{"id":2669,"date":"2022-06-13T21:06:59","date_gmt":"2022-06-13T21:06:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2669"},"modified":"2026-01-18T00:41:47","modified_gmt":"2026-01-18T00:41:47","slug":"7-2-transformations-of-the-rational-functions-fx1-x","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/7-2-transformations-of-the-rational-functions-fx1-x\/","title":{"raw":"7.2: Transformations of the Rational Function","rendered":"7.2: Transformations of the Rational 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\nFor the rational parent function [latex]f(x)=\\dfrac{1}{x}[\/latex],\r\n<ul>\r\n \t<li>Perform vertical and horizontal shifts<\/li>\r\n \t<li>Perform vertical stretches and compressions<\/li>\r\n \t<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\r\n \t<li>Perform reflections across the [latex]y[\/latex]-axis<\/li>\r\n \t<li>Determine the transformations performed on the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] to get the rational function [latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"entry-content\">\r\n<h2>Vertical Shifts<\/h2>\r\n<p id=\"fs-id1165137770279\">If we shift the graph of the rational function [latex]f(x)=\\dfrac{1}{x}[\/latex] up 5 units, all of the points on the graph increase their [latex]y[\/latex]-coordinates by 5, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=\\dfrac{1}{x}+5[\/latex]. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 68.81378216718218%; height: 206px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\r\n<th style=\"width: 29.2773%; height: 10px; text-align: right;\" scope=\"row\">[latex]\\dfrac{1}{x}+5[\/latex]<\/th>\r\n<td style=\"width: 21.566613864306788%; height: 206px;\" rowspan=\"9\">\r\n<div id=\"attachment_1823\" class=\"wp-caption aligncenter\" style=\"width: 408px;\">\r\n\r\n<img class=\"aligncenter wp-image-2787\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/18214620\/7-2-ShiftUp1-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 5 units above the blue graph.\" width=\"408\" height=\"408\" \/>\r\n\r\nFigure 1. Shifting the graph of the function\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] up 5 units.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]\\dfrac{9}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\\dfrac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\\dfrac{16}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 54.80468613569322%;\" colspan=\"3\">Table 1. [latex]f(x)=\\frac{1}{x}[\/latex] is transformed to[latex]f(x)=\\frac{1}{x}+5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=\\frac{1}{x}[\/latex] down 8 units, all of the points on the graph decrease their [latex]y[\/latex]-coordinates by 8, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\frac{1}{x}[\/latex] after it has been shifted down 8 units transforms to [latex]f(x)=\\frac{1}{x}-8[\/latex]. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 71.47138613569322%; height: 222px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\r\n<th style=\"width: 29.2773%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)=\\dfrac{1}{x}-8[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 222px;\" rowspan=\"9\">\r\n<div id=\"attachment_1825\" class=\"wp-caption aligncenter\" style=\"width: 389px;\">\r\n\r\n<img class=\"aligncenter wp-image-2788\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/18214643\/7-2-ShiftDown2-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 8 units below the blue graph.\" width=\"389\" height=\"389\" \/>\r\n\r\nFigure 2. Shifting the graph of the function\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] down 8\u00a0units.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]-\\dfrac{17}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]\u20139[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]\u201310[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\u20136[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\u20137[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]-\\frac{15}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]-\\frac{23}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 54.80468613569322%; height: 34px;\" colspan=\"3\">Table 2.\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] is transformed to \u00a0[latex]f(x)=\\frac{1}{x}-8[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">\r\n<h3>Vertical shifts<\/h3>\r\nWe can represent a vertical shift of the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] by adding a constant, [latex]k[\/latex], to the function:\r\n<p style=\"text-align: center;\">[latex]f(x)=\\dfrac{1}{x}+k[\/latex]<\/p>\r\n\u00a0If [latex]k&gt;0[\/latex], the graph shifts upwards and if [latex]k&lt;0[\/latex] the graph shifts downwards.\r\n\r\n<\/div>\r\nManipulate the graph in figure 3 to shift the graph vertically. Pay attention to what happens with the function as [latex]k[\/latex] changes value. Also, watch what happens with the asymptotes.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/q62i2yudlk?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 3. Vertical Transformations<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\n<ol>\r\n \t<li>Use figure 3 to graph the function [latex]f(x)=\\frac{1}{x}-6[\/latex]. What is the horizontal asymptote of the function? What is the relationship between the value of [latex]k[\/latex] and the horizontal asymptote? What is the vertical asymptote of the function?<\/li>\r\n \t<li>Without graphing the function, what is the horizontal asymptote of the function [latex]g(x)=\\frac{1}{x}+5[\/latex]? What is the vertical asymptote?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n1.<img class=\"aligncenter wp-image-3280 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-300x300.png\" alt=\"Graph of g(x)=1\/x -6\" width=\"300\" height=\"300\" \/>\r\n\r\nThe horizontal asymptote is the line [latex]y=-6[\/latex]. The value of [latex]k[\/latex] is [latex]-6[\/latex] so the horizontal asymptote mimics the value of [latex]k[\/latex].\r\n\r\nThe vertical asymptote is the line [latex]x=0[\/latex].\r\n\r\n2. Since [latex]k=5[\/latex] the graph of [latex]f(x)=\\frac{1}{x}[\/latex] gets shifted up by 5 units. This means that the horizontal asymptote is shifted up 5 units to [latex]y=5[\/latex]. The vertical asymptote does not move so is still [latex]x=0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\n<ol>\r\n \t<li>Use figure 3 to graph the function [latex]f(x)=\\frac{1}{x}+3[\/latex]. What is the horizontal asymptote of the function? What is the relationship between the value of [latex]k[\/latex] and the horizontal asymptote? What is the vertical asymptote of the function?<\/li>\r\n \t<li>Without graphing the function, what is the horizontal asymptote of the function [latex]g(x)=\\frac{1}{x}-7[\/latex]? What is the vertical asymptote?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm613\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm613\"]\r\n\r\n1.\u00a0<img class=\"aligncenter wp-image-3284 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-300x300.png\" alt=\"graph of f(x)=1\/x+3\" width=\"300\" height=\"300\" \/>\r\n\r\nThe horizontal asymptote is the line [latex]y=3[\/latex]. The value of [latex]k[\/latex] is [latex]3[\/latex] so the horizontal asymptote mimics the value of [latex]k[\/latex].\r\n\r\nThe vertical asymptote is the line [latex]x=0[\/latex].\r\n\r\n2. The horizontal asymptote is [latex]y=-7[\/latex]. The vertical asymptote is [latex]x=0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nThe graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=9[\/latex] and a vertical asymptote at [latex]x=0[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.\r\n<h4>Solution<\/h4>\r\nSince the vertical asymptote is at [latex]x=0[\/latex] there is no horizontal shift from the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. Since the horizontal asymptote is at [latex]y=9[\/latex], there has been a vertical shift of 9 units up from the\u00a0parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. This means that [latex]k=9[\/latex], and the function is [latex]f(x)=\\dfrac{1}{x}+9[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nThe graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=-12[\/latex] and a vertical asymptote at [latex]x=0[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.\r\n\r\n[reveal-answer q=\"hjm004\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm004\"][latex]f(x)=\\dfrac{1}{x}-12[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Horizontal Shifts<\/h2>\r\n<\/div>\r\nIf we shift the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] right 7 units, all of the points on the graph increase their [latex]x[\/latex]-coordinates by 7, but their [latex]y[\/latex]-coordinates remain the same. The point (1, 1) in the original graph is moved to (8, 1). Any point [latex](x, y)[\/latex] on the original graph is moved to [latex](x+7, y)[\/latex](figure 4).\r\n\r\nBut what happens to the original function [latex]f(x)=\\dfrac{1}{x}[\/latex]? An automatic assumption may be that since [latex]x[\/latex] moves to [latex]x+7[\/latex] that the function will become [latex]f(x)=\\dfrac{1}{x+7}[\/latex]. But that is NOT the case. Remember that the [latex]x[\/latex]-intercept is moved to (8, 1) and if we substitute [latex]x=8[\/latex] into the function [latex]f(x)=\\dfrac{1}{x+7}[\/latex] we get [latex]f(x)=\\dfrac{1}{15} \\neq 1[\/latex]!! The way to get a function value of 1 is for the transformed function to be [latex]f(x)=\\dfrac{1}{x-7}[\/latex]. Then[latex]f(8)=\\dfrac{1}{8-7}=1[\/latex]. So the function [latex]f(x)=\\dfrac{1}{x}[\/latex] transforms to [latex]f(x)=\\dfrac{1}{x-7}[\/latex] after being shifted 7 units to the right. The reason is that when we move the function 7 units to the right, the [latex]x[\/latex]-value increases by 7 and to keep the corresponding [latex]y[\/latex]-coordinate the same in the transformed function, the [latex]x[\/latex]-coordinate of the transformed function needs to subtract 7 to get back to the original [latex]x[\/latex] that is associated with the original [latex]y[\/latex]-value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 4.