{"id":1805,"date":"2022-04-25T16:40:00","date_gmt":"2022-04-25T16:40:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1805"},"modified":"2026-01-22T17:50:02","modified_gmt":"2026-01-22T17:50:02","slug":"4-2-transformations-of-the-quadratic-function-latexfxx2-latex","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/4-2-transformations-of-the-quadratic-function-latexfxx2-latex\/","title":{"raw":"4.2: Transformations of the Quadratic Function","rendered":"4.2: Transformations of the Quadratic Function"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\nFor the quadratic function [latex]f(x)=x^2[\/latex],\r\n<ul>\r\n \t<li>Perform vertical and horizontal shifts<\/li>\r\n \t<li>Perform vertical compressions and\u00a0stretches<\/li>\r\n \t<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\r\n \t<li>Explain the vertex form of a quadratic function<\/li>\r\n \t<li>Write the equation of a transformed quadratic function using the vertex form<\/li>\r\n \t<li>Determine the equation of a quadratic function given its vertex and a point on the graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Vertical Shifts<\/h2>\r\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=x^2[\/latex] up 5 units, all of the points on the graph increase their [latex]y[\/latex]-coordinates by 5, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=x^2[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=x^2+5[\/latex]. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 200%; height: 161px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 16.5612%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 20.1476%; height: 10px;\" scope=\"row\">[latex]x^2[\/latex]<\/th>\r\n<th style=\"width: 6.43459%; height: 10px;\" scope=\"row\">[latex]x^2+5[\/latex]<\/th>\r\n<td style=\"width: 56.8566%; height: 161px;\" rowspan=\"9\">[caption id=\"attachment_2707\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2707\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16192739\/desmos-graph-2022-06-16T132718.562-300x300.png\" alt=\"Shifting up by 5 units\" width=\"380\" height=\"380\" \/> Figure 1. Shifting the graph of [latex]f(x)=x^2[\/latex] up 5 units. Every point [latex](x, y)[\/latex] moves to ([latex](x, y+5)[\/latex].[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">9<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">14<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">4<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">1<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">0<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">0<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">1<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">1<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">2<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">4<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 16.5612%; height: 19px;\">3<\/td>\r\n<td style=\"width: 20.1476%; height: 19px;\">9<\/td>\r\n<td style=\"width: 6.43459%; height: 19px;\">14<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 43.1434%;\" colspan=\"3\">Table 1. [latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=x^2+5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=x^2[\/latex] down 3 units, all of the points on the graph decrease their [latex]y[\/latex]-coordinates by 3, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=x^2[\/latex] after it has been shifted down 3 units transforms to [latex]f(x)=x^2-3[\/latex]. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 197.812%; height: 332px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 60px;\">\r\n<th style=\"width: 16.7721%; height: 60px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 14.135%; height: 60px;\">[latex]x^2[\/latex]<\/th>\r\n<th style=\"width: 16.8776%; height: 60px;\">[latex]x^2-3[\/latex]<\/th>\r\n<td style=\"width: 52.2152%; height: 332px;\" rowspan=\"9\">[caption id=\"attachment_2708\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2708 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16193706\/desmos-graph-2022-06-16T133643.553-300x300.png\" alt=\"Shifting down b y 3 units\" width=\"380\" height=\"380\" \/> Figure 2. Shifting the graph of [latex]f(x)=x^2[\/latex] down 3 units. Every point [latex](x, y)[\/latex] moves to [latex](x, y-3)[\/latex].[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">-3<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">9<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">-2<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">4<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">-1<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">1<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">0<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">0<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">-3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">1<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">1<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">2<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">4<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 16.7721%; height: 34px;\">3<\/td>\r\n<td style=\"width: 14.135%; height: 34px;\">9<\/td>\r\n<td style=\"width: 16.8776%; height: 34px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 47.7847%; height: 34px;\" colspan=\"3\">Table 2.\u00a0[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=x^2-3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>Vertical shifts<\/h3>\r\nWe can represent a vertical shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]k[\/latex], to the function:\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 upwards and if [latex]k&lt;0[\/latex] the graph shifts downwards.\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">Change the value of [latex]k[\/latex] in the interactive desmos graph in figure 3 by moving the vertex up or down.<\/p>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/lymcpwcjki?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 3. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=x^2+k[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Every point [latex](x, y)[\/latex] moves to [latex](x, y+k)[\/latex].<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<h3>Example 1<\/h3>\r\nIf the function [latex]f(x)=x^2[\/latex] is shifted down 7 units, what is the equation of the transformed function?\r\n<h4>Solution<\/h4>\r\nWhen a function is shifted down by 7 units, all of the function values are decreased by 7.\r\n\r\n[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=x^2-7[\/latex]\r\n<div id=\"q941360\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nThe 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<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nIf the function [latex]f(x)=x^2[\/latex] is shifted up 4 units, what is the equation of the transformed function?\r\n\r\n[reveal-answer q=\"hjm387\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm387\"][latex]f(x)=x^2+4[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Horizontal Shifts<\/h2>\r\n<\/div>\r\nIf we shift the graph of the function [latex]f(x)=x^2[\/latex] right 4 units, all of the points on the graph increase their [latex]x[\/latex]-coordinates by 4, but their [latex]y[\/latex]-coordinates remain the same. The vertex (0, 0) in the original graph is moved to (4, 0) (figure 3). Any point [latex](x, y)[\/latex] on the original graph is moved to [latex](x+4, y)[\/latex].\r\n\r\nBut what happens to the original function [latex]f(x)=x^2[\/latex]? An automatic assumption may be that since [latex]x[\/latex] moves to [latex]x+4[\/latex] that the function will become [latex]f(x)=(x+4)^2[\/latex]. But that is NOT the case. Remember that the new vertex is (4, 0) and if we substitute [latex]x=4[\/latex] into the function\u00a0[latex]f(x)=(x+4)^2[\/latex] we get\u00a0[latex]f(4)=(4+4)^2=64\\ne0[\/latex]!! The way to get a function value of zero is for the transformed function to be [latex]f(x)=(x-4)^2[\/latex]. Then [latex]f(4)=(4-4)^2=0[\/latex]. So the function [latex]f(x)=x^2[\/latex] transforms to [latex]f(x)=(x-4)^2[\/latex] after being shifted 4 units to the right. The reason is that when we move the function 4 units to the right, the [latex]x[\/latex]-value increases by 4 and to keep the corresponding [latex]y[\/latex]-coordinate the same in the transformed function, the [latex]x[\/latex]-coordinate of the transformed function needs to subtract 4 to get back to the original [latex]x[\/latex] that is associated with the original [latex]y[\/latex]-value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 4.