{"id":1674,"date":"2016-11-02T17:53:30","date_gmt":"2016-11-02T17:53:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1674"},"modified":"2017-04-04T19:04:50","modified_gmt":"2017-04-04T19:04:50","slug":"transformations-of-quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/transformations-of-quadratic-functions\/","title":{"raw":"Transformations of Quadratic Functions","rendered":"Transformations of Quadratic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Graph vertical and horizontal shifts of quadratic functions<\/li>\r\n \t<li>Graph vertical compressions and\u00a0stretches of quadratic functions<\/li>\r\n \t<li>Write the equation of a transformed quadratic function using the vertex form<\/li>\r\n \t<li>Identify the vertex and axis of symmetry for a given quadratic function in vertex form<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n\r\n<div id=\"fs-id1165135320100\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\r\n<p id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\r\nThe standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. The figure below\u00a0is the graph of this basic function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201255\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" data-media-type=\"image\/jpg\" \/>\r\n\r\n&nbsp;\r\n<h2>Shift Up and Down by Changing the value of k<\/h2>\r\n<p id=\"fs-id1165137770279\">You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, k.<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=x^2 + k[\/latex]<\/p>\r\n\u00a0If [latex]k&gt;0[\/latex], the graph shifts upward, whereas if [latex]k&lt;0[\/latex], the graph shifts downward.\r\n\r\n<strong><span style=\"text-decoration: underline;\">Instructions:<\/span><\/strong>\r\n<ol>\r\n \t<li>Use the slider in the interactive below to shift the graph of [latex]f(x)=x^2[\/latex] down 4 units, then up 4 units.<\/li>\r\n \t<li>Use the textbox below the graph to write both transformed equations.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/fpatj6tbcn\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Make Note<\/h3>\r\nWrite the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted up 4 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow check yourself!\r\n[reveal-answer q=\"725488\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725488\"]The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units is\r\n\r\n[latex]f(x)=x^2+4[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units is\r\n\r\n[latex]f(x)=x^2-4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Shift left and right by changing the value of h.<\/h3>\r\n<p id=\"fs-id1165137770279\">You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, h, to the variable x, before squaring.<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=(x-h)^2 [\/latex]<\/p>\r\nIf [latex]h&gt;0[\/latex], the graph shifts toward the right and if [latex]h&lt;0[\/latex], the graph shifts to the left.\r\n\r\n<strong><span style=\"text-decoration: underline;\">Instructions:<\/span><\/strong>\r\n<ol>\r\n \t<li>Use the interactive graph below to shift the graph of [latex]f(x)=x^2[\/latex] 2 units to the right, then 2 units to the left.<\/li>\r\n \t<li>Use the textbox below the graph to write both transformed equations<\/li>\r\n<\/ol>\r\nhttps:\/\/www.desmos.com\/calculator\/5g3xfhkklq\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Make Note<\/h3>\r\nWrite the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted right 2 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted left 2 units in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow check yourself!\r\n[reveal-answer q=\"725588\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725588\"]The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is\r\n\r\n[latex]f(x)=(x-2)^2[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is\r\n\r\n[latex]f(x)=(x+2)^2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Stretch or compress by changing the value of a.<\/h3>\r\n<p id=\"fs-id1165137770279\">You can represent a stretch or compression (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, a.<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=ax^2 [\/latex]<\/p>\r\nThe magnitude of <em>a<\/em>\u00a0indicates the stretch of the graph. If [latex]|a|&gt;1[\/latex], the point associated with a particular <em>x<\/em>-value shifts farther from the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|&lt;1[\/latex], the point associated with a particular <em>x<\/em>-value shifts closer to the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression.\r\n\r\n<strong><span style=\"text-decoration: underline;\">Instructions:<\/span><\/strong>\r\n<ol>\r\n \t<li>Use the interactive graph below to make a graph of the function\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex],<\/li>\r\n \t<li>And another that has been vertically stretched by a factor of 3.<\/li>\r\n \t<li>Use the textbox below the graph to write your equations.