\r\n<table style=\"border-collapse: collapse; width: 71.471386%; height: 415px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 20.0263%; height: 10px; text-align: right;\">[latex]x-7[\/latex]<\/th>\r\n<th style=\"width: 22.0147%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)=\\dfrac{1}{x-7}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 415px;\" rowspan=\"9\"><img class=\"alignnone wp-image-3009\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/27032844\/7-2-ShiftRightNew-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 7 units to the right of the blue graph.\" width=\"409\" height=\"409\" \/>\r\n<p id=\"attachment_1828\" class=\"wp-caption aligncenter\" style=\"width: 397px;\">Figure 4. Shifting the graph of the function [latex]f(x)=\\frac{1}{x}[\/latex] right 7 units.<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 95px; text-align: right;\">[latex]\\frac{13}{2}[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 95px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 95px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{15}{2}[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 166px;\">\r\n<td style=\"width: 54.80468613569322%; height: 166px;\" colspan=\"3\">Table 3. Shifting the graph right by 7 units transforms [latex]f(x)=\\frac{1}{x}[\/latex] into [latex]f(x)=\\frac{1}{x-7}[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn the other hand, if we shift the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] left by 11 units, all of the points on the graph decrease their [latex]x[\/latex]-coordinates by 11, but their [latex]y[\/latex]-coordinates remain the same. So any point [latex](x, y)[\/latex] on the original graph moves to [latex](x-11, y)[\/latex]. Consequently, to keep the same [latex]y[\/latex]-values we need to increase the [latex]x[\/latex]-value by 11 in the transformed function. The equation of the function after being shifted left 11 units is [latex]f(x)=\\dfrac{1}{x+11}[\/latex]. Table 4 shows the changes to specific values of this function, and the graph is shown in figure 5.\r\n<table style=\"border-collapse: collapse; width: 71.4714%; height: 402px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 20.0263%; height: 10px; text-align: right;\">[latex]x+11[\/latex]<\/th>\r\n<th style=\"width: 22.0147%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)=\\dfrac{1}{x+11}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 402px;\" rowspan=\"9\">\r\n<div id=\"attachment_1830\" class=\"wp-caption aligncenter\" style=\"width: 407px;\">\r\n\r\n<img class=\"aligncenter wp-image-3010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/27032919\/7-2-ShiftLeftNew-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 11 units to the left of the blue graph.\" width=\"407\" height=\"407\" \/>\r\n\r\nFigure 5. Shifting the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0left 11 units.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u201313[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-12[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{23}{2}[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-\\frac{21}{2}[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-10[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-9[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-8[\/latex]<\/td>\r\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 214px;\">\r\n<td style=\"width: 54.8047%; height: 214px;\" colspan=\"3\">Table 4. Shifting the graph left by 11 units transforms [latex]f(x)=\\dfrac{1}{x}[\/latex] into [latex]f(x)=\\dfrac{1}{x+11}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>horizontal shifts<\/h3>\r\n<p id=\"fs-id1165137770279\">We can represent a horizontal shift of the graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] by subtracting a constant, [latex]h[\/latex], from the variable [latex]x[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=\\dfrac{1}{x-h}[\/latex]<\/p>\r\nIf [latex]h&gt;0[\/latex] the graph shifts toward the right and if [latex]h&lt;0[\/latex] the graph shifts to the left.\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">Manipulate the graph in figure 6 to shift the graph horizontally. Pay attention to what happens with the function as [latex]h[\/latex] changes value. Also, watch what happens with the asymptotes.<\/p>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/zhifvzvdfk?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 6. Horizontal shifts<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\n<ol>\r\n \t<li>Use figure 6 to graph the function [latex]f(x)=\\frac{1}{x+6}[\/latex]. What is the vertical asymptote of the function? What is the relationship between the value of [latex]h[\/latex] and the vertical asymptote? What is the horizontal asymptote of the function?<\/li>\r\n \t<li>Without graphing the function, what is the vertical asymptote of the function [latex]g(x)=\\frac{1}{x+3}[\/latex]? What is the horizontal asymptote?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n1.\u00a0<img class=\"aligncenter size-medium wp-image-3282\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/14225109\/desmos-graph-2022-07-14T165048.115-300x300.png\" alt=\"graph of f(x)=1\/(x+6)\" width=\"300\" height=\"300\" \/>\r\n\r\nThe vertical asymptote is the line [latex]x=-6[\/latex]. The value of [latex]h[\/latex] is [latex]-6[\/latex] so the vertical asymptote mimics the value of [latex]h[\/latex].\r\n\r\nThe horizontal asymptote is the line [latex]y=0[\/latex].\r\n\r\n2. Since [latex]h=-3[\/latex] the graph of [latex]f(x)=\\frac{1}{x}[\/latex] gets shifted left by 3 units. This means that the vertical asymptote is shifted left 3 units to [latex]x=-3[\/latex]. The horizontal asymptote does not move so is still [latex]y=0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\n<ol>\r\n \t<li>Use figure 6 to graph the function [latex]f(x)=\\frac{1}{x+3}[\/latex]. What is the vertical asymptote of the function? What is the relationship between the value of [latex]h[\/latex] and the vertical asymptote? What is the horizontal asymptote of the function?<\/li>\r\n \t<li>Without graphing the function, what is the vertical asymptote of the function [latex]g(x)=\\frac{1}{x-7}[\/latex]? What is the horizontal asymptote?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm614\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm614\"]\r\n\r\n1.\r\n<p style=\"text-align: center;\">\u00a0<img class=\"alignnone wp-image-3411 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-300x300.png\" alt=\"Graph of the function f(x)=1\/(x+3)\" width=\"300\" height=\"300\" \/><\/p>\r\nThe vertical asymptote is the line [latex]x=-3[\/latex]. The value of [latex]h[\/latex] is [latex]-3[\/latex] so the vertical asymptote mimics the value of [latex]h[\/latex].\r\n\r\nThe horizontal asymptote is the line [latex]y=0[\/latex].\r\n\r\n2. The vertical asymptote is [latex]x=7[\/latex]. The horizontal asymptote is [latex]y=0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nThe graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=0[\/latex] and a vertical asymptote at [latex]x=5[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.\r\n<h4>Solution<\/h4>\r\nSince the horizontal asymptote is at [latex]y=0[\/latex] there is no vertical shift from the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. Since the vertical asymptote is at [latex]x=5[\/latex], there has been a horizontal shift of 5 units right from the\u00a0parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. This means that [latex]h=5[\/latex], and the function is [latex]f(x)=\\dfrac{1}{x-5}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nThe graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=0[\/latex] and a vertical asymptote at [latex]x=-9[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.\r\n\r\n[reveal-answer q=\"hjm400\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm400\"][latex]f(x)=\\dfrac{1}{x+9}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Combining Vertical and Horizontal Shifts<\/h2>\r\n<div class=\"textbox shaded\">\r\n<h3>vertical and horizontal shifts<\/h3>\r\n<p id=\"fs-id1165137770279\">We can represent both a vertical and a horizontal shift of the graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] by the transformed function<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=\\dfrac{1}{x-h}+k[\/latex]<\/p>\r\nIf [latex]h&gt;0[\/latex] the graph shifts toward the right and if [latex]h&lt;0[\/latex] the graph shifts to the left.\r\n\r\nIf [latex]k&gt;0[\/latex] the graph shifts upwards and if [latex]k&lt;0[\/latex] the graph shifts downwards.\r\n\r\n<\/div>\r\nWhen the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is shifted both vertically and horizontally, there will be non-zero values for both [latex]h[\/latex] and\u00a0[latex]k[\/latex]. The graph in figure 7 can be manipulated both vertically and horizontally. Pay attention to what happens to the function as the values for\u00a0[latex]h[\/latex] and\u00a0[latex]k[\/latex] change. Also notice what happens to the asymptotes.\r\n\r\n<\/div>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/uaf2m6vgf0?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 7. Vertical and horizontal shifts<\/p>\r\n\r\n<div class=\"entry-content\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nThe graph of the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is shifted up by 4 units and left by 7 units.\r\n<ol>\r\n \t<li>Determine the equation of the transformed function.<\/li>\r\n \t<li>Determine the vertical asymptote.<\/li>\r\n \t<li>Determine the horizontal asymptote.<\/li>\r\n \t<li>The point [latex](2, \\frac{1}{2})[\/latex] lies on the parent function. Where does it end up after the transformation?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>Shifted up by 4 units means [latex]k=4[\/latex] and shifted left by 7 units\u00a0means [latex]h=-7[\/latex]. Therefore the function is [latex]f(x)=\\dfrac{1}{x-h}+k=\\dfrac{1}{x+7}+4[\/latex].<\/li>\r\n \t<li>Since the graph is shifted left 7 units, the vertical asymptote is also shifted left 7 units: [latex]x=-7[\/latex].<\/li>\r\n \t<li>Since the graph is shifted up 4 units, the horizontal asymptote is also shifted up 4 units: [latex]y=4[\/latex].<\/li>\r\n \t<li>Since the graph is shifted left 7 units, the [latex]x[\/latex]-coordinate is shifted left 7 units to 2 \u2013 7 = \u20135.