{"id":620,"date":"2022-01-19T17:47:00","date_gmt":"2022-01-19T17:47:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=620"},"modified":"2026-01-17T00:13:36","modified_gmt":"2026-01-17T00:13:36","slug":"1-3-the-transformation-and-inverse-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-3-the-transformation-and-inverse-of-a-function\/","title":{"raw":"1.3: Transformations and the Inverse of a Function","rendered":"1.3: Transformations and the Inverse of a 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=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Recognize graphical transformations:\r\n<ul>\r\n \t<li>Shifting a graph vertically and horizontally<\/li>\r\n \t<li>Reflecting a graph across the [latex]x[\/latex]-axis or [latex]y[\/latex]-axis<\/li>\r\n \t<li>Stretching a graph vertically<\/li>\r\n \t<li>Compressing a graph vertically<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Determine an inverse function by switching [latex]x[\/latex] and [latex]y[\/latex] in coordinate pairs<\/li>\r\n \t<li>Graph an inverse function using symmetry<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Transformations<\/h2>\r\nThe transformation of the graph of a function means shifting, flipping, stretching, or compressing a graph. The transformation keeps the same basic shape of the original graph; it has just been manipulated.\r\n<h2>Shifting a Graph Vertically<\/h2>\r\n<p style=\"text-align: left;\">In Figure 1, the graph with turning point at (1, 1) is shifted up 4 units so that the new turning point is (1, 5).\u00a0 In Figure 2, the graph with turning point at (0, 0) is shifted down 4 units so that the new turning point is (0, \u20134).\u00a0 Notice, in both cases, the [latex]x[\/latex]-values do not change, only the [latex]y[\/latex]-values shift up (Figure 1) or down (Figure 2). The graph stays exactly the same shape but has just been moved vertically. Each point on the graph keeps the same [latex]x[\/latex]-value but move either up of down the same amount. In figure 1, shifting the graph up by 4 units means all points [latex](x, y)[\/latex] move to [latex](x, y+4)[\/latex]. In figure 2, moving the graph down 4 units means all points\u00a0[latex](x, y)[\/latex] move to\u00a0[latex](x, y-4)[\/latex].<\/p>\r\n\u200b\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Adding 4<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Subtracting 4<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1731\" align=\"aligncenter\" width=\"362\"]<img class=\"wp-image-1731\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22193624\/desmos-graph-53-300x300.png\" alt=\"Every point on the original graph moves up 4 units.\" width=\"362\" height=\"362\" \/> Figure 1. Shifting up.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n<div class=\"mceTemp\"><\/div>\r\n\r\n[caption id=\"attachment_3609\" align=\"alignnone\" width=\"363\"]<img class=\"wp-image-3609\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/27151309\/1-3-ShiftDown-300x300.png\" alt=\"Every point on the original graph moves down 4 units.\" width=\"363\" height=\"363\" \/> FIgure 2. Shifting down.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nExplain which transformation the original (blue) graph has undergone to become the green graph. Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.\r\n\r\n<img class=\"aligncenter wp-image-816 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up-146x300.png\" alt=\"Every point on the original blue graph has moved up 10 units.\" width=\"146\" height=\"300\" \/>\r\n<h4>Solution<\/h4>\r\nThe original graph has been shifted up by 10 units. The turning point (0, 0) moves to (0, 10).\r\n\r\nAn original point [latex](x, y)[\/latex] moves to\u00a0[latex](x, y+10)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nExplain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.\r\n\r\n<img class=\"aligncenter wp-image-817 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-129x300.png\" alt=\"Every point on the original blue graph has moved down 7 units.\" width=\"129\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm874\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm874\"]\r\n\r\nThe original graph has been shifted down by 7 units. The turning point at (0, 0) moved to (0, \u20137).\r\n\r\nAn original point [latex](x, y)[\/latex] will move to\u00a0[latex](x, y-7)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Shifting a Graph Horizontally<\/h2>\r\nShifting a graph vertically means to move the graph up or down without changing the shape of the graph. Similarly, shifting a graph horizontally means to move the graph left or right without changing the shape of the graph. In Figure 3, the graph with turning point (1, 1) is shifted to the right 5 units. The new turning point is (6, 1).\u00a0 Similarly, in Figure 4, the graph with turning point (1, 1) is shifted to the left 8 units. The new turning point is (-7, 1). Notice that when shifting horizontally, the graph stays the same shape and the [latex]y[\/latex]-values do not change; only the [latex]x[\/latex]-values move left (Figure 3) or right (Figure 4).\u00a0 A general point [latex](x, y)[\/latex] on the graph in figure 3 moves to\u00a0[latex](x+5, y)[\/latex]. In figure 6,\u00a0[latex](x, y)[\/latex] moves to\u00a0[latex](x-8, y)[\/latex].\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\"><em>Subtracting<\/em> 5<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\"><em>Adding<\/em> 8<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_694\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-694 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-300x300.png\" alt=\"Each point on the original red graph has moved right five units.\" width=\"300\" height=\"300\" \/> Figure 3. Shifting to the right.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_695\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-695 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-300x300.png\" alt=\"Each point on the original red graph has moved left eight units.\" width=\"300\" height=\"300\" \/> Figure 4. Shifting to the left.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nExplain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.\r\n\r\n<img class=\"aligncenter wp-image-814 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right-250x300.png\" alt=\"The vertex of an original parabola has moved from (0,0) to (7,0).\" width=\"250\" height=\"300\" \/>\r\n<h4>Solution<\/h4>\r\nThe original graph has been shifted right by 7 units. The turning point (0, 0) moved to (7, 0).\r\n\r\nAn original point [latex](x, y)[\/latex] will move to\u00a0[latex](x+7, y)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nExplain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.\r\n\r\n<img class=\"aligncenter wp-image-815 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-278x300.png\" alt=\"The parabola representing f of x equals x squared has been transformed by shifting every point ten units to the left.\" width=\"278\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm814\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm814\"]\r\n\r\nThe original graph has been shifted left by 10 units. The turning point (0, 0) has moved to (\u201310 0).\r\n\r\nAn original point [latex](x, y)[\/latex] will move to\u00a0[latex](x-10, y)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Shifting a Graph Vertically and Horizontally<\/h2>\r\nWe can shift a graph either horizontally <em>or<\/em> vertically, but we can also shift a graph horizontally <em>and<\/em> vertically. Consider the graphs in Figures 5 and 6. The graph in Figure 5 has been shifted up by 4 units and left by 3 units. This is shown by the turning point moving from (0, 0) to (\u20133, 4). Moving up 4 units adds 4 to the [latex]y[\/latex]-coordinates, while moving left\u00a0<span style=\"font-size: 1em;\">3 units<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">adds \u20133 to the [latex]x[\/latex]-coordinates. This means that a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](x-3, y+4)[\/latex]. Notice we would get the same result if we completed the transformations in the reverse order.<\/span>\r\n\r\nThe graph in Figure 6 has been shifted right by 2 units and up by 3 units. The turning point moves from (0, 0) to (2, 3). Notice that all points on the graph move exactly the same way; right 2, up 3. For example, the point (\u20132, 4) moves to (0, 7). Moving 2 units right adds 2 to the [latex]x[\/latex]-coordinates, while moving up 3 units adds 3 to the [latex]y[\/latex]-coordinates. (\u20132, 4) moves to (\u20132 + 2, 4 + 3) = (0, 7).\u00a0This means that a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](x+2, y+3)[\/latex].\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Shifting vertically and horizontally<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Shifting horizontally and vertically<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_941\" align=\"aligncenter\" width=\"254\"]<img class=\"wp-image-941 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/02230702\/left-and-up-254x300.png\" alt=\"graph showing vertical and horizontal transformations\" width=\"254\" height=\"300\" \/> Figure 5. Vertical and horizontal shifts.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_942\" align=\"aligncenter\" width=\"277\"]<img class=\"wp-image-942 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/02231555\/right-and-up-277x300.png\" alt=\"Graph showing horizontal and vertical transformations\" width=\"277\" height=\"300\" \/> Figure 6. Vertical and horizontal shifts.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nExplain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-944 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-197x300.png\" alt=\"An original graph of a parabola left one unit and down seven units.\" width=\"197\" height=\"300\" \/><\/p>\r\nThe original graph has been shifted left by 1 units and down by 7 units. The turning point at (0, 0) has moved to (\u20131 \u20137).