r\n<table id=\"fs-id1169149016716\" class=\"unnumbered unstyled\" summary=\"This figure gives step-by-step instructions for completing the square with a quadratic function with a coefficient with the x squared term. Given the function f of x equals negative 3 times x squared minus 6 times x minus 1, separate the x terms from the constant to get f of x equals negative 3 times x squared, space, minus 6 times x minus 1. Next, factor the coefficient of x squared, which is negative 3, to get f of x equals negative 3 times the quantity of x squared plus 2 times x minus 1. Then, prepare to complete the square by leaving space between the 2 times x and parentheses. Then take half of 2 and then square it to complete the square, the quantity one-half times 2 squared equals 1. The constant 1 completes the square in the parentheses, but the parentheses is multiplied by negative 3. So we are really adding negative 3. We must then add 3 to not change the value of the function to get f of x equals negative 3 times the quantity of x squared plus 2 times x plus 1 minus 1 plus 3. Rewrite the trinomial as a square and subtract the constants to get f of x equals negative 3 times the quantity of x plus 1 squared plus 2. The function is now in the f of x equals a times the quantity x minus h squared plus k form.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146742130\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Separate the <em data-effect=\"italics\">x<\/em> terms from the constant.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147132205\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Factor the coefficient of \\({x}^{2}\\), \\(-3\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149030949\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Prepare to complete the square.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146656082\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Take half of 2 and then square it to complete the\r\n<div data-type=\"newline\"><\/div>\r\nsquare. \\({\\left(\\frac{1}{2}\u00b72\\right)}^{2}=1\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The constant 1 completes the square in the\r\n<div data-type=\"newline\"><\/div>\r\nparentheses, but the parentheses is multiplied by\r\n<div data-type=\"newline\"><\/div>\r\n\\(-3\\). So we are really adding \\(-3\\) We must then\r\n<div data-type=\"newline\"><\/div>\r\nadd 3 to not change the value of the function.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144375648\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial as a square and subtract the\r\n<div data-type=\"newline\"><\/div>\r\nconstants.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148894430\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The function is now in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\)\r\n<div data-type=\"newline\"><\/div>\r\nform.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149025514\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148987017\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148963136\">\r\n<div data-type=\"problem\" id=\"fs-id1169149011009\">\r\n<p id=\"fs-id1169146644111\">Rewrite \\(f\\left(x\\right)=-4{x}^{2}-8x+1\\) in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146647640\">\r\n<p id=\"fs-id1169147029305\">\\(f\\left(x\\right)=-4{\\left(x+1\\right)}^{2}+5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148998662\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149193773\">\r\n<div data-type=\"problem\" id=\"fs-id1169144813293\">\r\n<p id=\"fs-id1169148910881\">Rewrite \\(f\\left(x\\right)=2{x}^{2}-8x+3\\) in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148848860\">\r\n<p id=\"fs-id1169149016360\">\\(f\\left(x\\right)=2{\\left(x-2\\right)}^{2}-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146625591\">Once we put the function into the \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}+k\\) form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149195601\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149375092\">\r\n<div data-type=\"problem\" id=\"fs-id1169146655550\">\r\n<p id=\"fs-id1169146658046\">Graph \\(f\\left(x\\right)={x}^{2}+6x+5\\) by using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149296594\">\r\n<p id=\"fs-id1169149008581\"><strong data-effect=\"bold\">Step 1.<\/strong> Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) vertex form by completing the square.<\/p>\r\n\r\n<table id=\"fs-id1169149007262\" class=\"unnumbered unstyled\" summary=\"This figure gives step-by-step instructions for completing the square with a quadratic function with no coefficient with the x squared term. Given the function f of x equals x squared plus 6 times x plus 5, separate the x terms from the constant to get f of x equals x squared, space, plus 6 times x plus 5. Then take half of 6 and then square it to complete the square, the quantity one-half times 6 squared equals 9. We must then add 9 and subtract 9 to not change the value of the function to get f of x equals x squared plus 6 times x plus 9 plus 5 minus 9. Rewrite the trinomial as a square and subtract the constants to get f of x equals the quantity of x plus 3 squared minus 4. The function is now in the f of x equals a times the quantity x minus h squared plus k form.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149040281\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Separate the <em data-effect=\"italics\">x<\/em> terms from the constant.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144375967\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Take half of 6 and then square it to complete the square.\r\n<div data-type=\"newline\"><\/div>\r\n\\({\\left(\\frac{1}{2}\u00b76\\right)}^{2}=9\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">We both add 9 and subtract 9 to not change the value of the function.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149096773\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial as a square and subtract the constants.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148958083\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The function is now in the \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}+k\\) form.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148825054\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169149033195\"><strong data-effect=\"bold\">Step 2:<\/strong> Graph the function using transformations.<\/p>\r\n<p id=\"fs-id1169148895304\">Looking at the <em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em> values, we see the graph will take the graph of \\(f\\left(x\\right)={x}^{2}\\) and shift it to the left 3 units and down 4 units.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169146643276\" data-alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared minus 4. The next lines say h equals negative 3 which means shift left 3 unit and k equals negative 4 which means shift down 4 units\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared minus 4. The next lines say h equals negative 3 which means shift left 3 unit and k equals negative 4 which means shift down 4 units\" \/><\/span>\r\n<p id=\"fs-id1169148870087\">We first draw the graph of \\(f\\left(x\\right)={x}^{2}\\) on the grid.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169144378557\" data-alt=\"To graph f of x equals the quantity x plus 3 squared, shift the graph of f of x equals x squares to the left 3 units. To graph f of x equals the quantity x plus 3 squared minus 4, shift the graph the quantity x plus 3 squared down 4 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To graph f of x equals the quantity x plus 3 squared, shift the graph of f of x equals x squares to the left 3 units. To graph f of x equals the quantity x plus 3 squared minus 4, shift the graph the quantity x plus 3 squared down 4 units.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149122194\" data-alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 3, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left 3 units to produce f of x equals the quantity of x plus 3 squared. By moving f of x equals the quantity of x plus 3 squared down 2, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 3 squared, then another curve moved down 4 to produce f of x equals the quantity of x plus 1 squared minus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 3, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left 3 units to produce f of x equals the quantity of x plus 3 squared. By moving f of x equals the quantity of x plus 3 squared down 2, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 3 squared, then another curve moved down 4 to produce f of x equals the quantity of x plus 1 squared minus 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148837576\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149144148\">\r\n<div data-type=\"problem\" id=\"fs-id1169149149874\">\r\n<p id=\"fs-id1169146654089\">Graph \\(f\\left(x\\right)={x}^{2}+2x-3\\) by using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144730925\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149008752\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the left has been moved 1 unit to the left to produce f of x equals the quantity of x plus 1 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x plus 1 squared minus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the left has been moved 1 unit to the left to produce f of x equals the quantity of x plus 1 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x plus 1 squared minus 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149240506\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149040152\">\r\n<div data-type=\"problem\" id=\"fs-id1169149220551\">\r\n<p id=\"fs-id1169149023966\">Graph \\(f\\left(x\\right)={x}^{2}-8x+12\\) by using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148880803\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148866779\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the right has been moved 4 units to the right to produce f of x equals the quantity of x minus 4 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x minus 4 squared minus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the right has been moved 4 units to the right to produce f of x equals the quantity of x minus 4 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x minus 4 squared minus 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148859649\">We list the steps to take to graph a quadratic function using transformations here.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169149025824\" class=\"howto\">\r\n<div data-type=\"title\">Graph a quadratic function using transformations.<\/div>\r\n<ol id=\"fs-id1169148944333\" class=\"stepwise\" type=\"1\">\r\n \t<li>Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/li>\r\n \t<li>Graph the function using transformations.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1169144555491\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169144602993\">\r\n<div data-type=\"problem\" id=\"fs-id1169149109915\">\r\n<p id=\"fs-id1169149342531\">Graph \\(f\\left(x\\right)=-2{x}^{2}-4x+2\\) by using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146745591\">\r\n<p id=\"fs-id1169146664809\"><strong data-effect=\"bold\">Step 1.<\/strong> Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) vertex form by completing the square.<\/p>\r\n\r\n<table id=\"fs-id1169148938631\" class=\"unnumbered unstyled\" summary=\"This figure gives step-by-step instructions for completing the square with a quadratic function with a coefficient with the x squared term. Given the function f of x equals negative 2 times x squared minus 4 times x plus 2, separate the x terms from the constant to leave space for completing the square. We need the coefficient of x squared to be one. We factor negative 2 from the x-terms to get f of x equals negative 2 times the quantity of x squared plus 2 times x plus 2. Take half of 2 and then square it to complete the square, the quantity of one-half times 2 squared equals 1. We add 1 to complete the square in the parentheses, but the parentheses is multiplied by negative 2. Se we are really adding negative 2. To not change the value of the function we add 2 to get f of x equals negative 2 times the quantity of x squared plus 2 times x plus 1 squared plus 2 plus 2. Rewrite the trinomial as a square and subtract the constants to get f of x equals negative 2 times the quantity of x plus 1 squared plus 4. The function is now in the f of x equals a times the quantity x minus h squared plus k form.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149328740\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Separate the <em data-effect=\"italics\">x<\/em> terms from the constant.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149018093\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">We need the coefficient of \\({x}^{2}\\) to be one.\r\n<div data-type=\"newline\"><\/div>\r\nWe factor \\(-2\\) from the <em data-effect=\"italics\">x<\/em>-terms.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149329700\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Take half of 2 and then square it to complete the square.\r\n<div data-type=\"newline\"><\/div>\r\n\\({\\left(\\frac{1}{2}\u00b72\\right)}^{2}=1\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">We add 1 to complete the square in the parentheses, but the parentheses is multiplied by \\(-2\\). Se we are really adding \\(-2\\). To not change the value of the function we add 2.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149311090\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial as a square and subtract the constants.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144382686\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The function is now in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144729567\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169148964046\"><strong data-effect=\"bold\">Step 2.<\/strong> Graph the function using transformations.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148869574\" data-alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals negative 2 times x squared with an arrow coming from it pointing to f of x equals negative 2 times the quantity x plus 1 squared. An arrow come from it to point to f of x equals negative 2 times the quantity x plus 1 squared plus 4. The next line says a equals negative 2 which means multiply the y-values by negative 2, then h equals negative 1 which means shift left 1 unit and k equals 4 which means shift up 4 units\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals negative 2 times x squared with an arrow coming from it pointing to f of x equals negative 2 times the quantity x plus 1 squared. An arrow come from it to point to f of x equals negative 2 times the quantity x plus 1 squared plus 4. The next line says a equals negative 2 which means multiply the y-values by negative 2, then h equals negative 1 which means shift left 1 unit and k equals 4 which means shift up 4 units\" \/><\/span>\r\n<p id=\"fs-id1169148947726\">We first draw the graph of \\(f\\left(x\\right)={x}^{2}\\) on the grid.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149169235\" data-alt=\"To graph f of x equals negative 2 times x squared, multiply the y-values in parabola of f of x equals x squared by negative 2. To graph f of x equals negative 2 times the quantity x plus 1 squared, shift the graph of f of x equals negative 2 times x squared to the left 1 unit. To graph f of x equals negative 2 times the quantity x plus 1 squared plus 4, shift the graph of f of x equals negative 2 times the quantity x plus 1 squared up 4 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To graph f of x equals negative 2 times x squared, multiply the y-values in parabola of f of x equals x squared by negative 2. To graph f of x equals negative 2 times the quantity x plus 1 squared, shift the graph of f of x equals negative 2 times x squared to the left 1 unit. To graph f of x equals negative 2 times the quantity x plus 1 squared plus 4, shift the graph of f of x equals negative 2 times the quantity x plus 1 squared up 4 units.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169147089647\" data-alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By multiplying by negative 2, move to the next graph showing the original f of x equals x squared and the new slimmer and flipped graph of f of x equals negative 2 x squared. By shifting that graph of f of x equals negative 2 times x squared left 1, we move to the next graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and then another curve moved left 1 unit to produce f of x equals negative 2 times the quantity of x plus 1 squared. By moving f of x equals negative 2 times the quantity of x plus 1 squared up 4, we move to the final graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and the f of x equals negative 2 times the quantity of x plus 1 squared, then another curve moved up 4 to produce f of x equals negative 2 times the quantity of x plus 1 squared plus 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By multiplying by negative 2, move to the next graph showing the original f of x equals x squared and the new slimmer and flipped graph of f of x equals negative 2 x squared. By shifting that graph of f of x equals negative 2 times x squared left 1, we move to the next graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and then another curve moved left 1 unit to produce f of x equals negative 2 times the quantity of x plus 1 squared. By moving f of x equals negative 2 times the quantity of x plus 1 squared up 4, we move to the final graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and the f of x equals negative 2 times the quantity of x plus 1 squared, then another curve moved up 4 to produce f of x equals negative 2 times the quantity of x plus 1 squared plus 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149289706\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146621161\">\r\n<div data-type=\"problem\" id=\"fs-id1169146621164\">\r\n<p id=\"fs-id1169148958057\">Graph \\(f\\left(x\\right)=-3{x}^{2}+12x-4\\) by using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149220390\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169144523011\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149293431\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149023639\">\r\n<div data-type=\"problem\" id=\"fs-id1169146645023\">\r\n<p id=\"fs-id1169149152543\">Graph \\(f\\left(x\\right)=-2{x}^{2}+12x-9\\) by using transformations.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149286365\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149037643\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149017496\">Now that we have completed the square to put a quadratic function into \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form, we can also use this technique to graph the function using its properties as in the previous section.<\/p>\r\n<p id=\"fs-id1169144682919\">If we look back at the last few examples, we see that the vertex is related to the constants <em data-effect=\"italics\">h<\/em> and <em data-effect=\"italics\">k<\/em>.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169146628097\" data-alt=\"The first graph shows an upward-opening parabola on the x y-coordinate plane with a vertex of (negative 3, negative 4) with other points of (0, negative 5) and (0, negative 1). Underneath the graph, it shows the standard form of a parabola, f of x equals the quantity x minus h squared plus k, with the equation of the parabola f of x equals the quantity of x plus 3 squared minus 4 where h equals negative 3 and k equals negative 4. The second graph shows a downward-opening parabola on the x y-coordinate plane with a vertex of (negative 1, 4) and other points of (0,2) and (negative 2,2). Underneath the graph, it shows the standard form of a parabola, f of x equals a times the quantity x minus h squared plus k, with the equation of the parabola f of x equals negative 2 times the quantity of x plus 1 squared plus 4 where h equals negative 1 and k equals 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows an upward-opening parabola on the x y-coordinate plane with a vertex of (negative 3, negative 4) with other points of (0, negative 5) and (0, negative 1). Underneath the graph, it shows the standard form of a parabola, f of x equals the quantity x minus h squared plus k, with the equation of the parabola f of x equals the quantity of x plus 3 squared minus 4 where h equals negative 3 and k equals negative 4. The second graph shows a downward-opening parabola on the x y-coordinate plane with a vertex of (negative 1, 4) and other points of (0,2) and (negative 2,2). Underneath the graph, it shows the standard form of a parabola, f of x equals a times the quantity x minus h squared plus k, with the equation of the parabola f of x equals negative 2 times the quantity of x plus 1 squared plus 4 where h equals negative 1 and k equals 4.\" \/><\/span>\r\n<p id=\"fs-id1169146652019\">In each case, the vertex is (<em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>). Also the <span data-type=\"term\" class=\"no-emphasis\">axis of symmetry<\/span> is the line <em data-effect=\"italics\">x<\/em> = <em data-effect=\"italics\">h<\/em>.<\/p>\r\n<p id=\"fs-id1169146724594\">We rewrite our steps for graphing a quadratic function using properties for when the function is in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169149027551\" class=\"howto\">\r\n<div data-type=\"title\">Graph a quadratic function in the form \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) using properties.<\/div>\r\n<ol id=\"fs-id1169149219887\" class=\"stepwise\" type=\"1\">\r\n \t<li>Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form.<\/li>\r\n \t<li>Determine whether the parabola opens upward, <em data-effect=\"italics\">a<\/em> &gt; 0, or downward, <em data-effect=\"italics\">a<\/em> &lt; 0.<\/li>\r\n \t<li>Find the axis of symmetry, <em data-effect=\"italics\">x<\/em> = <em data-effect=\"italics\">h<\/em>.<\/li>\r\n \t<li>Find the vertex, (<em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>).<\/li>\r\n \t<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>\r\n \t<li>Find the <em data-effect=\"italics\">x<\/em>-intercepts.<\/li>\r\n \t<li>Graph the parabola.