{"id":2150,"date":"2022-05-17T18:14:14","date_gmt":"2022-05-17T18:14:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2150"},"modified":"2026-01-22T19:31:04","modified_gmt":"2026-01-22T19:31:04","slug":"5-2-transformations-of-the-exponential-function-fxrx","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/5-2-transformations-of-the-exponential-function-fxrx\/","title":{"raw":"5.2.1: Transformations of the Exponential Function\u2013\u2013Vertical and Horizontal Shifts","rendered":"5.2.1: Transformations of the Exponential Function\u2013\u2013Vertical and Horizontal Shifts"},"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-1805\" class=\"standard post-1805 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\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 exponential function [latex]f(x)=r^x[\/latex],\r\n<ul>\r\n \t<li>Perform vertical and horizontal shifts<\/li>\r\n \t<li>Determine the equation of a transformed function<\/li>\r\n \t<li>Determine the transformations of the exponential function\u00a0[latex]f(x)=r^{(x-h)}+k[\/latex]<\/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 exponential function [latex]f(x)=2^x[\/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)=2^x[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=2^x+5[\/latex]. Table 1 shows the changes to specific values of this function, which are graphed in figure 1.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 23.6287%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 25.5274%; height: 10px;\" scope=\"row\">[latex]2^x[\/latex]<\/th>\r\n<th style=\"width: 25%; height: 10px;\" scope=\"row\">[latex]2^x+5[\/latex]<\/th>\r\n<td style=\"width: 25.8439%; height: 152px;\" rowspan=\"9\">[caption id=\"attachment_2734\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2734\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/16234833\/desmos-graph-2022-06-16T174751.615-300x300.png\" alt=\"y=2^x+5 compared to y=2^x\" width=\"380\" height=\"380\" \/> Figure 1. Shifting the graph of [latex]f(x)=2^x[\/latex] up 5 units.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 25.5274%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 19px;\">[latex]\\dfrac{41}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 25.5274%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 19px;\">[latex]\\dfrac{21}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 25.5274%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 25%; height: 19px;\">[latex]\\dfrac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 10px;\">0<\/td>\r\n<td style=\"width: 25.5274%; height: 10px;\">1<\/td>\r\n<td style=\"width: 25%; height: 10px;\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 19px;\">1<\/td>\r\n<td style=\"width: 25.5274%; height: 19px;\">2<\/td>\r\n<td style=\"width: 25%; height: 19px;\">7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 19px;\">2<\/td>\r\n<td style=\"width: 25.5274%; height: 19px;\">4<\/td>\r\n<td style=\"width: 25%; height: 19px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 23.6287%; height: 19px;\">3<\/td>\r\n<td style=\"width: 25.5274%; height: 19px;\">8<\/td>\r\n<td style=\"width: 25%; height: 19px;\">13<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 74.1561%;\" colspan=\"3\">Table 1. [latex]f(x)=2^x[\/latex] is transformed to [latex]f(x)=2^x+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)=2^x[\/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)=2^x[\/latex] after it has been shifted down 3 units transforms to [latex]f(x)=2^x-3[\/latex]. Table 2 shows the changes to specific values of this function, which are graphed in figure 2.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%; height: 332px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 60px;\">\r\n<th style=\"width: 24.789%; height: 60px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 29.7468%; height: 60px;\">[latex]2^x[\/latex]<\/th>\r\n<th style=\"width: 28.7975%; height: 60px;\">[latex]2^x-3[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 332px;\" rowspan=\"9\">[caption id=\"attachment_2735\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2735\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/16235621\/desmos-graph-2022-06-16T175552.964-300x300.png\" alt=\"y=2^x-3 compared to y=2^x\" width=\"380\" height=\"380\" \/> Figure 2. Shifting the graph of [latex]f(x)=2^x[\/latex] down 3 units.