\r\n\r\nAn original point [latex](x, y)[\/latex] will move to\u00a0[latex](x-1, y-7)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nExplain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-945 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-239x300.png\" alt=\"An original parabola shifted right 4 units and down 5 units.\" width=\"239\" height=\"300\" \/><\/p>\r\n[reveal-answer q=\"hjm504\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm504\"]\r\n\r\nThe original graph has been shifted right by 4 units and down by 5 units. The turning point at (0, 0) has moved to (4, \u20135).\r\n\r\nAn original point [latex](x, y)[\/latex] will move to\u00a0[latex](x+4, y-5)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflecting a Graph Across the [latex]x[\/latex]-axis<\/h2>\r\nWhen a graph is reflected across the [latex]x[\/latex]-axis, the [latex]y[\/latex]-coordinates change sign while the [latex]x[\/latex]-coordinates stay the same. So a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](x, -y)[\/latex]. Notice in figure 7 only (0, 0) stays the same, and in Figure 8 only (\u20132, 0) stays the same.\u00a0 This is because their [latex]y[\/latex]-coordinates are zero. In figure 8, the point (0, 4) moves to (0, \u20134) showing that the\u00a0[latex]y[\/latex]-coordinates change sign while the [latex]x[\/latex]-coordinates stay the same.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Reflection of a parabola<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Reflection of a line<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_947\" align=\"aligncenter\" width=\"157\"]<img class=\"wp-image-947 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-157x300.png\" alt=\"Graph of a parabola opening up that has been flipped across the x-axis becomes a parabola opening down.\" width=\"157\" height=\"300\" \/> Figure 7. Reflecting across the [latex]x[\/latex]-axis.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_966\" align=\"aligncenter\" width=\"202\"]<img class=\"wp-image-966 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-202x300.png\" alt=\"A line with positive slope becomes a line with negative slope with the same x intercept.\" width=\"202\" height=\"300\" \/> Figure 8. Reflecting across [latex]x[\/latex]-axis.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nA point (3, 4) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is reflected across the [latex]x[\/latex]-axis, what happens to the point (3, 4)?\r\n<h4><strong>Solution<\/strong><\/h4>\r\nReflecting a graph across the [latex]x[\/latex]-axis results in the [latex]y[\/latex]-coordinates changing sign while the [latex]x[\/latex]-coordinates stay the same. So, (3, 4) moves to (3, \u20134).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nA point (\u20133, \u20137) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is reflected across the [latex]x[\/latex]-axis, what happens to the point (\u20133, \u20137)?\r\n\r\n[reveal-answer q=\"hjm945\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm945\"]\r\n\r\nReflecting a graph across the [latex]x[\/latex]-axis results in the [latex]y[\/latex]-coordinates changing sign while the [latex]x[\/latex]-coordinates stay the same. So, (\u20133, \u20137) moves to (\u20133, 7).[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflecting a Graph Across the [latex]y[\/latex]-axis<\/h2>\r\nWhen a graph is reflected across the [latex]y[\/latex]-axis, the [latex]x[\/latex]-coordinates change sign while the [latex]y[\/latex]-coordinates stay the same. So a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](-x, y)[\/latex]. This is shown in figure 9 where the turning point (4, 0) moves to (\u20134, 0), and the point (7, 10) moves to (\u20137, 10).\r\n\r\n[caption id=\"attachment_948\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-948 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-300x284.png\" alt=\"Graph that has been flipped across the y-axis showing every point has the opposite x value, but the y value stays the same.\" width=\"300\" height=\"284\" \/> Figure 9. Reflecting across the [latex]y[\/latex]-axis.[\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nA point (1, \u20135) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is flipped across the [latex]y[\/latex]-axis, what happens to the point (1, \u20135)?\r\n<h4><strong>Solution<\/strong><\/h4>\r\nFlipping a graph across the [latex]y[\/latex]-axis results in the [latex]x[\/latex]-coordinates changing sign while the [latex]y[\/latex]-coordinates stay the same. So, (1, \u20135) moves to (\u20131, \u20135).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nA point (\u20133, 4) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is flipped across the [latex]y[\/latex]-axis, what happens to the point (\u20133, 4)?\r\n\r\n[reveal-answer q=\"hjm352\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm352\"]\r\n\r\n(\u20133, 4) moves to (3, 4).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Stretching a Graph<\/h2>\r\nWhen we stretch a graph we stretch it vertically so that it becomes longer and thinner. The only points that stay in the same position when we stretch a graph are the points on the [latex]x[\/latex]-axis where [latex]f(x)=0[\/latex]. Positive function values get larger (i.e., more positive) and negative function values get smaller (i.e., more negative). For example, figure 10 shows a graph that has been stretched by a factor of 3. The point (2, 4)\u00a0 on the original blue graph moves to (2, 12) on the green graph. This means that the [latex]y[\/latex]-value in the stretched graph is 3 times greater than the\u00a0[latex]y[\/latex]-value in the original graph, when the [latex]x[\/latex]-value stays the same. So a general point [latex](x, y)[\/latex] moves to [latex](x, 3y)[\/latex].\r\n\r\n[caption id=\"attachment_949\" align=\"aligncenter\" width=\"180\"]<img class=\"wp-image-949 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-180x300.png\" alt=\"stretched graph showing for each x value, the y value is multiplied by the stretch factor.\" width=\"180\" height=\"300\" \/> Figure 10. Stretched graph.[\/caption]\r\n\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nA point (2, \u20135) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is stretched by a factor of 3, what happens to the point (2, \u20135)?\r\n<h4><strong>Solution<\/strong><\/h4>\r\nStretching a graph by a factor of 3 means that the [latex]y[\/latex]-coordinate gets multiplied by 3. So, (2, \u20135) moves to (2, \u201315).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nA point (3, \u20134) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is stretched by a factor of 4, what happens to the point (3, \u20134)?\r\n\r\n[reveal-answer q=\"hjm982\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm982\"]\r\n\r\n(3, \u20134) moves to (3, \u201316)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Compressing a Graph<\/h2>\r\nCompressing a graph is related to stretching a graph, but instead of pulling the graph vertically, we squish it in a vertical direction (from top and bottom) so that it becomes shorter and fatter. Just like stretching, the only points on the graph that do not move are those on the [latex]x[\/latex]-axis with a function value of zero. With compression, the positive function values will get smaller (but stay positive), and the negative function values will get less negative (but stay negative). For example, figure 11 shows a graph that has been compressed down to [latex]\\frac{1}{4}[\/latex] of its original height. This means that the [latex]y[\/latex]-value in the compressed graph will be [latex]\\frac{1}{4}[\/latex] of the\u00a0[latex]y[\/latex]-value in the original graph. So a general point [latex](x, y)[\/latex] will move to\u00a0[latex](x, \\frac{1}{4}y)[\/latex].\r\n\r\n[caption id=\"attachment_950\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-950 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-300x291.png\" alt=\"A graph showing an original parabola and it's compressed parabola where the x value stays the same and the y value is one fourth its original height.\" width=\"300\" height=\"291\" \/> Figure 11. Compressed graph.[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nA point (\u20132, 3) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is compressed to [latex]\\dfrac{1}{3}[\/latex] of its original height, what happens to the point (\u20132, 3)?\r\n<h4><strong>Solution<\/strong><\/h4>\r\nCompressing a graph to [latex]\\frac{1}{3}[\/latex] of its original height means that the [latex]y[\/latex]-coordinate gets multiplied by [latex]\\frac{1}{3}[\/latex]. So, (\u20132, 3) moves to (\u20132, 1).\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nA point (9, \u20136) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is compressed to [latex]\\frac{1}{2}[\/latex] of its original height, what happens to the point (9, \u20136)?\r\n\r\n[reveal-answer q=\"hjm635\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm635\"]\r\n\r\n(9, \u20136) moves to (9, \u20133).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAlthough we have mostly used similar graphs to show the different transformations, these transformations can be applied to the graph of any function.\r\n\r\nFor example, the blue function [latex]y=f(x)[\/latex] is stretched by a factor of 5 in figure 12, and compressed to [latex]\\dfrac{1}{2}[\/latex] it's height in figure 13. Every function value on the green stretched graph in figure 12 is 5 times the original function value. SImilarly, every function value on the green compressed graph in figure 13 is half of the original function value.The only points that do not move are those on the [latex]x[\/latex]-axis where the function value is zero. Positive function values stay positive and negative function values stay negative.