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1169146638071\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146638073\">\r\n<div data-type=\"problem\" id=\"fs-id1169146626688\">\r\n<p id=\"fs-id1169146626690\"><span class=\"token\">\u24d0<\/span> Rewrite \\(f\\left(x\\right)=2{x}^{2}+4x+5\\) in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph the function using properties.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146742819\">\r\n<table id=\"fs-id1169146814819\" class=\"unnumbered unstyled can-break\" summary=\"This is figure gives step-by-step instructions on how to graph a function using properties of the equation. Rewrite the function in f of x equals a times the quantity x minus h squared plus k form by completing the square. The function is f of x equals 2 x squared plus 4 times x plus 5. Follow the process to complete the square: f of x equals 2 times the quantity of x squared plus 2 x plus 5, f of x equals 2 times the quantity of x squared plus 2 x plus 1 plus 5 minus 2, and f of x equals 2 times the quantity of x plus 1 squared plus 3. Identify the constants a, h, k, to get a equals 2, h equals negative 1, and k equals 3. Since a equals 2, the parabola opens upward. A small picture of an upward-facing parabola is shown. The axis of symmetry is x equals h, so x equals negative 1. The vertex is (h, k) so (negative 1, 3). Find the y-intercept by finding f of 0. F of 0 equals 2 times 0 squared plus 4 times 0 plus 5, so f of o equals 5, so the y-intercept is (0,5). Find the point symmetric to (0,5) across the axis of symmetry which is (2,5). Find the x-intercepts. Since the discriminant is negative, so there are no x-intercepts. Graph the parabola. The graph shown is an upward facing parabola with vertex (negative 1, 3) and y-intercept (0,5). The axis of symmetry is shown, x equals negative 1.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\)\r\n<div data-type=\"newline\"><\/div>\r\nform by completing the square.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)=2{x}^{2}+4x+5\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)=2\\left({x}^{2}+2x\\right)+5\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)=2\\left({x}^{2}+2x+1\\right)+5-2\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(f\\left(x\\right)=2{\\left(x+1\\right)}^{2}+3\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Identify the constants \\(a,h,k.\\)<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.2em}{0ex}}a=2\\phantom{\\rule{1em}{0ex}}h=-1\\phantom{\\rule{1em}{0ex}}k=3\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Since \\(a=2\\), the parabola opens upward.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169144421651\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_027a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The axis of symmetry is \\(x=h\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The axis of symmetry is \\(x=-1\\).<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(h,k\\right)\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The vertex is \\(\\left(-1,3\\right)\\).<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">y<\/em>-intercept by finding \\(f\\left(0\\right)\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(f\\left(0\\right)=2\\cdot {0}^{2}+4\\cdot 0+5\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(f\\left(0\\right)=5\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-intercept \\(\\left(0,5\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Find the point symmetric to \\(\\left(0,5\\right)\\) across the\r\n<div data-type=\"newline\"><\/div>\r\naxis of symmetry.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(-2,\\phantom{\\rule{0.2em}{0ex}}5\\right)\\)<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercepts.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The discriminant negative, so there are\r\n<div data-type=\"newline\"><\/div>\r\nno <em data-effect=\"italics\">x<\/em>-intercepts. Graph the parabola.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149286236\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_027j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149285302\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149285306\">\r\n<div data-type=\"problem\" id=\"fs-id1169149285308\">\r\n<p id=\"fs-id1169149285310\"><span class=\"token\">\u24d0<\/span> Rewrite \\(f\\left(x\\right)=3{x}^{2}-6x+5\\) in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph the function using properties.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146643464\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=3{\\left(x-1\\right)}^{2}+2\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149329160\" data-alt=\"The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149329177\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149329181\">\r\n<div data-type=\"problem\" id=\"fs-id1169149329183\">\r\n<p id=\"fs-id1169149329185\"><span class=\"token\">\u24d0<\/span> Rewrite \\(f\\left(x\\right)=-2{x}^{2}+8x-7\\) in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph the function using properties.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148985233\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-2{\\left(x-2\\right)}^{2}+1\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149195445\" data-alt=\"The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149194051\">\r\n<h3 data-type=\"title\">Find a Quadratic Function from its Graph<\/h3>\r\n<p id=\"fs-id1169149194056\">So far we have started with a function and then found its graph.<\/p>\r\n<p id=\"fs-id1169149194059\">Now we are going to reverse the process. Starting with the graph, we will find the function.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149194063\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149194066\">\r\n<div data-type=\"problem\" id=\"fs-id1169149194068\">\r\n<p id=\"fs-id1169149194070\">Determine the quadratic function whose graph is shown.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149194073\" data-alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_028_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0, 7).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149214515\">\r\n<p id=\"fs-id1169149214518\">\\(\\begin{array}{cccccc}\\text{Since it is quadratic, we start with the}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\phantom{\\rule{0.2em}{0ex}}\\text{form.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\text{The vertex,}\\phantom{\\rule{0.2em}{0ex}}\\left(h,k\\right),\\phantom{\\rule{0.2em}{0ex}}\\text{is}\\phantom{\\rule{0.2em}{0ex}}\\left(-2,-1\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}h=-2\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}k=-1.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}f\\left(x\\right)&amp; =\\hfill &amp; a{\\left(x-\\left(-2\\right)\\right)}^{2}-1\\hfill \\\\ \\text{To find}\\phantom{\\rule{0.2em}{0ex}}a\\text{, we use the}\\phantom{\\rule{0.2em}{0ex}}y\\text{-intercept,}\\phantom{\\rule{0.2em}{0ex}}\\left(0,7\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\text{So}\\phantom{\\rule{0.2em}{0ex}}f\\left(0\\right)=7.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}7&amp; =\\hfill &amp; a{\\left(0+2\\right)}^{2}-1\\hfill \\\\ \\text{Solve for}\\phantom{\\rule{0.2em}{0ex}}a.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}7&amp; =\\hfill &amp; 4a-1\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}8&amp; =\\hfill &amp; 4a\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}2&amp; =\\hfill &amp; a\\hfill \\\\ \\text{Write the function.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}f\\left(x\\right)&amp; =\\hfill &amp; a{\\left(x-h\\right)}^{2}+k\\hfill \\\\ \\text{Substitute in}\\phantom{\\rule{0.2em}{0ex}}h=-2,k=-1\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}a=2.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}f\\left(x\\right)&amp; =\\hfill &amp; 2{\\left(x+2\\right)}^{2}-1\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148957428\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148957432\">\r\n<div data-type=\"problem\" id=\"fs-id1169148957434\">\r\n<p id=\"fs-id1169148957436\">Write the quadratic function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form whose graph is shown.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169146607084\" data-alt=\"The graph shown is an upward facing parabola with vertex (3, negative 4) and y-intercept (0, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_029_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (3, negative 4) and y-intercept (0, 5).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146607098\">\r\n<p id=\"fs-id1169146607100\">\\(f\\left(x\\right)={\\left(x-3\\right)}^{2}-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169146816985\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146816989\">\r\n<div data-type=\"problem\" id=\"fs-id1169146816991\">\r\n\r\nDetermine the quadratic function whose graph is shown.\r\n\r\n<span data-type=\"media\" id=\"fs-id1169146816996\" data-alt=\"The graph shown is an upward facing parabola with vertex (negative 3, negative 1) and y-intercept (0, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_030_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (negative 3, negative 1) and y-intercept (0, 8).\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149096712\">\r\n<p id=\"fs-id1169149096714\">\\(f\\left(x\\right)={\\left(x+3\\right)}^{2}-1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149096760\" class=\"media-2\">\r\n<p id=\"fs-id1169144768434\">Access these online resources for additional instruction and practice with graphing quadratic functions using transformations.<\/p>\r\n\r\n<ul id=\"fs-id1163870666809\" data-display=\"block\">\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran1\">Function Shift Rules Applied to Quadratic Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran2\">Changing a Quadratic from Standard Form to Vertex Form<\/a><\/li>\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran3\">Using Transformations to Graph Quadratic Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran4\">Finding Quadratic Equation in Vertex Form from Graph<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169144768465\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1169144768473\" data-bullet-style=\"bullet\">\r\n \t<li>Graph a Quadratic Function of the form \\(f\\left(x\\right)={x}^{2}+k\\) Using a Vertical Shift\r\n<ul id=\"fs-id1169144377944\" data-bullet-style=\"bullet\">\r\n \t<li>The graph of \\(f\\left(x\\right)={x}^{2}+k\\) shifts the graph of \\(f\\left(x\\right)={x}^{2}\\) vertically k units.\r\n<ul id=\"fs-id1169138762012\" data-bullet-style=\"bullet\">\r\n \t<li>If <em data-effect=\"italics\">k<\/em> &gt; 0, shift the parabola vertically up <em data-effect=\"italics\">k<\/em> units.<\/li>\r\n \t<li>If <em data-effect=\"italics\">k<\/em> &lt; 0, shift the parabola vertically down \\(|k|\\) units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Graph a Quadratic Function of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\) Using a Horizontal Shift\r\n<ul id=\"fs-id1169148828155\" data-bullet-style=\"bullet\">\r\n \t<li>The graph of \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\) shifts the graph of \\(f\\left(x\\right)={x}^{2}\\) horizontally h units.\r\n<ul id=\"fs-id1169148872363\" data-bullet-style=\"bullet\">\r\n \t<li>If <em data-effect=\"italics\">h<\/em> &gt; 0, shift the parabola horizontally left <em data-effect=\"italics\">h<\/em> units.<\/li>\r\n \t<li>If <em data-effect=\"italics\">h<\/em> &lt; 0, shift the parabola horizontally right \\(|h|\\) units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Graph of a Quadratic Function of the form \\(f\\left(x\\right)=a{x}^{2}\\)\r\n<ul id=\"fs-id1169149297377\" data-bullet-style=\"bullet\">\r\n \t<li>The coefficient <em data-effect=\"italics\">a<\/em> in the function \\(f\\left(x\\right)=a{x}^{2}\\) affects the graph of \\(f\\left(x\\right)={x}^{2}\\) by stretching or compressing it.\r\n<div data-type=\"newline\"><\/div>\r\nIf \\(0&lt;|a|&lt;1,\\) then the graph of \\(f\\left(x\\right)=a{x}^{2}\\) will be \u201cwider\u201d than the graph of \\(f\\left(x\\right)={x}^{2}.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf \\(|a|&gt;1,\\) then the graph of \\(f\\left(x\\right)=a{x}^{2}\\) will be \u201cskinnier\u201d than the graph of \\(f\\left(x\\right)={x}^{2}.\\)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>How to graph a quadratic function using transformations\r\n<ol id=\"fs-id1169149369519\" class=\"stepwise\" type=\"1\">\r\n \t<li>Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/li>\r\n \t<li>Graph the function using transformations.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Graph a quadratic function in the vertex form \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) using properties\r\n<ol id=\"fs-id1169149346092\" class=\"stepwise\" type=\"1\">\r\n \t<li>Rewrite the function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form.<\/li>\r\n \t<li>Determine whether the parabola opens upward, <em data-effect=\"italics\">a<\/em> &gt; 0, or downward, a &lt; 0.<\/li>\r\n \t<li>Find the axis of symmetry, <em data-effect=\"italics\">x<\/em> = <em data-effect=\"italics\">h<\/em>.<\/li>\r\n \t<li>Find the vertex, (<em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>).<\/li>\r\n \t<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>\r\n \t<li>Find the <em data-effect=\"italics\">x<\/em>-intercepts, if possible.<\/li>\r\n \t<li>Graph the parabola.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169144374336\">\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169144374340\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<p id=\"fs-id1169144374347\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form \\(f\\left(x\\right)={x}^{2}+k\\)<\/strong><\/p>\r\n<p id=\"fs-id1169144378724\">In the following exercises, <span class=\"token\">\u24d0<\/span> graph the quadratic functions on the same rectangular coordinate system and <span class=\"token\">\u24d1<\/span> describe what effect adding a constant, <em data-effect=\"italics\">k<\/em>, to the function has on the basic parabola.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144378742\">\r\n<div data-type=\"problem\" id=\"fs-id1169144378744\">\r\n<p id=\"fs-id1169144378747\">\\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={x}^{2}+4,\\) and \\(h\\left(x\\right)={x}^{2}-4.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146612344\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> The graph of \\(g\\left(x\\right)={x}^{2}+4\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted up 4 units. The graph of \\(h\\left(x\\right)={x}^{2}-4\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shift down 4 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144451007\">\r\n<div data-type=\"problem\" id=\"fs-id1169144451009\">\r\n<p id=\"fs-id1169144451011\">\\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={x}^{2}+7,\\) and \\(h\\left(x\\right)={x}^{2}-7.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144823918\">In the following exercises, graph each function using a vertical shift.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144823921\">\r\n<div data-type=\"problem\" id=\"fs-id1169144823923\">\r\n<p id=\"fs-id1169144823925\">\\(f\\left(x\\right)={x}^{2}+3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144724402\"><span data-type=\"media\" id=\"fs-id1169144724406\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144724421\">\r\n<div data-type=\"problem\" id=\"fs-id1169144724423\">\r\n<p id=\"fs-id1169144724425\">\\(f\\left(x\\right)={x}^{2}-7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149196553\">\r\n<div data-type=\"problem\" id=\"fs-id1169149196555\">\r\n<p id=\"fs-id1169149196557\">\\(g\\left(x\\right)={x}^{2}+2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144555525\"><span data-type=\"media\" id=\"fs-id1169144555529\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144555545\">\r\n<div data-type=\"problem\" id=\"fs-id1169144555547\">\r\n\r\n\\(g\\left(x\\right)={x}^{2}+5\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146626127\">\r\n<div data-type=\"problem\" id=\"fs-id1169146626129\">\r\n<p id=\"fs-id1169146626132\">\\(h\\left(x\\right)={x}^{2}-4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146669847\"><span data-type=\"media\" id=\"fs-id1169146669852\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146669868\">\r\n<div data-type=\"problem\" id=\"fs-id1169146669870\">\r\n<p id=\"fs-id1169146669872\">\\(h\\left(x\\right)={x}^{2}-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146638897\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form \\(f\\left(x\\right)={\\left(x-h\\right)}^{2}\\)<\/strong><\/p>\r\n<p id=\"fs-id1163870660975\">In the following exercises, <span class=\"token\">\u24d0<\/span> graph the quadratic functions on the same rectangular coordinate system and <span class=\"token\">\u24d1<\/span> describe what effect adding a constant, \\(h\\), inside the parentheses has<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149101378\">\r\n<div data-type=\"problem\" id=\"fs-id1169149101380\">\r\n<p id=\"fs-id1169149101383\">\\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={\\left(x-3\\right)}^{2},\\) and \\(h\\left(x\\right)={\\left(x+3\\right)}^{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144451066\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169144451076\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The graph to the right is shifted 3 units to the right to produce g of x equals the quantity of x minus 3 squared. The graph the left is shifted 3 units to the left to produce h of x equals the quantity of x plus 3 squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The graph to the right is shifted 3 units to the right to produce g of x equals the quantity of x minus 3 squared. The graph the left is shifted 3 units to the left to produce h of x equals the quantity of x plus 3 squared.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> The graph of \\(g\\left(x\\right)={\\left(x-3\\right)}^{2}\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted right 3 units. The graph of \\(h\\left(x\\right)={\\left(x+3\\right)}^{2}\\) is the same as the graph of \\(f\\left(x\\right)={x}^{2}\\) but shifted left 3 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148957288\">\r\n<div data-type=\"problem\" id=\"fs-id1169148957290\">\r\n<p id=\"fs-id1169148957292\">\\(f\\left(x\\right)={x}^{2},\\phantom{\\rule{0.2em}{0ex}}g\\left(x\\right)={\\left(x+4\\right)}^{2},\\) and \\(h\\left(x\\right)={\\left(x-4\\right)}^{2}.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146631834\">In the following exercises, graph each function using a horizontal shift.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146631837\">\r\n<div data-type=\"problem\" id=\"fs-id1169146631839\">\r\n<p id=\"fs-id1169146631842\">\\(f\\left(x\\right)={\\left(x-2\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144876562\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169144876566\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144876581\">\r\n<div data-type=\"problem\" id=\"fs-id1169144876583\">\r\n<p id=\"fs-id1169144876586\">\\(f\\left(x\\right)={\\left(x-1\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146639448\">\r\n<div data-type=\"problem\" id=\"fs-id1169146639450\">\r\n<p id=\"fs-id1169146639452\">\\(f\\left(x\\right)={\\left(x+5\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149360868\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149360872\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149360888\">\r\n<div data-type=\"problem\" id=\"fs-id1169149360890\">\r\n<p id=\"fs-id1169149360892\">\\(f\\left(x\\right)={\\left(x+3\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144892933\">\r\n<div data-type=\"problem\" id=\"fs-id1169144892935\">\r\n<p id=\"fs-id1169144892937\">\\(f\\left(x\\right)={\\left(x-5\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144566243\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169144566248\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_334_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144566263\">\r\n<div data-type=\"problem\" id=\"fs-id1169144556471\">\r\n<p id=\"fs-id1169144556473\">\\(f\\left(x\\right)={\\left(x+2\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149194316\">In the following exercises, graph each function using transformations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149194319\">\r\n<div data-type=\"problem\" id=\"fs-id1169149194321\">\r\n<p id=\"fs-id1169149194323\">\\(f\\left(x\\right)={\\left(x+2\\right)}^{2}+1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149194365\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146665353\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146665368\">\r\n<div data-type=\"problem\" id=\"fs-id1169146665370\">\r\n<p id=\"fs-id1169146665372\">\\(f\\left(x\\right)={\\left(x+4\\right)}^{2}+2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169139960649\">\r\n<div data-type=\"problem\" id=\"fs-id1169139960651\">\r\n<p id=\"fs-id1169139960653\">\\(f\\left(x\\right)={\\left(x-1\\right)}^{2}+5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149293770\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149293774\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149293789\">\r\n<div data-type=\"problem\" id=\"fs-id1169149293791\">\r\n<p id=\"fs-id1169149293793\">\\(f\\left(x\\right)={\\left(x-3\\right)}^{2}+4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146621931\">\r\n<div data-type=\"problem\" id=\"fs-id1169146621933\">\r\n<p id=\"fs-id1169146621935\">\\(f\\left(x\\right)={\\left(x+3\\right)}^{2}-1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149011458\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149011462\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149303573\">\r\n<div data-type=\"problem\" id=\"fs-id1169149303575\">\r\n<p id=\"fs-id1169149303577\">\\(f\\left(x\\right)={\\left(x+5\\right)}^{2}-2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146623143\">\r\n<div data-type=\"problem\" id=\"fs-id1169146623145\">\r\n<p id=\"fs-id1169146623148\">\\(f\\left(x\\right)={\\left(x-4\\right)}^{2}-3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146623189\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146937025\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_342_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146937041\">\r\n<div data-type=\"problem\" id=\"fs-id1169146937043\">\r\n<p id=\"fs-id1169146937045\">\\(f\\left(x\\right)={\\left(x-6\\right)}^{2}-2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144603085\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form \\(f\\left(x\\right)=a{x}^{2}\\)<\/strong><\/p>\r\n<p id=\"fs-id1169149043369\">In the following exercises, graph each function.