{"id":2358,"date":"2022-05-25T16:55:00","date_gmt":"2022-05-25T16:55:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2358"},"modified":"2026-01-17T03:24:45","modified_gmt":"2026-01-17T03:24:45","slug":"6-2-transformations-of-the-function-fxlog_bx","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/6-2-transformations-of-the-function-fxlog_bx\/","title":{"raw":"6.2: Transformations of the Logarithmic Function","rendered":"6.2: Transformations of the Logarithmic 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 logarithmic function [latex]f(x)=\\log_b{x}[\/latex],\r\n<ul>\r\n \t<li>Perform vertical and horizontal shifts<\/li>\r\n \t<li>Perform vertical compressions and\u00a0stretches<\/li>\r\n \t<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\r\n \t<li>Perform reflections across the [latex]y[\/latex]-axis<\/li>\r\n \t<li>Determine the transformations of the logarithmic function\u00a0[latex]f(x)=a\\log_b{(x-h)}+k[\/latex]<\/li>\r\n \t<li>Determine the equation of a function given the transformations<\/li>\r\n \t<li>Determine what happens to the vertical asymptote as transformations are made<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<h2>Vertical Shifts<\/h2>\r\n<p id=\"fs-id1165137770279\">If we shift the graph of the logarithmic function [latex]f(x)=\\log_2{x}[\/latex] up 5 units, all of the points on the graph increase their [latex]y[\/latex]-coordinates by 5, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\log_2{x}[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=\\log_2{x}+5[\/latex]. The vertical asymptote at [latex]x=0[\/latex] remains the same. 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;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\r\n<th style=\"width: 29.277286135693217%; height: 10px;\" scope=\"row\">[latex]\\log_2{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: 389px;\">\r\n\r\n<img class=\"aligncenter wp-image-2752 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-300x300.png\" alt=\"A green curve is 5 units above the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,y+5) on the green curve. \" width=\"300\" height=\"300\" \/>\r\n<p class=\"wp-caption-text\">Figure 1. Shifting the graph of [latex]f(x)=\\log_2{x}[\/latex] up 5 units.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]8[\/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)=\\log_2{x}[\/latex] is transformed to [latex]f(x)=\\log_2{x}+5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=\\log_2{x}[\/latex] down 6 units, all of the points on the graph decrease their [latex]y[\/latex]-coordinates by 6, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\log_2{x}[\/latex] after it has been shifted down 6 units transforms to [latex]f(x)=\\log_2{x}-6[\/latex].The vertical asymptote at [latex]x=0[\/latex] remains the same. 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: 71.4714%; height: 222px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\r\n<th style=\"width: 29.277286135693217%; height: 10px;\" scope=\"row\">[latex]\\log_2{x}-6[\/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: 370px;\">\r\n\r\n<img class=\"aligncenter wp-image-2755 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-300x300.png\" alt=\"A green curve is 6 units below the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,y-6) on the green curve. \" width=\"300\" height=\"300\" \/>\r\n<p class=\"wp-caption-text\">Figure 2. Shifting the graph of [latex]f(x)=\\log_2{x}[\/latex] down 6 units.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]-8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-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)=\\log_2{x}[\/latex] is transformed to [latex]f(x)=\\log_2{x}-6[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that vertical shifts up or down do not change the vertical asymptote.\r\n\r\nMove the red dots in manipulation 1 to change the values of [latex]b[\/latex] and [latex]k[\/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]k[\/latex] and the transformed function.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/n6lahppwjk?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><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;\">Manipulation 1. Vertical shifts<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Vertical shifts<\/h3>\r\nWe can represent a vertical shift of the graph of [latex]f(x)=\\log_2{x}[\/latex] by adding or subtracting a constant, [latex]k[\/latex], to the function:\r\n<p style=\"text-align: center;\">[latex]f(x)=\\log_2{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\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is shifted vertically up by 7 units, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is shifted vertically down by 4 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5x+9[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2x-3[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWith vertical shifts, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=\\log_bx+k[\/latex].\r\n<ol>\r\n \t<li>[latex]k=7[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3x+7[\/latex]<\/li>\r\n \t<li>[latex]k=-4[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7x-4[\/latex]<\/li>\r\n \t<li>[latex]k=9[\/latex] so the transformation was a vertical shift up by 9 units.<\/li>\r\n \t<li>[latex]k=-3[\/latex] so the transformation was a vertical shift down by 3 units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is shifted vertically up by 2 units, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is shifted vertically down by 9 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5x+3[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2x-8[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm624\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm624\"]\r\n<ol>\r\n \t<li>[latex]k=2[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3x+2[\/latex]<\/li>\r\n \t<li>[latex]k=-9[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7x-9[\/latex]<\/li>\r\n \t<li>[latex]k=3[\/latex] so the transformation was a vertical shift up by 3 units.