\u00a0Since the graph is shifted up 4 units, the [latex]y[\/latex]-coordinate is shifted up 4 units to [latex]\\frac{1}{2}+4=\\frac{9}{2}[\/latex]. Consequently,[latex](2, \\frac{1}{2})[\/latex] moves to\u00a0[latex](-5, \\frac{9}{2})[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nThe graph of the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is shifted down by 3 units and right by 2 units.\r\n<ol>\r\n \t<li>Determine the equation of the transformed function.<\/li>\r\n \t<li>Determine the vertical asymptote.<\/li>\r\n \t<li>Determine the horizontal asymptote.The point [latex](2, \\frac{1}{2})[\/latex] lies on the parent function. Where does it end up after the transformation?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm730\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm730\"]\r\n<ol>\r\n \t<li>[latex]f(x)=\\dfrac{1}{x-2}-3[\/latex]<\/li>\r\n \t<li>[latex]x=2[\/latex]<\/li>\r\n \t<li>[latex]y=-3[\/latex]<\/li>\r\n \t<li>[latex](4, -\\frac{5}{2})[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Stretching and Compressing<\/h2>\r\nIf we vertically stretch the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] by a factor of 7, all of the[latex]y[\/latex]-coordinates of the points on the graph are multiplied by 7, but their [latex]x[\/latex]-coordinates remain the same. The equation of the function after the graph is stretched is [latex]f(x)=7\\times\\dfrac{1}{x}=\\dfrac{7}{x}[\/latex]. The reason for multiplying [latex]f(x)=\\dfrac{1}{x}[\/latex] by 7 is that each [latex]y[\/latex]-coordinate is made 7 times larger, and since [latex]y=\\dfrac{1}{x}[\/latex], [latex]\\dfrac{1}{x}[\/latex] is also made 7 times larger. Table 5 shows this change and the graph is shown in figure 8.\r\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 365px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]f(x)=\\dfrac{7}{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 365px;\" rowspan=\"9\">\r\n<div id=\"attachment_1832\" class=\"wp-caption aligncenter\" style=\"width: 416px;\">\r\n\r\n<img class=\"aligncenter wp-image-2797\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/19202358\/7-2-Stretch1-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a vertically stretched green graph. One green branch opens to the left and downward, lying 7 times below the blue graph, while the other branch opens to the right and upward, lying 7 times above the blue graph.\" width=\"416\" height=\"416\" \/>\r\n\r\nFigure 8. Stretching the graph vertically.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\frac{7}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\frac{7}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\frac{7}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 177px;\">\r\n<td style=\"width: 42.1941%; height: 177px;\" colspan=\"3\">Table 5.\u00a0Stretching the graph vertically 7 times transforms [latex]\\dfrac{1}{x}[\/latex] into [latex]\\dfrac{7}{x}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn the other hand, if we vertically compress the graph of the function [latex]\\dfrac{1}{x}[\/latex] into [latex]\\dfrac{1}{5}[\/latex] of its original height, all of the [latex]y[\/latex]-coordinates of the points on the graph are divided by 5, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are multiplied by [latex]\\dfrac{1}{5}[\/latex]. The equation of the function after being compressed is [latex]f(x)=\\dfrac{1}{5}\\times\\dfrac{1}{x}=\\dfrac{1}{5x}[\/latex]. The reason for multiplying [latex]f(x)=\\dfrac{1}{x}[\/latex] by [latex]\\dfrac{1}{5}[\/latex] is that each [latex]y[\/latex]-coordinate becomes [latex]\\dfrac{1}{5}[\/latex]\u00a0of the original value. Table 6 shows this change and the graph is shown in figure 9.\r\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 152px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]\\dfrac{1}{5x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%;\" rowspan=\"9\">\r\n<div id=\"attachment_1866\" class=\"wp-caption aligncenter\" style=\"width: 419px;\">\r\n\r\n<img class=\"aligncenter wp-image-2798\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/19202431\/7-2-Compress-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a vertically compressed green graph. One green branch opens to the left and downward and lies at one-fifth the height of the blue graph, while the other branch opens to the right and upward and lies at one-fifth the height of the blue graph.\" width=\"419\" height=\"419\" \/>\r\n\r\nFigure 9. Compress the graph into [latex]\\frac{1}{5}[\/latex] of the original height.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{10}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\dfrac{2}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{2}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{10}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{15}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 42.1941%;\" colspan=\"3\">Table 6.\u00a0Compressing the graph vertically into [latex]\\frac{1}{5}[\/latex] of the original height transforms [latex]f(x)=\\frac{1}{x}[\/latex] into [latex]f(x)=\\frac{1}{5x}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>vertical stretching and compressing<\/h3>\r\n<p id=\"fs-id1165137770279\">A vertical stretch or compression of the graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] can be represented by\u00a0multiplying the function by a constant, [latex]a&gt;0[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=a\\dfrac{1}{x}[\/latex]<\/p>\r\nThe magnitude of [latex]a[\/latex]\u00a0indicates the stretch\/compression of the graph. If [latex]a&gt;1[\/latex], the graph is stretched. If [latex]0&lt;a&lt;1[\/latex], the graph is compressed.\r\n\r\n<\/div>\r\nMove the red dot in figure 10 to change the value of [latex]a[\/latex]. Notice whether the graph is stretching or compressing depending on the value of [latex]a[\/latex]. Notice also what happens to the function.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/fhnblbhdvm?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">FIgure 10. Stretching and compressing<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\n<ol>\r\n \t<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is stretched by a factor of 5. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/li>\r\n \t<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is compressed to [latex]\\frac{1}{2}[\/latex] its size. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>Stretched by a factor of 5 means [latex]a=5[\/latex], therefore, the transformed function is [latex]f(x)=5\\left(\\dfrac{1}{x}\\right)[\/latex]. This can also be written as [latex]f(x)=\\dfrac{5}{x}[\/latex]. When a function is stretched its [latex]x[\/latex]-value stays the same while the [latex]y[\/latex]-value is multiplied by the stretch factor. So, (1, 1) moves to (1, 5).<\/li>\r\n \t<li>Compressed to [latex]\\frac{1}{2}[\/latex] its size means [latex]a=\\dfrac{1}{2}[\/latex], therefore, the transformed function is [latex]f(x)=\\dfrac{1}{2}\\left(\\dfrac{1}{x}\\right)[\/latex]. This can also be written as [latex]f(x)=\\dfrac{1}{2x}[\/latex].<\/li>\r\n<\/ol>\r\nWhen a function is compressed its [latex]x[\/latex]-value stays the same while the [latex]y[\/latex]-value is multiplied by the compression. So, (1, 1) moves to [latex](1, \\frac{1}{5})[\/latex].\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\n<ol>\r\n \t<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is stretched by a factor of 4. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/li>\r\n \t<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is compressed to [latex]\\frac{1}{8}[\/latex]th its size. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/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>[latex]f(x)=4\\left(\\dfrac{1}{x}\\right)=\\dfrac{4}{x}\\;\\;\\;\\;(1, 4)[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{1}{8}\\left(\\dfrac{1}{x}\\right)=\\dfrac{1}{8x}\\;\\;\\;\\;(1, \\frac{1}{8})[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflections<\/h2>\r\n<h3>across the [latex]x[\/latex]-axis<\/h3>\r\nWhen the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] is reflected across the\u00a0[latex]x[\/latex]-axis, the\u00a0[latex]y[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values or from negative to positive values, while the [latex]x[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\\dfrac{1}{x}[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-\\dfrac{1}{x}[\/latex]. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 11.\r\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 342px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]-\\dfrac{1}{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 342px;\" rowspan=\"9\">\r\n<div id=\"attachment_1868\" class=\"wp-caption aligncenter\" style=\"width: 434px;\">\r\n\r\n<img class=\"aligncenter wp-image-2803\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/20001957\/7-2-ReflectionX1-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a reflected green graph across the x-axis. One green branch opens to the left and upward, while the other branch opens to the right and downward.\" width=\"434\" height=\"434\" \/>\r\n\r\nFigure 11. Reflecting the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0across the [latex]x[\/latex]-axis.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20132<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20131<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">3<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\dfrac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 154px;\">\r\n<td style=\"width: 42.1941%; height: 154px;\" colspan=\"3\">Table 7.\u00a0Reflecting the graph of [latex]f(x)=\\dfrac{1}{x}[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=\\dfrac{1}{x}[\/latex] into [latex]f(x)=-\\dfrac{1}{x}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>across the [latex]y[\/latex]-axis<\/h3>\r\nWhen the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0is reflected across the\u00a0[latex]y[\/latex]-axis, the\u00a0[latex]x[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values, or from negative to positive values, while the [latex]y[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0is reflected across the [latex]y[\/latex]-axis is [latex]f(x)=\\dfrac{1}{-x}=-\\dfrac{1}{x}[\/latex]. This equation is the same as the equation after being reflected across the [latex]x[\/latex]-axis. Therefore, its graph is the same as the graph after being reflected across the [latex]x[\/latex]-axis, even though it got there by a different route (note the differences in figures 11 and 13).