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149043372\">\r\n<div data-type=\"problem\" id=\"fs-id1169149043375\">\r\n<p id=\"fs-id1169149043377\">\\(f\\left(x\\right)=-2{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149043403\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149043408\" data-alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_344_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149240401\">\r\n<div data-type=\"problem\" id=\"fs-id1169149240404\">\r\n<p id=\"fs-id1169149240406\">\\(f\\left(x\\right)=4{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149240453\">\r\n<div data-type=\"problem\" id=\"fs-id1169149240455\">\r\n<p id=\"fs-id1169144538642\">\\(f\\left(x\\right)=-4{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144538669\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169144538674\" data-alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_346_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144538690\">\r\n<div data-type=\"problem\" id=\"fs-id1169144538692\">\r\n<p id=\"fs-id1169144538694\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144365536\">\r\n<div data-type=\"problem\" id=\"fs-id1169144365538\">\r\n<p id=\"fs-id1169144365540\">\\(f\\left(x\\right)=\\frac{1}{2}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147027577\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169147027582\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_348_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147027597\">\r\n<div data-type=\"problem\" id=\"fs-id1169147027600\">\r\n<p id=\"fs-id1169147027602\">\\(f\\left(x\\right)=\\frac{1}{3}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146640150\">\r\n<div data-type=\"problem\" id=\"fs-id1169146640152\">\r\n<p id=\"fs-id1169146640154\">\\(f\\left(x\\right)=\\frac{1}{4}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147086861\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169147086866\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_350_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147086881\">\r\n<div data-type=\"problem\" id=\"fs-id1169147086883\">\r\n<p id=\"fs-id1169147086885\">\\(f\\left(x\\right)=-\\frac{1}{2}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146813405\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Transformations<\/strong><\/p>\r\n<p id=\"fs-id1169146813412\">In the following exercises, rewrite each function in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149290730\">\r\n<div data-type=\"problem\" id=\"fs-id1169149290732\">\r\n<p id=\"fs-id1169149290734\">\\(f\\left(x\\right)=-3{x}^{2}-12x-5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149195302\">\r\n<p id=\"fs-id1169149195304\">\\(f\\left(x\\right)=-3{\\left(x+2\\right)}^{2}+7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149195345\">\r\n<div data-type=\"problem\" id=\"fs-id1169149195348\">\r\n<p id=\"fs-id1169149195350\">\\(f\\left(x\\right)=2{x}^{2}-12x+7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149328890\">\r\n<div data-type=\"problem\" id=\"fs-id1169149328892\">\r\n<p id=\"fs-id1169149328894\">\\(f\\left(x\\right)=3{x}^{2}+6x-1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149328929\">\r\n<p id=\"fs-id1169149328932\">\\(f\\left(x\\right)=3{\\left(x+1\\right)}^{2}-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146594120\">\r\n<div data-type=\"problem\" id=\"fs-id1169146594122\">\r\n<p id=\"fs-id1169146594124\">\\(f\\left(x\\right)=-4{x}^{2}-16x-9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144451625\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph it by using transformations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144451677\">\r\n<div data-type=\"problem\" id=\"fs-id1169149000795\">\r\n<p id=\"fs-id1169149000798\">\\(f\\left(x\\right)={x}^{2}+6x+5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149000831\">\r\n<p id=\"fs-id1169149000833\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)={\\left(x+3\\right)}^{2}-4\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149357656\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_352_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149357673\">\r\n<div data-type=\"problem\" id=\"fs-id1169149357675\">\r\n<p id=\"fs-id1169149357677\">\\(f\\left(x\\right)={x}^{2}+4x-12\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144365120\">\r\n<div data-type=\"problem\" id=\"fs-id1169144365122\">\r\n<p id=\"fs-id1169144365124\">\\(f\\left(x\\right)={x}^{2}+4x-12\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149222676\">\r\n<p id=\"fs-id1169149222678\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)={\\left(x+2\\right)}^{2}-1\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149280707\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_354_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149280723\">\r\n<div data-type=\"problem\" id=\"fs-id1169149280725\">\r\n<p id=\"fs-id1169149280727\">\\(f\\left(x\\right)={x}^{2}-6x+8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146644568\">\r\n<div data-type=\"problem\" id=\"fs-id1169146644570\">\r\n<p id=\"fs-id1169146644572\">\\(f\\left(x\\right)={x}^{2}-6x+15\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149194498\">\r\n<p id=\"fs-id1169149194500\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)={\\left(x-3\\right)}^{2}+6\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149194555\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_356_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149204947\">\r\n<div data-type=\"problem\" id=\"fs-id1169149204949\">\r\n<p id=\"fs-id1169149204952\">\\(f\\left(x\\right)={x}^{2}+8x+10\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144606276\">\r\n<div data-type=\"problem\" id=\"fs-id1169144606278\">\r\n<p id=\"fs-id1169144606280\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+8x-16\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144606316\">\r\n<p id=\"fs-id1169144606318\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=\\text{\u2212}{\\left(x-4\\right)}^{2}+0\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149348888\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_358_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149348904\">\r\n<div data-type=\"problem\" id=\"fs-id1169149041020\">\r\n<p id=\"fs-id1169149041022\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+2x-7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146732866\">\r\n<div data-type=\"problem\" id=\"fs-id1169146732868\">\r\n<p id=\"fs-id1169146732870\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}-4x+2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146818042\">\r\n<p id=\"fs-id1169146818045\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=\\text{\u2212}{\\left(x+2\\right)}^{2}+6\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169147085292\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_360_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147085309\">\r\n<div data-type=\"problem\" id=\"fs-id1169147085311\">\r\n<p id=\"fs-id1169147085313\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+4x-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146738012\">\r\n<div data-type=\"problem\" id=\"fs-id1169146738014\">\r\n<p id=\"fs-id1169146738016\">\\(f\\left(x\\right)=5{x}^{2}-10x+8\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144896828\">\r\n<p id=\"fs-id1169144896831\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=5{\\left(x-1\\right)}^{2}+3\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148958663\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_362_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148958679\">\r\n<div data-type=\"problem\" id=\"fs-id1169148958681\">\r\n<p id=\"fs-id1169148958683\">\\(f\\left(x\\right)=3{x}^{2}+18x+20\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144417671\">\r\n<div data-type=\"problem\" id=\"fs-id1169144417673\">\r\n<p id=\"fs-id1169144417675\">\\(f\\left(x\\right)=2{x}^{2}-4x+1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144717022\">\r\n<p id=\"fs-id1169144717024\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=2{\\left(x-1\\right)}^{2}-1\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169144717080\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_364_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146738134\">\r\n<div data-type=\"problem\" id=\"fs-id1169146738136\">\r\n<p id=\"fs-id1169146738138\">\\(f\\left(x\\right)=3{x}^{2}-6x-1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147106529\">\r\n<div data-type=\"problem\" id=\"fs-id1169147106531\">\r\n<p id=\"fs-id1169147106534\">\\(f\\left(x\\right)=-2{x}^{2}+8x-10\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147106569\">\r\n<p id=\"fs-id1169147106571\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-2{\\left(x-2\\right)}^{2}-2\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149190625\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_366_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149190641\">\r\n<div data-type=\"problem\" id=\"fs-id1169149190643\">\r\n<p id=\"fs-id1169149190645\">\\(f\\left(x\\right)=-3{x}^{2}+6x+1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169147034644\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph it using properties.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149023584\">\r\n<div data-type=\"problem\" id=\"fs-id1169149023586\">\r\n<p id=\"fs-id1169149023588\">\\(f\\left(x\\right)=2{x}^{2}+4x+6\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144451862\">\r\n<p id=\"fs-id1169144451864\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=2{\\left(x+1\\right)}^{2}+4\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149224312\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_368_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149224328\">\r\n<div data-type=\"problem\" id=\"fs-id1169149224330\">\r\n<p id=\"fs-id1169149224332\">\\(f\\left(x\\right)=3{x}^{2}-12x+7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144420817\">\r\n<div data-type=\"problem\" id=\"fs-id1169144420819\">\r\n<p id=\"fs-id1169144420822\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}+2x-4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144420857\">\r\n<p id=\"fs-id1169144420859\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=\\text{\u2212}{\\left(x-1\\right)}^{2}-3\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149288478\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_370_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144877272\">\r\n<div data-type=\"problem\" id=\"fs-id1169144877274\">\r\n<p id=\"fs-id1169144877276\">\\(f\\left(x\\right)=-2{x}^{2}-4x-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144796116\"><strong data-effect=\"bold\">Matching<\/strong><\/p>\r\n<p id=\"fs-id1169146814380\">In the following exercises, match the graphs to one of the following functions: <span class=\"token\">\u24d0<\/span> \\(f\\left(x\\right)={x}^{2}+4\\) <span class=\"token\">\u24d1<\/span> \\(f\\left(x\\right)={x}^{2}-4\\) <span class=\"token\">\u24d2<\/span> \\(f\\left(x\\right)={\\left(x+4\\right)}^{2}\\) <span class=\"token\">\u24d3<\/span> \\(f\\left(x\\right)={\\left(x-4\\right)}^{2}\\) <span class=\"token\">\u24d4<\/span> \\(f\\left(x\\right)={\\left(x+4\\right)}^{2}-4\\) <span class=\"token\">\u24d5<\/span> \\(f\\left(x\\right)={\\left(x+4\\right)}^{2}+4\\) <span class=\"token\">\u24d6<\/span> \\(f\\left(x\\right)={\\left(x-4\\right)}^{2}-4\\) <span class=\"token\">\u24d7<\/span> \\(f\\left(x\\right)={\\left(x-4\\right)}^{2}+4\\)<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146613391\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169146613393\"><span data-type=\"media\" id=\"fs-id1169146613395\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144685085\">\r\n<p id=\"fs-id1169144685087\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144685095\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169144685097\"><span data-type=\"media\" id=\"fs-id1169144685100\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144685124\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169144685126\"><span data-type=\"media\" id=\"fs-id1169144685128\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144685142\">\r\n<p id=\"fs-id1169149367373\"><span class=\"token\">\u24d4<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149367381\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169149367383\"><span data-type=\"media\" id=\"fs-id1169149367386\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 4) and other points (negative 6, 8) and (negative 2, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 4) and other points (negative 6, 8) and (negative 2, 8).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149367410\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169149367412\"><span data-type=\"media\" id=\"fs-id1169149367414\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149367428\">\r\n<p id=\"fs-id1169149367430\"><span class=\"token\">\u24d3<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146621060\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169146621062\"><span data-type=\"media\" id=\"fs-id1169146621064\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 4) and other points (negative 2, 8) and (2, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 4) and other points (negative 2, 8) and (2, 8).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146621089\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169146621091\"><span data-type=\"media\" id=\"fs-id1169146621093\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146621107\">\r\n<p id=\"fs-id1169146621109\"><span class=\"token\">\u24d6<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146621117\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169146621119\"><span data-type=\"media\" id=\"fs-id1169144829486\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 4) and other points (2,8) and (6,8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 4) and other points (2,8) and (6,8).\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144829510\"><strong data-effect=\"bold\">Find a Quadratic Function from its Graph<\/strong><\/p>\r\n<p id=\"fs-id1171780911535\">In the following exercises, write the quadratic function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form whose graph is shown.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144384693\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169144384696\"><span data-type=\"media\" id=\"fs-id1169144384698\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_209_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144384711\">\r\n<p id=\"fs-id1169144384714\">\\(f\\left(x\\right)={\\left(x+1\\right)}^{2}-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144538707\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169144538710\"><span data-type=\"media\" id=\"fs-id1169144538712\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_210_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144421503\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169144421505\"><span data-type=\"media\" id=\"fs-id1169144421507\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_211_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144421521\">\r\n<p id=\"fs-id1169144421523\">\\(f\\left(x\\right)=2{\\left(x-1\\right)}^{2}-3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146628627\" class=\"material-set-2\">\r\n<div data-type=\"problem\" id=\"fs-id1169146628629\"><span data-type=\"media\" id=\"fs-id1169146628632\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_212_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149339224\">\r\n<h4 data-type=\"title\">Writing Exercise<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1169149339232\">\r\n<div data-type=\"problem\" id=\"fs-id1169149339234\">\r\n<p id=\"fs-id1169149339236\">Graph the quadratic function \\(f\\left(x\\right)={x}^{2}+4x+5\\) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149339272\">\r\n<p id=\"fs-id1169146627952\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146627958\">\r\n<div data-type=\"problem\" id=\"fs-id1169146627960\">\r\n<p id=\"fs-id1169146627962\">Graph the quadratic function \\(f\\left(x\\right)=2{x}^{2}-4x-3\\) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169146628009\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1169146628014\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149189348\" data-alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is graph Quadratic Functions of the form f of x equals x squared plus k; graph Quadratic Functions of the form f of x equals the quantity x minus h squared; graph Quadratic functions of the form f of x equals a times x squared; graph Quadratic Functions Using Transformations; find a Quadratic Function from its Graph. The other columns are left blank for you to check you understanding.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_213_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is graph Quadratic Functions of the form f of x equals x squared plus k; graph Quadratic Functions of the form f of x equals the quantity x minus h squared; graph Quadratic functions of the form f of x equals a times x squared; graph Quadratic Functions Using Transformations; find a Quadratic Function from its Graph. The other columns are left blank for you to check you understanding.\" \/><\/span>\r\n<p id=\"fs-id1169149189358\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\r\n\r\n<\/div>\r\n<\/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 quadratic functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Graph quadratic functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Graph quadratic functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Graph quadratic functions using transformations<\/li>\n<li>Find a quadratic function from its graph<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146948008\" class=\"be-prepared\">\n<p id=\"fs-id1169146616422\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1169149034590\" type=\"1\">\n<li>Graph the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by plotting points.\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/da9d6ce0-a078-4ca2-97af-8cb374f040f5#fs-id1167836683384\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor completely: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb05ef9baba18724fede8d1bfee811fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#121;&#43;&#52;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358#fs-id1167835345249\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor completely: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ecbf5e319cc87c523fd058577506a70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#43;&#51;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358#fs-id1167834396304\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149172251\">\n<h3 data-type=\"title\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/h3>\n<p id=\"fs-id1169146658984\">In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and \u2018moving\u2019 it according to information given in the function equation. We call this graphing quadratic functions using transformations.<\/p>\n<p id=\"fs-id1169149155466\">In the first example, we will graph the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by plotting points. Then we will see what effect adding a constant, <em data-effect=\"italics\">k<\/em>, to the equation will have on the graph of the new function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b832bdc6202008c160d6da129b4a3576_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1169149113341\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148867308\">\n<div data-type=\"problem\" id=\"fs-id1169148958428\">\n<p id=\"fs-id1169149123092\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70f6ecc68d3e35f59b7371258a77780e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35e451d741f48346a590142ec8bd1f8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/> on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148959807\">\n<p id=\"fs-id1169144560376\">Plotting points will help us see the effect of the constants on the basic <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> graph. We fill in the chart for all three functions.<\/p>\n<p><span data-type=\"media\" data-alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals x squared plus 2, the ordered pair (x, g of x), h of x equals x squared minus 2, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals x squared plus 2 column contains the expressions 9 plus 2, 4 plus 2, 1 plus 2, 0 plus 2, 1 plus 2, 4 plus 2, and 9 plus 2. The (x, g of x) column has the ordered pairs of (negative 3, 11), (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), (2, 6), and (3, 11). In the h of x equals x squared minus 2 column, the expressions given are 9 minus 2, 4 minus 2, 1 minus 2, 0 minus 2, 1 minus 2, 4 minus 2, and 9 minus 2. In last column, (x, h of x), contains the ordered pairs (negative 3, 7), (negative 2, 2), (negative 1, negative 1), (0, negative 2), (1, negative 1), (2, 2), and (3, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals x squared plus 2, the ordered pair (x, g of x), h of x equals x squared minus 2, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals x squared plus 2 column contains the expressions 9 plus 2, 4 plus 2, 1 plus 2, 0 plus 2, 1 plus 2, 4 plus 2, and 9 plus 2. The (x, g of x) column has the ordered pairs of (negative 3, 11), (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), (2, 6), and (3, 11). In the h of x equals x squared minus 2 column, the expressions given are 9 minus 2, 4 minus 2, 1 minus 2, 0 minus 2, 1 minus 2, 4 minus 2, and 9 minus 2. In last column, (x, h of x), contains the ordered pairs (negative 3, 7), (negative 2, 2), (negative 1, negative 1), (0, negative 2), (1, negative 1), (2, 2), and (3, 7).