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">-3<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">[latex]-\\dfrac{23}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">-2<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">[latex]-\\dfrac{11}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">-1<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">[latex]-\\dfrac{5}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">0<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">1<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">1<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">2<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">2<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">4<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 24.789%; height: 34px;\">3<\/td>\r\n<td style=\"width: 29.7468%; height: 34px;\">8<\/td>\r\n<td style=\"width: 28.7975%; height: 34px;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 83.33330000000001%; height: 34px;\" colspan=\"3\">Table 2.\u00a0[latex]f(x)=2^x[\/latex] is transformed to [latex]f(x)=2^x-3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese vertical shifts can be applied to any exponential function. Change the values of [latex]r[\/latex] and [latex]k[\/latex] in figure 3 by moving the green and red dots and note the changes that happen.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/vvyju3otio?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. Manipulation of [latex]f(x)=r^x+k[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Vertical shifts<\/h3>\r\nA vertical shift of the graph of [latex]f(x)=r^x[\/latex] with [latex]r&gt;0,\\;r\\neq1[\/latex], adds a constant, [latex]k[\/latex], resulting in the transformed function:\r\n<p style=\"text-align: center;\">[latex]f(x)=r^x + k[\/latex]<\/p>\r\n\u00a0If [latex]k&gt;0[\/latex], the graph shifts upwards by [latex]k[\/latex] units and if [latex]k&lt;0[\/latex] the graph shifts downwards by [latex]k[\/latex] units<span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 1<\/h3>\r\nManipulate the values of [latex]r[\/latex] and [latex]k[\/latex] in figure 3 to answer the following questions:\r\n<ol>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[\/latex] when [latex]k=5[\/latex]?<\/li>\r\n \t<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=2^x[\/latex] when [latex]k=5[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[\/latex] when [latex]k=-2[\/latex]?<\/li>\r\n \t<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=3^x[\/latex] when [latex]k=-2[\/latex]?<\/li>\r\n \t<li>What happens to every point on the graph of [latex]y=r^x[\/latex] for [latex]r&gt;0,\\;r\\neq1[\/latex] when [latex]k=4[\/latex]?<\/li>\r\n \t<li>What happens to the horizontal asymptote of the graph of [latex]y=r^x[\/latex] for [latex]r&gt;0,\\;r\\neq1[\/latex] when [latex]k=-3[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm619\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm619\"]\r\n<ol>\r\n \t<li>(0, 1) moves up 5 units to (0, 6).<\/li>\r\n \t<li>The asymptote [latex]y=0[\/latex] moves up 5 units to the asymptote [latex]y=5[\/latex].<\/li>\r\n \t<li>(0, 1) moves down 2 units to (0, \u20131).<\/li>\r\n \t<li>The asymptote [latex]y=0[\/latex] moves down 2 units to the asymptote [latex]y=-2[\/latex].<\/li>\r\n \t<li>The [latex]y[\/latex]-value of every point increases by 4 while the [latex]x[\/latex]-value stays the same.<\/li>\r\n \t<li>The asymptote [latex]y=0[\/latex] moves down 3 units to the asymptote [latex]y=-3[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nDetermine the exponential function that comes from transforming the parent function [latex]f(x)=2^x[\/latex]:\r\n<ol>\r\n \t<li>3 units up<\/li>\r\n \t<li>7 units down<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm451\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm451\"]\r\n<ol>\r\n \t<li>[latex]f(x)=2^x+3[\/latex]<\/li>\r\n \t<li>[latex]f(x)=2^x-7[\/latex]<\/li>\r\n<\/ol>\r\n[\/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)=2^x[\/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 [latex]y[\/latex]-intercept (0, 1) in the original graph moves to (4, 1) (figure 4). 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)=2^x[\/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)=2^{x+4}[\/latex]. But that is NOT the case. Remember that the [latex]y[\/latex]-intercept is moved to (4, 1) and if we substitute [latex]x=4[\/latex] into the function\u00a0[latex]f(x)=2^{x+4}[\/latex] we get\u00a0[latex]f(4)=2^(4+4)=256 \\neq 1[\/latex]!! The way to get a function value of 1 is for the transformed function to be [latex]f(x)=2^{x-4}[\/latex]. Then [latex]f(4)=2^{4-4}=2^0=1[\/latex]. So the function [latex]f(x)=2^x[\/latex] transforms to [latex]f(x)=2^{x-4}[\/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: 100%; height: 170px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 21.519%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 30.0632%; height: 19px;\">[latex]x-4[\/latex]<\/th>\r\n<th style=\"width: 31.7511%; height: 19px;\">[latex]2^{x-4}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 170px;\" rowspan=\"9\">&nbsp;\r\n\r\n[caption id=\"attachment_2738\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2738\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/17004534\/desmos-graph-2022-06-16T184455.268-300x300.png\" alt=\"shifting right\" 