\u00a0Table 8 shows the effect of such a reflection on the functions values and the graph is shown in figure 13.\r\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 683px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]\\dfrac{1}{-x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 306px;\" rowspan=\"9\">\r\n<div id=\"attachment_1868\" class=\"wp-caption aligncenter\" style=\"width: 437px;\">\r\n\r\n<img class=\"aligncenter wp-image-2802\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/20001933\/7-2-ReflectionY-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a reflected green graph across the y-axis. One green branch opens to the left and upward, while the other branch opens to the right and downward.\" width=\"437\" height=\"437\" \/>\r\n\r\nFigure 13. Reflecting the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0across the [latex]y[\/latex]-axis.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\r\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20132<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20131<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">3<\/td>\r\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\frac{1}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 154px;\">\r\n<td style=\"width: 42.1941%; height: 154px;\" colspan=\"3\">Table 8.\u00a0Reflecting the graph of [latex]f(x)=\\dfrac{1}{x}[\/latex] across the [latex]y[\/latex]-axis transforms [latex]f(x)=\\dfrac{1}{x}[\/latex] into [latex]f(x)=\\dfrac{1}{-x}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nExplain why the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed to [latex]f(x)=-\\dfrac{1}{x}[\/latex] when it is reflected across the [latex]x[\/latex]-axis.\r\n<h4>Solution<\/h4>\r\nWith reflection across the [latex]x[\/latex]-axis, the [latex]x[\/latex] values stay the same and the [latex]y[\/latex]-values change sign. So, [latex]y=\\dfrac{1}{x}[\/latex] transforms to [latex]-y=\\dfrac{1}{x}[\/latex]. If we multiply (or divide) both sides of the equation by \u20131 we get, [latex]y=-\\dfrac{1}{x}[\/latex]. Consequently, the transformed function is [latex]f(x)=-\\dfrac{1}{x}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nExplain why the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed to [latex]f(x)=-\\dfrac{1}{x}[\/latex] when it is reflected across the [latex]y[\/latex]-axis.\r\n\r\n[reveal-answer q=\"hjm518\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm518\"]\r\n\r\nWith reflection across the [latex]y[\/latex]-axis, the [latex]y[\/latex] values stay the same and the [latex]x[\/latex]-values change sign. So, [latex]y=\\dfrac{1}{x}[\/latex] transforms to [latex]y=\\dfrac{1}{-x}[\/latex], which simplifies to [latex]y=-\\dfrac{1}{x}[\/latex]. Consequently, the transformed function is [latex]f(x)=-\\dfrac{1}{x}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Combining Transformations<\/h2>\r\nNow that we have learned all the transformations for the function [latex]f(x)=\\dfrac{1}{x}[\/latex], we should be able to write the transformed function equation given specific transformations, and determine what transformations have been performed on the function [latex]f(x)=\\dfrac{1}{x}[\/latex], given an arbitrary transformed function [latex]f(x)=a\\dfrac{1}{x-h}+k[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Combining Transformations<\/h3>\r\n<p id=\"fs-id1165137770279\">The graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] can be shifted horizontally ([latex]h[\/latex]-value), shifted vertically\u00a0([latex]k[\/latex]-value), stretched or compressed ([latex]a[\/latex]-value), reflected across the [latex]x[\/latex]-axis (negative function value} or reflected across the [latex]y[\/latex]-axis (negative [latex]x[\/latex]-value) and can be represented by\u00a0the function<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\nThe parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed. Write the equation of the transformed function.\r\n<ol>\r\n \t<li>Stretched by a factor of 2; shifted vertically up 3 units; shifted horizontally 5 units right.<\/li>\r\n \t<li>Shifted horizontally left by 7 units; reflected across the [latex]x[\/latex]-axis.<\/li>\r\n \t<li>Compressed to [latex]\\frac{1}{3}[\/latex]rd its height; reflected across the [latex]y[\/latex]-axis; shifted vertically down by 4 units.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]a=2,\\;k=3,\\;h=5[\/latex] so function is [latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k=2\\left(\\dfrac{1}{x-5}\\right)+3[\/latex]<\/li>\r\n \t<li>[latex]a=-1,\\;k=0,\\;h=-7[\/latex] so function is [latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k=-\\dfrac{1}{x+7}[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{3},\\;x\\rightarrow -x,\\;k=-4[\/latex] so function is [latex]f(x)=\\dfrac{1}{3}\\left(\\dfrac{1}{-x}\\right)-4=-\\dfrac{1}{3}\\left(\\dfrac{1}{x}\\right)-4[\/latex]. This can also be written as [latex]f(x)=-\\dfrac{1}{3x}-4[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 8<\/h3>\r\nThe parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed. Write the equation of the transformed function.\r\n<ol>\r\n \t<li>Stretched by a factor of 4; shifted vertically down by 7 units; shifted horizontally 3 units left.<\/li>\r\n \t<li>Shifted horizontally left by 8 units; reflected across the [latex]x[\/latex]-axis.<\/li>\r\n \t<li>Compressed to [latex]\\frac{1}{5}[\/latex]th its height; reflected across the [latex]x[\/latex]-axis; shifted vertically down by 4 units; shifted right by 2 units.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm236\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm236\"]\r\n<ol>\r\n \t<li>[latex]f(x)=4\\left(\\dfrac{1}{x+3}\\right)-7[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(x)=-\\dfrac{1}{x+8}[\/latex]<\/li>\r\n \t<li>\u00a0[latex]f(x)=-\\frac{1}{5}\\left(\\dfrac{1}{x-2}\\right)-4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 9<\/h3>\r\nDetermine the transformations made to the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] to get the transformed function [latex]f(x)=-\\dfrac{7}{x-4}+3[\/latex].\r\n<h4>Solution<\/h4>\r\n[latex]f(x)=-\\dfrac{7}{x-4}+3[\/latex] can be written as [latex]f(x)=-7\\left(\\dfrac{1}{x-4}\\right)+3[\/latex].\r\n\r\nComparing\u00a0[latex]f(x)=-7\\left(\\dfrac{1}{x-4}\\right)+3[\/latex] to\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex], we get [latex]a=-7,\\;h=4,\\;k=3[\/latex], which translated to a reflection across the [latex]x[\/latex]-axis, a stretch by a factor of 7, a horizontal shift to the right by 4 units, and a vertical shift up by 3 units.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 9<\/h3>\r\nDetermine the transformations made to the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] to get the transformed function [latex]f(x)=-\\dfrac{5}{3x}-7[\/latex].\r\n\r\n[reveal-answer q=\"hjm245\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm245\"]\r\n\r\nReflection across the [latex]x[\/latex]-axis (negative sign in front)\r\n\r\nStretched to\u00a0[latex]\\frac{5}{3}[\/latex]rds its height\r\n\r\nShifted down by 7 units\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Formats of Rational Functions<\/h2>\r\nYou may be wondering why we defined a rational function as [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] when we write a rational function using transformations in the form\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex].\r\n\r\nIt is simply that rational functions in the form\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex] are based on the parent rational function [latex]f(x)=\\dfrac{1}{x}[\/latex], and a special case of\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex].\r\n\r\nWe could take the function\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex] and write it as a single fraction by using a common denominator:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(x)&amp;=a\\left(\\dfrac{1}{x-h}\\right)+k\\\\\\\\&amp;=a\\left(\\dfrac{1}{x-h}\\right)+k\\color{blue}{\\cdot \\dfrac{x-h}{x-h}}\\\\\\\\&amp;=\\dfrac{a+k(x-h)}{x-h}\\\\\\\\&amp;=\\dfrac{kx+(a-hk)}{x-h}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\nNow the function is written in standard form with [latex]P(x)=kx+(a-hk)[\/latex] and [latex]Q(x)=x-h[\/latex]. Both [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] have degree 1.\r\n\r\nConsequently, a rational function in the transformed form\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex] is a special case of the standard form\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] with\u00a0[latex]P(x)[\/latex] and [latex]Q(x)[\/latex] having degree 1.\r\n\r\nWe leave the more in depth study of rational functions in the form\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] to College Algebra.\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<p>For the rational parent function [latex]f(x)=\\dfrac{1}{x}[\/latex],<\/p>\n<ul>\n<li>Perform vertical and horizontal shifts<\/li>\n<li>Perform vertical stretches and compressions<\/li>\n<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\n<li>Perform reflections across the [latex]y[\/latex]-axis<\/li>\n<li>Determine the transformations performed on the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] to get the rational function [latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"entry-content\">\n<h2>Vertical Shifts<\/h2>\n<p id=\"fs-id1165137770279\">If we shift the graph of the rational function [latex]f(x)=\\dfrac{1}{x}[\/latex] up 5 units, all of the points on the graph increase their [latex]y[\/latex]-coordinates by 5, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=\\dfrac{1}{x}+5[\/latex]. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.<\/p>\n<table style=\"border-collapse: collapse; width: 68.81378216718218%; height: 206px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\n<th style=\"width: 29.2773%; height: 10px; text-align: right;\" scope=\"row\">[latex]\\dfrac{1}{x}+5[\/latex]<\/th>\n<td style=\"width: 21.566613864306788%; height: 206px;\" rowspan=\"9\">\n<div id=\"attachment_1823\" class=\"wp-caption aligncenter\" style=\"width: 408px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2787\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/18214620\/7-2-ShiftUp1-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 5 units above the blue graph.