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Stretch<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Compression<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_4382\" align=\"aligncenter\" width=\"360\"]<img class=\"wp-image-4382\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/20204404\/fxsinx-stretched.png\" alt=\"Graph of a stretched wave function. Each y value has been multiplied by 5.\" width=\"360\" height=\"360\" \/> Figure 12. The parent function [latex]y=f(x)[\/latex] is stretch by a factor of 5.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_4381\" align=\"aligncenter\" width=\"360\"]<img class=\"wp-image-4381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/20204400\/ysinx-compressed-300x300.png\" alt=\"Graph of a compressed wave function. Each y value has been multiplied by one half.\" width=\"360\" height=\"360\" \/> Figure 13. The parent function [latex]y=f(x)[\/latex] is compressed to one-half its height.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Combining Transformations<\/h2>\r\n<h3>The order of transformations<\/h3>\r\nTransformations can be done in any order we want, but the order affects the result. If we are determining in which order to do them in order to transform a function into another specific function, the order matters. There are two types of transformations; vertical transformations that affect the function value and horizontal transformations that affect the independent variable ([latex]x[\/latex]). Vertical and horizontal transformations are completed independently of one another. Horizontal transformations can be completed in any order. It is when we are completing vertical transformations when we have to be careful of the order. For example, shifting right by 3 units and up by 2 units combines a horizontal transformation with a vertical transformation, and they can be done in any order. Likewise, horizontal transformations can be completed in any order. For example, any function [latex]y=f(x)[\/latex] can be transformed into [latex]y=f(-x+3)[\/latex] by either reflecting the function [latex]y=f(x)[\/latex] across the [latex]y[\/latex]-axis then shifting the resulting function left by 3 units, or vice versa. Either way the result is the same. But if we were to combine only vertical transformations we have to be careful of the order in which they are completed. For example, if we shifted the function 4 units up then scaled the function by stretching it by a factor of 3, the [latex]y=f(x)[\/latex] would transform first to [latex]y=f(x)+4[\/latex] and then to [latex]y=3(f(x)+4)=3f(x)+12[\/latex]. We would get a completely different result if we first stretched the function then shifted the function up, as [latex]y=f(x)[\/latex] would transform first to [latex]y=3f(x)[\/latex] and then to [latex]y=3f(x)+4[\/latex].\r\n\r\nThe <strong>order of operations<\/strong> determines the <strong>order of the transformations<\/strong>.\r\n\r\nA function may be shifted up or down, left or right, but such vertical or horizontal shifts should not be performed before the transformations of stretch (or compress) and reflection.\r\n\r\nConsidering points on the original function, if a point (1,1) is shifted up 6 units, it becomes (1, 7). If it is then stretched by a factor of 2, then the position of the point become (1, 14). On the other hand, if the point is first stretched by a factor of 2, it becomes (1, 2). When we then shift the graph up 6 units, it becomes (1, 8). You may notice that the result of the former (1, 14) is different from the result of the latter (1, 8). The former is stretched twice the size from its new vertical position (i.e., [latex]7\\times2[\/latex]). This means the shift has been stretched. Whereas the latter is shifted up after being stretched twice the size from its original vertical position (i.e., [latex]1\\times2[\/latex]). Therefore, to keep the correct scale of a stretch, stretch must be performed before vertical shifts.\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>The Order of transformations<\/h3>\r\n<p class=\"\">We may be asked to write or graph a function with multiple transformations. In this case, we follow something similar to the order of operations.<\/p>\r\nVertical Transformations are completed before horizontal transformations, with stretched, compressions and reflections completed before shifts:\r\n<ol class=\"\">\r\n \t<li class=\"\">Multiplication to function: vertical stretches, compressions, and reflections,\u00a0<span class=\"mathfont\">a<\/span><\/li>\r\n \t<li class=\"\">Addition to function: vertical shifts, k<\/li>\r\n \t<li>Multiplication inside function (reflections across the [latex]y[\/latex]-axis, [latex]b=-1[\/latex])<\/li>\r\n \t<li class=\"\">Addition inside Function: horizontal shifts, h<\/li>\r\n<\/ol>\r\nThis shifts the original function [latex]y=f(x)[\/latex] to [latex]y=a \\cdot f(bx-h)+k[\/latex]\r\n\r\n<\/div>\r\n<span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">Combining Transformations based on the Order of Transformations<\/span>\r\n\r\n<\/div>\r\nFor example, the graph of a function could be stretched by a factor of 2, moved horizontally right by 3 units, then moved vertically down by 6 units. Figure 14 shows a point (1, 1) on the blue curve being transformed first to (1, 2) when the graph is stretched by a factor of 2. The point (1, 2) is then transformed to (4, 2) when the graph is shifted horizontally right by 3 units. FInally the point (4, 2) gets shifted down by 6 units to (4, \u20134). If every point on the blue curve undergoes the same transformations, we will end up with the green curve.\r\n\r\n[caption id=\"attachment_2017\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2017 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-300x300.png\" alt=\"An original polynomial graph transformed by Stretching by a factor of 2, then shifted horizontally right 3 units, then shifted vertically  down 6 units.\" width=\"300\" height=\"300\" \/> Figure 14. Multiple transformations[\/caption]\r\n\r\nAny point [latex](x, y)[\/latex] on the original green curve will first move to [latex](x,2y)[\/latex] when the graph is stretched by a factor of 2, then that point will move to [latex](x+3,2y)[\/latex] when the graph is moved horizontally right by 3 units. Finally, moving this point vertically down by 6 units will take the point to [latex](x+3, 2y-6)[\/latex]. We can check this with the point (1, 1):\u00a0\u00a0[latex](x+3, 2y-6)=(1+3, 2(1)-6)=(4,-4)[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\nThe graph of a function is transformed by first compressing it to [latex]\\frac{1}{4}[\/latex] of its original height, then moving it vertically up by 3 units, then horizontally left by 5 units.\r\n<ol>\r\n \t<li>\u00a0What happens to the point (1, 4), which is on the original graph, after the transformations?<\/li>\r\n \t<li>\u00a0What happens to a general point [latex](x, y)[\/latex] on the original graph after the transformations?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>(1, 4) first moves to (1, 1) as compressing to [latex]\\frac{1}{4}[\/latex] of its original height, means we have to multiply the [latex]y[\/latex]-values by\u00a0[latex]\\frac{1}{4}[\/latex]. Then (1, 1) moves to (1, 4) when we move up by 3 units as we have to add 3 to the [latex]y[\/latex]-coordinate. Finally, we have to subtract 5 from the [latex]x[\/latex]-coordinate to move left by 5 units to (-4, 4).<\/li>\r\n \t<li>[latex](x, y)[\/latex] first moves to [latex]\\left (x, \\dfrac{y}{4}\\right)[\/latex]. [latex]\\left (x, \\dfrac{y}{4}\\right)[\/latex] moves to [latex]\\left (x, \\dfrac{y}{4}+3\\right)[\/latex], which then moves to [latex]\\left (x-5, \\dfrac{y}{4}+3\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 8<\/h3>\r\nThe graph of a function is transformed by first reflecting it across the [latex]y[\/latex]-axis, moving it vertically down by 3 units, then horizontally left by 7 units.\r\n<ol>\r\n \t<li>\u00a0What happens to the point (2, 5), which is on the original graph, after the transformations?<\/li>\r\n \t<li>\u00a0What happens to a general point [latex](x, y)[\/latex] on the original graph after the transformations?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm208\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm208\"]\r\n<ol>\r\n \t<li>(2, 5) moves to (-9, 2)<\/li>\r\n \t<li>[latex](x, y)[\/latex] moves to [latex](-x-7, y-3)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Graphing Inverse Functions<\/h2>\r\nWe have already learned that an inverse function [latex]{f}^{-1}(x)[\/latex] exists if the function [latex]f(x)[\/latex] is one-to-one. We also learned that the range of the function\u00a0[latex]f(x)[\/latex] becomes the domain of the inverse function\u00a0[latex]{f}^{-1}(x)[\/latex], and the domain of\u00a0[latex]f(x)[\/latex] becomes the range of the inverse function\u00a0[latex]{f}^{-1}(x)[\/latex].\r\n\r\n[caption id=\"attachment_784\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-784 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-1024x333.png\" alt=\"Inverse function mapping, described above and below.\" width=\"1024\" height=\"333\" \/> Figure 15. A mapping and its inverse.[\/caption]\r\n\r\nThis implies that when an inverse function is a set of ordered pairs with a one-to-one mapping, we find the inverse function by switching the [latex]x[\/latex]- and [latex]y[\/latex]-values. For example, if the original function is [latex]\\{(1, 2), (2, 4), (3, 6), (4, 8), ...\\}[\/latex], the inverse function is\u00a0[latex]\\{(2, 1), (4, 2), (6, 3), (8, 4), ...\\}[\/latex]. Graphically, this function is an infinite set of coordinate points, so to graph the inverse function, we simply switch all the coordinate points.\r\n\r\n[caption id=\"attachment_801\" align=\"aligncenter\" width=\"298\"]<img class=\"wp-image-801 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-298x300.png\" alt=\"graph of inverse points found above.