\r\n<table style=\"border-collapse: collapse; width: 197.812%; height: 294px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 10.3376%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.5526%; height: 19px;\">[latex]x-4[\/latex]<\/th>\r\n<th style=\"width: 19.7259%; height: 19px;\">[latex](x-4)^2[\/latex]<\/th>\r\n<td style=\"width: 57.3839%; height: 294px;\" rowspan=\"9\">&nbsp;\r\n\r\n[caption id=\"attachment_2711\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2711 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16200803\/desmos-graph-2022-06-16T140737.172-300x300.png\" alt=\"Shifting right when h = 4\" width=\"380\" height=\"380\" \/> Figure 4. Shift the graph right 4 units.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">1<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">2<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">3<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">4<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">0<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">5<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">1<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">6<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">2<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 10.3376%; height: 19px;\">7<\/td>\r\n<td style=\"width: 12.5526%; height: 19px;\">3<\/td>\r\n<td style=\"width: 19.7259%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 142px;\">\r\n<td style=\"width: 42.6161%; height: 142px;\" colspan=\"3\">Table 3. Shifting the graph right by 4 units transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=(x-4)^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn the other hand, if we shift the graph of the function [latex]f(x)=x^2[\/latex] left by 7 units, all of the points on the graph decrease their [latex]x[\/latex]-coordinates by 7, but their [latex]y[\/latex]-coordinates remain the same. So any point [latex](x, y)[\/latex] on the original graph moves to [latex](x-7, y[\/latex]. Consequently, to keep the same [latex]y[\/latex]-values we need to increase the [latex]x[\/latex]-value by 7 in the transformed function. The equation of the function after being shifted left 7 units is [latex]f(x)=(x+7)^2[\/latex]. Table 4 shows the changes to specific values of this function, and the graph is shown in figure 5.\r\n<table style=\"border-collapse: collapse; width: 197.812%; height: 340px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 11.0759%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 13.0801%; height: 19px;\">[latex]x+7[\/latex]<\/th>\r\n<th style=\"width: 22.4684%; height: 19px;\">[latex](x+7)^2[\/latex]<\/th>\r\n<td style=\"width: 53.3756%; height: 340px;\" rowspan=\"9\">&nbsp;\r\n\r\n[caption id=\"attachment_2713\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2713 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16201844\/desmos-graph-2022-06-16T141821.783-300x300.png\" alt=\"Shifting left\" width=\"380\" height=\"380\" \/> Figure 5. Shifting [latex]f(x)=x^2[\/latex] left by 7 units to get [latex]f(x)=(x+7)^2[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-10<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-9<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-8<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-7<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">0<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-6<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">1<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-5<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">2<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 11.0759%; height: 19px;\">-4<\/td>\r\n<td style=\"width: 13.0801%; height: 19px;\">3<\/td>\r\n<td style=\"width: 22.4684%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 188px;\">\r\n<td style=\"width: 46.6244%; height: 188px;\" colspan=\"3\">Table 4. Shifting the graph left by 7 units transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=(x+7)^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"entry-content\">\r\n\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3>horizontal shifts<\/h3>\r\n<p id=\"fs-id1165137770279\">We can represent a horizontal shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex], 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<\/div>\r\n<p style=\"text-align: left;\">Change the value of [latex]h[\/latex] in the interactive desmos graph in figure 6 by moving the vertex left or right.<\/p>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/xscqzchbvu?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 6. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=(x-h)^2[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>tip for success<\/h3>\r\nRemember that the negative sign inside the argument of the vertex form of a parabola (in the parentheses with the variable [latex]x[\/latex] ) is part of the formula [latex]f(x)=(x-h)^2 +k[\/latex].\r\n\r\nIf [latex]h&gt;0[\/latex], we have\u00a0[latex]f(x)=(x-h)^2+k=(x-\\text{positive number})^2 +k[\/latex]. You\u2019ll see the negative sign, but the graph will shift right.\r\n\r\nIf\u00a0 [latex]h&lt;0[\/latex], we have\u00a0[latex]f(x)=(x-h)^2+k=(x-\\text{negative number})^2 +k \\rightarrow f(x)=(x+\\text{positive number})^2+k[\/latex], since subtracting a negative number is equivalent to adding a positive number. You\u2019ll see the positive sign, but the graph will shift left.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nIf the function [latex]f(x)=x^2[\/latex] is shifted left by 5 units, what is the equation of the transformed function?\r\n<h4>Solution<\/h4>\r\nWhen a function is shifted left by 5 units, all of the [latex]x[\/latex]-values are decreased by 5. So we have to add 5 to the [latex]x[\/latex]-values in the transformed function to keep the correct function values.\r\n\r\n[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=(x+5)^2[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<div id=\"q978434\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nThe equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is\r\n<p style=\"text-align: center;\">[latex]f(x)=(x-2)^2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=(x+2)^2[\/latex]<\/p>\r\n<p style=\"text-align: left;\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nIf the function [latex]f(x)=x^2[\/latex] is shifted right by 8 units, what is the equation of the transformed function?\r\n\r\n[reveal-answer q=\"hjm476\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm476\"][latex]f(x)=(x-8)^2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Stretching Up and Compressing Down<\/h2>\r\nIf we vertically stretch the graph of the function [latex]f(x)=x^2[\/latex] by a factor of 2, all of the [latex]y[\/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[\/latex]-coordinates remain the same. Since the vertex of [latex]f(x)=x^2[\/latex] is (0, 0), it transforms to (0, 0). That is, [latex]2\\times0^2=2\\times0=0[\/latex]. The vertex, since it is an [latex]x[\/latex]-intercept, stays at the same location (0, 0). In other words, the vertex is the anchor point in the stretching process. The equation of the function after the graph is stretched by a factor of 2 is [latex]f(x)=2x^2[\/latex]. The reason for multiplying\u00a0 [latex]x^2[\/latex] by 2 is that each [latex]y[\/latex]-coordinate is doubled, and since[latex]y=x^2[\/latex], [latex]x^2[\/latex] is doubled. Table 5 shows this change and the graph is shown in figure 7.\u200b\r\n<table style=\"border-collapse: collapse; width: 197.812%; height: 170px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 15.7172%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 14.8734%; height: 19px;\">[latex]x^2[\/latex]<\/th>\r\n<th style=\"width: 17.8271%; height: 19px;\">[latex]2x^2[\/latex]<\/th>\r\n<td style=\"width: 51.5823%; height: 170px;\" rowspan=\"9\">&nbsp;\r\n\r\n[caption id=\"attachment_2714\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2714\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16202411\/desmos-graph-2022-06-16T142347.297-300x300.png\" alt=\"Stretching the graph\" width=\"380\" height=\"380\" \/> Figure 7. [latex]f(x)=x^2[\/latex] transformed to [latex]f(x)=2x^2[\/latex]. Each [latex]y[\/latex]-value is doubled.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">9<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">18<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">4<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">1<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">0<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">0<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">1<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">1<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">2<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">4<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 15.7172%; height: 19px;\">3<\/td>\r\n<td style=\"width: 14.8734%; height: 19px;\">9<\/td>\r\n<td style=\"width: 17.8271%; height: 19px;\">18<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 48.4177%; height: 18px;\" colspan=\"3\">Table 5.