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: 21.519%; height: 19px;\">1<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.519%; height: 19px;\">2<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.519%; height: 19px;\">3<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.519%; height: 19px;\">4<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">0<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.519%; height: 19px;\">5<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">1<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.519%; height: 19px;\">6<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">2<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.519%; height: 19px;\">7<\/td>\r\n<td style=\"width: 30.0632%; height: 19px;\">3<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 83.3333%; height: 18px;\" colspan=\"3\">Table 3. Shifting the graph right by 4 units transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2^{x-4}[\/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)=2^x[\/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)=2^{x+7}[\/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: 100%; height: 170px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 21.9409%; height: 19px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 29.4303%; height: 19px;\">[latex]x+7[\/latex]<\/th>\r\n<th style=\"width: 31.9621%; height: 19px;\">[latex]2^{x+7}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 170px;\" rowspan=\"9\">&nbsp;\r\n\r\n[caption id=\"attachment_2740\" align=\"aligncenter\" width=\"380\"]<img class=\"wp-image-2740\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/17005255\/desmos-graph-2022-06-16T185222.154-300x300.png\" alt=\"shifting left\" width=\"380\" height=\"380\" \/> Figure 5. Shifting the graph left 7 units.[\/caption]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-10<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">-3<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-9<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">-2<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-8<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">-1<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-7<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">0<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-6<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">1<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-5<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">2<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 21.9409%; height: 19px;\">-4<\/td>\r\n<td style=\"width: 29.4303%; height: 19px;\">3<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 83.3333%; height: 18px;\" colspan=\"3\">Table 4. Shifting the graph left by 7 units transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2^{x+7}[\/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\">A horizontal shift of the graph of [latex]f(x)=r^x[\/latex] subtracts a constant, [latex]h[\/latex], from the variable [latex]x[\/latex] resulting in the transformed function<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=r^{x-h} [\/latex]<\/p>\r\nIf [latex]h&gt;0[\/latex] the graph shifts to the right and if [latex]h&lt;0[\/latex] the graph shifts to the left.\r\n\r\n<\/div>\r\n<\/div>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/bb0k4fx1zx?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.\u00a0Manipulation of [latex]f(x)=r^{x-h}[\/latex]<\/p>\r\n\r\n<div class=\"entry-content\">\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 3<\/h3>\r\nManipulate the values of [latex]r[\/latex] and [latex]h[\/latex] in figure 6 to answer the following questions:\r\n<ol>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[\/latex] when [latex]h=5[\/latex]?<\/li>\r\n \t<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=2^x[\/latex] when [latex]h=5[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[\/latex] when [latex]h=-2[\/latex]?<\/li>\r\n \t<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=3^x[\/latex] when [latex]h=-2[\/latex]?<\/li>\r\n \t<li>What happens to every point on the graph of [latex]y=r^x[\/latex] for [latex]r&gt;0,\\;r\\neq1[\/latex] when [latex]h=4[\/latex]?<\/li>\r\n \t<li>What happens to every point on the graph of [latex]y=r^x[\/latex] for [latex]r&gt;0,\\;r\\neq1[\/latex] when [latex]h=-5[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm618\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm618\"]\r\n<ol>\r\n \t<li>(0, 1) moves right 5 units to (5, 1).<\/li>\r\n \t<li>The asymptote [latex]y=0[\/latex] does not change.<\/li>\r\n \t<li>(0, 1) moves left 2 units to (\u20132, 1).<\/li>\r\n \t<li>The asymptote [latex]y=0[\/latex] does not change.<\/li>\r\n \t<li>The [latex]x[\/latex]-value of every point increases by 4 while the [latex]y[\/latex]-value stays the same.