\" width=\"298\" height=\"300\" \/> Figure 16. A function and its inverse.[\/caption]\r\n\r\nThe black dots represent the original function, while the purple dots represent the inverse function.\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>Graphing an inverse function<\/strong>\r\n\r\nTo graph the inverse function [latex]{f}^{-1}(x)[\/latex], switch all coordinate pairs [latex](x, y)[\/latex] where [latex]y=f(x)[\/latex], to [latex](y, x)[\/latex].\r\n\r\n<\/div>\r\nConsider the graphed one-to-one functions and their inverses in figure 17. The original function is graphed in blue, with the inverse function graphed in green.\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">Line<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">Curve<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">Curve<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-802 size-medium\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-300x300.png\" alt=\"Inverse linear functions symmetric around y = x\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-804 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-803 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-805 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-298x300.png\" alt=\"Inverse linear functions symmetric around y = x\" width=\"298\" height=\"300\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-806 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"aligncenter wp-image-807 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\" colspan=\"3\">Figure 17. Functions and their inverses.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat do you notice about these graphs? Is there any type of symmetry?\r\n\r\nThe graph of every one-to-one function and its inverse function are symmetric about the line [latex]y=x[\/latex]. The line\u00a0[latex]y=x[\/latex] is a line of symmetry for inverse functions. Reflecting across the line\u00a0[latex]y=x[\/latex] causes the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates to switch places, which is exactly what happens with a function and its inverse (figure 18).\r\n\r\n[caption id=\"attachment_820\" align=\"aligncenter\" width=\"646\"]<img class=\"wp-image-820\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18230335\/inverse-symmetry-1024x906.png\" alt=\"Two inverse linear functions demonstrating that for any point on one function the coordinates have switched position on the other.\" width=\"646\" height=\"571\" \/> Figure 18. A function and its inverse function.[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 9<\/h3>\r\nUse the graph of the function to sketch its inverse function.\r\n\r\n<img class=\"aligncenter wp-image-822 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-168x300.png\" alt=\"Graph of a function Passing through the points (-2,-8), (-1,-1), (0,0), (1,1), and (2,8).\" width=\"168\" height=\"300\" \/>\r\n<h4>Solution<\/h4>\r\nStart by sketching the line [latex]y=x[\/latex], then plot the given points with the coordinates switched. Finish by joining the points with a smooth curve mimicking the original graph as a mirror image.\r\n\r\n<img class=\"aligncenter wp-image-823 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-300x300.png\" alt=\"Graph of the function with the points above, and the inverse function with points (-8,-2), (-1,-1), (0,0), (1,1), and (8,2).\" width=\"300\" height=\"300\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 9<\/h3>\r\nUse the graph of the function to sketch its inverse function.\r\n\r\n<img class=\"aligncenter wp-image-824 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-300x240.png\" alt=\"graph of quarter circle in the first quadrant, starting at (2,6) and ending at (8,0).\" width=\"300\" height=\"240\" \/>\r\n\r\n[reveal-answer q=\"hjm307\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm307\"]\r\n\r\n<img class=\"aligncenter wp-image-825 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-300x264.png\" alt=\"graph of quarter circle starting at (2,6) and ending at (8,0) together with its inverse function, a quarter circle starting at (0,8) and ending at (6,2)\" width=\"300\" height=\"264\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIt is very important to remember that <strong>an inverse function exists only if the original function is a one-to-one function.<\/strong> If the original function is not a one-to-one function, it has no inverse function.\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=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Recognize graphical transformations:\n<ul>\n<li>Shifting a graph vertically and horizontally<\/li>\n<li>Reflecting a graph across the [latex]x[\/latex]-axis or [latex]y[\/latex]-axis<\/li>\n<li>Stretching a graph vertically<\/li>\n<li>Compressing a graph vertically<\/li>\n<\/ul>\n<\/li>\n<li>Determine an inverse function by switching [latex]x[\/latex] and [latex]y[\/latex] in coordinate pairs<\/li>\n<li>Graph an inverse function using symmetry<\/li>\n<\/ul>\n<\/div>\n<h2>Transformations<\/h2>\n<p>The transformation of the graph of a function means shifting, flipping, stretching, or compressing a graph. The transformation keeps the same basic shape of the original graph; it has just been manipulated.<\/p>\n<h2>Shifting a Graph Vertically<\/h2>\n<p style=\"text-align: left;\">In Figure 1, the graph with turning point at (1, 1) is shifted up 4 units so that the new turning point is (1, 5).\u00a0 In Figure 2, the graph with turning point at (0, 0) is shifted down 4 units so that the new turning point is (0, \u20134).\u00a0 Notice, in both cases, the [latex]x[\/latex]-values do not change, only the [latex]y[\/latex]-values shift up (Figure 1) or down (Figure 2). The graph stays exactly the same shape but has just been moved vertically. Each point on the graph keeps the same [latex]x[\/latex]-value but move either up of down the same amount. In figure 1, shifting the graph up by 4 units means all points [latex](x, y)[\/latex] move to [latex](x, y+4)[\/latex]. In figure 2, moving the graph down 4 units means all points\u00a0[latex](x, y)[\/latex] move to\u00a0[latex](x, y-4)[\/latex].<\/p>\n<p>\u200b<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Adding 4<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Subtracting 4<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1731\" style=\"width: 372px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1731\" class=\"wp-image-1731\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22193624\/desmos-graph-53-300x300.png\" alt=\"Every point on the original graph moves up 4 units.\" width=\"362\" height=\"362\" \/><\/p>\n<p id=\"caption-attachment-1731\" class=\"wp-caption-text\">Figure 1. Shifting up.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div class=\"mceTemp\"><\/div>\n<div id=\"attachment_3609\" style=\"width: 373px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3609\" class=\"wp-image-3609\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/27151309\/1-3-ShiftDown-300x300.png\" alt=\"Every point on the original graph moves down 4 units.\" width=\"363\" height=\"363\" \/><\/p>\n<p id=\"caption-attachment-3609\" class=\"wp-caption-text\">FIgure 2. Shifting down.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Explain which transformation the original (blue) graph has undergone to become the green graph. Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-816 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up-146x300.png\" alt=\"Every point on the original blue graph has moved up 10 units.\" width=\"146\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up-146x300.png 146w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up-65x134.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up-225x462.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up-350x719.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-up.png 436w\" sizes=\"auto, (max-width: 146px) 100vw, 146px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The original graph has been shifted up by 10 units. The turning point (0, 0) moves to (0, 10).<\/p>\n<p>An original point [latex](x, y)[\/latex] moves to\u00a0[latex](x, y+10)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Explain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-817 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-129x300.png\" alt=\"Every point on the original blue graph has moved down 7 units.\" width=\"129\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-129x300.png 129w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-440x1024.png 440w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-65x151.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-225x523.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down-350x814.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-down.png 486w\" sizes=\"auto, (max-width: 129px) 100vw, 129px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm874\">Show Answer<\/span><\/p>\n<div id=\"qhjm874\" class=\"hidden-answer\" style=\"display: none\">\n<p>The original graph has been shifted down by 7 units. The turning point at (0, 0) moved to (0, \u20137).<\/p>\n<p>An original point [latex](x, y)[\/latex] will move to\u00a0[latex](x, y-7)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Shifting a Graph Horizontally<\/h2>\n<p>Shifting a graph vertically means to move the graph up or down without changing the shape of the graph. Similarly, shifting a graph horizontally means to move the graph left or right without changing the shape of the graph. In Figure 3, the graph with turning point (1, 1) is shifted to the right 5 units. The new turning point is (6, 1).\u00a0 Similarly, in Figure 4, the graph with turning point (1, 1) is shifted to the left 8 units. The new turning point is (-7, 1). Notice that when shifting horizontally, the graph stays the same shape and the [latex]y[\/latex]-values do not change; only the [latex]x[\/latex]-values move left (Figure 3) or right (Figure 4).