\u00a0Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=2x^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn the other hand, if our factor is [latex]\\dfrac{1}{2}[\/latex] and we vertically compress the graph of the function [latex]f(x)=x^2[\/latex] all of the [latex]y[\/latex]-coordinates of the points on the graph are halved, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are\u00a0<span style=\"font-size: 1em;\">multiplied by [latex]\\dfrac{1}{2}[\/latex], or\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">divided by 2. The [latex]x[\/latex]-intercept (0, 0) of [latex]f(x)=x^2[\/latex] is the only point that doesn't move. In other words, the [latex]x[\/latex]-intercept is the anchor point in the compressing process. The equation of the function after being compressed is [latex]f(x)=\\dfrac{1}{2}x^2[\/latex]. The reason for multiplying [latex]x^2[\/latex] by [latex]\\dfrac{1}{2}[\/latex] is that each [latex]y[\/latex]-coordinate becomes half of the original value when it is divided by 2. Table 6 shows this change and the graph is shown in figure 8.<\/span>\r\n<table style=\"border-collapse: collapse; width: 197.812%; height: 152px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 20.9915%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 27.9536%; height: 19px;\">[latex]x^2[\/latex]<\/th>\r\n<th style=\"width: 34.3882%; height: 19px;\">[latex]\\frac{1}{2}x^2[\/latex]<\/th>\r\n<td style=\"width: 16.6667%;\" rowspan=\"9\">&nbsp;\r\n\r\n[caption id=\"attachment_2715\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2715\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16204218\/desmos-graph-2022-06-16T144146.437-300x300.png\" alt=\"Compressing a graph\" width=\"380\" height=\"380\" \/> Figure 8. Compressing the graph when the factor is [latex]\\frac{1}{2}[\/latex]. Each [latex]y[\/latex]-value is divided by 2.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">9<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">4.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">4<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">1<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">0.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">0<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">0<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">1<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">1<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">0.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">2<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">4<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 20.9915%; height: 19px;\">3<\/td>\r\n<td style=\"width: 27.9536%; height: 19px;\">9<\/td>\r\n<td style=\"width: 34.3882%; height: 19px;\">4.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 83.3333%;\" colspan=\"3\">Table 6.\u00a0When the factor is [latex]\\dfrac{1}{2}[\/latex] the graph is compressed and transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=\\dfrac{1}{2}x^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nChange the value of [latex]a[\/latex] in the interactive desmos graph in figure 9 by moving the point [latex](1, a)[\/latex] vertically.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/w9mhtmlhf0?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 9.\u00a0Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=ax^2[\/latex]<\/p>\r\n\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>vertical stretching and compressing<\/h3>\r\n<p id=\"fs-id1165137770279\">A stretch or compression of the graph of [latex]f(x)=x^2[\/latex] can be represented by\u00a0multiplying the [latex]x^2[\/latex] by a constant, [latex]a&gt;0[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=ax^2 [\/latex]<\/p>\r\nThe magnitude of [latex]a[\/latex]\u00a0indicates the stretch\/compression of the graph. If [latex]a&gt;1[\/latex], the graph is stretched vertically by a factor of [latex]a.[\/latex] If [latex]0&lt;a&lt;1[\/latex], the graph is compressed vertically.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nIf the function [latex]f(x)=x^2[\/latex] is compressed to one-quarter of its original height, what is the equation of the transformed function? What happens to the points (0, 0) and (1, 1) on the graph of the original function?\r\n<h4>Solution<\/h4>\r\nWhen a function is compressed to one-quarter of its original height, [latex]a=\\dfrac{1}{4}[\/latex] in the transformed equation\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]f(x)=ax^2[\/latex]. So [latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=\\dfrac{1}{4}x^2[\/latex].<\/span>\r\n\r\nThe point (0, 0) stays the same and the point (1, 1) moves to [latex]\\left (1,\\dfrac{1}{4}\\right )[\/latex] since the [latex]y[\/latex]-values are divided by 4.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nIf the function [latex]f(x)=x^2[\/latex] is stretched by a factor of 9, what is the equation of the transformed function? What happens to the points (0, 0) and (1, 1) on the graph of the original function?\r\n\r\n[reveal-answer q=\"hjm713\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm713\"]\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=9x^2[\/latex].<\/span>\r\n\r\n(0, 0) stays the same.\r\n\r\n(1, 1) moves to (1, 9).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflection across the [latex]x[\/latex]-axis<\/h2>\r\nWhen the graph of the function [latex]f(x)=x^2[\/latex] is reflected across the\u00a0[latex]x[\/latex]-axis, the\u00a0[latex]y[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values or from negative to positive values, while the [latex]x[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=x^2[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-x^2[\/latex]. The graph changes from opening upwards to opening downwards. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 10.\r\n<table style=\"border-collapse: collapse; width: 197.812%; height: 294px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x^2[\/latex]<\/th>\r\n<th style=\"width: 16.6667%; height: 19px;\">[latex]-x^2[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 294px;\" rowspan=\"9\">[caption id=\"attachment_2703\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2703 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16175348\/desmos-graph-2022-06-16T115234.073-300x300.png\" alt=\"Reflection across x axis\" width=\"300\" height=\"300\" \/> Figure 10. Reflection across [latex]x[\/latex]=axis. [latex](x, y)[\/latex] transforms to [latex](x, -y)[\/latex].[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">9<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">-9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">4<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">-4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">1<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">0<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">0<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">1<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">1<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">2<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">4<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">-4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 33.3333%; height: 19px;\">3<\/td>\r\n<td style=\"width: 33.3333%; height: 19px;\">9<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">-9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 142px;\">\r\n<td style=\"width: 83.3333%; height: 142px;\" colspan=\"3\">Table 7.\u00a0Reflecting the graph of [latex]f(x)=x^2[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=-x^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Vertex Form of a Quadratic Function<\/h2>\r\nAll of the transformations we have applied to the function [latex]f(x)=x^2[\/latex] can be combined. The result is the function [latex]f(x)=a(x-h)^2+k[\/latex], where [latex]a[\/latex] represents vertical stretching ([latex]a&gt;1[\/latex]) and compression ([latex]0&lt;a&lt;1[\/latex]) as well as reflection across the [latex]x[\/latex]-axis ([latex]a[\/latex] is negative); [latex]h[\/latex] represents a horizontal shift right when [latex]h&gt;0[\/latex] and left when [latex]h&lt;0[\/latex]; and [latex]k[\/latex] represents a vertical shift up when [latex]k&gt;0[\/latex] and down when [latex]k&lt;0[\/latex]. In addition, the vertex will move from (0, 0) to [latex](h, k)[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>VERTEX form of a quadratic equation<\/h3>\r\n<p id=\"fs-id1165137676320\">The\u00a0<strong><em>vertex form of a quadratic function<\/em>\u00a0<\/strong>is<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\n<p style=\"text-align: left;\">where [latex]\\left(h, k\\right)[\/latex] is the vertex, and [latex]a[\/latex] represents stretching ([latex]a&gt;1[\/latex]), compression ([latex]0&lt;a&lt;1[\/latex]), or reflection across the [latex]x[\/latex]-axis ([latex]a[\/latex] is negative).