<\/li>\r\n \t<li>The [latex]x[\/latex]-value of every point decreases by 5 while the [latex]y[\/latex]-value stays the same.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can now combine vertical and horizontal transformations.\r\n<div class=\"textbox shaded\">\r\n<h3>Horizontal and vertical shifts<\/h3>\r\nThe parent function [latex]f(x)=r^x[\/latex], [latex]r&gt;0, r\\neq1[\/latex], transforms to [latex]f(x)=r^{x-h}+k[\/latex] when it is moved [latex]h[\/latex] units horizontally and [latex]k[\/latex] units vertically.\r\n\r\n<\/div>\r\nIn figure 7, we can manipulate the values of [latex]r[\/latex], [latex]h[\/latex], and [latex]k[\/latex] to determine what happens to the graph of [latex]f(x)=r^x[\/latex].\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive\" src=\"https:\/\/www.desmos.com\/calculator\/qdn8sl6ecy?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 7.\u00a0Animation of [latex]f(x)=r^{x-h}+k[\/latex]<\/p>\r\n\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nUse the animation in figure 7 to determine the following:\r\n<ol>\r\n \t<li>What happens to the point (0, 1) on the function [latex]f(x)=2^x[\/latex] when [latex]h=4[\/latex] and [latex]k=5[\/latex]?<\/li>\r\n \t<li>What happens to the point (1, 2) on the function [latex]f(x)=2^x[\/latex] when [latex]h=4[\/latex] and [latex]k=5[\/latex]?<\/li>\r\n \t<li>What happens to any point [latex](x, y)[\/latex] on the function [latex]f(x)=2^x[\/latex] when [latex]h=4[\/latex] and [latex]k=5[\/latex]?<\/li>\r\n \t<li>What happens to the point (0, 1) on the function [latex]f(x)=3^x[\/latex] when [latex]h=-3[\/latex] and [latex]k=-2[\/latex]?<\/li>\r\n \t<li>What happens to the point (1, 2) on the function [latex]f(x)=2^x[\/latex] when [latex]h=-3[\/latex] and [latex]k=-2[\/latex]?<\/li>\r\n \t<li>What happens to any point [latex](x, y)[\/latex] on the function [latex]f(x)=2^x[\/latex] when [latex]h=-3[\/latex] and [latex]k=-2[\/latex]?<\/li>\r\n \t<li>What happens to any point [latex](x, y)[\/latex] on the function [latex]f(x)=2^x[\/latex] when it is shifted [latex]h[\/latex] units horizontally and [latex]k[\/latex] units vertically?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm052\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm052\"]\r\n<ol>\r\n \t<li>(0, 1) moves to (4, 6)<\/li>\r\n \t<li>(1, 2) moves to (5, 7)<\/li>\r\n \t<li>[latex](x, y)[\/latex] moves to\u00a0[latex](x+4, y+5)[\/latex]<\/li>\r\n \t<li>(0, 1) moves to (\u20133, \u20131)<\/li>\r\n \t<li>(1, 2) moves to (\u20132, 0)<\/li>\r\n \t<li>[latex](x, y)[\/latex] moves to\u00a0[latex](x-3, y-2)[\/latex]<\/li>\r\n \t<li>[latex](x, y)[\/latex] moves to\u00a0[latex](x+h, y+k)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nExplain the transformations that need to happen to the function [latex]f(x)=3^x[\/latex] to get the function [latex]f(x)=3^{x+5}-7[\/latex].\r\n<h4>Solution<\/h4>\r\nFirst we identify [latex]h[\/latex] and [latex]k[\/latex] in the function [latex]f(x)=3^{x-h}+k[\/latex]: [latex]h=-5[\/latex] since [latex]x+5=x-(-5)[\/latex]. [latex]k=-7[\/latex].\r\n\r\nThe function\u00a0[latex]f(x)=3^x[\/latex] is shifted horizontal left by 5 units and vertically down by 7 units.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nExplain the transformations that need to happen to the function [latex]f(x)=4^x[\/latex] to get the function [latex]f(x)=4^{x-7}-2[\/latex].\r\n\r\n[reveal-answer q=\"hjm997\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm997\"]\r\n\r\n[latex]f(x)=4^x[\/latex] is shifted right 7 units and down 2 units.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nDetermine the transformed function after [latex]g(x)=2^x[\/latex] is shifted right 4 units and up 3 units.\r\n<h4>Solution<\/h4>\r\nShifting right 4 units means [latex]h=4[\/latex]. Shifting up 3 units means [latex]k=3[\/latex].\r\n\r\nSo the transformed function is [latex]g(x)=2^{x-4}+3[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nDetermine the transformed function after [latex]g(x)=7^x[\/latex] is shifted left 6 units and down 2 units.\r\n\r\n[reveal-answer q=\"hjm616\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm616\"]\r\n\r\n[latex]g(x)=7^{x+6}-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>\r\n<\/div>\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\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-1805\" class=\"standard post-1805 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\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 exponential function [latex]f(x)=r^x[\/latex],<\/p>\n<ul>\n<li>Perform vertical and horizontal shifts<\/li>\n<li>Determine the equation of a transformed function<\/li>\n<li>Determine the transformations of the exponential function\u00a0[latex]f(x)=r^{(x-h)}+k[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Vertical Shifts<\/h2>\n<p id=\"fs-id1165137770279\">If we shift the graph of the exponential function [latex]f(x)=2^x[\/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)=2^x[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=2^x+5[\/latex]. Table 1 shows the changes to specific values of this function, which are graphed in figure 1.