\" \/><\/span><\/p>\n<p id=\"fs-id1169149356096\">The <em data-effect=\"italics\">g<\/em>(<em data-effect=\"italics\">x<\/em>) values are two more than the <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) values. Also, the <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">x<\/em>) values are two less than the <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) values. Now we will graph all three functions on the same rectangular coordinate system.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147088946\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle is the graph of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top parabola has been moved up 2 units, and the bottom has been moved down 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle is the graph of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top parabola has been moved up 2 units, and the bottom has been moved down 2 units.\" \/><\/span><\/p>\n<p id=\"fs-id1169149348416\">The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ded55672642c130f77e58b8af6918e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted up 2 units.<\/p>\n<p id=\"fs-id1169149114126\">The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35e451d741f48346a590142ec8bd1f8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted down 2 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144768997\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149017068\">\n<div data-type=\"problem\" id=\"fs-id1169149197830\">\n<p id=\"fs-id1169148984114\"><span class=\"token\">\u24d0<\/span> Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62a0b841b72d9b749be5fec052771d8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6074f220ec7c56fc8301baf7ab27b91c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/> on the same rectangular coordinate system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144374509\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149306542\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_302_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-209c9a78496fb97a4a1da75ba3595cff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted up 1 unit. The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6074f220ec7c56fc8301baf7ab27b91c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted down 1 unit.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149218298\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146639326\">\n<div data-type=\"problem\" id=\"fs-id1169144729632\">\n<p id=\"fs-id1169146632351\"><span class=\"token\">\u24d0<\/span> Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-701e1cae2b0919cbc1cf39194493d75a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-960a45d1c0bbf3137ec46b418034d994_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/> on the same rectangular coordinate system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149308582\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149367723\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_303_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-419f70bcd7ef4dcbb90f8e093fc90d1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted up 6 units. The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-960a45d1c0bbf3137ec46b418034d994_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted down 6 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144359256\">The last example shows us that to graph a quadratic function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-901f1db8664049de819e20cf7f43703f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/> we take the basic parabola graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and vertically shift it up <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c2cb6665fa1cc0ae7991ed30eca5f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#62;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> or shift it down <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a884521c64a87e5be919e7a3bb0c3a9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#60;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"fs-id1169149003818\"><em data-effect=\"italics\">This transformation is called a vertical shift.<\/em><\/p>\n<div data-type=\"note\" id=\"fs-id1169149027327\">\n<div data-type=\"title\">Graph a Quadratic Function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> Using a Vertical Shift<\/div>\n<p id=\"fs-id1169148952364\">The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> shifts the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> vertically <em data-effect=\"italics\">k<\/em> units.<\/p>\n<ul id=\"fs-id1169148960205\" data-bullet-style=\"bullet\">\n<li>If <em data-effect=\"italics\">k<\/em> &gt; 0, shift the parabola vertically up <em data-effect=\"italics\">k<\/em> units.<\/li>\n<li>If <em data-effect=\"italics\">k &lt;<\/em> 0, shift the parabola vertically down <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-717afd6bd88914fd87d415dcab93ad94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#107;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"16\" style=\"vertical-align: -4px;\" \/> units.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1169149329648\">Now that we have seen the effect of the constant, <em data-effect=\"italics\">k<\/em>, it is easy to graph functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b832bdc6202008c160d6da129b4a3576_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/> We just start with the basic parabola of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and then shift it up or down.<\/p>\n<p id=\"fs-id1169149008306\">It may be helpful to practice sketching <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> quickly. We know the values and can sketch the graph from there.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169146643835\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane, with vertex (0, 0). Other points on the curve are located at (negative 4, 16), (negative 3, 9), (negative 2, 4), (negative 1, 1), (1, 1), (2, 4), (3, 9), and (4, 16).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane, with vertex (0, 0). Other points on the curve are located at (negative 4, 16), (negative 3, 9), (negative 2, 4), (negative 1, 1), (1, 1), (2, 4), (3, 9), and (4, 16).\" \/><\/span><\/p>\n<p id=\"fs-id1169149015752\">Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148860112\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169146631695\">\n<div data-type=\"problem\" id=\"fs-id1169146642824\">\n<p id=\"fs-id1169144377376\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27d02dbda05c8bfd937503f2f2145f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> using a vertical shift.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149025527\">\n<table id=\"fs-id1169149149956\" class=\"unnumbered unstyled can-break\" summary=\".\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We first draw the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> on<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the grid.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148828285\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane with a vertex of (0, 0) with other points on the curve located at (negative 1, 1) and (1, 1). It is the graph of f of x equals x squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane with a vertex of (0, 0) with other points on the curve located at (negative 1, 1) and (1, 1). It is the graph of f of x equals x squared.\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Determine <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149040105\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004b_img-1.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-id1169149173152\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_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\">Shift the graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> down 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148952589\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 3 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 3 units.\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146655572\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148952706\">\n<div data-type=\"problem\" id=\"fs-id1169149016289\">\n<p id=\"fs-id1169148861142\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-501c0e1208ee956b2061f0162b4da809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/> using a vertical shift.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144604928\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149086768\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 5 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 5 units.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149178424\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149336332\">\n<div data-type=\"problem\" id=\"fs-id1169148837434\">\n<p id=\"fs-id1169149296930\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3778aabd324e9d8ba9953c304348d46d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> using a vertical shift.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149156241\"><span data-type=\"media\" id=\"fs-id1169148964091\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/h3>\n<p id=\"fs-id1169148861412\">In the first example, we graphed the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by plotting points and then saw the effect of adding a constant <em data-effect=\"italics\">k<\/em> to the function had on the resulting graph of the new function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b832bdc6202008c160d6da129b4a3576_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169144565259\">We will now explore the effect of subtracting a constant, <em data-effect=\"italics\">h<\/em>, from <em data-effect=\"italics\">x<\/em> has on the resulting graph of the new function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d314ffb78d9269d058259c4111ed2b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1169149218779\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149194685\">\n<div data-type=\"problem\" id=\"fs-id1169149306119\">\n<p id=\"fs-id1169146593912\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce3397b3487edec4207f3b082b0db73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#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;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"221\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21852c377b299ae01b7bcc9df2c671ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/> on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149016255\">\n<p id=\"fs-id1169148860835\">Plotting points will help us see the effect of the constants on the basic <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> graph. We fill in the chart for all three functions.<\/p>\n<p><span data-type=\"media\" data-alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals the quantity of x minus 1 squared, the ordered pair (x, g of x), h of x equals the quantity of x plus 1 squared, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals the quantity of x minus 1 squared column contains the values of 16, 9, 4, 1, 0, 1, and 4. The (x, g of x) column has the ordered pairs of (negative 3, 1), (negative 2, 9), (negative 1, 4), (0, 1), (1, 0), (2, 1), and (3, 4). In the h of x equals the quantity of x plus 1 squared, the values given are 4, 1, 0, 1, 4, 9, and 16. In last column, (x, h of x), contains the ordered pairs (negative 3, 4), (negative 2, 1), (negative 1, 0), (0, 4), (1, negative 1), (2, 9), and (3, 16).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals the quantity of x minus 1 squared, the ordered pair (x, g of x), h of x equals the quantity of x plus 1 squared, and the ordered pair (x, h of x). In the x column, the values given are negative 3, negative 2, negative 1, 0, 1, 2, and 3. In the f of x equals x squared column, the values are 9, 4, 1, 0, 1, 4, and 9. In the (x, f of x) column, the ordered pairs (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9) are given. The g of x equals the quantity of x minus 1 squared column contains the values of 16, 9, 4, 1, 0, 1, and 4. The (x, g of x) column has the ordered pairs of (negative 3, 1), (negative 2, 9), (negative 1, 4), (0, 1), (1, 0), (2, 1), and (3, 4). In the h of x equals the quantity of x plus 1 squared, the values given are 4, 1, 0, 1, 4, 9, and 16. In last column, (x, h of x), contains the ordered pairs (negative 3, 4), (negative 2, 1), (negative 1, 0), (0, 4), (1, negative 1), (2, 9), and (3, 16).\" \/><\/span><\/p>\n<p id=\"fs-id1169144769382\">The <em data-effect=\"italics\">g<\/em>(<em data-effect=\"italics\">x<\/em>) values and the <em data-effect=\"italics\">h<\/em>(<em data-effect=\"italics\">x<\/em>) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169146630286\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 1 unit, and the right curve has been moved to the right 1 unit.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 1 unit, and the right curve has been moved to the right 1 unit.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148874734\" data-alt=\"The figure says on the first line that the graph of g of x equals the quantity x minus 1 square is the same as the graph of f of x equals x squared but shifted right 1 unit. The second line states that the graph of h of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The third line of the figure says g of x equals the quantity x minus 1 squared with an arrow underneath it pointing to the right with 1 unit written beside it. Finally, it gives h of x equals the quantity of x plus 1 squared with an arrow underneath it pointing to the left with 1 unit written beside it.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure says on the first line that the graph of g of x equals the quantity x minus 1 square is the same as the graph of f of x equals x squared but shifted right 1 unit. The second line states that the graph of h of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The third line of the figure says g of x equals the quantity x minus 1 squared with an arrow underneath it pointing to the right with 1 unit written beside it. Finally, it gives h of x equals the quantity of x plus 1 squared with an arrow underneath it pointing to the left with 1 unit written beside it.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169139944015\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148991807\">\n<div data-type=\"problem\" id=\"fs-id1169149345242\">\n<p id=\"fs-id1169148869913\"><span class=\"token\">\u24d0<\/span> Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cdfd598f1ed87d978ec6d0d743165e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#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;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"221\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-847f073aa16bbf88e8060610a4763b26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/> on the same rectangular coordinate system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149112894\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146646903\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 2 units, and the right curve has been moved to the right 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 2 units, and the right curve has been moved to the right 2 units.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2230f2c07cf725b2148af821b9e60c62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"121\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted left 2 units. The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-847f073aa16bbf88e8060610a4763b26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shift right 2 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146648281\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146741856\">\n<div data-type=\"problem\" id=\"fs-id1169148985707\">\n<p><span class=\"token\">\u24d0<\/span> Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-500985ccd8ad6d01b5b966b6f6a1cc30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#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;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa3a89c06dc5c021da0679655f961963_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/> on the same rectangular coordinate system.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Describe what effect adding a constant to the function has on the basic parabola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146644234\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149298000\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 5 units, and the right curve has been moved to the right 5 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 5 units, and the right curve has been moved to the right 5 units.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56eba79859b137fdbd17a0fd876daedf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"121\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted left 5 units. The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53782f526a772fb930e2c3682689b2d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted right 5 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149026032\">The last example shows us that to graph a quadratic function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b902779b7d7002168fed2420aac616e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"129\" style=\"vertical-align: -4px;\" \/> we take the basic parabola graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and shift it left (<em data-effect=\"italics\">h<\/em> &gt; 0) or shift it right (<em data-effect=\"italics\">h<\/em> &lt; 0).<\/p>\n<p id=\"fs-id1169148992304\"><em data-effect=\"italics\">This transformation is called a horizontal shift<\/em>.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Graph a Quadratic Function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/> Using a Horizontal Shift<\/div>\n<p id=\"fs-id1169148926201\">The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/> shifts the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> horizontally <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> units.<\/p>\n<ul id=\"fs-id1169149033754\" data-bullet-style=\"bullet\">\n<li>If <em data-effect=\"italics\">h<\/em> &gt; 0, shift the parabola horizontally left <em data-effect=\"italics\">h<\/em> units.<\/li>\n<li>If <em data-effect=\"italics\">h<\/em> &lt; 0, shift the parabola horizontally right <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-314088aa97d0b7aa21d8f48d8317a8e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#104;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"16\" style=\"vertical-align: -4px;\" \/> units.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1169148909281\">Now that we have seen the effect of the constant, <em data-effect=\"italics\">h<\/em>, it is easy to graph functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d314ffb78d9269d058259c4111ed2b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"129\" style=\"vertical-align: -4px;\" \/> We just start with the basic parabola of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and then shift it left or right.<\/p>\n<p id=\"fs-id1169149296601\">The next example will require a horizontal shift.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149144219\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169148849971\">\n<p id=\"fs-id1169149214640\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f104a046f254adff4c1162ec520459c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/> using a horizontal shift.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146612670\">\n<table id=\"fs-id1169148910136\" class=\"unnumbered unstyled can-break\" summary=\"Graph f of x equals the quantity x minus 6 squared by using a horizontal shift. First draw the graph f of x equals x squared on a grid. This graph shows an upward-opening parabola on the x y-coordinate plane with a vertex of (0, 0) with other points on the curve located at (negative 1, 1) and (1, 1). It is the graph of f of x equals x squared. Determine h. F of x equals the quantity x minus h squared. F of x equals the quantity x minus 6 squared, so h is equal to 6. Shift the graph f of x equals x squared to the right by 6 units. This graph shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved to the right 6 units.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We first draw the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> on<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the grid.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149173075\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Determine <em data-effect=\"italics\">h<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149285273\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008b_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-id1169146665752\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Shift the graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> to the right 6 units.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148924608\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_008d_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-id1169149023374\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169148859376\">\n<p id=\"fs-id1169147089300\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d655291eb61514946fa4d8bd33384b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/> using a horizontal shift.<\/p>\n<\/div>\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1169149007400\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved right 4 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved right 4 units.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148843373\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148821362\">\n<div data-type=\"problem\" id=\"fs-id1169148838444\">\n<p id=\"fs-id1169149002072\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfa99fe13460d3ea03084c809a3c19b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/> using a horizontal shift.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144376177\"><span data-type=\"media\" id=\"fs-id1169148934004\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The right curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 6 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The right curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 6 units.