<\/li>\r\n<\/ol>\r\nhttps:\/\/www.desmos.com\/calculator\/ha6gh59rq7\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Make Note<\/h3>\r\nWrite the equation for the graph of [latex]f(x)=x^2[\/latex] that has been has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] in the textbox below.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nThen, \u00a0write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.[practice-area rows=\"2\"][\/practice-area]\r\n\r\nNow check yourself!\r\n[reveal-answer q=\"725489\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"725489\"]The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex]\r\n\r\n[latex]f(x)=\\frac{1}{2}x^2[\/latex]\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.is\r\n\r\n[latex]f(x)=3x^2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135353112\">The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\r\n\r\n<div id=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}a{\\left(x-h\\right)}^{2}+k=a{x}^{2}+bx+c\\hfill \\\\ a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)=a{x}^{2}+bx+c\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137409211\">For the linear terms to be equal, the coefficients must be equal.<\/p>\r\n\r\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]-2ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex].<\/div>\r\n<p id=\"fs-id1165134118295\">This is the <strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal:<\/p>\r\n\r\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}a{h}^{2}+k=c\\hfill \\\\ \\text{ }k=c-a{h}^{2}\\hfill \\\\ \\text{ }=c-a-{\\left(\\frac{b}{2a}\\right)}^{2}\\hfill \\\\ \\text{ }=c-\\frac{{b}^{2}}{4a}\\hfill \\end{cases}[\/latex]<\/div>\r\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em data-effect=\"italics\">k<\/em> is the output value of the function when the input is <em>h<\/em>, so [latex]f\\left(h\\right)=k[\/latex].<\/p>\r\nNow you try it.\r\n\r\nUse the interactive graph below to define two quadratic functions whose axis of symmetry is x = -3, and whose vertex is (-3, 2). Use the sliders for a, h, k below to help you.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/pimelalx4i\r\n\r\nNow answer the following questions about the graphs you made.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Take Note<\/h3>\r\nHow many potential values are there for h in this scenario?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\nHow about k?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\nHow about a?\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n[reveal-answer q=\"349748\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"349748\"]\r\n\r\nThere is only one [latex](h,k)[\/latex] pair that will satisfy these conditions,\u00a0[latex](-3,2)[\/latex]. \u00a0The value of a does not affect the line of symmetry or the vertex of a quadratic graph, so a can be an infinite number of values.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Challenge Problem<\/h2>\r\nDefine a function whose axis of symmetry is x = -3, and whose vertex is (-3,2) and has an average rate of change of 3\/2 on the interval [-2,0].\r\n\r\nHere is the interactive graph again, if it helps.\r\n\r\n&nbsp;\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/pimelalx4i\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170344\/CNX_Precalc_Figure_03_02_0082.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" data-media-type=\"image\/jpg\" \/> (credit: modification of work by Dan Meyer)[\/caption]\r\n\r\n[reveal-answer q=\"283194\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"283194\"]\r\n\r\nThe path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right)[\/latex], so [latex]\\left(h\\right)x=-\\frac{7}{16}{\\left(x+4\\right)}^{2}+7[\/latex]. To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(-7.5\\right)\\approx 1.64[\/latex]; he doesn\u2019t make it.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Graph vertical and horizontal shifts of quadratic functions<\/li>\n<li>Graph vertical compressions and\u00a0stretches of quadratic functions<\/li>\n<li>Write the equation of a transformed quadratic function using the vertex form<\/li>\n<li>Identify the vertex and axis of symmetry for a given quadratic function in vertex form<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<div id=\"fs-id1165135320100\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\n<p id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\n<p>The standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. The figure below\u00a0is the graph of this basic function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201255\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>&nbsp;<\/p>\n<h2>Shift Up and Down by Changing the value of k<\/h2>\n<p id=\"fs-id1165137770279\">You can represent a vertical (up, down) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, k.