<\/p>\r\n<p style=\"text-align: left;\">If we were to multiply and simplify the vertex form of a quadratic function, the quadratic function would turn into <strong><em>standard<\/em><\/strong>\u00a0<em><strong>form<\/strong>: <\/em>[latex]f(x)=ax^2+bx+c[\/latex], where [latex]a, b, c[\/latex] are real numbers. the value of [latex]a[\/latex] in standard form is the same value of [latex]a[\/latex] in vertex form.<\/p>\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">Move the red dots in figure 11 to change the values of [latex]a, h[\/latex], and [latex]k[\/latex], and watch the transformation unfold.<\/p>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/zu0q4pzuvl?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem;\">Figure 11.\u00a0Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=a(x-h)^2+k[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\" style=\"text-align: left;\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nThe function [latex]f(x)=x^2[\/latex] is transformed by shifting the graph to the right by 4 units then down by 3 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span>\r\n<h4>Solution<\/h4>\r\nMoving the graph right by 4 units means that [latex]h=4[\/latex].\r\n\r\nMoving the graph down by 3 units means tha [latex]k=-3[\/latex]\r\n\r\nVertex form is [latex]f(x)=a(x-h)^2+k[\/latex] so the transformed function is [latex]f(x)=(x-4)^2-3[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nThe function [latex]f(x)=x^2[\/latex] is transformed by reflecting the graph across the [latex]x[\/latex]-axis, stretching the graph by a factor of 4, shifting the graph to the left by 1 unit then up by 7 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span>\r\n<h4>Solution<\/h4>\r\nReflecting the graph across the [latex]x[\/latex]-axis and stretching the graph by a factor of 4 means that [latex]a=-4[\/latex].\r\n\r\nMoving the graph left by 1 unit means that [latex]h=-1[\/latex].\r\n\r\nMoving the graph up by 7 units means that [latex]k=7[\/latex]\r\n\r\nVertex form is [latex]f(x)=a(x-h)^2+k[\/latex] so the transformed function is [latex]f(x)=-4(x+1)^2+7[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nThe function [latex]f(x)=x^2[\/latex] is transformed by reflecting the graph across the [latex]x[\/latex]-axis and shifting the graph to the left by 3 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span>\r\n\r\n[reveal-answer q=\"hjm212\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm212\"][latex]f(x)=-(x+3)^2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nThe function [latex]f(x)=x^2[\/latex] is transformed by compressing the graph to one-half of its original height, shifting the graph to the right by 5 units then down 4 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span>\r\n\r\n[reveal-answer q=\"hjm992\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm992\"][latex]f(x)=\\dfrac{1}{2}(x-5)^2-4[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Determining the Equation of a Quadratic Function<\/h2>\r\nThe vertex form of a quadratic function is [latex]f(x)=a(x-h)^2+k[\/latex], where [latex]a, h, k[\/latex] are real numbers and [latex]a\\ne 0[\/latex]. So, if we know the values of\u00a0[latex]a, h, k[\/latex], we can write the corresponding function. For example, if we are told that\u00a0 [latex]h=2,\\;k=5,\\;a=3[\/latex]. Then the function is [latex]f(x)=3(x-2)^2+5[\/latex]. Figure 12 shows the graph.\r\n\r\n[caption id=\"attachment_2716\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2716 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-300x300.png\" alt=\"f(x)=3(x-2)^2+5 graphed\" width=\"300\" height=\"300\" \/> Figure 12. Graph of\u00a0[latex]f(x)=3(x-2)^2+5[\/latex][\/caption]What if we were told instead that the vertex is (2, 5) and the graph passes through the point (1, 8)? Can we find the equation of the function knowing this information?<\/div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<div class=\"entry-content\" style=\"text-align: left;\">First off, we know that by definition the vertex gives us the values of [latex]h=2[\/latex] and [latex]k=5[\/latex]. This means that the function is [latex]f(x)=a(x-2)^2+5[\/latex]. Now all we need is the value of [latex]a[\/latex].Since the point (1, 8) lies on the graph, it must be a solution of the equation that represents the function, [latex]y=a(x-2)^2+5[\/latex]. Substituting [latex]x=1,\\;y=8[\/latex] leaves [latex]a[\/latex] as the only unknown so we can solve for [latex]a[\/latex]:<\/div>\r\n<div class=\"entry-content\" style=\"text-align: center;\">[latex]\\begin{aligned}y&amp;=a(x-2)^2+5\\\\8&amp;=a(1-2)^2+5\\\\8&amp;=a+5\\\\3&amp;=a\\end{aligned}[\/latex]<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<div>Now we know [latex]h=2,\\;k=5,\\;a=3[\/latex] so [latex]f(x)=3(x-2)^2+5[\/latex].<\/div>\r\n<div><\/div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nDetermine the function whose graph has a vertex at (\u20134, 1) and that passes through the point (0, 2).\r\n<h4>Solution<\/h4>\r\nSince we know the vertex, we know [latex]h=-4,\\;k=1[\/latex].\r\n\r\nTherefore, [latex]f(x)=a(x+4)^2+1[\/latex].\r\n\r\nTo find [latex]a[\/latex], we substitute the point (0, 2) for [latex](x, y)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}y&amp;=a(x+4)^2+1\\\\2&amp;=a(0+4)^2+1\\\\2&amp;=16a+1\\\\1&amp;=16a\\\\\\dfrac{1}{16}&amp;=a\\end{aligned}[\/latex]<\/p>\r\nConsequently, [latex]f(x)=\\dfrac{1}{16}\\left (x+4\\right )^2+1[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nDetermine the function whose graph has a vertex at (6, \u20132) and that passes through the point (1, \u20135).\r\n\r\n[reveal-answer q=\"hjm383\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm383\"]\r\n\r\n[latex]f(x)=-\\dfrac{3}{25}(x-6)^2-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<p>For the quadratic function [latex]f(x)=x^2[\/latex],<\/p>\n<ul>\n<li>Perform vertical and horizontal shifts<\/li>\n<li>Perform vertical compressions and\u00a0stretches<\/li>\n<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\n<li>Explain the vertex form of a quadratic function<\/li>\n<li>Write the equation of a transformed quadratic function using the vertex form<\/li>\n<li>Determine the equation of a quadratic function given its vertex and a point on the graph.<\/li>\n<\/ul>\n<\/div>\n<h2>Vertical Shifts<\/h2>\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=x^2[\/latex] up 5 units, all of the points on the graph increase their [latex]y[\/latex]-coordinates by 5, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=x^2[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=x^2+5[\/latex]. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.<\/p>\n<table style=\"border-collapse: collapse; width: 200%; height: 161px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 16.5612%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 20.1476%; height: 10px;\" scope=\"row\">[latex]x^2[\/latex]<\/th>\n<th style=\"width: 6.43459%; height: 10px;\" scope=\"row\">[latex]x^2+5[\/latex]<\/th>\n<td style=\"width: 56.8566%; height: 161px;\" rowspan=\"9\">\n<div id=\"attachment_2707\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2707\" class=\"wp-image-2707\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16192739\/desmos-graph-2022-06-16T132718.562-300x300.png\" alt=\"Shifting up by 5 units\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2707\" class=\"wp-caption-text\">Figure 1. Shifting the graph of [latex]f(x)=x^2[\/latex] up 5 units. Every point [latex](x, y)[\/latex] moves to ([latex](x, y+5)[\/latex].<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">-3<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">9<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">14<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">-2<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">4<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">-1<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">1<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">0<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">0<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">5<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">1<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">1<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">2<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">4<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 16.5612%; height: 19px;\">3<\/td>\n<td style=\"width: 20.1476%; height: 19px;\">9<\/td>\n<td style=\"width: 6.43459%; height: 19px;\">14<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 43.1434%;\" colspan=\"3\">Table 1. [latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=x^2+5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=x^2[\/latex] down 3 units, all of the points on the graph decrease their [latex]y[\/latex]-coordinates by 3, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=x^2[\/latex] after it has been shifted down 3 units transforms to [latex]f(x)=x^2-3[\/latex]. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.<\/p>\n<table style=\"border-collapse: collapse; width: 197.812%; height: 332px;\">\n<tbody>\n<tr style=\"height: 60px;\">\n<th style=\"width: 16.7721%; height: 60px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 14.135%; height: 60px;\">[latex]x^2[\/latex]<\/th>\n<th style=\"width: 16.8776%; height: 60px;\">[latex]x^2-3[\/latex]<\/th>\n<td style=\"width: 52.2152%; height: 332px;\" rowspan=\"9\">\n<div id=\"attachment_2708\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2708\" class=\"wp-image-2708\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16193706\/desmos-graph-2022-06-16T133643.553-300x300.png\" alt=\"Shifting down b y 3 units\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2708\" class=\"wp-caption-text\">Figure 2. Shifting the graph of [latex]f(x)=x^2[\/latex] down 3 units. Every point [latex](x, y)[\/latex] moves to [latex](x, y-3)[\/latex].<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">-3<\/td>\n<td style=\"width: 14.135%; height: 34px;\">9<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">-2<\/td>\n<td style=\"width: 14.135%; height: 34px;\">4<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">-1<\/td>\n<td style=\"width: 14.135%; height: 34px;\">1<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">0<\/td>\n<td style=\"width: 14.135%; height: 34px;\">0<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">-3<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">1<\/td>\n<td style=\"width: 14.135%; height: 34px;\">1<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">2<\/td>\n<td style=\"width: 14.135%; height: 34px;\">4<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 16.7721%; height: 34px;\">3<\/td>\n<td style=\"width: 14.135%; height: 34px;\">9<\/td>\n<td style=\"width: 16.8776%; height: 34px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 47.7847%; height: 34px;\" colspan=\"3\">Table 2.\u00a0[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=x^2-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>Vertical shifts<\/h3>\n<p>We can represent a vertical shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]k[\/latex], to the function:<\/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 upwards and if [latex]k<0[\/latex] the graph shifts downwards.\n\n<\/div>\n<p style=\"text-align: left;\">Change the value of [latex]k[\/latex] in the interactive desmos graph in figure 3 by moving the vertex up or down.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/lymcpwcjki?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 3. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=x^2+k[\/latex]<\/p>\n<p style=\"text-align: center;\">Every point [latex](x, y)[\/latex] moves to [latex](x, y+k)[\/latex].<\/p>\n<div class=\"textbox examples\">\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<h3>Example 1<\/h3>\n<p>If the function [latex]f(x)=x^2[\/latex] is shifted down 7 units, what is the equation of the transformed function?<\/p>\n<h4>Solution<\/h4>\n<p>When a function is shifted down by 7 units, all of the function values are decreased by 7.<\/p>\n<p>[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=x^2-7[\/latex]<\/p>\n<div id=\"q941360\" class=\"hidden-answer\" style=\"display: none;\">\n<p>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<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>If the function [latex]f(x)=x^2[\/latex] is shifted up 4 units, what is the equation of the transformed function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm387\">Show Answer<\/span><\/p>\n<div id=\"qhjm387\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=x^2+4[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Horizontal Shifts<\/h2>\n<\/div>\n<p>If we shift the graph of the function [latex]f(x)=x^2[\/latex] right 4 units, all of the points on the graph increase their [latex]x[\/latex]-coordinates by 4, but their [latex]y[\/latex]-coordinates remain the same. The vertex (0, 0) in the original graph is moved to (4, 0) (figure 3). Any point [latex](x, y)[\/latex] on the original graph is moved to [latex](x+4, y)[\/latex].<\/p>\n<p>But what happens to the original function [latex]f(x)=x^2[\/latex]? An automatic assumption may be that since [latex]x[\/latex] moves to [latex]x+4[\/latex] that the function will become [latex]f(x)=(x+4)^2[\/latex]. But that is NOT the case. Remember that the new vertex is (4, 0) and if we substitute [latex]x=4[\/latex] into the function\u00a0[latex]f(x)=(x+4)^2[\/latex] we get\u00a0[latex]f(4)=(4+4)^2=64\\ne0[\/latex]!! The way to get a function value of zero is for the transformed function to be [latex]f(x)=(x-4)^2[\/latex]. Then [latex]f(4)=(4-4)^2=0[\/latex]. So the function [latex]f(x)=x^2[\/latex] transforms to [latex]f(x)=(x-4)^2[\/latex] after being shifted 4 units to the right. The reason is that when we move the function 4 units to the right, the [latex]x[\/latex]-value increases by 4 and to keep the corresponding [latex]y[\/latex]-coordinate the same in the transformed function, the [latex]x[\/latex]-coordinate of the transformed function needs to subtract 4 to get back to the original [latex]x[\/latex] that is associated with the original [latex]y[\/latex]-value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 4.<\/p>\n<table style=\"border-collapse: collapse; width: 197.812%; height: 294px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 10.3376%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.5526%; height: 19px;\">[latex]x-4[\/latex]<\/th>\n<th style=\"width: 19.7259%; height: 19px;\">[latex](x-4)^2[\/latex]<\/th>\n<td style=\"width: 57.3839%; height: 294px;\" rowspan=\"9\">&nbsp;<\/p>\n<div id=\"attachment_2711\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2711\" class=\"wp-image-2711\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16200803\/desmos-graph-2022-06-16T140737.172-300x300.png\" alt=\"Shifting right when h = 4\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2711\" class=\"wp-caption-text\">Figure 4. Shift the graph right 4 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">1<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">-3<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">2<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">-2<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">3<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">-1<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">4<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">0<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">5<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">1<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">6<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">2<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 10.3376%; height: 19px;\">7<\/td>\n<td style=\"width: 12.5526%; height: 19px;\">3<\/td>\n<td style=\"width: 19.7259%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 142px;\">\n<td style=\"width: 42.6161%; height: 142px;\" colspan=\"3\">Table 3. Shifting the graph right by 4 units transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=(x-4)^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>On the other hand, if we shift the graph of the function [latex]f(x)=x^2[\/latex] left by 7 units, all of the points on the graph decrease their [latex]x[\/latex]-coordinates by 7, but their [latex]y[\/latex]-coordinates remain the same. So any point [latex](x, y)[\/latex] on the original graph moves to [latex](x-7, y[\/latex]. Consequently, to keep the same [latex]y[\/latex]-values we need to increase the [latex]x[\/latex]-value by 7 in the transformed function. The equation of the function after being shifted left 7 units is [latex]f(x)=(x+7)^2[\/latex]. Table 4 shows the changes to specific values of this function, and the graph is shown in figure 5.