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148878550\">Now that we know the effect of the constants <em data-effect=\"italics\">h<\/em> and <em data-effect=\"italics\">k<\/em>, we will graph a quadratic function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a05eae74c76937d1efe50d7791db2460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#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=\"155\" style=\"vertical-align: -4px;\" \/> by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148912189\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149344893\">\n<div data-type=\"problem\" id=\"fs-id1169149196976\">\n<p id=\"fs-id1169146814186\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0bf6017bf1f4730ea32278b1b5058d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/> using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149344188\">\n<p id=\"fs-id1169146744941\">This function will involve two transformations and we need a plan.<\/p>\n<p id=\"fs-id1169149000682\">Let\u2019s first identify the constants <em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148929946\" data-alt=\"F of x equals the quantity x plush 1 squared minus 2 is given on the top line with f of x equals the quanitity x minus h squared minis k on the second line. The given equation was changed to f of x equals the quantity of x minus negative 1 squared plush negative 2 on the third line. The final line says h equals negative 1 and k equals negative 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals the quantity x plush 1 squared minus 2 is given on the top line with f of x equals the quanitity x minus h squared minis k on the second line. The given equation was changed to f of x equals the quantity of x minus negative 1 squared plush negative 2 on the third line. The final line says h equals negative 1 and k equals negative 2.\" \/><\/span><\/p>\n<p id=\"fs-id1169149114301\">The <em data-effect=\"italics\">h<\/em> constant gives us a horizontal shift and the <em data-effect=\"italics\">k<\/em> gives us a vertical shift.<\/p>\n<p><span data-type=\"media\" data-alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared minus 2. The next lines say h equals negative 1 which means shift left 1 unit and k equals negative 2 which means shift down 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared minus 2. The next lines say h equals negative 1 which means shift left 1 unit and k equals negative 2 which means shift down 2 units.\" \/><\/span><\/p>\n<p id=\"fs-id1169148959791\">We first draw the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> on the grid.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169149013677\" data-alt=\"The figure says on the first line that the graph of f of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The second line states that the graph of f of x equals the quantity x plus 1 squared minus 2 is the same as the graph of f of x equals the quantity x plus 1 squared but shifted down 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure says on the first line that the graph of f of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The second line states that the graph of f of x equals the quantity x plus 1 squared minus 2 is the same as the graph of f of x equals the quantity x plus 1 squared but shifted down 2 units.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149349732\" data-alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 1, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left one unit to produce f of x equals the quantity of x plus 1 squared. By moving f of x equals the quantity of x plus 1 squared down 1, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 1, then another curve moved down 1 to produce f of x equals the quantity of x plus 1 squared minus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 1, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left one unit to produce f of x equals the quantity of x plus 1 squared. By moving f of x equals the quantity of x plus 1 squared down 1, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 1, then another curve moved down 1 to produce f of x equals the quantity of x plus 1 squared minus 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149309956\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149089502\">\n<div data-type=\"problem\" id=\"fs-id1169148924313\">\n<p id=\"fs-id1169148909943\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b77f33df43b12bf52e09b1856f0b450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/> using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149161091\"><span data-type=\"media\" id=\"fs-id1169149009344\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 2 units to the left to produce f of x equals the quantity of x plus 2 squared. The final curve is produced by moving down 3 units to produce f of x equals the quantity of x plus 2 squared minus 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 2 units to the left to produce f of x equals the quantity of x plus 2 squared. The final curve is produced by moving down 3 units to produce f of x equals the quantity of x plus 2 squared minus 3.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149349373\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148957608\">\n<div data-type=\"problem\" id=\"fs-id1169148933961\">\n<p id=\"fs-id1169149223960\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f77c8342107ca1ee642d492a8cc4dbd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/> using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148870526\"><span data-type=\"media\" id=\"fs-id1169149117732\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 3 units to the right to produce f of x equals the quantity of x minus 3 squared. The final curve is produced by moving up 1 unit to produce f of x equals the quantity of x minus 3squared plus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 3 units to the right to produce f of x equals the quantity of x minus 3 squared. The final curve is produced by moving up 1 unit to produce f of x equals the quantity of x minus 3squared plus 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148968555\">\n<h3 data-type=\"title\">Graph Quadratic Functions of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/h3>\n<p id=\"fs-id1169146652112\">So far we graphed the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and then saw the effect of including a constant <em data-effect=\"italics\">h<\/em> or <em data-effect=\"italics\">k<\/em> in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient <em data-effect=\"italics\">a<\/em> on the resulting graph of the new function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b488e3096a4e4fcf89e131ed853db96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148924245\" data-alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals 2 times x squared, the ordered pair (x, g of x), h of x equals one-half times x squared, and the ordered pair (x, h of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared column, the values are 4, 1, 0, 1, and 4. In the (x, f of x) column, the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 2 times x squared column contains the expressions 2 times 4, 2 times 1, 2 times 0, 2 times 1, and 2 times 4. The (x, g of x) column has the ordered pairs of (negative 2, 8), (negative 1, 2), (0, 0), (1, 2), and (2,8). In the h of x equals one-half times x squared, the expressions given are one-half times 4, one-half times 1, one-half times 0, one-half times 1, and one-half times 4. In last column, (x, h of x), contains the ordered pairs (negative 2, 2), (negative 1, one-half), (0, 0), (1, one-half), and (2, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A table depicting the effect of constants on the basic function of x squared. The table has seven columns labeled x, f of x equals x squared, the ordered pair (x, f of x), g of x equals 2 times x squared, the ordered pair (x, g of x), h of x equals one-half times x squared, and the ordered pair (x, h of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared column, the values are 4, 1, 0, 1, and 4. In the (x, f of x) column, the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 2 times x squared column contains the expressions 2 times 4, 2 times 1, 2 times 0, 2 times 1, and 2 times 4. The (x, g of x) column has the ordered pairs of (negative 2, 8), (negative 1, 2), (0, 0), (1, 2), and (2,8). In the h of x equals one-half times x squared, the expressions given are one-half times 4, one-half times 1, one-half times 0, one-half times 1, and one-half times 4. In last column, (x, h of x), contains the ordered pairs (negative 2, 2), (negative 1, one-half), (0, 0), (1, one-half), and (2, 2).\" \/><\/span><\/p>\n<p id=\"fs-id1169149279870\">If we graph these functions, we can see the effect of the constant <em data-effect=\"italics\">a<\/em>, assuming <em data-effect=\"italics\">a<\/em> &gt; 0.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169149113873\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of g of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half). The wider curve, h of x equals one-half x squared, has a vertex at (0,0) and other points of (negative 2, 2) and (2,2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of g of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half). The wider curve, h of x equals one-half x squared, has a vertex at (0,0) and other points of (negative 2, 2) and (2,2).\" \/><\/span><\/p>\n<p id=\"fs-id1169149042294\">To graph a function with constant <em data-effect=\"italics\">a<\/em> it is easiest to choose a few points on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and multiply the <em data-effect=\"italics\">y<\/em>-values by <em data-effect=\"italics\">a<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149003706\">\n<div data-type=\"title\">Graph of a Quadratic Function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1169149157942\">The coefficient <em data-effect=\"italics\">a<\/em> in the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> affects the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by stretching or compressing it.<\/p>\n<ul id=\"fs-id1169144874738\" data-bullet-style=\"bullet\">\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd7b16bfa2f7f7e1ac6d82c152b28183_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#124;&#97;&#124;&#60;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> will be \u201cwider\u201d than the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac78978801d31220977b3a0e9009f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc1bf1075fe2fcde93c78040804c407a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#97;&#124;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/>, the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> will be \u201cskinnier\u201d than the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac78978801d31220977b3a0e9009f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169138882032\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149027796\">\n<div data-type=\"problem\" id=\"fs-id1169148881265\">\n<p id=\"fs-id1169148969134\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-040c84b27423f27534853f0757d67cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148872077\">\n<p id=\"fs-id1169149004284\">We will graph the functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23c5436cf5f3b44cda277f0be3ac43a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -4px;\" \/> on the same grid. We will choose a few points on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and then multiply the <em data-effect=\"italics\">y<\/em>-values by 3 to get the points for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c39965f6f51f2db8a1f84b3fdddf4ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1169149041586\" data-alt=\"The table depicts the effect of constants on the basic function of x squared. The table has 3 columns labeled x, f of x equals x squared with the ordered pair (x, f of x), and g of x equals 3 times x squared with the ordered pair (x, g of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared with the ordered pair (x, f of x), the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 3 times x squared with the ordered pair (x, g of x) column has the ordered pairs of (negative 2, 12) because 3 times 4 equals 12, (negative 1, 3) because 3 times 1 equals 3, (0, 0) because 3 times 0 equals 0, (1, 3) because 3 times 1 equals 3, and (2,12) because 3 times 4 equals 12. The graph beside the table shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). The slimmer curve of g of x equals 3 times x squared has a vertex at (0,0) and other points given of (negative 2, 12), (negative 1, 3), (1, 3), and (2,12).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table depicts the effect of constants on the basic function of x squared. The table has 3 columns labeled x, f of x equals x squared with the ordered pair (x, f of x), and g of x equals 3 times x squared with the ordered pair (x, g of x). In the x column, the values given are negative 2, negative 1, 0, 1, and 2. In the f of x equals x squared with the ordered pair (x, f of x), the ordered pairs (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4) are given. The g of x equals 3 times x squared with the ordered pair (x, g of x) column has the ordered pairs of (negative 2, 12) because 3 times 4 equals 12, (negative 1, 3) because 3 times 1 equals 3, (0, 0) because 3 times 0 equals 0, (1, 3) because 3 times 1 equals 3, and (2,12) because 3 times 4 equals 12. The graph beside the table shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). The slimmer curve of g of x equals 3 times x squared has a vertex at (0,0) and other points given of (negative 2, 12), (negative 1, 3), (1, 3), and (2,12).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149188800\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149150153\">\n<div data-type=\"problem\" id=\"fs-id1169148952025\">\n<p id=\"fs-id1169144645882\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8d5973503902f1313f518519a9fda07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146652009\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149026718\" data-alt=\"The graph shows the upward-opening parabola on the x y-coordinate plane of f of x equals x squared that has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). Also shown is a downward-opening parabola of f of x equals negative 3 times x squared. It has a vertex of (0,0) with other points at (negative 1, negative 3) and (1, negative 3)\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the upward-opening parabola on the x y-coordinate plane of f of x equals x squared that has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). Also shown is a downward-opening parabola of f of x equals negative 3 times x squared. It has a vertex of (0,0) with other points at (negative 1, negative 3) and (1, negative 3)\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148988111\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148943450\">\n<div data-type=\"problem\" id=\"fs-id1169148917724\">\n<p id=\"fs-id1169148964704\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-678f72f91e18a3734808cba56bd86995_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149223770\"><span data-type=\"media\" id=\"fs-id1169144382865\" data-alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_313_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149065202\">\n<h3 data-type=\"title\">Graph Quadratic Functions Using Transformations<\/h3>\n<p id=\"fs-id1169149095024\">We have learned how the constants <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">h<\/em>, and <em data-effect=\"italics\">k<\/em> in the functions, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeb1b6873d380660f27a0686c90c7b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"250\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> affect their graphs. We can now put this together and graph quadratic functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ff39a37a230408f7c9a6410a33dbe1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/> by first putting them into the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> by completing the square. This form is sometimes known as the vertex form or standard form.<\/p>\n<p id=\"fs-id1169149087320\">We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148938795\" data-alt=\"This figure shows the difference when completing the square with a quadratic equation and a quadratic function. For the quadratic equation, start with x squared plus 8 times x plus 6 equals zero. Subtract 6 from both sides to get x squared plus 8 times x equals negative 6 while leaving space to complete the square. Then, complete the square by adding 16 to both sides to get x squared plush 8 times x plush 16 equals negative 6 plush 16. Factor to get the quantity x plus 4 squared equals 10. For the quadratic function, start with f of x equals x squared plus 8 times x plus 6. The second line shows to leave space between the 8 times x and the 6 in order to complete the square. Complete the square by adding 16 and subtracting 16 on the same side to get f of x equals x squared plus 8 times x plush 16 plus 6 minus 16. Factor to get f of x equals the quantity of x plush 4 squared minus 10.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the difference when completing the square with a quadratic equation and a quadratic function. For the quadratic equation, start with x squared plus 8 times x plus 6 equals zero. Subtract 6 from both sides to get x squared plus 8 times x equals negative 6 while leaving space to complete the square. Then, complete the square by adding 16 to both sides to get x squared plush 8 times x plush 16 equals negative 6 plush 16. Factor to get the quantity x plus 4 squared equals 10. For the quadratic function, start with f of x equals x squared plus 8 times x plus 6. The second line shows to leave space between the 8 times x and the 6 in order to complete the square. Complete the square by adding 16 and subtracting 16 on the same side to get f of x equals x squared plus 8 times x plush 16 plus 6 minus 16. Factor to get f of x equals the quantity of x plush 4 squared minus 10.\" \/><\/span><\/p>\n<p id=\"fs-id1169148956985\">When we complete the square in a function with a coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> that is not one, we have to factor that coefficient from just the <em data-effect=\"italics\">x<\/em>-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the <em data-effect=\"italics\">x<\/em>-terms.<\/p>\n<p id=\"fs-id1169149156980\">Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149374763\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169144563134\">\n<div data-type=\"problem\" id=\"fs-id1169149370896\">\n<p id=\"fs-id1169148889707\">Rewrite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-742da98a993f349066ce1c9143c6b0a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148968618\">\n<table id=\"fs-id1169149016716\" class=\"unnumbered unstyled\" summary=\"This figure gives step-by-step instructions for completing the square with a quadratic function with a coefficient with the x squared term. Given the function f of x equals negative 3 times x squared minus 6 times x minus 1, separate the x terms from the constant to get f of x equals negative 3 times x squared, space, minus 6 times x minus 1. Next, factor the coefficient of x squared, which is negative 3, to get f of x equals negative 3 times the quantity of x squared plus 2 times x minus 1. Then, prepare to complete the square by leaving space between the 2 times x and parentheses. Then take half of 2 and then square it to complete the square, the quantity one-half times 2 squared equals 1. The constant 1 completes the square in the parentheses, but the parentheses is multiplied by negative 3. So we are really adding negative 3. We must then add 3 to not change the value of the function to get f of x equals negative 3 times the quantity of x squared plus 2 times x plus 1 minus 1 plus 3. Rewrite the trinomial as a square and subtract the constants to get f of x equals negative 3 times the quantity of x plus 1 squared plus 2. The function is now in the f of x equals a times the quantity x minus h squared plus k form.\" 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-id1169146742130\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Separate the <em data-effect=\"italics\">x<\/em> terms from the constant.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147132205\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017b_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 coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149030949\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Prepare to complete the square.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146656082\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take half of 2 and then square it to complete the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>square. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5913c2545028166b587c769a9b3fd0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/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 constant 1 completes the square in the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>parentheses, but the parentheses is multiplied by<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>. So we are really adding <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> We must then<\/p>\n<div data-type=\"newline\"><\/div>\n<p>add 3 to not change the value of the function.