\u00a0 A general point [latex](x, y)[\/latex] on the graph in figure 3 moves to\u00a0[latex](x+5, y)[\/latex]. In figure 6,\u00a0[latex](x, y)[\/latex] moves to\u00a0[latex](x-8, y)[\/latex].<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\"><em>Subtracting<\/em> 5<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\"><em>Adding<\/em> 8<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_694\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-694\" class=\"wp-image-694 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-300x300.png\" alt=\"Each point on the original red graph has moved right five units.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-768x767.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-1024x1022.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftiRight-350x349.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-694\" class=\"wp-caption-text\">Figure 3. Shifting to the right.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_695\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-695\" class=\"wp-image-695 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-300x300.png\" alt=\"Each point on the original red graph has moved left eight units.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-768x767.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-1024x1022.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-3-1-ShiftLeft-350x349.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-695\" class=\"wp-caption-text\">Figure 4. Shifting to the left.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Explain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-814 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right-250x300.png\" alt=\"The vertex of an original parabola has moved from (0,0) to (7,0).\" width=\"250\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right-250x300.png 250w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right-65x78.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right-225x270.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right-350x420.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-right.png 696w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>The original graph has been shifted right by 7 units. The turning point (0, 0) moved to (7, 0).<\/p>\n<p>An original point [latex](x, y)[\/latex] will move to\u00a0[latex](x+7, y)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Explain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-815 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-278x300.png\" alt=\"The parabola representing f of x equals x squared has been transformed by shifting every point ten units to the left.\" width=\"278\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-278x300.png 278w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-768x829.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-65x70.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-225x243.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left-350x378.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/shifted-left.png 826w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm814\">Show Answer<\/span><\/p>\n<div id=\"qhjm814\" class=\"hidden-answer\" style=\"display: none\">\n<p>The original graph has been shifted left by 10 units. The turning point (0, 0) has moved to (\u201310 0).<\/p>\n<p>An original point [latex](x, y)[\/latex] will move to\u00a0[latex](x-10, y)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Shifting a Graph Vertically and Horizontally<\/h2>\n<p>We can shift a graph either horizontally <em>or<\/em> vertically, but we can also shift a graph horizontally <em>and<\/em> vertically. Consider the graphs in Figures 5 and 6. The graph in Figure 5 has been shifted up by 4 units and left by 3 units. This is shown by the turning point moving from (0, 0) to (\u20133, 4). Moving up 4 units adds 4 to the [latex]y[\/latex]-coordinates, while moving left\u00a0<span style=\"font-size: 1em;\">3 units<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">adds \u20133 to the [latex]x[\/latex]-coordinates. This means that a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](x-3, y+4)[\/latex]. Notice we would get the same result if we completed the transformations in the reverse order.<\/span><\/p>\n<p>The graph in Figure 6 has been shifted right by 2 units and up by 3 units. The turning point moves from (0, 0) to (2, 3). Notice that all points on the graph move exactly the same way; right 2, up 3. For example, the point (\u20132, 4) moves to (0, 7). Moving 2 units right adds 2 to the [latex]x[\/latex]-coordinates, while moving up 3 units adds 3 to the [latex]y[\/latex]-coordinates. (\u20132, 4) moves to (\u20132 + 2, 4 + 3) = (0, 7).\u00a0This means that a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](x+2, y+3)[\/latex].<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Shifting vertically and horizontally<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Shifting horizontally and vertically<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_941\" style=\"width: 264px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-941\" class=\"wp-image-941 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/02230702\/left-and-up-254x300.png\" alt=\"graph showing vertical and horizontal transformations\" width=\"254\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-941\" class=\"wp-caption-text\">Figure 5. Vertical and horizontal shifts.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_942\" style=\"width: 287px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-942\" class=\"wp-image-942 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/02231555\/right-and-up-277x300.png\" alt=\"Graph showing horizontal and vertical transformations\" width=\"277\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-942\" class=\"wp-caption-text\">Figure 6. Vertical and horizontal shifts.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Explain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-944 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-197x300.png\" alt=\"An original graph of a parabola left one unit and down seven units.\" width=\"197\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-197x300.png 197w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-768x1168.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-673x1024.png 673w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-65x99.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-225x342.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7-350x532.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/xy-to-x-1-y-7.png 926w\" sizes=\"auto, (max-width: 197px) 100vw, 197px\" \/><\/p>\n<p>The original graph has been shifted left by 1 units and down by 7 units. The turning point at (0, 0) has moved to (\u20131 \u20137).<\/p>\n<p>An original point [latex](x, y)[\/latex] will move to\u00a0[latex](x-1, y-7)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Explain which transformation the original (blue) graph has undergone to become the green graph.\u00a0Write the new coordinates of a general point [latex](x,y)[\/latex] after the transformation.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-945 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-239x300.png\" alt=\"An original parabola shifted right 4 units and down 5 units.\" width=\"239\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-239x300.png 239w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-768x963.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-817x1024.png 817w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-65x81.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-225x282.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5-350x439.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/x-y-to-x4-y-5.png 908w\" sizes=\"auto, (max-width: 239px) 100vw, 239px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm504\">Show Answer<\/span><\/p>\n<div id=\"qhjm504\" class=\"hidden-answer\" style=\"display: none\">\n<p>The original graph has been shifted right by 4 units and down by 5 units. The turning point at (0, 0) has moved to (4, \u20135).<\/p>\n<p>An original point [latex](x, y)[\/latex] will move to\u00a0[latex](x+4, y-5)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflecting a Graph Across the [latex]x[\/latex]-axis<\/h2>\n<p>When a graph is reflected across the [latex]x[\/latex]-axis, the [latex]y[\/latex]-coordinates change sign while the [latex]x[\/latex]-coordinates stay the same. So a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](x, -y)[\/latex]. Notice in figure 7 only (0, 0) stays the same, and in Figure 8 only (\u20132, 0) stays the same.\u00a0 This is because their [latex]y[\/latex]-coordinates are zero. In figure 8, the point (0, 4) moves to (0, \u20134) showing that the\u00a0[latex]y[\/latex]-coordinates change sign while the [latex]x[\/latex]-coordinates stay the same.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Reflection of a parabola<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Reflection of a line<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_947\" style=\"width: 167px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-947\" class=\"wp-image-947 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-157x300.png\" alt=\"Graph of a parabola opening up that has been flipped across the x-axis becomes a parabola opening down.\" width=\"157\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-157x300.png 157w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-537x1024.png 537w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-65x124.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-225x429.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis-350x667.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-x-axis.png 728w\" sizes=\"auto, (max-width: 157px) 100vw, 157px\" \/><\/p>\n<p id=\"caption-attachment-947\" class=\"wp-caption-text\">Figure 7. Reflecting across the [latex]x[\/latex]-axis.