\" width=\"408\" height=\"408\" \/><\/p>\n<p>Figure 1. Shifting the graph of the function\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] up 5 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]\\dfrac{9}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\\dfrac{11}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\\dfrac{16}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 54.80468613569322%;\" colspan=\"3\">Table 1. [latex]f(x)=\\frac{1}{x}[\/latex] is transformed to[latex]f(x)=\\frac{1}{x}+5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=\\frac{1}{x}[\/latex] down 8 units, all of the points on the graph decrease their [latex]y[\/latex]-coordinates by 8, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\frac{1}{x}[\/latex] after it has been shifted down 8 units transforms to [latex]f(x)=\\frac{1}{x}-8[\/latex]. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.<\/p>\n<table style=\"border-collapse: collapse; width: 71.47138613569322%; height: 222px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\n<th style=\"width: 29.2773%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)=\\dfrac{1}{x}-8[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 222px;\" rowspan=\"9\">\n<div id=\"attachment_1825\" class=\"wp-caption aligncenter\" style=\"width: 389px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2788\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/18214643\/7-2-ShiftDown2-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 8 units below the blue graph.\" width=\"389\" height=\"389\" \/><\/p>\n<p>Figure 2. Shifting the graph of the function\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] down 8\u00a0units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]-\\dfrac{17}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]\u20139[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 34px; text-align: right;\">[latex]\u201310[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\u20136[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]\u20137[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]-\\frac{15}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\n<td style=\"width: 29.2773%; height: 19px; text-align: right;\">[latex]-\\frac{23}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 54.80468613569322%; height: 34px;\" colspan=\"3\">Table 2.\u00a0[latex]f(x)=\\frac{1}{x}[\/latex] is transformed to \u00a0[latex]f(x)=\\frac{1}{x}-8[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">\n<h3>Vertical shifts<\/h3>\n<p>We can represent a vertical shift of the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] by adding a constant, [latex]k[\/latex], to the function:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\dfrac{1}{x}+k[\/latex]<\/p>\n<p>\u00a0If [latex]k>0[\/latex], the graph shifts upwards and if [latex]k<0[\/latex] the graph shifts downwards.\n\n<\/div>\n<p>Manipulate the graph in figure 3 to shift the graph vertically. Pay attention to what happens with the function as [latex]k[\/latex] changes value. Also, watch what happens with the asymptotes.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/q62i2yudlk?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 3. Vertical Transformations<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<ol>\n<li>Use figure 3 to graph the function [latex]f(x)=\\frac{1}{x}-6[\/latex]. What is the horizontal asymptote of the function? What is the relationship between the value of [latex]k[\/latex] and the horizontal asymptote? What is the vertical asymptote of the function?<\/li>\n<li>Without graphing the function, what is the horizontal asymptote of the function [latex]g(x)=\\frac{1}{x}+5[\/latex]? What is the vertical asymptote?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>1.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3280 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-300x300.png\" alt=\"Graph of g(x)=1\/x -6\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T161001.019.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The horizontal asymptote is the line [latex]y=-6[\/latex]. The value of [latex]k[\/latex] is [latex]-6[\/latex] so the horizontal asymptote mimics the value of [latex]k[\/latex].<\/p>\n<p>The vertical asymptote is the line [latex]x=0[\/latex].<\/p>\n<p>2. Since [latex]k=5[\/latex] the graph of [latex]f(x)=\\frac{1}{x}[\/latex] gets shifted up by 5 units. This means that the horizontal asymptote is shifted up 5 units to [latex]y=5[\/latex]. The vertical asymptote does not move so is still [latex]x=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<ol>\n<li>Use figure 3 to graph the function [latex]f(x)=\\frac{1}{x}+3[\/latex]. What is the horizontal asymptote of the function? What is the relationship between the value of [latex]k[\/latex] and the horizontal asymptote? What is the vertical asymptote of the function?<\/li>\n<li>Without graphing the function, what is the horizontal asymptote of the function [latex]g(x)=\\frac{1}{x}-7[\/latex]? What is the vertical asymptote?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm613\">Show Answer<\/span><\/p>\n<div id=\"qhjm613\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3284 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-300x300.png\" alt=\"graph of f(x)=1\/x+3\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/desmos-graph-2022-07-14T171609.382.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The horizontal asymptote is the line [latex]y=3[\/latex]. The value of [latex]k[\/latex] is [latex]3[\/latex] so the horizontal asymptote mimics the value of [latex]k[\/latex].<\/p>\n<p>The vertical asymptote is the line [latex]x=0[\/latex].<\/p>\n<p>2. The horizontal asymptote is [latex]y=-7[\/latex]. The vertical asymptote is [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>The graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=9[\/latex] and a vertical asymptote at [latex]x=0[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.<\/p>\n<h4>Solution<\/h4>\n<p>Since the vertical asymptote is at [latex]x=0[\/latex] there is no horizontal shift from the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. Since the horizontal asymptote is at [latex]y=9[\/latex], there has been a vertical shift of 9 units up from the\u00a0parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. This means that [latex]k=9[\/latex], and the function is [latex]f(x)=\\dfrac{1}{x}+9[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>The graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=-12[\/latex] and a vertical asymptote at [latex]x=0[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm004\">Show Answer<\/span><\/p>\n<div id=\"qhjm004\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=\\dfrac{1}{x}-12[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Horizontal Shifts<\/h2>\n<\/div>\n<p>If we shift the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] right 7 units, all of the points on the graph increase their [latex]x[\/latex]-coordinates by 7, but their [latex]y[\/latex]-coordinates remain the same. The point (1, 1) in the original graph is moved to (8, 1). Any point [latex](x, y)[\/latex] on the original graph is moved to [latex](x+7, y)[\/latex](figure 4).<\/p>\n<p>But what happens to the original function [latex]f(x)=\\dfrac{1}{x}[\/latex]? An automatic assumption may be that since [latex]x[\/latex] moves to [latex]x+7[\/latex] that the function will become [latex]f(x)=\\dfrac{1}{x+7}[\/latex]. But that is NOT the case. Remember that the [latex]x[\/latex]-intercept is moved to (8, 1) and if we substitute [latex]x=8[\/latex] into the function [latex]f(x)=\\dfrac{1}{x+7}[\/latex] we get [latex]f(x)=\\dfrac{1}{15} \\neq 1[\/latex]!! The way to get a function value of 1 is for the transformed function to be [latex]f(x)=\\dfrac{1}{x-7}[\/latex]. Then[latex]f(8)=\\dfrac{1}{8-7}=1[\/latex]. So the function [latex]f(x)=\\dfrac{1}{x}[\/latex] transforms to [latex]f(x)=\\dfrac{1}{x-7}[\/latex] after being shifted 7 units to the right. The reason is that when we move the function 7 units to the right, the [latex]x[\/latex]-value increases by 7 and to keep the corresponding [latex]y[\/latex]-coordinate the same in the transformed function, the [latex]x[\/latex]-coordinate of the transformed function needs to subtract 7 to get back to the original [latex]x[\/latex] that is associated with the original [latex]y[\/latex]-value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 4.<\/p>\n<table style=\"border-collapse: collapse; width: 71.471386%; height: 415px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 20.0263%; height: 10px; text-align: right;\">[latex]x-7[\/latex]<\/th>\n<th style=\"width: 22.0147%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)=\\dfrac{1}{x-7}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 415px;\" rowspan=\"9\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3009\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/27032844\/7-2-ShiftRightNew-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 7 units to the right of the blue graph.\" width=\"409\" height=\"409\" \/><\/p>\n<p id=\"attachment_1828\" class=\"wp-caption aligncenter\" style=\"width: 397px;\">Figure 4. Shifting the graph of the function [latex]f(x)=\\frac{1}{x}[\/latex] right 7 units.<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 95px; text-align: right;\">[latex]\\frac{13}{2}[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 95px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 95px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{15}{2}[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 166px;\">\n<td style=\"width: 54.80468613569322%; height: 166px;\" colspan=\"3\">Table 3. Shifting the graph right by 7 units transforms [latex]f(x)=\\frac{1}{x}[\/latex] into [latex]f(x)=\\frac{1}{x-7}[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>On the other hand, if we shift the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] left by 11 units, all of the points on the graph decrease their [latex]x[\/latex]-coordinates by 11, but their [latex]y[\/latex]-coordinates remain the same. So any point [latex](x, y)[\/latex] on the original graph moves to [latex](x-11, y)[\/latex]. Consequently, to keep the same [latex]y[\/latex]-values we need to increase the [latex]x[\/latex]-value by 11 in the transformed function. The equation of the function after being shifted left 11 units is [latex]f(x)=\\dfrac{1}{x+11}[\/latex]. Table 4 shows the changes to specific values of this function, and the graph is shown in figure 5.