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144375648\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017e_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 trinomial as a square and subtract the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>constants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148894430\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017f_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 function is now in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149025514\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_017g_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-id1169148987017\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148963136\">\n<div data-type=\"problem\" id=\"fs-id1169149011009\">\n<p id=\"fs-id1169146644111\">Rewrite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fbf1fd7e03268dace5446908996bcfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146647640\">\n<p id=\"fs-id1169147029305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-462508cd0a156aad0c8683006a01a1ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148998662\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149193773\">\n<div data-type=\"problem\" id=\"fs-id1169144813293\">\n<p id=\"fs-id1169148910881\">Rewrite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b63104260d8a81f9f35229957ca43a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148848860\">\n<p id=\"fs-id1169149016360\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2490d05d8bb0e09c3a507fe7880b69b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146625591\">Once we put the function into the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a05eae74c76937d1efe50d7791db2460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#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=\"155\" style=\"vertical-align: -4px;\" \/> form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149195601\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149375092\">\n<div data-type=\"problem\" id=\"fs-id1169146655550\">\n<p id=\"fs-id1169146658046\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7764e8629197413d70a63c624bc5a664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/> by using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149296594\">\n<p id=\"fs-id1169149008581\"><strong data-effect=\"bold\">Step 1.<\/strong> Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> vertex form by completing the square.<\/p>\n<table id=\"fs-id1169149007262\" class=\"unnumbered unstyled\" summary=\"This figure gives step-by-step instructions for completing the square with a quadratic function with no coefficient with the x squared term. Given the function f of x equals x squared plus 6 times x plus 5, separate the x terms from the constant to get f of x equals x squared, space, plus 6 times x plus 5. Then take half of 6 and then square it to complete the square, the quantity one-half times 6 squared equals 9. We must then add 9 and subtract 9 to not change the value of the function to get f of x equals x squared plus 6 times x plus 9 plus 5 minus 9. Rewrite the trinomial as a square and subtract the constants to get f of x equals the quantity of x plus 3 squared minus 4. The function is now in the f of x equals a times the quantity x minus h squared plus k form.\" 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-id1169149040281\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Separate the <em data-effect=\"italics\">x<\/em> terms from the constant.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144375967\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take half of 6 and then square it to complete the square.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f2e1424c2f2f16903b328e6d45fd0a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"74\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We both add 9 and subtract 9 to not change the value of the function.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149096773\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial as a square and subtract the constants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148958083\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The function is now in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a05eae74c76937d1efe50d7791db2460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#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=\"155\" style=\"vertical-align: -4px;\" \/> form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148825054\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149033195\"><strong data-effect=\"bold\">Step 2:<\/strong> Graph the function using transformations.<\/p>\n<p id=\"fs-id1169148895304\">Looking at the <em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em> values, we see the graph will take the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> and shift it to the left 3 units and down 4 units.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169146643276\" data-alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared minus 4. The next lines say h equals negative 3 which means shift left 3 unit and k equals negative 4 which means shift down 4 units\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared minus 4. The next lines say h equals negative 3 which means shift left 3 unit and k equals negative 4 which means shift down 4 units\" \/><\/span><\/p>\n<p id=\"fs-id1169148870087\">We first draw the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> on the grid.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169144378557\" data-alt=\"To graph f of x equals the quantity x plus 3 squared, shift the graph of f of x equals x squares to the left 3 units. To graph f of x equals the quantity x plus 3 squared minus 4, shift the graph the quantity x plus 3 squared down 4 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To graph f of x equals the quantity x plus 3 squared, shift the graph of f of x equals x squares to the left 3 units. To graph f of x equals the quantity x plus 3 squared minus 4, shift the graph the quantity x plus 3 squared down 4 units.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149122194\" data-alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 3, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left 3 units to produce f of x equals the quantity of x plus 3 squared. By moving f of x equals the quantity of x plus 3 squared down 2, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 3 squared, then another curve moved down 4 to produce f of x equals the quantity of x plus 1 squared minus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_021_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By shifting that graph of f of x equals x squared left 3, we move to the next graph, which shows the original f of x equals x squared and then another curve moved left 3 units to produce f of x equals the quantity of x plus 3 squared. By moving f of x equals the quantity of x plus 3 squared down 2, we move to the final graph, which shows the original f of x equals x squared and the f of x equals the quantity of x plus 3 squared, then another curve moved down 4 to produce f of x equals the quantity of x plus 1 squared minus 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148837576\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149144148\">\n<div data-type=\"problem\" id=\"fs-id1169149149874\">\n<p id=\"fs-id1169146654089\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d2ed58c3ac928ee400d7cec57cc91ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/> by using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144730925\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149008752\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the left has been moved 1 unit to the left to produce f of x equals the quantity of x plus 1 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x plus 1 squared minus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the left has been moved 1 unit to the left to produce f of x equals the quantity of x plus 1 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x plus 1 squared minus 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149240506\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149040152\">\n<div data-type=\"problem\" id=\"fs-id1169149220551\">\n<p id=\"fs-id1169149023966\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-777de80a31fa84cf7c175b771dbb98d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> by using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148880803\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148866779\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the right has been moved 4 units to the right to produce f of x equals the quantity of x minus 4 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x minus 4 squared minus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_315_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the right has been moved 4 units to the right to produce f of x equals the quantity of x minus 4 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x minus 4 squared minus 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148859649\">We list the steps to take to graph a quadratic function using transformations here.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149025824\" class=\"howto\">\n<div data-type=\"title\">Graph a quadratic function using transformations.<\/div>\n<ol id=\"fs-id1169148944333\" class=\"stepwise\" type=\"1\">\n<li>Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/li>\n<li>Graph the function using transformations.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169144555491\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169144602993\">\n<div data-type=\"problem\" id=\"fs-id1169149109915\">\n<p id=\"fs-id1169149342531\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd9b69739c77180b91b83b1a49eb75d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> by using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146745591\">\n<p id=\"fs-id1169146664809\"><strong data-effect=\"bold\">Step 1.<\/strong> Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> vertex form by completing the square.<\/p>\n<table id=\"fs-id1169148938631\" class=\"unnumbered unstyled\" summary=\"This figure gives step-by-step instructions for completing the square with a quadratic function with a coefficient with the x squared term. Given the function f of x equals negative 2 times x squared minus 4 times x plus 2, separate the x terms from the constant to leave space for completing the square. We need the coefficient of x squared to be one. We factor negative 2 from the x-terms to get f of x equals negative 2 times the quantity of x squared plus 2 times x plus 2. Take half of 2 and then square it to complete the square, the quantity of one-half times 2 squared equals 1. We add 1 to complete the square in the parentheses, but the parentheses is multiplied by negative 2. Se we are really adding negative 2. To not change the value of the function we add 2 to get f of x equals negative 2 times the quantity of x squared plus 2 times x plus 1 squared plus 2 plus 2. Rewrite the trinomial as a square and subtract the constants to get f of x equals negative 2 times the quantity of x plus 1 squared plus 4. The function is now in the f of x equals a times the quantity x minus h squared plus k form.\" 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-id1169149328740\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Separate the <em data-effect=\"italics\">x<\/em> terms from the constant.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149018093\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We need the coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> to be one.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>We factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> from the <em data-effect=\"italics\">x<\/em>-terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149329700\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take half of 2 and then square it to complete the square.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5913c2545028166b587c769a9b3fd0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We add 1 to complete the square in the parentheses, but the parentheses is multiplied by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>. Se we are really adding <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>. To not change the value of the function we add 2.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149311090\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial as a square and subtract the constants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144382686\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The function is now in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144729567\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_022g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148964046\"><strong data-effect=\"bold\">Step 2.<\/strong> Graph the function using transformations.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148869574\" data-alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals negative 2 times x squared with an arrow coming from it pointing to f of x equals negative 2 times the quantity x plus 1 squared. An arrow come from it to point to f of x equals negative 2 times the quantity x plus 1 squared plus 4. The next line says a equals negative 2 which means multiply the y-values by negative 2, then h equals negative 1 which means shift left 1 unit and k equals 4 which means shift up 4 units\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"F of x equals x squared is given with an arrow coming from it pointing to f of x equals negative 2 times x squared with an arrow coming from it pointing to f of x equals negative 2 times the quantity x plus 1 squared. An arrow come from it to point to f of x equals negative 2 times the quantity x plus 1 squared plus 4. The next line says a equals negative 2 which means multiply the y-values by negative 2, then h equals negative 1 which means shift left 1 unit and k equals 4 which means shift up 4 units\" \/><\/span><\/p>\n<p id=\"fs-id1169148947726\">We first draw the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> on the grid.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169149169235\" data-alt=\"To graph f of x equals negative 2 times x squared, multiply the y-values in parabola of f of x equals x squared by negative 2. To graph f of x equals negative 2 times the quantity x plus 1 squared, shift the graph of f of x equals negative 2 times x squared to the left 1 unit. To graph f of x equals negative 2 times the quantity x plus 1 squared plus 4, shift the graph of f of x equals negative 2 times the quantity x plus 1 squared up 4 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_024_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"To graph f of x equals negative 2 times x squared, multiply the y-values in parabola of f of x equals x squared by negative 2. To graph f of x equals negative 2 times the quantity x plus 1 squared, shift the graph of f of x equals negative 2 times x squared to the left 1 unit. To graph f of x equals negative 2 times the quantity x plus 1 squared plus 4, shift the graph of f of x equals negative 2 times the quantity x plus 1 squared up 4 units.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169147089647\" data-alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By multiplying by negative 2, move to the next graph showing the original f of x equals x squared and the new slimmer and flipped graph of f of x equals negative 2 x squared. By shifting that graph of f of x equals negative 2 times x squared left 1, we move to the next graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and then another curve moved left 1 unit to produce f of x equals negative 2 times the quantity of x plus 1 squared. By moving f of x equals negative 2 times the quantity of x plus 1 squared up 4, we move to the final graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and the f of x equals negative 2 times the quantity of x plus 1 squared, then another curve moved up 4 to produce f of x equals negative 2 times the quantity of x plus 1 squared plus 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_025_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows 1 upward-opening parabola on the x y-coordinate plane. It is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). By multiplying by negative 2, move to the next graph showing the original f of x equals x squared and the new slimmer and flipped graph of f of x equals negative 2 x squared. By shifting that graph of f of x equals negative 2 times x squared left 1, we move to the next graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and then another curve moved left 1 unit to produce f of x equals negative 2 times the quantity of x plus 1 squared. By moving f of x equals negative 2 times the quantity of x plus 1 squared up 4, we move to the final graph, which shows the original f of x equals x squared, f of x equals negative 2 x squared, and the f of x equals negative 2 times the quantity of x plus 1 squared, then another curve moved up 4 to produce f of x equals negative 2 times the quantity of x plus 1 squared plus 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149289706\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146621161\">\n<div data-type=\"problem\" id=\"fs-id1169146621164\">\n<p id=\"fs-id1169148958057\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c114fb28bcefc3c1643291ca1d3ce28b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -4px;\" \/> by using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149220390\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169144523011\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149293431\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149023639\">\n<div data-type=\"problem\" id=\"fs-id1169146645023\">\n<p id=\"fs-id1169149152543\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10cf24b3ecf44f5cb9840e584ca1a988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -4px;\" \/> by using transformations.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149286365\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149037643\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_317_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149017496\">Now that we have completed the square to put a quadratic function into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form, we can also use this technique to graph the function using its properties as in the previous section.<\/p>\n<p id=\"fs-id1169144682919\">If we look back at the last few examples, we see that the vertex is related to the constants <em data-effect=\"italics\">h<\/em> and <em data-effect=\"italics\">k<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169146628097\" data-alt=\"The first graph shows an upward-opening parabola on the x y-coordinate plane with a vertex of (negative 3, negative 4) with other points of (0, negative 5) and (0, negative 1). Underneath the graph, it shows the standard form of a parabola, f of x equals the quantity x minus h squared plus k, with the equation of the parabola f of x equals the quantity of x plus 3 squared minus 4 where h equals negative 3 and k equals negative 4. The second graph shows a downward-opening parabola on the x y-coordinate plane with a vertex of (negative 1, 4) and other points of (0,2) and (negative 2,2). Underneath the graph, it shows the standard form of a parabola, f of x equals a times the quantity x minus h squared plus k, with the equation of the parabola f of x equals negative 2 times the quantity of x plus 1 squared plus 4 where h equals negative 1 and k equals 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_026_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph shows an upward-opening parabola on the x y-coordinate plane with a vertex of (negative 3, negative 4) with other points of (0, negative 5) and (0, negative 1). Underneath the graph, it shows the standard form of a parabola, f of x equals the quantity x minus h squared plus k, with the equation of the parabola f of x equals the quantity of x plus 3 squared minus 4 where h equals negative 3 and k equals negative 4. The second graph shows a downward-opening parabola on the x y-coordinate plane with a vertex of (negative 1, 4) and other points of (0,2) and (negative 2,2). Underneath the graph, it shows the standard form of a parabola, f of x equals a times the quantity x minus h squared plus k, with the equation of the parabola f of x equals negative 2 times the quantity of x plus 1 squared plus 4 where h equals negative 1 and k equals 4.\" \/><\/span><\/p>\n<p id=\"fs-id1169146652019\">In each case, the vertex is (<em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>). Also the <span data-type=\"term\" class=\"no-emphasis\">axis of symmetry<\/span> is the line <em data-effect=\"italics\">x<\/em> = <em data-effect=\"italics\">h<\/em>.<\/p>\n<p id=\"fs-id1169146724594\">We rewrite our steps for graphing a quadratic function using properties for when the function is in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149027551\" class=\"howto\">\n<div data-type=\"title\">Graph a quadratic function in the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> using properties.<\/div>\n<ol id=\"fs-id1169149219887\" class=\"stepwise\" type=\"1\">\n<li>Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form.<\/li>\n<li>Determine whether the parabola opens upward, <em data-effect=\"italics\">a<\/em> &gt; 0, or downward, <em data-effect=\"italics\">a<\/em> &lt; 0.<\/li>\n<li>Find the axis of symmetry, <em data-effect=\"italics\">x<\/em> = <em data-effect=\"italics\">h<\/em>.<\/li>\n<li>Find the vertex, (<em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>).<\/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<div data-type=\"example\" id=\"fs-id1169146638071\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169146638073\">\n<div data-type=\"problem\" id=\"fs-id1169146626688\">\n<p id=\"fs-id1169146626690\"><span class=\"token\">\u24d0<\/span> Rewrite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40c427fb21444b9adbc1aa2b74fe8f54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph the function using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146742819\">\n<table id=\"fs-id1169146814819\" class=\"unnumbered unstyled can-break\" summary=\"This is figure gives step-by-step instructions on how to graph a function using properties of the equation. Rewrite the function in f of x equals a times the quantity x minus h squared plus k form by completing the square. The function is f of x equals 2 x squared plus 4 times x plus 5. Follow the process to complete the square: f of x equals 2 times the quantity of x squared plus 2 x plus 5, f of x equals 2 times the quantity of x squared plus 2 x plus 1 plus 5 minus 2, and f of x equals 2 times the quantity of x plus 1 squared plus 3. Identify the constants a, h, k, to get a equals 2, h equals negative 1, and k equals 3. Since a equals 2, the parabola opens upward. A small picture of an upward-facing parabola is shown. The axis of symmetry is x equals h, so x equals negative 1. The vertex is (h, k) so (negative 1, 3). Find the y-intercept by finding f of 0. F of 0 equals 2 times 0 squared plus 4 times 0 plus 5, so f of o equals 5, so the y-intercept is (0,5). Find the point symmetric to (0,5) across the axis of symmetry which is (2,5). Find the x-intercepts. Since the discriminant is negative, so there are no x-intercepts. Graph the parabola. The graph shown is an upward facing parabola with vertex (negative 1, 3) and y-intercept (0,5). The axis of symmetry is shown, x equals negative 1.