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 23.6287%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 25.5274%; height: 10px;\" scope=\"row\">[latex]2^x[\/latex]<\/th>\n<th style=\"width: 25%; height: 10px;\" scope=\"row\">[latex]2^x+5[\/latex]<\/th>\n<td style=\"width: 25.8439%; height: 152px;\" rowspan=\"9\">\n<div id=\"attachment_2734\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2734\" class=\"wp-image-2734\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/16234833\/desmos-graph-2022-06-16T174751.615-300x300.png\" alt=\"y=2^x+5 compared to y=2^x\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2734\" class=\"wp-caption-text\">Figure 1. Shifting the graph of [latex]f(x)=2^x[\/latex] up 5 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 19px;\">-3<\/td>\n<td style=\"width: 25.5274%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 25%; height: 19px;\">[latex]\\dfrac{41}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 19px;\">-2<\/td>\n<td style=\"width: 25.5274%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 25%; height: 19px;\">[latex]\\dfrac{21}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 19px;\">-1<\/td>\n<td style=\"width: 25.5274%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 25%; height: 19px;\">[latex]\\dfrac{11}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 10px;\">0<\/td>\n<td style=\"width: 25.5274%; height: 10px;\">1<\/td>\n<td style=\"width: 25%; height: 10px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 19px;\">1<\/td>\n<td style=\"width: 25.5274%; height: 19px;\">2<\/td>\n<td style=\"width: 25%; height: 19px;\">7<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 19px;\">2<\/td>\n<td style=\"width: 25.5274%; height: 19px;\">4<\/td>\n<td style=\"width: 25%; height: 19px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 23.6287%; height: 19px;\">3<\/td>\n<td style=\"width: 25.5274%; height: 19px;\">8<\/td>\n<td style=\"width: 25%; height: 19px;\">13<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 74.1561%;\" colspan=\"3\">Table 1. [latex]f(x)=2^x[\/latex] is transformed to [latex]f(x)=2^x+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)=2^x[\/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)=2^x[\/latex] after it has been shifted down 3 units transforms to [latex]f(x)=2^x-3[\/latex]. Table 2 shows the changes to specific values of this function, which are graphed in figure 2.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 332px;\">\n<tbody>\n<tr style=\"height: 60px;\">\n<th style=\"width: 24.789%; height: 60px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 29.7468%; height: 60px;\">[latex]2^x[\/latex]<\/th>\n<th style=\"width: 28.7975%; height: 60px;\">[latex]2^x-3[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 332px;\" rowspan=\"9\">\n<div id=\"attachment_2735\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2735\" class=\"wp-image-2735\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/16235621\/desmos-graph-2022-06-16T175552.964-300x300.png\" alt=\"y=2^x-3 compared to y=2^x\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2735\" class=\"wp-caption-text\">Figure 2. Shifting the graph of [latex]f(x)=2^x[\/latex] down 3 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">-3<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">[latex]-\\dfrac{23}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">-2<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">[latex]-\\dfrac{11}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">-1<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">[latex]-\\dfrac{5}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">0<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">1<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">1<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">2<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">2<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">4<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 24.789%; height: 34px;\">3<\/td>\n<td style=\"width: 29.7468%; height: 34px;\">8<\/td>\n<td style=\"width: 28.7975%; height: 34px;\">5<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 83.33330000000001%; height: 34px;\" colspan=\"3\">Table 2.\u00a0[latex]f(x)=2^x[\/latex] is transformed to [latex]f(x)=2^x-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These vertical shifts can be applied to any exponential function. Change the values of [latex]r[\/latex] and [latex]k[\/latex] in figure 3 by moving the green and red dots and note the changes that happen.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/vvyju3otio?