<\/p>\n<table style=\"border-collapse: collapse; width: 71.4714%; height: 402px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 20.0263%; height: 10px; text-align: right;\">[latex]x+11[\/latex]<\/th>\n<th style=\"width: 22.0147%; height: 10px; text-align: right;\" scope=\"row\">[latex]f(x)=\\dfrac{1}{x+11}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 402px;\" rowspan=\"9\">\n<div id=\"attachment_1830\" class=\"wp-caption aligncenter\" style=\"width: 407px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/27032919\/7-2-ShiftLeftNew-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, with an identical green graph 11 units to the left of the blue graph.\" width=\"407\" height=\"407\" \/><\/p>\n<p>Figure 5. Shifting the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0left 11 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]\u201313[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-12[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{23}{2}[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-\\frac{21}{2}[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-10[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-9[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]-8[\/latex]<\/td>\n<td style=\"width: 20.0263%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 22.0147%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 214px;\">\n<td style=\"width: 54.8047%; height: 214px;\" colspan=\"3\">Table 4. Shifting the graph left by 11 units transforms [latex]f(x)=\\dfrac{1}{x}[\/latex] into [latex]f(x)=\\dfrac{1}{x+11}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>horizontal shifts<\/h3>\n<p id=\"fs-id1165137770279\">We can represent a horizontal shift of the graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] by subtracting a constant, [latex]h[\/latex], from the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\dfrac{1}{x-h}[\/latex]<\/p>\n<p>If [latex]h>0[\/latex] the graph shifts toward the right and if [latex]h<0[\/latex] the graph shifts to the left.\n\n<\/div>\n<p style=\"text-align: left;\">Manipulate the graph in figure 6 to shift the graph horizontally. Pay attention to what happens with the function as [latex]h[\/latex] changes value. Also, watch what happens with the asymptotes.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/zhifvzvdfk?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 6. Horizontal shifts<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<ol>\n<li>Use figure 6 to graph the function [latex]f(x)=\\frac{1}{x+6}[\/latex]. What is the vertical asymptote of the function? What is the relationship between the value of [latex]h[\/latex] and the vertical asymptote? What is the horizontal asymptote of the function?<\/li>\n<li>Without graphing the function, what is the vertical asymptote of the function [latex]g(x)=\\frac{1}{x+3}[\/latex]? What is the horizontal asymptote?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>1.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-3282\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/14225109\/desmos-graph-2022-07-14T165048.115-300x300.png\" alt=\"graph of f(x)=1\/(x+6)\" width=\"300\" height=\"300\" \/><\/p>\n<p>The vertical asymptote is the line [latex]x=-6[\/latex]. The value of [latex]h[\/latex] is [latex]-6[\/latex] so the vertical asymptote mimics the value of [latex]h[\/latex].<\/p>\n<p>The horizontal asymptote is the line [latex]y=0[\/latex].<\/p>\n<p>2. Since [latex]h=-3[\/latex] the graph of [latex]f(x)=\\frac{1}{x}[\/latex] gets shifted left by 3 units. This means that the vertical asymptote is shifted left 3 units to [latex]x=-3[\/latex]. The horizontal asymptote does not move so is still [latex]y=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<ol>\n<li>Use figure 6 to graph the function [latex]f(x)=\\frac{1}{x+3}[\/latex]. What is the vertical asymptote of the function? What is the relationship between the value of [latex]h[\/latex] and the vertical asymptote? What is the horizontal asymptote of the function?<\/li>\n<li>Without graphing the function, what is the vertical asymptote of the function [latex]g(x)=\\frac{1}{x-7}[\/latex]? What is the horizontal asymptote?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm614\">Show Answer<\/span><\/p>\n<div id=\"qhjm614\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p style=\"text-align: center;\">\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3411 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-300x300.png\" alt=\"Graph of the function f(x)=1\/(x+3)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/06\/7-2-TryIt3.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The vertical asymptote is the line [latex]x=-3[\/latex]. The value of [latex]h[\/latex] is [latex]-3[\/latex] so the vertical asymptote mimics the value of [latex]h[\/latex].<\/p>\n<p>The horizontal asymptote is the line [latex]y=0[\/latex].<\/p>\n<p>2. The vertical asymptote is [latex]x=7[\/latex]. The horizontal asymptote is [latex]y=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>The graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=0[\/latex] and a vertical asymptote at [latex]x=5[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.<\/p>\n<h4>Solution<\/h4>\n<p>Since the horizontal asymptote is at [latex]y=0[\/latex] there is no vertical shift from the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. Since the vertical asymptote is at [latex]x=5[\/latex], there has been a horizontal shift of 5 units right from the\u00a0parent function [latex]f(x)=\\dfrac{1}{x}[\/latex]. This means that [latex]h=5[\/latex], and the function is [latex]f(x)=\\dfrac{1}{x-5}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>The graph of the rational function [latex]y=f(x)[\/latex] has a horizontal asymptote at [latex]y=0[\/latex] and a vertical asymptote at [latex]x=-9[\/latex]. Determine a rational functional [latex]f(x)[\/latex] whose graph meets these criteria.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm400\">Show Answer<\/span><\/p>\n<div id=\"qhjm400\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=\\dfrac{1}{x+9}[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2>Combining Vertical and Horizontal Shifts<\/h2>\n<div class=\"textbox shaded\">\n<h3>vertical and horizontal shifts<\/h3>\n<p id=\"fs-id1165137770279\">We can represent both a vertical and a horizontal shift of the graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] by the transformed function<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\dfrac{1}{x-h}+k[\/latex]<\/p>\n<p>If [latex]h>0[\/latex] the graph shifts toward the right and if [latex]h<0[\/latex] the graph shifts to the left.\n\nIf [latex]k>0[\/latex] the graph shifts upwards and if [latex]k<0[\/latex] the graph shifts downwards.\n\n<\/div>\n<p>When the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is shifted both vertically and horizontally, there will be non-zero values for both [latex]h[\/latex] and\u00a0[latex]k[\/latex]. The graph in figure 7 can be manipulated both vertically and horizontally. Pay attention to what happens to the function as the values for\u00a0[latex]h[\/latex] and\u00a0[latex]k[\/latex] change. Also notice what happens to the asymptotes.<\/p>\n<\/div>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/uaf2m6vgf0?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 7. Vertical and horizontal shifts<\/p>\n<div class=\"entry-content\">\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>The graph of the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is shifted up by 4 units and left by 7 units.<\/p>\n<ol>\n<li>Determine the equation of the transformed function.<\/li>\n<li>Determine the vertical asymptote.<\/li>\n<li>Determine the horizontal asymptote.<\/li>\n<li>The point [latex](2, \\frac{1}{2})[\/latex] lies on the parent function. Where does it end up after the transformation?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>Shifted up by 4 units means [latex]k=4[\/latex] and shifted left by 7 units\u00a0means [latex]h=-7[\/latex]. Therefore the function is [latex]f(x)=\\dfrac{1}{x-h}+k=\\dfrac{1}{x+7}+4[\/latex].<\/li>\n<li>Since the graph is shifted left 7 units, the vertical asymptote is also shifted left 7 units: [latex]x=-7[\/latex].<\/li>\n<li>Since the graph is shifted up 4 units, the horizontal asymptote is also shifted up 4 units: [latex]y=4[\/latex].<\/li>\n<li>Since the graph is shifted left 7 units, the [latex]x[\/latex]-coordinate is shifted left 7 units to 2 \u2013 7 = \u20135.\u00a0Since the graph is shifted up 4 units, the [latex]y[\/latex]-coordinate is shifted up 4 units to [latex]\\frac{1}{2}+4=\\frac{9}{2}[\/latex]. Consequently,[latex](2, \\frac{1}{2})[\/latex] moves to\u00a0[latex](-5, \\frac{9}{2})[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>The graph of the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is shifted down by 3 units and right by 2 units.<\/p>\n<ol>\n<li>Determine the equation of the transformed function.<\/li>\n<li>Determine the vertical asymptote.<\/li>\n<li>Determine the horizontal asymptote.The point [latex](2, \\frac{1}{2})[\/latex] lies on the parent function. Where does it end up after the transformation?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm730\">Show Answer<\/span><\/p>\n<div id=\"qhjm730\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=\\dfrac{1}{x-2}-3[\/latex]<\/li>\n<li>[latex]x=2[\/latex]<\/li>\n<li>[latex]y=-3[\/latex]<\/li>\n<li>[latex](4, -\\frac{5}{2})[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Stretching and Compressing<\/h2>\n<p>If we vertically stretch the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] by a factor of 7, all of the[latex]y[\/latex]-coordinates of the points on the graph are multiplied by 7, but their [latex]x[\/latex]-coordinates remain the same. The equation of the function after the graph is stretched is [latex]f(x)=7\\times\\dfrac{1}{x}=\\dfrac{7}{x}[\/latex]. The reason for multiplying [latex]f(x)=\\dfrac{1}{x}[\/latex] by 7 is that each [latex]y[\/latex]-coordinate is made 7 times larger, and since [latex]y=\\dfrac{1}{x}[\/latex], [latex]\\dfrac{1}{x}[\/latex] is also made 7 times larger. Table 5 shows this change and the graph is shown in figure 8.