\" 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-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>form 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-40c427fb21444b9adbc1aa2b74fe8f54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" 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-bf4c64d5e4ba6cfcb2449a5e2c64c0f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"177\" 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-d118498ade49fec13111c12c0daa7a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"238\" 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-28bf1dda728dc04e2c0d5ea508544f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#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=\"163\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Identify the constants <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5dc300a884e0460e07a1b13767e717a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#104;&#44;&#107;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" 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-27799520e3333a491a648eda3884f3d1_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;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#61;&#50;&#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;&#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;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"177\" 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-df08b703a4783940e7a806c467f2bc12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/>, the parabola opens upward.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169144421651\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_027a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The axis of symmetry is <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<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-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The vertex is <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<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-f2347704000ad2e9ae878a8427611b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#51;&#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>-intercept by finding <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d73c012bc8b5a6f560bea30840502b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-563ef505e70e594356ee8078fd2b6303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#48;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" 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-51adc3a2480d8665f7083283e42019ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" 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<\/p>\n<div data-type=\"newline\"><\/div>\n<p>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-c0ded887d7338dbffa5082717180f3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" 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\">The discriminant negative, so there are<\/p>\n<div data-type=\"newline\"><\/div>\n<p>no <em data-effect=\"italics\">x<\/em>-intercepts. Graph the parabola.<\/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-id1169149286236\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_027j_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-id1169149285302\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149285306\">\n<div data-type=\"problem\" id=\"fs-id1169149285308\">\n<p id=\"fs-id1169149285310\"><span class=\"token\">\u24d0<\/span> Rewrite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0f1b522567b5b487dcea91ee2893f42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/> in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph the function using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146643464\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c490648f440f20890250656b34a3d9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#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=\"162\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149329160\" data-alt=\"The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149329177\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149329181\">\n<div data-type=\"problem\" id=\"fs-id1169149329183\">\n<p id=\"fs-id1169149329185\"><span class=\"token\">\u24d0<\/span> Rewrite <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50097936e8ab6a3cf44f956e0b637b67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" style=\"vertical-align: -4px;\" \/> in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph the function using properties.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148985233\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1ea7848520773aaae8ecd87d1446124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#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=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149195445\" data-alt=\"The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_319_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149194051\">\n<h3 data-type=\"title\">Find a Quadratic Function from its Graph<\/h3>\n<p id=\"fs-id1169149194056\">So far we have started with a function and then found its graph.<\/p>\n<p id=\"fs-id1169149194059\">Now we are going to reverse the process. Starting with the graph, we will find the function.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149194063\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149194066\">\n<div data-type=\"problem\" id=\"fs-id1169149194068\">\n<p id=\"fs-id1169149194070\">Determine the quadratic function whose graph is shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169149194073\" data-alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_028_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0, 7).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149214515\">\n<p id=\"fs-id1169149214518\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23e4112bb3ce0bdac0c864f91cb86802_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#105;&#116;&#32;&#105;&#115;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#44;&#32;&#119;&#101;&#32;&#115;&#116;&#97;&#114;&#116;&#32;&#119;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#118;&#101;&#114;&#116;&#101;&#120;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#61;&#45;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#61;&#45;&#49;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#119;&#101;&#32;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#43;&#50;&#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;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#97;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#97;&#123;&#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;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#61;&#45;&#50;&#44;&#107;&#61;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#61;&#50;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#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=\"221\" width=\"730\" style=\"vertical-align: -106px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148957428\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148957432\">\n<div data-type=\"problem\" id=\"fs-id1169148957434\">\n<p id=\"fs-id1169148957436\">Write the quadratic function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form whose graph is shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169146607084\" data-alt=\"The graph shown is an upward facing parabola with vertex (3, negative 4) and y-intercept (0, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_029_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (3, negative 4) and y-intercept (0, 5).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146607098\">\n<p id=\"fs-id1169146607100\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6093954125aee491f2b18267b9276f81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146816985\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146816989\">\n<div data-type=\"problem\" id=\"fs-id1169146816991\">\n<p>Determine the quadratic function whose graph is shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169146816996\" data-alt=\"The graph shown is an upward facing parabola with vertex (negative 3, negative 1) and y-intercept (0, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_030_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (negative 3, negative 1) and y-intercept (0, 8).\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149096712\">\n<p id=\"fs-id1169149096714\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afeb024b83b15ad699a8fd2a59a39c9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149096760\" class=\"media-2\">\n<p id=\"fs-id1169144768434\">Access these online resources for additional instruction and practice with graphing quadratic functions using transformations.<\/p>\n<ul id=\"fs-id1163870666809\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran1\">Function Shift Rules Applied to Quadratic Functions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran2\">Changing a Quadratic from Standard Form to Vertex Form<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran3\">Using Transformations to Graph Quadratic Functions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadFuncTran4\">Finding Quadratic Equation in Vertex Form from Graph<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169144768465\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169144768473\" data-bullet-style=\"bullet\">\n<li>Graph a Quadratic Function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> Using a Vertical Shift\n<ul id=\"fs-id1169144377944\" data-bullet-style=\"bullet\">\n<li>The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> shifts the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> vertically k units.\n<ul id=\"fs-id1169138762012\" data-bullet-style=\"bullet\">\n<li>If <em data-effect=\"italics\">k<\/em> &gt; 0, shift the parabola vertically up <em data-effect=\"italics\">k<\/em> units.<\/li>\n<li>If <em data-effect=\"italics\">k<\/em> &lt; 0, shift the parabola vertically down <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-717afd6bd88914fd87d415dcab93ad94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#107;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"16\" style=\"vertical-align: -4px;\" \/> units.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Graph a Quadratic Function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/> Using a Horizontal Shift\n<ul id=\"fs-id1169148828155\" data-bullet-style=\"bullet\">\n<li>The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/> shifts the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> horizontally h units.\n<ul id=\"fs-id1169148872363\" data-bullet-style=\"bullet\">\n<li>If <em data-effect=\"italics\">h<\/em> &gt; 0, shift the parabola horizontally left <em data-effect=\"italics\">h<\/em> units.<\/li>\n<li>If <em data-effect=\"italics\">h<\/em> &lt; 0, shift the parabola horizontally right <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-314088aa97d0b7aa21d8f48d8317a8e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#104;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"16\" style=\"vertical-align: -4px;\" \/> units.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Graph of a Quadratic Function of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/>\n<ul id=\"fs-id1169149297377\" data-bullet-style=\"bullet\">\n<li>The coefficient <em data-effect=\"italics\">a<\/em> in the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> affects the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> by stretching or compressing it.\n<div data-type=\"newline\"><\/div>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd7b16bfa2f7f7e1ac6d82c152b28183_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#124;&#97;&#124;&#60;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> then the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> will be \u201cwider\u201d than the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac78978801d31220977b3a0e9009f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6809a464885b2b33f3e6143baabeddc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#97;&#124;&#62;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> then the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/> will be \u201cskinnier\u201d than the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac78978801d31220977b3a0e9009f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li>How to graph a quadratic function using transformations\n<ol id=\"fs-id1169149369519\" class=\"stepwise\" type=\"1\">\n<li>Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/li>\n<li>Graph the function using transformations.<\/li>\n<\/ol>\n<\/li>\n<li>Graph a quadratic function in the vertex form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> using properties\n<ol id=\"fs-id1169149346092\" class=\"stepwise\" type=\"1\">\n<li>Rewrite the function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form.<\/li>\n<li>Determine whether the parabola opens upward, <em data-effect=\"italics\">a<\/em> &gt; 0, or downward, a &lt; 0.<\/li>\n<li>Find the axis of symmetry, <em data-effect=\"italics\">x<\/em> = <em data-effect=\"italics\">h<\/em>.<\/li>\n<li>Find the vertex, (<em data-effect=\"italics\">h<\/em>, <em data-effect=\"italics\">k<\/em>).<\/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, if possible.<\/li>\n<li>Graph the parabola.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169144374336\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169144374340\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169144374347\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1169144378724\">In the following exercises, <span class=\"token\">\u24d0<\/span> graph the quadratic functions on the same rectangular coordinate system and <span class=\"token\">\u24d1<\/span> describe what effect adding a constant, <em data-effect=\"italics\">k<\/em>, to the function has on the basic parabola.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144378742\">\n<div data-type=\"problem\" id=\"fs-id1169144378744\">\n<p id=\"fs-id1169144378747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afb8774046aad333df4d6834b3dfaada_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d40d56638436b585c70eb8652bc9003_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146612344\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13dcead0d565a36990c4814adee92a10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted up 4 units. The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e51751ea6f72137ac68f90d71cd13edf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shift down 4 units.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144451007\">\n<div data-type=\"problem\" id=\"fs-id1169144451009\">\n<p id=\"fs-id1169144451011\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca7966a311ee12764a36e04875322b6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbae2d6875ad064be4eca554d7d7780b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144823918\">In the following exercises, graph each function using a vertical shift.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144823921\">\n<div data-type=\"problem\" id=\"fs-id1169144823923\">\n<p id=\"fs-id1169144823925\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79d33ba440ae182f1cdc0f6b2ee89f49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144724402\"><span data-type=\"media\" id=\"fs-id1169144724406\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144724421\">\n<div data-type=\"problem\" id=\"fs-id1169144724423\">\n<p id=\"fs-id1169144724425\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5182687556ccb3a5483a0be8b47dd5e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149196553\">\n<div data-type=\"problem\" id=\"fs-id1169149196555\">\n<p id=\"fs-id1169149196557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ded55672642c130f77e58b8af6918e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144555525\"><span data-type=\"media\" id=\"fs-id1169144555529\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144555545\">\n<div data-type=\"problem\" id=\"fs-id1169144555547\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68d27bd3b56ec73ff8ac49d891f10434_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146626127\">\n<div data-type=\"problem\" id=\"fs-id1169146626129\">\n<p id=\"fs-id1169146626132\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e51751ea6f72137ac68f90d71cd13edf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146669847\"><span data-type=\"media\" id=\"fs-id1169146669852\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146669868\">\n<div data-type=\"problem\" id=\"fs-id1169146669870\">\n<p id=\"fs-id1169146669872\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa3a89c06dc5c021da0679655f961963_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146638897\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2ce6408f24f17a7235aee691fbaf9e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1163870660975\">In the following exercises, <span class=\"token\">\u24d0<\/span> graph the quadratic functions on the same rectangular coordinate system and <span class=\"token\">\u24d1<\/span> describe what effect adding a constant, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/>, inside the parentheses has<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149101378\">\n<div data-type=\"problem\" id=\"fs-id1169149101380\">\n<p id=\"fs-id1169149101383\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12f260df963b3d91eb5df1006d23ad25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"216\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05afe387703c33f21f61033de354493e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144451066\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169144451076\" data-alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The graph to the right is shifted 3 units to the right to produce g of x equals the quantity of x minus 3 squared. The graph the left is shifted 3 units to the left to produce h of x equals the quantity of x plus 3 squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The graph to the right is shifted 3 units to the right to produce g of x equals the quantity of x minus 3 squared. The graph the left is shifted 3 units to the left to produce h of x equals the quantity of x plus 3 squared.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caad0418523c5ad535f196eb8182b0f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"121\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted right 3 units. The graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50edbe340171682b7d0a09d706faaddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/> is the same as the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a982effe5b7adb50b49fa2be219fd43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/> but shifted left 3 units.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148957288\">\n<div data-type=\"problem\" id=\"fs-id1169148957290\">\n<p id=\"fs-id1169148957292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82a32652cc7d4efea5681abf9f8959f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"216\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-197c69bcf6ac25e9acf8c21455bf7ff6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146631834\">In the following exercises, graph each function using a horizontal shift.