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. Manipulation of [latex]f(x)=r^x+k[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Vertical shifts<\/h3>\n<p>A vertical shift of the graph of [latex]f(x)=r^x[\/latex] with [latex]r>0,\\;r\\neq1[\/latex], adds a constant, [latex]k[\/latex], resulting in the transformed function:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=r^x + k[\/latex]<\/p>\n<p>\u00a0If [latex]k>0[\/latex], the graph shifts upwards by [latex]k[\/latex] units and if [latex]k<0[\/latex] the graph shifts downwards by [latex]k[\/latex] units<span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 1<\/h3>\n<p>Manipulate the values of [latex]r[\/latex] and [latex]k[\/latex] in figure 3 to answer the following questions:<\/p>\n<ol>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[\/latex] when [latex]k=5[\/latex]?<\/li>\n<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=2^x[\/latex] when [latex]k=5[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[\/latex] when [latex]k=-2[\/latex]?<\/li>\n<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=3^x[\/latex] when [latex]k=-2[\/latex]?<\/li>\n<li>What happens to every point on the graph of [latex]y=r^x[\/latex] for [latex]r>0,\\;r\\neq1[\/latex] when [latex]k=4[\/latex]?<\/li>\n<li>What happens to the horizontal asymptote of the graph of [latex]y=r^x[\/latex] for [latex]r>0,\\;r\\neq1[\/latex] when [latex]k=-3[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm619\">Show Answer<\/span><\/p>\n<div id=\"qhjm619\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>(0, 1) moves up 5 units to (0, 6).<\/li>\n<li>The asymptote [latex]y=0[\/latex] moves up 5 units to the asymptote [latex]y=5[\/latex].<\/li>\n<li>(0, 1) moves down 2 units to (0, \u20131).<\/li>\n<li>The asymptote [latex]y=0[\/latex] moves down 2 units to the asymptote [latex]y=-2[\/latex].<\/li>\n<li>The [latex]y[\/latex]-value of every point increases by 4 while the [latex]x[\/latex]-value stays the same.<\/li>\n<li>The asymptote [latex]y=0[\/latex] moves down 3 units to the asymptote [latex]y=-3[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Determine the exponential function that comes from transforming the parent function [latex]f(x)=2^x[\/latex]:<\/p>\n<ol>\n<li>3 units up<\/li>\n<li>7 units down<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm451\">Show Answer<\/span><\/p>\n<div id=\"qhjm451\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=2^x+3[\/latex]<\/li>\n<li>[latex]f(x)=2^x-7[\/latex]<\/li>\n<\/ol>\n<\/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)=2^x[\/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 [latex]y[\/latex]-intercept (0, 1) in the original graph moves to (4, 1) (figure 4). 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)=2^x[\/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)=2^{x+4}[\/latex]. But that is NOT the case. Remember that the [latex]y[\/latex]-intercept is moved to (4, 1) and if we substitute [latex]x=4[\/latex] into the function\u00a0[latex]f(x)=2^{x+4}[\/latex] we get\u00a0[latex]f(4)=2^(4+4)=256 \\neq 1[\/latex]!! The way to get a function value of 1 is for the transformed function to be [latex]f(x)=2^{x-4}[\/latex]. Then [latex]f(4)=2^{4-4}=2^0=1[\/latex]. So the function [latex]f(x)=2^x[\/latex] transforms to [latex]f(x)=2^{x-4}[\/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: 100%; height: 170px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 21.519%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 30.0632%; height: 19px;\">[latex]x-4[\/latex]<\/th>\n<th style=\"width: 31.7511%; height: 19px;\">[latex]2^{x-4}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 170px;\" rowspan=\"9\">&nbsp;<\/p>\n<div id=\"attachment_2738\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2738\" class=\"wp-image-2738\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/17004534\/desmos-graph-2022-06-16T184455.268-300x300.png\" alt=\"shifting right\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2738\" 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: 21.519%; height: 19px;\">1<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">-3<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.519%; height: 19px;\">2<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">-2<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.519%; height: 19px;\">3<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">-1<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.519%; height: 19px;\">4<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">0<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.519%; height: 19px;\">5<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">1<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.519%; height: 19px;\">6<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">2<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.519%; height: 19px;\">7<\/td>\n<td