<\/p>\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 365px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]f(x)=\\dfrac{7}{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 365px;\" rowspan=\"9\">\n<div id=\"attachment_1832\" class=\"wp-caption aligncenter\" style=\"width: 416px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2797\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/19202358\/7-2-Stretch1-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a vertically stretched green graph. One green branch opens to the left and downward, lying 7 times below the blue graph, while the other branch opens to the right and upward, lying 7 times above the blue graph.\" width=\"416\" height=\"416\" \/><\/p>\n<p>Figure 8. Stretching the graph vertically.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\frac{7}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\frac{7}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\frac{7}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 177px;\">\n<td style=\"width: 42.1941%; height: 177px;\" colspan=\"3\">Table 5.\u00a0Stretching the graph vertically 7 times transforms [latex]\\dfrac{1}{x}[\/latex] into [latex]\\dfrac{7}{x}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>On the other hand, if we vertically compress the graph of the function [latex]\\dfrac{1}{x}[\/latex] into [latex]\\dfrac{1}{5}[\/latex] of its original height, all of the [latex]y[\/latex]-coordinates of the points on the graph are divided by 5, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are multiplied by [latex]\\dfrac{1}{5}[\/latex]. The equation of the function after being compressed is [latex]f(x)=\\dfrac{1}{5}\\times\\dfrac{1}{x}=\\dfrac{1}{5x}[\/latex]. The reason for multiplying [latex]f(x)=\\dfrac{1}{x}[\/latex] by [latex]\\dfrac{1}{5}[\/latex] is that each [latex]y[\/latex]-coordinate becomes [latex]\\dfrac{1}{5}[\/latex]\u00a0of the original value. Table 6 shows this change and the graph is shown in figure 9.<\/p>\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 152px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]\\dfrac{1}{5x}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%;\" rowspan=\"9\">\n<div id=\"attachment_1866\" class=\"wp-caption aligncenter\" style=\"width: 419px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2798\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/19202431\/7-2-Compress-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a vertically compressed green graph. One green branch opens to the left and downward and lies at one-fifth the height of the blue graph, while the other branch opens to the right and upward and lies at one-fifth the height of the blue graph.\" width=\"419\" height=\"419\" \/><\/p>\n<p>Figure 9. Compress the graph into [latex]\\frac{1}{5}[\/latex] of the original height.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{10}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{5}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]-\\dfrac{2}{5}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{2}{5}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{5}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{10}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{15}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.1941%;\" colspan=\"3\">Table 6.\u00a0Compressing the graph vertically into [latex]\\frac{1}{5}[\/latex] of the original height transforms [latex]f(x)=\\frac{1}{x}[\/latex] into [latex]f(x)=\\frac{1}{5x}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>vertical stretching and compressing<\/h3>\n<p id=\"fs-id1165137770279\">A vertical stretch or compression of the graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] can be represented by\u00a0multiplying the function by a constant, [latex]a>0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a\\dfrac{1}{x}[\/latex]<\/p>\n<p>The magnitude of [latex]a[\/latex]\u00a0indicates the stretch\/compression of the graph. If [latex]a>1[\/latex], the graph is stretched. If [latex]0<a<1[\/latex], the graph is compressed.\n\n<\/div>\n<p>Move the red dot in figure 10 to change the value of [latex]a[\/latex]. Notice whether the graph is stretching or compressing depending on the value of [latex]a[\/latex]. Notice also what happens to the function.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/fhnblbhdvm?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">FIgure 10. Stretching and compressing<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<ol>\n<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is stretched by a factor of 5. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/li>\n<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is compressed to [latex]\\frac{1}{2}[\/latex] its size. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>Stretched by a factor of 5 means [latex]a=5[\/latex], therefore, the transformed function is [latex]f(x)=5\\left(\\dfrac{1}{x}\\right)[\/latex]. This can also be written as [latex]f(x)=\\dfrac{5}{x}[\/latex]. When a function is stretched its [latex]x[\/latex]-value stays the same while the [latex]y[\/latex]-value is multiplied by the stretch factor. So, (1, 1) moves to (1, 5).<\/li>\n<li>Compressed to [latex]\\frac{1}{2}[\/latex] its size means [latex]a=\\dfrac{1}{2}[\/latex], therefore, the transformed function is [latex]f(x)=\\dfrac{1}{2}\\left(\\dfrac{1}{x}\\right)[\/latex]. This can also be written as [latex]f(x)=\\dfrac{1}{2x}[\/latex].<\/li>\n<\/ol>\n<p>When a function is compressed its [latex]x[\/latex]-value stays the same while the [latex]y[\/latex]-value is multiplied by the compression. So, (1, 1) moves to [latex](1, \\frac{1}{5})[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<ol>\n<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is stretched by a factor of 4. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/li>\n<li>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is compressed to [latex]\\frac{1}{8}[\/latex]th its size. Determine the equation of the transformed function. The point (1, 1) lies on the parent function. What happens to this point after the transformation?<\/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>[latex]f(x)=4\\left(\\dfrac{1}{x}\\right)=\\dfrac{4}{x}\\;\\;\\;\\;(1, 4)[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{1}{8}\\left(\\dfrac{1}{x}\\right)=\\dfrac{1}{8x}\\;\\;\\;\\;(1, \\frac{1}{8})[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflections<\/h2>\n<h3>across the [latex]x[\/latex]-axis<\/h3>\n<p>When the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex] is reflected across the\u00a0[latex]x[\/latex]-axis, the\u00a0[latex]y[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values or from negative to positive values, while the [latex]x[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\\dfrac{1}{x}[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-\\dfrac{1}{x}[\/latex]. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 11.<\/p>\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 342px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]-\\dfrac{1}{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 342px;\" rowspan=\"9\">\n<div id=\"attachment_1868\" class=\"wp-caption aligncenter\" style=\"width: 434px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2803\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/20001957\/7-2-ReflectionX1-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a reflected green graph across the x-axis. One green branch opens to the left and upward, while the other branch opens to the right and downward.\" width=\"434\" height=\"434\" \/><\/p>\n<p>Figure 11. Reflecting the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0across the [latex]x[\/latex]-axis.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20132<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20131<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">3<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\dfrac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 154px;\">\n<td style=\"width: 42.1941%; height: 154px;\" colspan=\"3\">Table 7.\u00a0Reflecting the graph of [latex]f(x)=\\dfrac{1}{x}[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=\\dfrac{1}{x}[\/latex] into [latex]f(x)=-\\dfrac{1}{x}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>across the [latex]y[\/latex]-axis<\/h3>\n<p>When the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0is reflected across the\u00a0[latex]y[\/latex]-axis, the\u00a0[latex]x[\/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values, or from negative to positive values, while the [latex]y[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0is reflected across the [latex]y[\/latex]-axis is [latex]f(x)=\\dfrac{1}{-x}=-\\dfrac{1}{x}[\/latex]. This equation is the same as the equation after being reflected across the [latex]x[\/latex]-axis. Therefore, its graph is the same as the graph after being reflected across the [latex]x[\/latex]-axis, even though it got there by a different route (note the differences in figures 11 and 13).\u00a0Table 8 shows the effect of such a reflection on the functions values and the graph is shown in figure 13.<\/p>\n<table style=\"border-collapse: collapse; width: 58.8608%; height: 683px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px; text-align: right;\">[latex]\\dfrac{1}{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 10px; text-align: right;\"><strong>[latex]\\dfrac{1}{-x}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 306px;\" rowspan=\"9\">\n<div id=\"attachment_1868\" class=\"wp-caption aligncenter\" style=\"width: 437px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2802\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/06\/20001933\/7-2-ReflectionY-300x300.png\" alt=\"A blue graph of the function f(x)=1\/x, along with a reflected green graph across the y-axis. One green branch opens to the left and upward, while the other branch opens to the right and downward.