<\/p>\n<table style=\"border-collapse: collapse; width: 197.812%; height: 340px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 11.0759%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 13.0801%; height: 19px;\">[latex]x+7[\/latex]<\/th>\n<th style=\"width: 22.4684%; height: 19px;\">[latex](x+7)^2[\/latex]<\/th>\n<td style=\"width: 53.3756%; height: 340px;\" rowspan=\"9\">&nbsp;<\/p>\n<div id=\"attachment_2713\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2713\" class=\"wp-image-2713\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16201844\/desmos-graph-2022-06-16T141821.783-300x300.png\" alt=\"Shifting left\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2713\" class=\"wp-caption-text\">Figure 5. Shifting [latex]f(x)=x^2[\/latex] left by 7 units to get [latex]f(x)=(x+7)^2[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-10<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">-3<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-9<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">-2<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-8<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">-1<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-7<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">0<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-6<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">1<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-5<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">2<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 11.0759%; height: 19px;\">-4<\/td>\n<td style=\"width: 13.0801%; height: 19px;\">3<\/td>\n<td style=\"width: 22.4684%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 188px;\">\n<td style=\"width: 46.6244%; height: 188px;\" colspan=\"3\">Table 4. Shifting the graph left by 7 units transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=(x+7)^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"entry-content\">\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3>horizontal shifts<\/h3>\n<p id=\"fs-id1165137770279\">We can represent a horizontal shift of the graph of [latex]f(x)=x^2[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex], 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<\/div>\n<p style=\"text-align: left;\">Change the value of [latex]h[\/latex] in the interactive desmos graph in figure 6 by moving the vertex left or right.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/xscqzchbvu?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 6. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=(x-h)^2[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>tip for success<\/h3>\n<p>Remember that the negative sign inside the argument of the vertex form of a parabola (in the parentheses with the variable [latex]x[\/latex] ) is part of the formula [latex]f(x)=(x-h)^2 +k[\/latex].<\/p>\n<p>If [latex]h>0[\/latex], we have\u00a0[latex]f(x)=(x-h)^2+k=(x-\\text{positive number})^2 +k[\/latex]. You\u2019ll see the negative sign, but the graph will shift right.<\/p>\n<p>If\u00a0 [latex]h<0[\/latex], we have\u00a0[latex]f(x)=(x-h)^2+k=(x-\\text{negative number})^2 +k \\rightarrow f(x)=(x+\\text{positive number})^2+k[\/latex], since subtracting a negative number is equivalent to adding a positive number. You\u2019ll see the positive sign, but the graph will shift left.\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>If the function [latex]f(x)=x^2[\/latex] is shifted left by 5 units, what is the equation of the transformed function?<\/p>\n<h4>Solution<\/h4>\n<p>When a function is shifted left by 5 units, all of the [latex]x[\/latex]-values are decreased by 5. So we have to add 5 to the [latex]x[\/latex]-values in the transformed function to keep the correct function values.<\/p>\n<p>[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=(x+5)^2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<div id=\"q978434\" class=\"hidden-answer\" style=\"display: none;\">\n<p>The equation for the graph of\u00a0[latex]f(x)=x^2[\/latex] that has been shifted right 2 units is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x-2)^2[\/latex]<\/p>\n<p style=\"text-align: left;\">The equation for the graph of\u00a0[latex]f(x)=^2[\/latex] that has been shifted left 2 units is<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=(x+2)^2[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>If the function [latex]f(x)=x^2[\/latex] is shifted right by 8 units, what is the equation of the transformed function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm476\">Show Answer<\/span><\/p>\n<div id=\"qhjm476\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=(x-8)^2[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Stretching Up and Compressing Down<\/h2>\n<p>If we vertically stretch the graph of the function [latex]f(x)=x^2[\/latex] by a factor of 2, all of the [latex]y[\/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[\/latex]-coordinates remain the same. Since the vertex of [latex]f(x)=x^2[\/latex] is (0, 0), it transforms to (0, 0). That is, [latex]2\\times0^2=2\\times0=0[\/latex]. The vertex, since it is an [latex]x[\/latex]-intercept, stays at the same location (0, 0). In other words, the vertex is the anchor point in the stretching process. The equation of the function after the graph is stretched by a factor of 2 is [latex]f(x)=2x^2[\/latex]. The reason for multiplying\u00a0 [latex]x^2[\/latex] by 2 is that each [latex]y[\/latex]-coordinate is doubled, and since[latex]y=x^2[\/latex], [latex]x^2[\/latex] is doubled. Table 5 shows this change and the graph is shown in figure 7.\u200b<\/p>\n<table style=\"border-collapse: collapse; width: 197.812%; height: 170px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 15.7172%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 14.8734%; height: 19px;\">[latex]x^2[\/latex]<\/th>\n<th style=\"width: 17.8271%; height: 19px;\">[latex]2x^2[\/latex]<\/th>\n<td style=\"width: 51.5823%; height: 170px;\" rowspan=\"9\">&nbsp;<\/p>\n<div id=\"attachment_2714\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2714\" class=\"wp-image-2714\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16202411\/desmos-graph-2022-06-16T142347.297-300x300.png\" alt=\"Stretching the graph\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2714\" class=\"wp-caption-text\">Figure 7. [latex]f(x)=x^2[\/latex] transformed to [latex]f(x)=2x^2[\/latex]. Each [latex]y[\/latex]-value is doubled.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">-3<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">9<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">18<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">-2<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">4<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">-1<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">1<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">0<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">0<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">1<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">1<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">2<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">4<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 15.7172%; height: 19px;\">3<\/td>\n<td style=\"width: 14.8734%; height: 19px;\">9<\/td>\n<td style=\"width: 17.8271%; height: 19px;\">18<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 48.4177%; height: 18px;\" colspan=\"3\">Table 5.\u00a0Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=2x^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>On the other hand, if our factor is [latex]\\dfrac{1}{2}[\/latex] and we vertically compress the graph of the function [latex]f(x)=x^2[\/latex] all of the [latex]y[\/latex]-coordinates of the points on the graph are halved, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are\u00a0<span style=\"font-size: 1em;\">multiplied by [latex]\\dfrac{1}{2}[\/latex], or\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">divided by 2. The [latex]x[\/latex]-intercept (0, 0) of [latex]f(x)=x^2[\/latex] is the only point that doesn&#8217;t move. In other words, the [latex]x[\/latex]-intercept is the anchor point in the compressing process. The equation of the function after being compressed is [latex]f(x)=\\dfrac{1}{2}x^2[\/latex]. The reason for multiplying [latex]x^2[\/latex] by [latex]\\dfrac{1}{2}[\/latex] is that each [latex]y[\/latex]-coordinate becomes half of the original value when it is divided by 2. Table 6 shows this change and the graph is shown in figure 8.<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 197.812%; height: 152px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 20.9915%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 27.9536%; height: 19px;\">[latex]x^2[\/latex]<\/th>\n<th style=\"width: 34.3882%; height: 19px;\">[latex]\\frac{1}{2}x^2[\/latex]<\/th>\n<td style=\"width: 16.6667%;\" rowspan=\"9\">&nbsp;<\/p>\n<div id=\"attachment_2715\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2715\" class=\"wp-image-2715\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16204218\/desmos-graph-2022-06-16T144146.437-300x300.png\" alt=\"Compressing a graph\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2715\" class=\"wp-caption-text\">Figure 8. Compressing the graph when the factor is [latex]\\frac{1}{2}[\/latex]. Each [latex]y[\/latex]-value is divided by 2.