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=x^2 + k[\/latex]<\/p>\n<p>\u00a0If [latex]k>0[\/latex], the graph shifts upward, whereas if [latex]k<0[\/latex], the graph shifts downward.\n\n<strong><span style=\"text-decoration: underline;\">Instructions:<\/span><\/strong><\/p>\n<ol>\n<li>Use the slider in the interactive below to shift the graph of [latex]f(x)=x^2[\/latex] down 4 units, then up 4 units.<\/li>\n<li>Use the textbox below the graph to write both transformed equations.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/fpatj6tbcn<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Make Note<\/h3>\n<p>Write the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted up 4 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now check yourself!<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725488\">Show Answer<\/span><\/p>\n<div id=\"q725488\" class=\"hidden-answer\" style=\"display: none\">The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted up 4 units is<\/p>\n<p>[latex]f(x)=x^2+4[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted down 4 units is<\/p>\n<p>[latex]f(x)=x^2-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Shift left and right by changing the value of h.<\/h3>\n<p id=\"fs-id1165137770279\">You can represent a horizontal (left, right) shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, h, to the variable x, before squaring.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x-h)^2[\/latex]<\/p>\n<p>If [latex]h>0[\/latex], the graph shifts toward the right and if [latex]h<0[\/latex], the graph shifts to the left.\n\n<strong><span style=\"text-decoration: underline;\">Instructions:<\/span><\/strong><\/p>\n<ol>\n<li>Use the interactive graph below to shift the graph of [latex]f(x)=x^2[\/latex] 2 units to the right, then 2 units to the left.<\/li>\n<li>Use the textbox below the graph to write both transformed equations<\/li>\n<\/ol>\n<p>https:\/\/www.desmos.com\/calculator\/5g3xfhkklq<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Make Note<\/h3>\n<p>Write the equation for the graph of [latex]f(x)=x^2[\/latex] that has been shifted right 2 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted left 2 units in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now check yourself!<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725588\">Show Answer<\/span><\/p>\n<div id=\"q725588\" class=\"hidden-answer\" style=\"display: none\">The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is<\/p>\n<p>[latex]f(x)=(x-2)^2[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is<\/p>\n<p>[latex]f(x)=(x+2)^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Stretch or compress by changing the value of a.<\/h3>\n<p id=\"fs-id1165137770279\">You can represent a stretch or compression (narrowing, widening)\u00a0of the graph of [latex]f(x)=x^2[\/latex] by\u00a0multiplying the squared variable by a constant, a.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ax^2[\/latex]<\/p>\n<p>The magnitude of <em>a<\/em>\u00a0indicates the stretch of the graph. If [latex]|a|>1[\/latex], the point associated with a particular <em>x<\/em>-value shifts farther from the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|<1[\/latex], the point associated with a particular <em>x<\/em>-value shifts closer to the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression.<\/p>\n<p><strong><span style=\"text-decoration: underline;\">Instructions:<\/span><\/strong><\/p>\n<ol>\n<li>Use the interactive graph below to make a graph of the function\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex],<\/li>\n<li>And another that has been vertically stretched by a factor of 3.<\/li>\n<li>Use the textbox below the graph to write your equations.<\/li>\n<\/ol>\n<p>https:\/\/www.desmos.com\/calculator\/ha6gh59rq7<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Make Note<\/h3>\n<p>Write the equation for the graph of [latex]f(x)=x^2[\/latex] that has been has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex] in the textbox below.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Then, \u00a0write the equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.<textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>Now check yourself!<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725489\">Show Answer<\/span><\/p>\n<div id=\"q725489\" class=\"hidden-answer\" style=\"display: none\">The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been compressed vertically by a factor of [latex]\\frac{1}{2}[\/latex]<\/p>\n<p>[latex]f(x)=\\frac{1}{2}x^2[\/latex]<\/p>\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been vertically stretched by a factor of 3.is<\/p>\n<p>[latex]f(x)=3x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135353112\">The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\n<div id=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}a{\\left(x-h\\right)}^{2}+k=a{x}^{2}+bx+c\\hfill \\\\ a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)=a{x}^{2}+bx+c\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137409211\">For the linear terms to be equal, the coefficients must be equal.