\" width=\"437\" height=\"437\" \/><\/p>\n<p>Figure 13. Reflecting the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex]\u00a0across the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20131<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px; text-align: right;\">\u20132<\/td>\n<td style=\"width: 16.6667%; height: 34px; text-align: right;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20132<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">1<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">\u20131<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">3<\/td>\n<td style=\"width: 12.7637%; height: 19px; text-align: right;\">[latex]\\frac{1}{3}[\/latex]<\/td>\n<td style=\"width: 16.6667%; height: 19px; text-align: right;\">[latex]-\\frac{1}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 154px;\">\n<td style=\"width: 42.1941%; height: 154px;\" colspan=\"3\">Table 8.\u00a0Reflecting the graph of [latex]f(x)=\\dfrac{1}{x}[\/latex] across the [latex]y[\/latex]-axis transforms [latex]f(x)=\\dfrac{1}{x}[\/latex] into [latex]f(x)=\\dfrac{1}{-x}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Explain why the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed to [latex]f(x)=-\\dfrac{1}{x}[\/latex] when it is reflected across the [latex]x[\/latex]-axis.<\/p>\n<h4>Solution<\/h4>\n<p>With reflection across the [latex]x[\/latex]-axis, the [latex]x[\/latex] values stay the same and the [latex]y[\/latex]-values change sign. So, [latex]y=\\dfrac{1}{x}[\/latex] transforms to [latex]-y=\\dfrac{1}{x}[\/latex]. If we multiply (or divide) both sides of the equation by \u20131 we get, [latex]y=-\\dfrac{1}{x}[\/latex]. Consequently, the transformed function is [latex]f(x)=-\\dfrac{1}{x}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>Explain why the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed to [latex]f(x)=-\\dfrac{1}{x}[\/latex] when it is reflected across the [latex]y[\/latex]-axis.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm518\">Show Answer<\/span><\/p>\n<div id=\"qhjm518\" class=\"hidden-answer\" style=\"display: none\">\n<p>With reflection across the [latex]y[\/latex]-axis, the [latex]y[\/latex] values stay the same and the [latex]x[\/latex]-values change sign. So, [latex]y=\\dfrac{1}{x}[\/latex] transforms to [latex]y=\\dfrac{1}{-x}[\/latex], which simplifies to [latex]y=-\\dfrac{1}{x}[\/latex]. Consequently, the transformed function is [latex]f(x)=-\\dfrac{1}{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Combining Transformations<\/h2>\n<p>Now that we have learned all the transformations for the function [latex]f(x)=\\dfrac{1}{x}[\/latex], we should be able to write the transformed function equation given specific transformations, and determine what transformations have been performed on the function [latex]f(x)=\\dfrac{1}{x}[\/latex], given an arbitrary transformed function [latex]f(x)=a\\dfrac{1}{x-h}+k[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Combining Transformations<\/h3>\n<p id=\"fs-id1165137770279\">The graph of the function\u00a0[latex]f(x)=\\dfrac{1}{x}[\/latex] can be shifted horizontally ([latex]h[\/latex]-value), shifted vertically\u00a0([latex]k[\/latex]-value), stretched or compressed ([latex]a[\/latex]-value), reflected across the [latex]x[\/latex]-axis (negative function value} or reflected across the [latex]y[\/latex]-axis (negative [latex]x[\/latex]-value) and can be represented by\u00a0the function<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex]<\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed. Write the equation of the transformed function.<\/p>\n<ol>\n<li>Stretched by a factor of 2; shifted vertically up 3 units; shifted horizontally 5 units right.<\/li>\n<li>Shifted horizontally left by 7 units; reflected across the [latex]x[\/latex]-axis.<\/li>\n<li>Compressed to [latex]\\frac{1}{3}[\/latex]rd its height; reflected across the [latex]y[\/latex]-axis; shifted vertically down by 4 units.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]a=2,\\;k=3,\\;h=5[\/latex] so function is [latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k=2\\left(\\dfrac{1}{x-5}\\right)+3[\/latex]<\/li>\n<li>[latex]a=-1,\\;k=0,\\;h=-7[\/latex] so function is [latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k=-\\dfrac{1}{x+7}[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{3},\\;x\\rightarrow -x,\\;k=-4[\/latex] so function is [latex]f(x)=\\dfrac{1}{3}\\left(\\dfrac{1}{-x}\\right)-4=-\\dfrac{1}{3}\\left(\\dfrac{1}{x}\\right)-4[\/latex]. This can also be written as [latex]f(x)=-\\dfrac{1}{3x}-4[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 8<\/h3>\n<p>The parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] is transformed. Write the equation of the transformed function.<\/p>\n<ol>\n<li>Stretched by a factor of 4; shifted vertically down by 7 units; shifted horizontally 3 units left.<\/li>\n<li>Shifted horizontally left by 8 units; reflected across the [latex]x[\/latex]-axis.<\/li>\n<li>Compressed to [latex]\\frac{1}{5}[\/latex]th its height; reflected across the [latex]x[\/latex]-axis; shifted vertically down by 4 units; shifted right by 2 units.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm236\">Show Answer<\/span><\/p>\n<div id=\"qhjm236\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=4\\left(\\dfrac{1}{x+3}\\right)-7[\/latex]<\/li>\n<li>\u00a0[latex]f(x)=-\\dfrac{1}{x+8}[\/latex]<\/li>\n<li>\u00a0[latex]f(x)=-\\frac{1}{5}\\left(\\dfrac{1}{x-2}\\right)-4[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 9<\/h3>\n<p>Determine the transformations made to the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] to get the transformed function [latex]f(x)=-\\dfrac{7}{x-4}+3[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>[latex]f(x)=-\\dfrac{7}{x-4}+3[\/latex] can be written as [latex]f(x)=-7\\left(\\dfrac{1}{x-4}\\right)+3[\/latex].<\/p>\n<p>Comparing\u00a0[latex]f(x)=-7\\left(\\dfrac{1}{x-4}\\right)+3[\/latex] to\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex], we get [latex]a=-7,\\;h=4,\\;k=3[\/latex], which translated to a reflection across the [latex]x[\/latex]-axis, a stretch by a factor of 7, a horizontal shift to the right by 4 units, and a vertical shift up by 3 units.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 9<\/h3>\n<p>Determine the transformations made to the parent function [latex]f(x)=\\dfrac{1}{x}[\/latex] to get the transformed function [latex]f(x)=-\\dfrac{5}{3x}-7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm245\">Show Answer<\/span><\/p>\n<div id=\"qhjm245\" class=\"hidden-answer\" style=\"display: none\">\n<p>Reflection across the [latex]x[\/latex]-axis (negative sign in front)<\/p>\n<p>Stretched to\u00a0[latex]\\frac{5}{3}[\/latex]rds its height<\/p>\n<p>Shifted down by 7 units<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Formats of Rational Functions<\/h2>\n<p>You may be wondering why we defined a rational function as [latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] when we write a rational function using transformations in the form\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex].<\/p>\n<p>It is simply that rational functions in the form\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex] are based on the parent rational function [latex]f(x)=\\dfrac{1}{x}[\/latex], and a special case of\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex].<\/p>\n<p>We could take the function\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex] and write it as a single fraction by using a common denominator:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}f(x)&=a\\left(\\dfrac{1}{x-h}\\right)+k\\\\\\\\&=a\\left(\\dfrac{1}{x-h}\\right)+k\\color{blue}{\\cdot \\dfrac{x-h}{x-h}}\\\\\\\\&=\\dfrac{a+k(x-h)}{x-h}\\\\\\\\&=\\dfrac{kx+(a-hk)}{x-h}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<p>Now the function is written in standard form with [latex]P(x)=kx+(a-hk)[\/latex] and [latex]Q(x)=x-h[\/latex]. Both [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] have degree 1.<\/p>\n<p>Consequently, a rational function in the transformed form\u00a0[latex]f(x)=a\\left(\\dfrac{1}{x-h}\\right)+k[\/latex] is a special case of the standard form\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] with\u00a0[latex]P(x)[\/latex] and [latex]Q(x)[\/latex] having degree 1.<\/p>\n<p>We leave the more in depth study of rational functions in the form\u00a0[latex]f(x)=\\dfrac{P(x)}{Q(x)}[\/latex] to College Algebra.<\/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-2669\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Transformations of the Rational 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=\"http:\/\/www.desmos.com\/calculator\">http:\/\/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><li>Examples and Try Its: hjm245; hjm236; hjm518; hjm208; hjm730; hjm400; hjm614; hjm004; hjm613. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Formats of Rational Functions. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Combining Vertical and Horizontal Shifts. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Transformations of the Rational Function\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"www.desmos.com\/calculator\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples and Try Its: hjm245; hjm236; hjm518; hjm208; hjm730; hjm400; hjm614; hjm004; hjm613\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Formats of Rational Functions\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Combining Vertical and Horizontal Shifts\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2669","chapter","type-chapter","status-publish","hentry"],"part":2581,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2669","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":52,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2669\/revisions"}],"predecessor-version":[{"id":4827,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2669\/revisions\/4827"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2581"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2669\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2669"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2669"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2669"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2669"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}