<\/li>\r\n \t<li>[latex]k=-8[\/latex] so the transformation was a vertical shift down by 8 units.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Horizontal Shifts<\/h2>\r\n<\/div>\r\nIf we shift the graph of the function [latex]f(x)=\\log_2{x}[\/latex] right 8 units, all of the points on the graph increase their [latex]x[\/latex]-coordinates by 8, but their [latex]y[\/latex]-coordinates remain the same. The [latex]x[\/latex]-intercept (1, 0) in the original graph is moved to (9, 0) (figure 3). The vertical asymptote at [latex]x=0[\/latex] shifts right by 8 units to [latex]x=8[\/latex]. Any point [latex](x, y)[\/latex] on the original graph is moved to [latex](x+8, y)[\/latex].\r\n\r\nBut what happens to the original function [latex]f(x)=\\log_2{x}[\/latex]? An automatic assumption may be that since [latex]x[\/latex] moves to [latex]x+8[\/latex] that the function will become [latex]f(x)=\\log_2{(x+8)}[\/latex]. But that is NOT the case. Remember that the [latex]x[\/latex]-intercept is moved to (9, 0) and if we substitute [latex]x=9[\/latex] into the function\u00a0[latex]f(x)=\\log_2{(x+8)}[\/latex] we get\u00a0[latex]f(9)=\\log_2{(9+8)}=4.0875 \\neq 1[\/latex]!! The way to get a function value of 0 is for the transformed function to be [latex]f(x)=\\log_2{(x-8)}[\/latex]. Then [latex]f(9)=\\log_2{(9-8)}=0[\/latex]. So the function [latex]f(x)=\\log_2{x}[\/latex] transforms to [latex]f(x)=\\log_2{(x-8)}[\/latex] after being shifted 8 units to the right. The reason is that when we move the function 8 units to the right, the [latex]x[\/latex]-value increases by 8 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 8 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 3.\r\n<table style=\"border-collapse: collapse; width: 73.9452%; height: 337px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]x-8[\/latex]<\/th>\r\n<td style=\"width: 31.7511%; height: 10px;\"><strong>[latex]\\log_2{(x-8)}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 337px;\" rowspan=\"9\">\r\n<div id=\"attachment_1828\" class=\"wp-caption aligncenter\" style=\"width: 425px;\">\r\n\r\n<img class=\"aligncenter wp-image-3030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/27152325\/6-2-ShiftRightNew3-300x300.png\" alt=\"A green curve is 8 units to the right of the blue curve for all y-values where each point (x,y) on the blue curve corresponds to a point (x+8,y) on the green curve. \" width=\"425\" height=\"425\" \/>\r\n<p class=\"wp-caption-text\">Figure 3. Shifting the graph right 8 units.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{65}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 34px;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{33}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 34px;\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{17}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 34px;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 31.7511%; height: 19px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 149px;\">\r\n<td style=\"width: 57.2785%; height: 149px;\" colspan=\"3\">Table 3. Shifting the graph right by 8 units transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\log_2{(x-8)}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that the vertical asymptote also shifts from [latex]x=0[\/latex] to [latex]x=8[\/latex].\r\n\r\nOn the other hand, if we shift the graph of the function [latex]f(x)=\\log_2{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)=\\log_2{(x+11)}[\/latex]. Table 4 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: 74.1562%; height: 296px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 19px;\">\r\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]x+11[\/latex]<\/th>\r\n<td style=\"width: 31.9621%; height: 10px;\"><strong>[latex]f(x)=\\log_2{(x+11)}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%; height: 296px;\" rowspan=\"9\">\r\n<div id=\"attachment_1830\" class=\"wp-caption aligncenter\" style=\"width: 439px;\">\r\n\r\n<img class=\"aligncenter wp-image-3031\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/27152356\/6-2-ShiftLeftNew3-300x300.png\" alt=\"A green curve is 11 units to the left of the blue curve for all y-values where each point (x,y) on the blue curve corresponds to a point (x-11,y) on the green curve. \" width=\"439\" height=\"439\" \/>\r\n<p class=\"wp-caption-text\">Figure 4. Shifting the graph left 11 units.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{87}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 31.9621%; height: 34px;\">\u20133<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{43}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 31.9621%; height: 34px;\">\u20132<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{21}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 31.9621%; height: 34px;\">\u20131<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u201310<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20139<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20137<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20133<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\r\n<td style=\"width: 31.9621%; height: 19px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 108px;\">\r\n<td style=\"width: 57.4895%; height: 108px;\" colspan=\"3\">Table 4. Shifting the graph left by 11 units transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\log_2{(x+11)}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"entry-content\">\r\n\r\nNotice that the vertical asymptote also shifts from [latex]x=0[\/latex] to [latex]x=-11[\/latex].\r\n\r\nMove the red dots in manipulation 2 to change the values of [latex]b[\/latex] and [latex]h[\/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]h[\/latex] and the transformed function.\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\/ukwqebsewa?