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146631837\">\n<div data-type=\"problem\" id=\"fs-id1169146631839\">\n<p id=\"fs-id1169146631842\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0324358718ab26a044f2f19526206b20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144876562\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169144876566\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144876581\">\n<div data-type=\"problem\" id=\"fs-id1169144876583\">\n<p id=\"fs-id1169144876586\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cedd37648950681118832ae7ff962d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146639448\">\n<div data-type=\"problem\" id=\"fs-id1169146639450\">\n<p id=\"fs-id1169146639452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d00dd19e5f46a7f012067797873458d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149360868\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149360872\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149360888\">\n<div data-type=\"problem\" id=\"fs-id1169149360890\">\n<p id=\"fs-id1169149360892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-faadc03c320187435de9710316791b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144892933\">\n<div data-type=\"problem\" id=\"fs-id1169144892935\">\n<p id=\"fs-id1169144892937\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a609504ea826e7ba2bf75640c64052df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144566243\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169144566248\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_334_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144566263\">\n<div data-type=\"problem\" id=\"fs-id1169144556471\">\n<p id=\"fs-id1169144556473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5aa20b5ce511bc013d9441795a0ee3f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149194316\">In the following exercises, graph each function using transformations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149194319\">\n<div data-type=\"problem\" id=\"fs-id1169149194321\">\n<p id=\"fs-id1169149194323\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9d869c4c6c5165469b26d8920190b5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149194365\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146665353\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146665368\">\n<div data-type=\"problem\" id=\"fs-id1169146665370\">\n<p id=\"fs-id1169146665372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1ab3f9fbf793996de3132ad84793c25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169139960649\">\n<div data-type=\"problem\" id=\"fs-id1169139960651\">\n<p id=\"fs-id1169139960653\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ca64e019faad2bec1fda22f347d0490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149293770\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149293774\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149293789\">\n<div data-type=\"problem\" id=\"fs-id1169149293791\">\n<p id=\"fs-id1169149293793\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed23ac339b6b0d46d3a8ad5471a112ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146621931\">\n<div data-type=\"problem\" id=\"fs-id1169146621933\">\n<p id=\"fs-id1169146621935\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afeb024b83b15ad699a8fd2a59a39c9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149011458\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149011462\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149303573\">\n<div data-type=\"problem\" id=\"fs-id1169149303575\">\n<p id=\"fs-id1169149303577\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c14e2bceffe41edf29ba3c261d43ea0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146623143\">\n<div data-type=\"problem\" id=\"fs-id1169146623145\">\n<p id=\"fs-id1169146623148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2e513acee98fc835ff36953d9069d6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146623189\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146937025\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_342_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146937041\">\n<div data-type=\"problem\" id=\"fs-id1169146937043\">\n<p id=\"fs-id1169146937045\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8fbee264210e4e61ba45a14f18b7edb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144603085\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1169149043369\">In the following exercises, graph each function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149043372\">\n<div data-type=\"problem\" id=\"fs-id1169149043375\">\n<p id=\"fs-id1169149043377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc824c3ab32a9545fc1e6b93adba8909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149043403\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149043408\" data-alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_344_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149240401\">\n<div data-type=\"problem\" id=\"fs-id1169149240404\">\n<p id=\"fs-id1169149240406\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35761561d886ce826de430fed463cd28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149240453\">\n<div data-type=\"problem\" id=\"fs-id1169149240455\">\n<p id=\"fs-id1169144538642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f127c3678637936ac270864407204078_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144538669\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169144538674\" data-alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_346_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144538690\">\n<div data-type=\"problem\" id=\"fs-id1169144538692\">\n<p id=\"fs-id1169144538694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1be1d57a2ab45be04aa1b45d2427657e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144365536\">\n<div data-type=\"problem\" id=\"fs-id1169144365538\">\n<p id=\"fs-id1169144365540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-395f327411241134f1dd2a8ec1f36227_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147027577\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169147027582\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_348_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147027597\">\n<div data-type=\"problem\" id=\"fs-id1169147027600\">\n<p id=\"fs-id1169147027602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86da4af9086fd15b7c78edd450d5543b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146640150\">\n<div data-type=\"problem\" id=\"fs-id1169146640152\">\n<p id=\"fs-id1169146640154\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0306d762c45b60447b58c6375f273b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147086861\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169147086866\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_350_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147086881\">\n<div data-type=\"problem\" id=\"fs-id1169147086883\">\n<p id=\"fs-id1169147086885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07274cb9949debc019c75208137fe9a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"103\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146813405\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Transformations<\/strong><\/p>\n<p id=\"fs-id1169146813412\">In the following exercises, rewrite each function in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149290730\">\n<div data-type=\"problem\" id=\"fs-id1169149290732\">\n<p id=\"fs-id1169149290734\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a36bb1e62f1b6c7787181f4bb1a5536_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149195302\">\n<p id=\"fs-id1169149195304\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f763c91a397e23fe0efd8687375c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149195345\">\n<div data-type=\"problem\" id=\"fs-id1169149195348\">\n<p id=\"fs-id1169149195350\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec581d44e31ac7e9d86bea17b858ad67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149328890\">\n<div data-type=\"problem\" id=\"fs-id1169149328892\">\n<p id=\"fs-id1169149328894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95aa7b75f8bcb5a60060957ef18ee8af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149328929\">\n<p id=\"fs-id1169149328932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-377a84bb0aa4049876b3cb67e8bf607a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146594120\">\n<div data-type=\"problem\" id=\"fs-id1169146594122\">\n<p id=\"fs-id1169146594124\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad7fac9c677412a5711bafefe7aed9f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144451625\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph it by using transformations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144451677\">\n<div data-type=\"problem\" id=\"fs-id1169149000795\">\n<p id=\"fs-id1169149000798\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7764e8629197413d70a63c624bc5a664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149000831\">\n<p id=\"fs-id1169149000833\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e80f2026e2be0bbe847c44fad9e772c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149357656\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_352_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149357673\">\n<div data-type=\"problem\" id=\"fs-id1169149357675\">\n<p id=\"fs-id1169149357677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1677c89a2469ba2687ac8f1ec0d9a646_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144365120\">\n<div data-type=\"problem\" id=\"fs-id1169144365122\">\n<p id=\"fs-id1169144365124\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1677c89a2469ba2687ac8f1ec0d9a646_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149222676\">\n<p id=\"fs-id1169149222678\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25b8572e6ccbaf2c22ece53fd757f820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149280707\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_354_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149280723\">\n<div data-type=\"problem\" id=\"fs-id1169149280725\">\n<p id=\"fs-id1169149280727\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f1b9b782091679ba0aa268bfd172d41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146644568\">\n<div data-type=\"problem\" id=\"fs-id1169146644570\">\n<p id=\"fs-id1169146644572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c71f4497839d851ef3ac3435ee02fb56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149194498\">\n<p id=\"fs-id1169149194500\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d6cf173e33b2c921bd6ca60cc81b3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149194555\" data-alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_356_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149204947\">\n<div data-type=\"problem\" id=\"fs-id1169149204949\">\n<p id=\"fs-id1169149204952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1aad49fd3a60de2ad354a51135fb71eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144606276\">\n<div data-type=\"problem\" id=\"fs-id1169144606278\">\n<p id=\"fs-id1169144606280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4af2e80e5072314de828d998cb00c262_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144606316\">\n<p id=\"fs-id1169144606318\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf6e7841abfb6466e2e6c08c8f87722d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149348888\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_358_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149348904\">\n<div data-type=\"problem\" id=\"fs-id1169149041020\">\n<p id=\"fs-id1169149041022\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e4d5ad6a1c51e6225ca932ad7df9244_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146732866\">\n<div data-type=\"problem\" id=\"fs-id1169146732868\">\n<p id=\"fs-id1169146732870\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6565b7b214cd33873fcbea9885d72441_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146818042\">\n<p id=\"fs-id1169146818045\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0798b5d72ba14e80f2d877b69fc786b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169147085292\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_360_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147085309\">\n<div data-type=\"problem\" id=\"fs-id1169147085311\">\n<p id=\"fs-id1169147085313\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7000534465bafd76e8064b1c8d7dc436_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146738012\">\n<div data-type=\"problem\" id=\"fs-id1169146738014\">\n<p id=\"fs-id1169146738016\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d387e1230499bacffd130fa18193c82f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144896828\">\n<p id=\"fs-id1169144896831\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53631a3a8966140773843b77aab2faa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148958663\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_362_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148958679\">\n<div data-type=\"problem\" id=\"fs-id1169148958681\">\n<p id=\"fs-id1169148958683\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26b4e0379d82ae427ec54a998e6595fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144417671\">\n<div data-type=\"problem\" id=\"fs-id1169144417673\">\n<p id=\"fs-id1169144417675\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24f3f708ca8a1fa02f814955fba31c4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144717022\">\n<p id=\"fs-id1169144717024\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4eb0e57390a346c04e2f2f22660467f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"162\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169144717080\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_364_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146738134\">\n<div data-type=\"problem\" id=\"fs-id1169146738136\">\n<p id=\"fs-id1169146738138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f15a9be6bdd54556c4507265027a79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147106529\">\n<div data-type=\"problem\" id=\"fs-id1169147106531\">\n<p id=\"fs-id1169147106534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6878d6a93409d773441810417e63e8f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147106569\">\n<p id=\"fs-id1169147106571\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2b6cfb38ba2a15951930e92a280c1e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149190625\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_366_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149190641\">\n<div data-type=\"problem\" id=\"fs-id1169149190643\">\n<p id=\"fs-id1169149190645\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eef4c6a1512c6d1bf452088b43000869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147034644\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph it using properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149023584\">\n<div data-type=\"problem\" id=\"fs-id1169149023586\">\n<p id=\"fs-id1169149023588\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8b3c6d94be5e15cc3037ab977b91fcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144451862\">\n<p id=\"fs-id1169144451864\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1be3d09803199c5e3c4f5c107797700_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149224312\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_368_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149224328\">\n<div data-type=\"problem\" id=\"fs-id1169149224330\">\n<p id=\"fs-id1169149224332\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb892015195b7a94a6365d651ca86282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144420817\">\n<div data-type=\"problem\" id=\"fs-id1169144420819\">\n<p id=\"fs-id1169144420822\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0c1b52f19c8895c4beddb3772d68711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144420857\">\n<p id=\"fs-id1169144420859\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7120171c535fba45409964e1b807eef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#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=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149288478\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_370_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144877272\">\n<div data-type=\"problem\" id=\"fs-id1169144877274\">\n<p id=\"fs-id1169144877276\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3b562e230b65194b838ca7625f036d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144796116\"><strong data-effect=\"bold\">Matching<\/strong><\/p>\n<p id=\"fs-id1169146814380\">In the following exercises, match the graphs to one of the following functions: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e2845f94686c9a1a153b6cfcb344420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a184c8d17ee498a4673704f6e611ed96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6babb8ff9527529ec9a529991fca7ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d655291eb61514946fa4d8bd33384b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d4<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13a78b4e76ab3802ea08f9cda5e8e3a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d5<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9894b0dbd8c45c0af5b66c2eb544c878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d6<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22d0c51e4c13952cf6a7b1f6683dc03c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d7<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a189ec1f152b4b2a45bb16e6cbbe3ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146613391\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146613393\"><span data-type=\"media\" id=\"fs-id1169146613395\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169144685085\">\n<p id=\"fs-id1169144685087\"><span class=\"token\">\u24d2<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144685095\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144685097\"><span data-type=\"media\" id=\"fs-id1169144685100\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144685124\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144685126\"><span data-type=\"media\" id=\"fs-id1169144685128\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169144685142\">\n<p id=\"fs-id1169149367373\"><span class=\"token\">\u24d4<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149367381\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149367383\"><span data-type=\"media\" id=\"fs-id1169149367386\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 4) and other points (negative 6, 8) and (negative 2, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 4) and other points (negative 6, 8) and (negative 2, 8).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149367410\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149367412\"><span data-type=\"media\" id=\"fs-id1169149367414\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169149367428\">\n<p id=\"fs-id1169149367430\"><span class=\"token\">\u24d3<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146621060\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146621062\"><span data-type=\"media\" id=\"fs-id1169146621064\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 4) and other points (negative 2, 8) and (2, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 4) and other points (negative 2, 8) and (2, 8).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146621089\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146621091\"><span data-type=\"media\" id=\"fs-id1169146621093\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_207_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169146621107\">\n<p id=\"fs-id1169146621109\"><span class=\"token\">\u24d6<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146621117\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146621119\"><span data-type=\"media\" id=\"fs-id1169144829486\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 4) and other points (2,8) and (6,8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_208_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 4) and other points (2,8) and (6,8).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1169144829510\"><strong data-effect=\"bold\">Find a Quadratic Function from its Graph<\/strong><\/p>\n<p id=\"fs-id1171780911535\">In the following exercises, write the quadratic function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_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;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form whose graph is shown.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144384693\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144384696\"><span data-type=\"media\" id=\"fs-id1169144384698\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_209_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169144384711\">\n<p id=\"fs-id1169144384714\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03edc2357a2e442a1d968de8ffc04044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144538707\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144538710\"><span data-type=\"media\" id=\"fs-id1169144538712\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_210_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144421503\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144421505\"><span data-type=\"media\" id=\"fs-id1169144421507\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_211_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169144421521\">\n<p id=\"fs-id1169144421523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9467dd7cf41818e0d41ed81228920ab4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#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=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146628627\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146628629\"><span data-type=\"media\" id=\"fs-id1169146628632\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_212_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149339224\">\n<h4 data-type=\"title\">Writing Exercise<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169149339232\">\n<div data-type=\"problem\" id=\"fs-id1169149339234\">\n<p id=\"fs-id1169149339236\">Graph the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a264c985a93316c6f5fda47b8e0a064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/> first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149339272\">\n<p id=\"fs-id1169146627952\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146627958\">\n<div data-type=\"problem\" id=\"fs-id1169146627960\">\n<p id=\"fs-id1169146627962\">Graph the quadratic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d21dc34104e0a9bb9fa6d628c2efd1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/> first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169146628009\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1169146628014\"><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-id1169149189348\" data-alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is graph Quadratic Functions of the form f of x equals x squared plus k; graph Quadratic Functions of the form f of x equals the quantity x minus h squared; graph Quadratic functions of the form f of x equals a times x squared; graph Quadratic Functions Using Transformations; find a Quadratic Function from its Graph. The other columns are left blank for you to check you understanding.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_07_213_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is graph Quadratic Functions of the form f of x equals x squared plus k; graph Quadratic Functions of the form f of x equals the quantity x minus h squared; graph Quadratic functions of the form f of x equals a times x squared; graph Quadratic Functions Using Transformations; find a Quadratic Function from its Graph. The other columns are left blank for you to check you understanding.\" \/><\/span><\/p>\n<p id=\"fs-id1169149189358\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4932","chapter","type-chapter","status-publish","hentry"],"part":3677,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4932","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":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4932\/revisions"}],"predecessor-version":[{"id":5083,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4932\/revisions\/5083"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3677"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4932\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=4932"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=4932"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=4932"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=4932"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}