<\/p>\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]-2ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex].<\/div>\n<p id=\"fs-id1165134118295\">This is the <strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal:<\/p>\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{cases}a{h}^{2}+k=c\\hfill \\\\ \\text{ }k=c-a{h}^{2}\\hfill \\\\ \\text{ }=c-a-{\\left(\\frac{b}{2a}\\right)}^{2}\\hfill \\\\ \\text{ }=c-\\frac{{b}^{2}}{4a}\\hfill \\end{cases}[\/latex]<\/div>\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em data-effect=\"italics\">k<\/em> is the output value of the function when the input is <em>h<\/em>, so [latex]f\\left(h\\right)=k[\/latex].<\/p>\n<p>Now you try it.<\/p>\n<p>Use the interactive graph below to define two quadratic functions whose axis of symmetry is x = -3, and whose vertex is (-3, 2). Use the sliders for a, h, k below to help you.<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/pimelalx4i<\/p>\n<p>Now answer the following questions about the graphs you made.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Take Note<\/h3>\n<p>How many potential values are there for h in this scenario?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>How about k?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<p>How about a?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q349748\">Show Answer<\/span><\/p>\n<div id=\"q349748\" class=\"hidden-answer\" style=\"display: none\">\n<p>There is only one [latex](h,k)[\/latex] pair that will satisfy these conditions,\u00a0[latex](-3,2)[\/latex]. \u00a0The value of a does not affect the line of symmetry or the vertex of a quadratic graph, so a can be an infinite number of values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Challenge Problem<\/h2>\n<p>Define a function whose axis of symmetry is x = -3, and whose vertex is (-3,2) and has an average rate of change of 3\/2 on the interval [-2,0].<\/p>\n<p>Here is the interactive graph again, if it helps.<\/p>\n<p>&nbsp;<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/pimelalx4i<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170344\/CNX_Precalc_Figure_03_02_0082.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\">(credit: modification of work by Dan Meyer)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283194\">Solution<\/span><\/p>\n<div id=\"q283194\" class=\"hidden-answer\" style=\"display: none\">\n<p>The path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right)[\/latex], so [latex]\\left(h\\right)x=-\\frac{7}{16}{\\left(x+4\\right)}^{2}+7[\/latex]. To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(-7.5\\right)\\approx 1.64[\/latex]; he doesn\u2019t make it.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1674\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Interactive: Transform Quadratic 1. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/fpatj6tbcn\">https:\/\/www.desmos.com\/calculator\/fpatj6tbcn<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 2. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/5g3xfhkklq\">https:\/\/www.desmos.com\/calculator\/5g3xfhkklq<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 3. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/ha6gh59rq7\">https:\/\/www.desmos.com\/calculator\/ha6gh59rq7<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 4. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/pimelalx4i\">https:\/\/www.desmos.com\/calculator\/pimelalx4i<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 1\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/fpatj6tbcn\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 2\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/5g3xfhkklq\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 3\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/ha6gh59rq7\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Interactive: Transform Quadratic 4\",\"author\":\"\",\"organization\":\"Lumen Learning (with Desmos)\",\"url\":\"https:\/\/www.desmos.com\/calculator\/pimelalx4i\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"}]","CANDELA_OUTCOMES_GUID":"6cf244ab-91f3-4f31-8bae-4831fa431b64","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1674","chapter","type-chapter","status-publish","hentry"],"part":764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1674","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1674\/revisions"}],"predecessor-version":[{"id":3890,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1674\/revisions\/3890"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1674\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=1674"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1674"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1674"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=1674"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}