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_966\" style=\"width: 212px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-966\" class=\"wp-image-966 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-202x300.png\" alt=\"A line with positive slope becomes a line with negative slope with the same x intercept.\" width=\"202\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-202x300.png 202w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-768x1142.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-689x1024.png 689w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-65x97.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-225x334.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis-350x520.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-flip-across-x-axis.png 1032w\" sizes=\"auto, (max-width: 202px) 100vw, 202px\" \/><\/p>\n<p id=\"caption-attachment-966\" class=\"wp-caption-text\">Figure 8. Reflecting across [latex]x[\/latex]-axis.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>A point (3, 4) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is reflected across the [latex]x[\/latex]-axis, what happens to the point (3, 4)?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Reflecting a graph across the [latex]x[\/latex]-axis results in the [latex]y[\/latex]-coordinates changing sign while the [latex]x[\/latex]-coordinates stay the same. So, (3, 4) moves to (3, \u20134).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>A point (\u20133, \u20137) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is reflected across the [latex]x[\/latex]-axis, what happens to the point (\u20133, \u20137)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm945\">Show Answer<\/span><\/p>\n<div id=\"qhjm945\" class=\"hidden-answer\" style=\"display: none\">\n<p>Reflecting a graph across the [latex]x[\/latex]-axis results in the [latex]y[\/latex]-coordinates changing sign while the [latex]x[\/latex]-coordinates stay the same. So, (\u20133, \u20137) moves to (\u20133, 7).<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Reflecting a Graph Across the [latex]y[\/latex]-axis<\/h2>\n<p>When a graph is reflected across the [latex]y[\/latex]-axis, the [latex]x[\/latex]-coordinates change sign while the [latex]y[\/latex]-coordinates stay the same. So a general point\u00a0[latex](x, y)[\/latex] will move to\u00a0[latex](-x, y)[\/latex]. This is shown in figure 9 where the turning point (4, 0) moves to (\u20134, 0), and the point (7, 10) moves to (\u20137, 10).<\/p>\n<div id=\"attachment_948\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-948\" class=\"wp-image-948 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-300x284.png\" alt=\"Graph that has been flipped across the y-axis showing every point has the opposite x value, but the y value stays the same.\" width=\"300\" height=\"284\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-300x284.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-768x726.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-1024x968.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-65x61.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-225x213.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis-350x331.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Flip-across-the-y-axis.png 1320w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-948\" class=\"wp-caption-text\">Figure 9. Reflecting across the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>A point (1, \u20135) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is flipped across the [latex]y[\/latex]-axis, what happens to the point (1, \u20135)?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Flipping a graph across the [latex]y[\/latex]-axis results in the [latex]x[\/latex]-coordinates changing sign while the [latex]y[\/latex]-coordinates stay the same. So, (1, \u20135) moves to (\u20131, \u20135).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>A point (\u20133, 4) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is flipped across the [latex]y[\/latex]-axis, what happens to the point (\u20133, 4)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm352\">Show Answer<\/span><\/p>\n<div id=\"qhjm352\" class=\"hidden-answer\" style=\"display: none\">\n<p>(\u20133, 4) moves to (3, 4).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Stretching a Graph<\/h2>\n<p>When we stretch a graph we stretch it vertically so that it becomes longer and thinner. The only points that stay in the same position when we stretch a graph are the points on the [latex]x[\/latex]-axis where [latex]f(x)=0[\/latex]. Positive function values get larger (i.e., more positive) and negative function values get smaller (i.e., more negative). For example, figure 10 shows a graph that has been stretched by a factor of 3. The point (2, 4)\u00a0 on the original blue graph moves to (2, 12) on the green graph. This means that the [latex]y[\/latex]-value in the stretched graph is 3 times greater than the\u00a0[latex]y[\/latex]-value in the original graph, when the [latex]x[\/latex]-value stays the same. So a general point [latex](x, y)[\/latex] moves to [latex](x, 3y)[\/latex].<\/p>\n<div id=\"attachment_949\" style=\"width: 190px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-949\" class=\"wp-image-949 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-180x300.png\" alt=\"stretched graph showing for each x value, the y value is multiplied by the stretch factor.\" width=\"180\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-180x300.png 180w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-768x1281.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-614x1024.png 614w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-65x108.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-225x375.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph-350x584.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/stretsch-graph.png 782w\" sizes=\"auto, (max-width: 180px) 100vw, 180px\" \/><\/p>\n<p id=\"caption-attachment-949\" class=\"wp-caption-text\">Figure 10. Stretched graph.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>A point (2, \u20135) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is stretched by a factor of 3, what happens to the point (2, \u20135)?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Stretching a graph by a factor of 3 means that the [latex]y[\/latex]-coordinate gets multiplied by 3. So, (2, \u20135) moves to (2, \u201315).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>A point (3, \u20134) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is stretched by a factor of 4, what happens to the point (3, \u20134)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm982\">Show Answer<\/span><\/p>\n<div id=\"qhjm982\" class=\"hidden-answer\" style=\"display: none\">\n<p>(3, \u20134) moves to (3, \u201316)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Compressing a Graph<\/h2>\n<p>Compressing a graph is related to stretching a graph, but instead of pulling the graph vertically, we squish it in a vertical direction (from top and bottom) so that it becomes shorter and fatter. Just like stretching, the only points on the graph that do not move are those on the [latex]x[\/latex]-axis with a function value of zero. With compression, the positive function values will get smaller (but stay positive), and the negative function values will get less negative (but stay negative). For example, figure 11 shows a graph that has been compressed down to [latex]\\frac{1}{4}[\/latex] of its original height. This means that the [latex]y[\/latex]-value in the compressed graph will be [latex]\\frac{1}{4}[\/latex] of the\u00a0[latex]y[\/latex]-value in the original graph. So a general point [latex](x, y)[\/latex] will move to\u00a0[latex](x, \\frac{1}{4}y)[\/latex].<\/p>\n<div id=\"attachment_950\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-950\" class=\"wp-image-950 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-300x291.png\" alt=\"A graph showing an original parabola and it's compressed parabola where the x value stays the same and the y value is one fourth its original height.\" width=\"300\" height=\"291\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-300x291.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-768x746.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-1024x994.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-65x63.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-225x219.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph-350x340.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Compressed-graph.png 1180w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-950\" class=\"wp-caption-text\">Figure 11. Compressed graph.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>A point (\u20132, 3) lies on the graph of [latex]y=f(x)[\/latex]. If the graph is compressed to [latex]\\dfrac{1}{3}[\/latex] of its original height, what happens to the point (\u20132, 3)?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Compressing a graph to [latex]\\frac{1}{3}[\/latex] of its original height means that the [latex]y[\/latex]-coordinate gets multiplied by [latex]\\frac{1}{3}[\/latex]. So, (\u20132, 3) moves to (\u20132, 1).<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>A point (9, \u20136) lies on the graph of [latex]y=g(x)[\/latex]. If the graph is compressed to [latex]\\frac{1}{2}[\/latex] of its original height, what happens to the point (9, \u20136)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm635\">Show Answer<\/span><\/p>\n<div id=\"qhjm635\" class=\"hidden-answer\" style=\"display: none\">\n<p>(9, \u20136) moves to (9, \u20133).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Although we have mostly used similar graphs to show the different transformations, these transformations can be applied to the graph of any function.