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">-3<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">9<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">4.5<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">-2<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">4<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">-1<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">1<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">0.5<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">0<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">0<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">1<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">1<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">0.5<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">2<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">4<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 20.9915%; height: 19px;\">3<\/td>\n<td style=\"width: 27.9536%; height: 19px;\">9<\/td>\n<td style=\"width: 34.3882%; height: 19px;\">4.5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 83.3333%;\" colspan=\"3\">Table 6.\u00a0When the factor is [latex]\\dfrac{1}{2}[\/latex] the graph is compressed and transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=\\dfrac{1}{2}x^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Change the value of [latex]a[\/latex] in the interactive desmos graph in figure 9 by moving the point [latex](1, a)[\/latex] vertically.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/w9mhtmlhf0?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 9.\u00a0Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=ax^2[\/latex]<\/p>\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>vertical stretching and compressing<\/h3>\n<p id=\"fs-id1165137770279\">A stretch or compression of the graph of [latex]f(x)=x^2[\/latex] can be represented by\u00a0multiplying the [latex]x^2[\/latex] by a constant, [latex]a>0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ax^2[\/latex]<\/p>\n<p>The magnitude of [latex]a[\/latex]\u00a0indicates the stretch\/compression of the graph. If [latex]a>1[\/latex], the graph is stretched vertically by a factor of [latex]a.[\/latex] If [latex]0<a<1[\/latex], the graph is compressed vertically.\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>If the function [latex]f(x)=x^2[\/latex] is compressed to one-quarter of its original height, what is the equation of the transformed function? What happens to the points (0, 0) and (1, 1) on the graph of the original function?<\/p>\n<h4>Solution<\/h4>\n<p>When a function is compressed to one-quarter of its original height, [latex]a=\\dfrac{1}{4}[\/latex] in the transformed equation\u00a0<span style=\"font-size: 1rem; text-align: initial;\">[latex]f(x)=ax^2[\/latex]. So [latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=\\dfrac{1}{4}x^2[\/latex].<\/span><\/p>\n<p>The point (0, 0) stays the same and the point (1, 1) moves to [latex]\\left (1,\\dfrac{1}{4}\\right )[\/latex] since the [latex]y[\/latex]-values are divided by 4.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>If the function [latex]f(x)=x^2[\/latex] is stretched by a factor of 9, what is the equation of the transformed function? What happens to the points (0, 0) and (1, 1) on the graph of the original function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm713\">Show Answer<\/span><\/p>\n<div id=\"qhjm713\" class=\"hidden-answer\" style=\"display: none\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">[latex]f(x)=x^2[\/latex] is transformed to [latex]f(x)=9x^2[\/latex].<\/span><\/p>\n<p>(0, 0) stays the same.<\/p>\n<p>(1, 1) moves to (1, 9).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflection across the [latex]x[\/latex]-axis<\/h2>\n<p>When the graph of the function [latex]f(x)=x^2[\/latex] is reflected across the\u00a0[latex]x[\/latex]-axis, the\u00a0[latex]y[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values or from negative to positive values, while the [latex]x[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=x^2[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-x^2[\/latex]. The graph changes from opening upwards to opening downwards. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 10.<\/p>\n<table style=\"border-collapse: collapse; width: 197.812%; height: 294px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 33.3333%; height: 19px;\">[latex]x^2[\/latex]<\/th>\n<th style=\"width: 16.6667%; height: 19px;\">[latex]-x^2[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 294px;\" rowspan=\"9\">\n<div id=\"attachment_2703\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2703\" class=\"wp-image-2703 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/16175348\/desmos-graph-2022-06-16T115234.073-300x300.png\" alt=\"Reflection across x axis\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-2703\" class=\"wp-caption-text\">Figure 10. Reflection across [latex]x[\/latex]=axis. [latex](x, y)[\/latex] transforms to [latex](x, -y)[\/latex].<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">-3<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">9<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">-9<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">-2<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">4<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">-4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">-1<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">1<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">0<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">0<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">1<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">1<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">2<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">4<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">-4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 33.3333%; height: 19px;\">3<\/td>\n<td style=\"width: 33.3333%; height: 19px;\">9<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">-9<\/td>\n<\/tr>\n<tr style=\"height: 142px;\">\n<td style=\"width: 83.3333%; height: 142px;\" colspan=\"3\">Table 7.\u00a0Reflecting the graph of [latex]f(x)=x^2[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=x^2[\/latex] into [latex]f(x)=-x^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Vertex Form of a Quadratic Function<\/h2>\n<p>All of the transformations we have applied to the function [latex]f(x)=x^2[\/latex] can be combined. The result is the function [latex]f(x)=a(x-h)^2+k[\/latex], where [latex]a[\/latex] represents vertical stretching ([latex]a>1[\/latex]) and compression ([latex]0<a<1[\/latex]) as well as reflection across the [latex]x[\/latex]-axis ([latex]a[\/latex] is negative); [latex]h[\/latex] represents a horizontal shift right when [latex]h>0[\/latex] and left when [latex]h<0[\/latex]; and [latex]k[\/latex] represents a vertical shift up when [latex]k>0[\/latex] and down when [latex]k<0[\/latex]. In addition, the vertex will move from (0, 0) to [latex](h, k)[\/latex].\n\n\n<div class=\"textbox shaded\">\n<h3>VERTEX form of a quadratic equation<\/h3>\n<p id=\"fs-id1165137676320\">The\u00a0<strong><em>vertex form of a quadratic function<\/em>\u00a0<\/strong>is<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p style=\"text-align: left;\">where [latex]\\left(h, k\\right)[\/latex] is the vertex, and [latex]a[\/latex] represents stretching ([latex]a>1[\/latex]), compression ([latex]0<a<1[\/latex]), or reflection across the [latex]x[\/latex]-axis ([latex]a[\/latex] is negative).<\/p>\n<p style=\"text-align: left;\">If we were to multiply and simplify the vertex form of a quadratic function, the quadratic function would turn into <strong><em>standard<\/em><\/strong>\u00a0<em><strong>form<\/strong>: <\/em>[latex]f(x)=ax^2+bx+c[\/latex], where [latex]a, b, c[\/latex] are real numbers. the value of [latex]a[\/latex] in standard form is the same value of [latex]a[\/latex] in vertex form.<\/p>\n<\/div>\n<p style=\"text-align: left;\">Move the red dots in figure 11 to change the values of [latex]a, h[\/latex], and [latex]k[\/latex], and watch the transformation unfold.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/zu0q4pzuvl?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem;\">Figure 11.