style=\"width: 30.0632%; height: 19px;\">3<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 83.3333%; height: 18px;\" colspan=\"3\">Table 3. Shifting the graph right by 4 units transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2^{x-4}[\/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)=2^x[\/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)=2^{x+7}[\/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: 100%; height: 170px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 21.9409%; height: 19px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 29.4303%; height: 19px;\">[latex]x+7[\/latex]<\/th>\n<th style=\"width: 31.9621%; height: 19px;\">[latex]2^{x+7}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 170px;\" rowspan=\"9\">&nbsp;<\/p>\n<div id=\"attachment_2740\" style=\"width: 390px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2740\" class=\"wp-image-2740\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/17005255\/desmos-graph-2022-06-16T185222.154-300x300.png\" alt=\"shifting left\" width=\"380\" height=\"380\" \/><\/p>\n<p id=\"caption-attachment-2740\" class=\"wp-caption-text\">Figure 5. Shifting the graph left 7 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-10<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">-3<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-9<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">-2<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-8<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">-1<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-7<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">0<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-6<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">1<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-5<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">2<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 21.9409%; height: 19px;\">-4<\/td>\n<td style=\"width: 29.4303%; height: 19px;\">3<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">8<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 83.3333%; height: 18px;\" colspan=\"3\">Table 4. Shifting the graph left by 7 units transforms [latex]f(x)=2^x[\/latex] into [latex]f(x)=2^{x+7}[\/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\">A horizontal shift of the graph of [latex]f(x)=r^x[\/latex] subtracts a constant, [latex]h[\/latex], from the variable [latex]x[\/latex] resulting in the transformed function<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=r^{x-h}[\/latex]<\/p>\n<p>If [latex]h>0[\/latex] the graph shifts to the right and if [latex]h<0[\/latex] the graph shifts to the left.\n\n<\/div>\n<\/div>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/bb0k4fx1zx?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.\u00a0Manipulation of [latex]f(x)=r^{x-h}[\/latex]<\/p>\n<div class=\"entry-content\">\n<div class=\"textbox tryit\">\n<h3>TRY IT 3<\/h3>\n<p>Manipulate the values of [latex]r[\/latex] and [latex]h[\/latex] in figure 6 to answer the following questions:<\/p>\n<ol>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[\/latex] when [latex]h=5[\/latex]?<\/li>\n<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=2^x[\/latex] when [latex]h=5[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[\/latex] when [latex]h=-2[\/latex]?<\/li>\n<li>What happens to the horizontal asymptote [latex]y=0[\/latex] of the function [latex]f(x)=3^x[\/latex] when [latex]h=-2[\/latex]?<\/li>\n<li>What happens to every point on the graph of [latex]y=r^x[\/latex] for [latex]r>0,\\;r\\neq1[\/latex] when [latex]h=4[\/latex]?<\/li>\n<li>What happens to every point on the graph of [latex]y=r^x[\/latex] for [latex]r>0,\\;r\\neq1[\/latex] when [latex]h=-5[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm618\">Show Answer<\/span><\/p>\n<div id=\"qhjm618\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>(0, 1) moves right 5 units to (5, 1).<\/li>\n<li>The asymptote [latex]y=0[\/latex] does not change.<\/li>\n<li>(0, 1) moves left 2 units to (\u20132, 1).<\/li>\n<li>The asymptote [latex]y=0[\/latex] does not change.<\/li>\n<li>The [latex]x[\/latex]-value of every point increases by 4 while the [latex]y[\/latex]-value stays the same.<\/li>\n<li>The [latex]x[\/latex]-value of every point decreases by 5 while the [latex]y[\/latex]-value stays the same.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>We can now combine vertical and horizontal transformations.<\/p>\n<div class=\"textbox shaded\">\n<h3>Horizontal and vertical shifts<\/h3>\n<p>The parent function [latex]f(x)=r^x[\/latex], [latex]r>0, r\\neq1[\/latex], transforms to [latex]f(x)=r^{x-h}+k[\/latex] when it is moved [latex]h[\/latex] units horizontally and [latex]k[\/latex] units vertically.