<\/p>\n<p>For example, the blue function [latex]y=f(x)[\/latex] is stretched by a factor of 5 in figure 12, and compressed to [latex]\\dfrac{1}{2}[\/latex] it&#8217;s height in figure 13. Every function value on the green stretched graph in figure 12 is 5 times the original function value. SImilarly, every function value on the green compressed graph in figure 13 is half of the original function value.The only points that do not move are those on the [latex]x[\/latex]-axis where the function value is zero. Positive function values stay positive and negative function values stay negative.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Stretch<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Compression<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_4382\" style=\"width: 370px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4382\" class=\"wp-image-4382\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/20204404\/fxsinx-stretched.png\" alt=\"Graph of a stretched wave function. Each y value has been multiplied by 5.\" width=\"360\" height=\"360\" \/><\/p>\n<p id=\"caption-attachment-4382\" class=\"wp-caption-text\">Figure 12. The parent function [latex]y=f(x)[\/latex] is stretch by a factor of 5.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_4381\" style=\"width: 370px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4381\" class=\"wp-image-4381\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/20204400\/ysinx-compressed-300x300.png\" alt=\"Graph of a compressed wave function. Each y value has been multiplied by one half.\" width=\"360\" height=\"360\" \/><\/p>\n<p id=\"caption-attachment-4381\" class=\"wp-caption-text\">Figure 13. The parent function [latex]y=f(x)[\/latex] is compressed to one-half its height.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Combining Transformations<\/h2>\n<h3>The order of transformations<\/h3>\n<p>Transformations can be done in any order we want, but the order affects the result. If we are determining in which order to do them in order to transform a function into another specific function, the order matters. There are two types of transformations; vertical transformations that affect the function value and horizontal transformations that affect the independent variable ([latex]x[\/latex]). Vertical and horizontal transformations are completed independently of one another. Horizontal transformations can be completed in any order. It is when we are completing vertical transformations when we have to be careful of the order. For example, shifting right by 3 units and up by 2 units combines a horizontal transformation with a vertical transformation, and they can be done in any order. Likewise, horizontal transformations can be completed in any order. For example, any function [latex]y=f(x)[\/latex] can be transformed into [latex]y=f(-x+3)[\/latex] by either reflecting the function [latex]y=f(x)[\/latex] across the [latex]y[\/latex]-axis then shifting the resulting function left by 3 units, or vice versa. Either way the result is the same. But if we were to combine only vertical transformations we have to be careful of the order in which they are completed. For example, if we shifted the function 4 units up then scaled the function by stretching it by a factor of 3, the [latex]y=f(x)[\/latex] would transform first to [latex]y=f(x)+4[\/latex] and then to [latex]y=3(f(x)+4)=3f(x)+12[\/latex]. We would get a completely different result if we first stretched the function then shifted the function up, as [latex]y=f(x)[\/latex] would transform first to [latex]y=3f(x)[\/latex] and then to [latex]y=3f(x)+4[\/latex].<\/p>\n<p>The <strong>order of operations<\/strong> determines the <strong>order of the transformations<\/strong>.<\/p>\n<p>A function may be shifted up or down, left or right, but such vertical or horizontal shifts should not be performed before the transformations of stretch (or compress) and reflection.<\/p>\n<p>Considering points on the original function, if a point (1,1) is shifted up 6 units, it becomes (1, 7). If it is then stretched by a factor of 2, then the position of the point become (1, 14). On the other hand, if the point is first stretched by a factor of 2, it becomes (1, 2). When we then shift the graph up 6 units, it becomes (1, 8). You may notice that the result of the former (1, 14) is different from the result of the latter (1, 8). The former is stretched twice the size from its new vertical position (i.e., [latex]7\\times2[\/latex]). This means the shift has been stretched. Whereas the latter is shifted up after being stretched twice the size from its original vertical position (i.e., [latex]1\\times2[\/latex]). Therefore, to keep the correct scale of a stretch, stretch must be performed before vertical shifts.<\/p>\n<div>\n<div class=\"textbox shaded\">\n<h3>The Order of transformations<\/h3>\n<p class=\"\">We may be asked to write or graph a function with multiple transformations. In this case, we follow something similar to the order of operations.<\/p>\n<p>Vertical Transformations are completed before horizontal transformations, with stretched, compressions and reflections completed before shifts:<\/p>\n<ol class=\"\">\n<li class=\"\">Multiplication to function: vertical stretches, compressions, and reflections,\u00a0<span class=\"mathfont\">a<\/span><\/li>\n<li class=\"\">Addition to function: vertical shifts, k<\/li>\n<li>Multiplication inside function (reflections across the [latex]y[\/latex]-axis, [latex]b=-1[\/latex])<\/li>\n<li class=\"\">Addition inside Function: horizontal shifts, h<\/li>\n<\/ol>\n<p>This shifts the original function [latex]y=f(x)[\/latex] to [latex]y=a \\cdot f(bx-h)+k[\/latex]<\/p>\n<\/div>\n<p><span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">Combining Transformations based on the Order of Transformations<\/span><\/p>\n<\/div>\n<p>For example, the graph of a function could be stretched by a factor of 2, moved horizontally right by 3 units, then moved vertically down by 6 units. Figure 14 shows a point (1, 1) on the blue curve being transformed first to (1, 2) when the graph is stretched by a factor of 2. The point (1, 2) is then transformed to (4, 2) when the graph is shifted horizontally right by 3 units. FInally the point (4, 2) gets shifted down by 6 units to (4, \u20134). If every point on the blue curve undergoes the same transformations, we will end up with the green curve.<\/p>\n<div id=\"attachment_2017\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2017\" class=\"wp-image-2017 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-300x300.png\" alt=\"An original polynomial graph transformed by Stretching by a factor of 2, then shifted horizontally right 3 units, then shifted vertically  down 6 units.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-2022-05-05T135114.405.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-2017\" class=\"wp-caption-text\">Figure 14. Multiple transformations<\/p>\n<\/div>\n<p>Any point [latex](x, y)[\/latex] on the original green curve will first move to [latex](x,2y)[\/latex] when the graph is stretched by a factor of 2, then that point will move to [latex](x+3,2y)[\/latex] when the graph is moved horizontally right by 3 units. Finally, moving this point vertically down by 6 units will take the point to [latex](x+3, 2y-6)[\/latex]. We can check this with the point (1, 1):\u00a0\u00a0[latex](x+3, 2y-6)=(1+3, 2(1)-6)=(4,-4)[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p>The graph of a function is transformed by first compressing it to [latex]\\frac{1}{4}[\/latex] of its original height, then moving it vertically up by 3 units, then horizontally left by 5 units.<\/p>\n<ol>\n<li>\u00a0What happens to the point (1, 4), which is on the original graph, after the transformations?<\/li>\n<li>\u00a0What happens to a general point [latex](x, y)[\/latex] on the original graph after the transformations?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>(1, 4) first moves to (1, 1) as compressing to [latex]\\frac{1}{4}[\/latex] of its original height, means we have to multiply the [latex]y[\/latex]-values by\u00a0[latex]\\frac{1}{4}[\/latex]. Then (1, 1) moves to (1, 4) when we move up by 3 units as we have to add 3 to the [latex]y[\/latex]-coordinate. Finally, we have to subtract 5 from the [latex]x[\/latex]-coordinate to move left by 5 units to (-4, 4).<\/li>\n<li>[latex](x, y)[\/latex] first moves to [latex]\\left (x, \\dfrac{y}{4}\\right)[\/latex]. [latex]\\left (x, \\dfrac{y}{4}\\right)[\/latex] moves to [latex]\\left (x, \\dfrac{y}{4}+3\\right)[\/latex], which then moves to [latex]\\left (x-5, \\dfrac{y}{4}+3\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 8<\/h3>\n<p>The graph of a function is transformed by first reflecting it across the [latex]y[\/latex]-axis, moving it vertically down by 3 units, then horizontally left by 7 units.<\/p>\n<ol>\n<li>\u00a0What happens to the point (2, 5), which is on the original graph, after the transformations?<\/li>\n<li>\u00a0What happens to a general point [latex](x, y)[\/latex] on the original graph after the transformations?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm208\">Show Answer<\/span><\/p>\n<div id=\"qhjm208\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>(2, 5) moves to (-9, 2)<\/li>\n<li>[latex](x, y)[\/latex] moves to [latex](-x-7, y-3)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Graphing Inverse Functions<\/h2>\n<p>We have already learned that an inverse function [latex]{f}^{-1}(x)[\/latex] exists if the function [latex]f(x)[\/latex] is one-to-one. We also learned that the range of the function\u00a0[latex]f(x)[\/latex] becomes the domain of the inverse function\u00a0[latex]{f}^{-1}(x)[\/latex], and the domain of\u00a0[latex]f(x)[\/latex] becomes the range of the inverse function\u00a0[latex]{f}^{-1}(x)[\/latex].