\u00a0Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=a(x-h)^2+k[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\" style=\"text-align: left;\">\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>The function [latex]f(x)=x^2[\/latex] is transformed by shifting the graph to the right by 4 units then down by 3 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span><\/p>\n<h4>Solution<\/h4>\n<p>Moving the graph right by 4 units means that [latex]h=4[\/latex].<\/p>\n<p>Moving the graph down by 3 units means tha [latex]k=-3[\/latex]<\/p>\n<p>Vertex form is [latex]f(x)=a(x-h)^2+k[\/latex] so the transformed function is [latex]f(x)=(x-4)^2-3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>The function [latex]f(x)=x^2[\/latex] is transformed by reflecting the graph across the [latex]x[\/latex]-axis, stretching the graph by a factor of 4, shifting the graph to the left by 1 unit then up by 7 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span><\/p>\n<h4>Solution<\/h4>\n<p>Reflecting the graph across the [latex]x[\/latex]-axis and stretching the graph by a factor of 4 means that [latex]a=-4[\/latex].<\/p>\n<p>Moving the graph left by 1 unit means that [latex]h=-1[\/latex].<\/p>\n<p>Moving the graph up by 7 units means that [latex]k=7[\/latex]<\/p>\n<p>Vertex form is [latex]f(x)=a(x-h)^2+k[\/latex] so the transformed function is [latex]f(x)=-4(x+1)^2+7[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>The function [latex]f(x)=x^2[\/latex] is transformed by reflecting the graph across the [latex]x[\/latex]-axis and shifting the graph to the left by 3 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm212\">Show Answer<\/span><\/p>\n<div id=\"qhjm212\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=-(x+3)^2[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>The function [latex]f(x)=x^2[\/latex] is transformed by compressing the graph to one-half of its original height, shifting the graph to the right by 5 units then down 4 units. Write the vertex form of the function<span style=\"font-size: 1rem; text-align: initial;\">\u00a0that represents this transformation.<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm992\">Show Answer<\/span><\/p>\n<div id=\"qhjm992\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=\\dfrac{1}{2}(x-5)^2-4[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Determining the Equation of a Quadratic Function<\/h2>\n<p>The vertex form of a quadratic function is [latex]f(x)=a(x-h)^2+k[\/latex], where [latex]a, h, k[\/latex] are real numbers and [latex]a\\ne 0[\/latex]. So, if we know the values of\u00a0[latex]a, h, k[\/latex], we can write the corresponding function. For example, if we are told that\u00a0 [latex]h=2,\\;k=5,\\;a=3[\/latex]. Then the function is [latex]f(x)=3(x-2)^2+5[\/latex]. Figure 12 shows the graph.<\/p>\n<div id=\"attachment_2716\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2716\" class=\"wp-image-2716 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-300x300.png\" alt=\"f(x)=3(x-2)^2+5 graphed\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-2022-06-16T145808.330.png 2000w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-2716\" class=\"wp-caption-text\">Figure 12. Graph of\u00a0[latex]f(x)=3(x-2)^2+5[\/latex]<\/p>\n<\/div>\n<p>What if we were told instead that the vertex is (2, 5) and the graph passes through the point (1, 8)? Can we find the equation of the function knowing this information?<\/p><\/div>\n<div style=\"text-align: center;\"><\/div>\n<div class=\"entry-content\" style=\"text-align: left;\">First off, we know that by definition the vertex gives us the values of [latex]h=2[\/latex] and [latex]k=5[\/latex]. This means that the function is [latex]f(x)=a(x-2)^2+5[\/latex]. Now all we need is the value of [latex]a[\/latex].Since the point (1, 8) lies on the graph, it must be a solution of the equation that represents the function, [latex]y=a(x-2)^2+5[\/latex]. Substituting [latex]x=1,\\;y=8[\/latex] leaves [latex]a[\/latex] as the only unknown so we can solve for [latex]a[\/latex]:<\/div>\n<div class=\"entry-content\" style=\"text-align: center;\">[latex]\\begin{aligned}y&=a(x-2)^2+5\\\\8&=a(1-2)^2+5\\\\8&=a+5\\\\3&=a\\end{aligned}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<div>Now we know [latex]h=2,\\;k=5,\\;a=3[\/latex] so [latex]f(x)=3(x-2)^2+5[\/latex].<\/div>\n<div><\/div>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Determine the function whose graph has a vertex at (\u20134, 1) and that passes through the point (0, 2).<\/p>\n<h4>Solution<\/h4>\n<p>Since we know the vertex, we know [latex]h=-4,\\;k=1[\/latex].<\/p>\n<p>Therefore, [latex]f(x)=a(x+4)^2+1[\/latex].<\/p>\n<p>To find [latex]a[\/latex], we substitute the point (0, 2) for [latex](x, y)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}y&=a(x+4)^2+1\\\\2&=a(0+4)^2+1\\\\2&=16a+1\\\\1&=16a\\\\\\dfrac{1}{16}&=a\\end{aligned}[\/latex]<\/p>\n<p>Consequently, [latex]f(x)=\\dfrac{1}{16}\\left (x+4\\right )^2+1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Determine the function whose graph has a vertex at (6, \u20132) and that passes through the point (1, \u20135).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm383\">Show Answer<\/span><\/p>\n<div id=\"qhjm383\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)=-\\dfrac{3}{25}(x-6)^2-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-1805\">\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>Transformations of the Quadratic Function . <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 3. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=x^2+b[\/latex]. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/k32ib1qwhr\">https:\/\/www.desmos.com\/calculator\/k32ib1qwhr<\/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>Figure 6. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=(x-h)^2[\/latex]. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/at1khozktd\">https:\/\/www.desmos.com\/calculator\/at1khozktd<\/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>Figure 9. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=ax^2[\/latex]. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/jyngdxpkzl\">https:\/\/www.desmos.com\/calculator\/jyngdxpkzl<\/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>Figure 11. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=a(x-h)^2+k[\/latex]. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/xhd1sutekk\">https:\/\/www.desmos.com\/calculator\/xhd1sutekk<\/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>Determining the Equation of a Quadratic Function. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Adaptation and Revision. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples 1 - 6 ; Try It: hjm387, hjm484, hjm713, hjm212, hjm992, hjm383. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <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":422608,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Transformations of the Quadratic Function \",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figure 3. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=x^2+b[\/latex]\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\/k32ib1qwhr\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figure 6. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=(x-h)^2[\/latex]\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\/at1khozktd\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figure 9. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=ax^2[\/latex]\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\/jyngdxpkzl\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Figure 11. Interactive transformation from [latex]f(x)=x^2[\/latex] to [latex]f(x)=a(x-h)^2+k[\/latex]\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\/xhd1sutekk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Determining the Equation of a Quadratic Function\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"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\"},{\"type\":\"original\",\"description\":\"Adaptation and Revision\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples 1 - 6 ; Try It: hjm387, hjm484, hjm713, hjm212, hjm992, hjm383\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1805","chapter","type-chapter","status-publish","hentry"],"part":1691,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1805","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":62,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1805\/revisions"}],"predecessor-version":[{"id":4850,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1805\/revisions\/4850"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/1691"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1805\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=1805"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1805"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=1805"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=1805"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}