<\/p>\n<\/div>\n<p>In figure 7, we can manipulate the values of [latex]r[\/latex], [latex]h[\/latex], and [latex]k[\/latex] to determine what happens to the graph of [latex]f(x)=r^x[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive\" src=\"https:\/\/www.desmos.com\/calculator\/qdn8sl6ecy?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 7.\u00a0Animation of [latex]f(x)=r^{x-h}+k[\/latex]<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Use the animation in figure 7 to determine the following:<\/p>\n<ol>\n<li>What happens to the point (0, 1) on the function [latex]f(x)=2^x[\/latex] when [latex]h=4[\/latex] and [latex]k=5[\/latex]?<\/li>\n<li>What happens to the point (1, 2) on the function [latex]f(x)=2^x[\/latex] when [latex]h=4[\/latex] and [latex]k=5[\/latex]?<\/li>\n<li>What happens to any point [latex](x, y)[\/latex] on the function [latex]f(x)=2^x[\/latex] when [latex]h=4[\/latex] and [latex]k=5[\/latex]?<\/li>\n<li>What happens to the point (0, 1) on the function [latex]f(x)=3^x[\/latex] when [latex]h=-3[\/latex] and [latex]k=-2[\/latex]?<\/li>\n<li>What happens to the point (1, 2) on the function [latex]f(x)=2^x[\/latex] when [latex]h=-3[\/latex] and [latex]k=-2[\/latex]?<\/li>\n<li>What happens to any point [latex](x, y)[\/latex] on the function [latex]f(x)=2^x[\/latex] when [latex]h=-3[\/latex] and [latex]k=-2[\/latex]?<\/li>\n<li>What happens to any point [latex](x, y)[\/latex] on the function [latex]f(x)=2^x[\/latex] when it is shifted [latex]h[\/latex] units horizontally and [latex]k[\/latex] units vertically?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm052\">Show Answer<\/span><\/p>\n<div id=\"qhjm052\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>(0, 1) moves to (4, 6)<\/li>\n<li>(1, 2) moves to (5, 7)<\/li>\n<li>[latex](x, y)[\/latex] moves to\u00a0[latex](x+4, y+5)[\/latex]<\/li>\n<li>(0, 1) moves to (\u20133, \u20131)<\/li>\n<li>(1, 2) moves to (\u20132, 0)<\/li>\n<li>[latex](x, y)[\/latex] moves to\u00a0[latex](x-3, y-2)[\/latex]<\/li>\n<li>[latex](x, y)[\/latex] moves to\u00a0[latex](x+h, y+k)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Explain the transformations that need to happen to the function [latex]f(x)=3^x[\/latex] to get the function [latex]f(x)=3^{x+5}-7[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>First we identify [latex]h[\/latex] and [latex]k[\/latex] in the function [latex]f(x)=3^{x-h}+k[\/latex]: [latex]h=-5[\/latex] since [latex]x+5=x-(-5)[\/latex]. [latex]k=-7[\/latex].<\/p>\n<p>The function\u00a0[latex]f(x)=3^x[\/latex] is shifted horizontal left by 5 units and vertically down by 7 units.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Explain the transformations that need to happen to the function [latex]f(x)=4^x[\/latex] to get the function [latex]f(x)=4^{x-7}-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm997\">Show Answer<\/span><\/p>\n<div id=\"qhjm997\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)=4^x[\/latex] is shifted right 7 units and down 2 units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Determine the transformed function after [latex]g(x)=2^x[\/latex] is shifted right 4 units and up 3 units.<\/p>\n<h4>Solution<\/h4>\n<p>Shifting right 4 units means [latex]h=4[\/latex]. Shifting up 3 units means [latex]k=3[\/latex].<\/p>\n<p>So the transformed function is [latex]g(x)=2^{x-4}+3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Determine the transformed function after [latex]g(x)=7^x[\/latex] is shifted left 6 units and down 2 units.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm616\">Show Answer<\/span><\/p>\n<div id=\"qhjm616\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)=7^{x+6}-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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-2150\">\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 Exponential Function f(x)=r^x. <strong>Authored by<\/strong>: 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><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: 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>\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 Exponential Function f(x)=r^x\",\"author\":\"Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"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-2150","chapter","type-chapter","status-publish","hentry"],"part":2116,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2150","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":38,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2150\/revisions"}],"predecessor-version":[{"id":4854,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2150\/revisions\/4854"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2116"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2150\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2150"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2150"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2150"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}