<\/p>\n<div id=\"attachment_784\" style=\"width: 1034px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-784\" class=\"wp-image-784 size-large\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-1024x333.png\" alt=\"Inverse function mapping, described above and below.\" width=\"1024\" height=\"333\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-1024x333.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-300x98.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-768x250.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-65x21.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-225x73.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1-350x114.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-function-example-1.png 1648w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p id=\"caption-attachment-784\" class=\"wp-caption-text\">Figure 15. A mapping and its inverse.<\/p>\n<\/div>\n<p>This implies that when an inverse function is a set of ordered pairs with a one-to-one mapping, we find the inverse function by switching the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values. For example, if the original function is [latex]\\{(1, 2), (2, 4), (3, 6), (4, 8), ...\\}[\/latex], the inverse function is\u00a0[latex]\\{(2, 1), (4, 2), (6, 3), (8, 4), ...\\}[\/latex]. Graphically, this function is an infinite set of coordinate points, so to graph the inverse function, we simply switch all the coordinate points.<\/p>\n<div id=\"attachment_801\" style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-801\" class=\"wp-image-801 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-298x300.png\" alt=\"graph of inverse points found above.\" width=\"298\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-298x300.png 298w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-225x227.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-350x353.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points.png 540w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<p id=\"caption-attachment-801\" class=\"wp-caption-text\">Figure 16. A function and its inverse.<\/p>\n<\/div>\n<p>The black dots represent the original function, while the purple dots represent the inverse function.<\/p>\n<div class=\"textbox shaded\">\n<p><strong>Graphing an inverse function<\/strong><\/p>\n<p>To graph the inverse function [latex]{f}^{-1}(x)[\/latex], switch all coordinate pairs [latex](x, y)[\/latex] where [latex]y=f(x)[\/latex], to [latex](y, x)[\/latex].<\/p>\n<\/div>\n<p>Consider the graphed one-to-one functions and their inverses in figure 17. The original function is graphed in blue, with the inverse function graphed in green.<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">Line<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">Curve<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">Curve<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-802 size-medium\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-300x300.png\" alt=\"Inverse linear functions symmetric around y = x\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-768x769.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2x-inverse.png 1496w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-804 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-768x767.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/y2^x-inverse.png 1498w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-803 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-768x766.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-225x224.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse-350x349.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^2-inverse.png 830w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-805 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-298x300.png\" alt=\"Inverse linear functions symmetric around y = x\" width=\"298\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-298x300.png 298w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-768x772.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-1019x1024.png 1019w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-225x226.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse-350x352.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/linear-function-with-inverse.png 1146w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-806 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-768x767.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-1024x1022.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse-350x349.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/cubic-inverse.png 1148w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-807 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-300x300.png\" alt=\"Inverse non-linear functions symmetric around y = x\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/tan-inverse.png 1148w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\" colspan=\"3\">Figure 17. Functions and their inverses.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What do you notice about these graphs? Is there any type of symmetry?<\/p>\n<p>The graph of every one-to-one function and its inverse function are symmetric about the line [latex]y=x[\/latex]. The line\u00a0[latex]y=x[\/latex] is a line of symmetry for inverse functions. Reflecting across the line\u00a0[latex]y=x[\/latex] causes the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates to switch places, which is exactly what happens with a function and its inverse (figure 18).<\/p>\n<div id=\"attachment_820\" style=\"width: 656px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-820\" class=\"wp-image-820\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18230335\/inverse-symmetry-1024x906.png\" alt=\"Two inverse linear functions demonstrating that for any point on one function the coordinates have switched position on the other.\" width=\"646\" height=\"571\" \/><\/p>\n<p id=\"caption-attachment-820\" class=\"wp-caption-text\">Figure 18. A function and its inverse function.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 9<\/h3>\n<p>Use the graph of the function to sketch its inverse function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-822 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-168x300.png\" alt=\"Graph of a function Passing through the points (-2,-8), (-1,-1), (0,0), (1,1), and (2,8).\" width=\"168\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-168x300.png 168w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-572x1024.png 572w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-65x116.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-225x403.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-350x627.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3.png 764w\" sizes=\"auto, (max-width: 168px) 100vw, 168px\" \/><\/p>\n<h4>Solution<\/h4>\n<p>Start by sketching the line [latex]y=x[\/latex], then plot the given points with the coordinates switched. Finish by joining the points with a smooth curve mimicking the original graph as a mirror image.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-823 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-300x300.png\" alt=\"Graph of the function with the points above, and the inverse function with points (-8,-2), (-1,-1), (0,0), (1,1), and (8,2).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-768x770.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-1021x1024.png 1021w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-225x226.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse-350x351.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/yx^3-and-inverse.png 1356w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 9<\/h3>\n<p>Use the graph of the function to sketch its inverse function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-824 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-300x240.png\" alt=\"graph of quarter circle in the first quadrant, starting at (2,6) and ending at (8,0).\" width=\"300\" height=\"240\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-300x240.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-768x615.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-1024x820.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-65x52.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-225x180.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-350x280.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle.png 1076w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm307\">Show Answer<\/span><\/p>\n<div id=\"qhjm307\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-825 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-300x264.png\" alt=\"graph of quarter circle starting at (2,6) and ending at (8,0) together with its inverse function, a quarter circle starting at (0,8) and ending at (6,2)\" width=\"300\" height=\"264\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-300x264.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-768x675.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-1024x900.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-65x57.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-225x198.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse-350x308.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/quarter-circle-inerse.png 1306w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>It is very important to remember that <strong>an inverse function exists only if the original function is a one-to-one function.<\/strong> If the original function is not a one-to-one function, it has no inverse function.<\/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-620\">\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 and the Inverse of a 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>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/a>. <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 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