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>\r\nManipulation 2. Horizontal shifts<\/p>\r\n\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 [latex]f(x)=\\log_2{x}[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=\\log_2{(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\nThe vertical asymptote [latex]x=0[\/latex] shifts to [latex]x=h[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is shifted right by 2 units, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 9 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x+3)}[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x-8)}[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWith horizontal shifts, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=\\log_b{(x-h)}[\/latex].\r\n<ol>\r\n \t<li>[latex]h=2[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-2)}[\/latex]<\/li>\r\n \t<li>[latex]h=-9[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+9)}[\/latex]<\/li>\r\n \t<li>[latex]h=-3[\/latex] so the transformation was a horizontal shift left by 3 units.<\/li>\r\n \t<li>[latex]h=8[\/latex] so the transformation was a horizontal shift right by 8 units<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is shifted right by 7 units, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 4 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x-5)}[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x+4)}[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm832\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm832\"]\r\n<ol>\r\n \t<li>[latex]h=7[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-7)}[\/latex]<\/li>\r\n \t<li>[latex]h=-4[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+4)}[\/latex]<\/li>\r\n \t<li>[latex]h=5[\/latex] so the transformation was a horizontal shift right by 5 units.<\/li>\r\n \t<li>[latex]h=-4[\/latex] so the transformation was a horizontal shift left by 4 units<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can combine vertical and horizontal shifts by transforming [latex]f(x)=\\log_b{(x-h)}+k[\/latex].\r\n\r\nMove the red dots in manipulation 3 to change the values of [latex]b, \\;h[\/latex] and [latex]k[\/latex]. Pay attention to what happens to the graph and the relationship between the values of [latex]h[\/latex] and[latex]h[\/latex] and the transformed function.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/tq5mj8l9xf?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><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\nManipulation 3. Vertical and horizontal shifts<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is shifted vertically down by 3 units and right by 2 units, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 9 units and up by 6 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x-2)}+7[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x+5)}-4[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWith horizontal and vertical shifts, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=\\log_b{(x-h)}+k[\/latex].\r\n<ol>\r\n \t<li>[latex]h=2,\\;k=-3[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-2)}-3[\/latex]<\/li>\r\n \t<li>[latex]h=-9,\\;k=6[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+9)}+6[\/latex]<\/li>\r\n \t<li>[latex]h=2,\\;k=7[\/latex] so the transformation was a horizontal shift right by 2 units and up by 7 units.<\/li>\r\n \t<li>[latex]h=-5,\\;k=-4[\/latex] so the transformation was a horizontal shift left by 5 units and down by 4 units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is shifted down by 7 units and right by 5 units, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 4 units and up by 3 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x-2)}-1[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x+4)}+7[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm302\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm302\"]\r\n<ol>\r\n \t<li>[latex]h=5,\\;k=-7[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-5)}-7[\/latex]<\/li>\r\n \t<li>[latex]h=-4,\\;k=3[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+4)}+3[\/latex]<\/li>\r\n \t<li>[latex]h=2,\\;k=-1[\/latex] so the transformation was a horizontal shift right by 2 units and down by 1 unit.<\/li>\r\n \t<li>[latex]h=-4,\\;k=7[\/latex] so the transformation was a horizontal shift left by 4 units and up by 7 units.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Vertical Stretching and Compressing<\/h2>\r\nIf we vertically stretch the graph of the function [latex]f(x)=\\log_2{x}[\/latex] by a factor of 2, all of the[latex]y[\/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[\/latex]-coordinates remain the same. The equation of the function after the graph is stretched up by a factor of 2 is [latex]f(x)=2\\log_2{x}[\/latex]. The reason for multiplying\u00a0 [latex]\\log_2{x}[\/latex] by 2 is that each [latex]y[\/latex]-coordinate is doubled, and since [latex]y=\\log_2{x}[\/latex], [latex]\\log_2{x}[\/latex] is doubled. Table 5 shows this change and the graph is shown in figure 5.\r\n<table style=\"border-collapse: collapse; width: 58.860800000000005%; height: 365px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 10px;\"><strong>[latex]f(x)=2\\log_2{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: 425px;\">\r\n\r\n<img class=\"aligncenter wp-image-3002 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-300x300.png\" alt=\"A green curve is twice as far above or below the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,2y) on the green curve. \" width=\"300\" height=\"300\" \/>\r\n<p class=\"wp-caption-text\">Figure 5. Stretching the graph vertically.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-3<\/td>\r\n<td style=\"width: 16.6667%; height: 34px;\">-6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-2<\/td>\r\n<td style=\"width: 16.6667%; height: 34px;\">-4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-1<\/td>\r\n<td style=\"width: 16.6667%; height: 34px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">4<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">6<\/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 by a factor of 2 transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=2\\log_2{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]f(x)=\\log_2{x}[\/latex] to half of its original height, we multiply the function by the factor [latex]\\dfrac{1}{2}[\/latex]. All of the [latex]y[\/latex]-coordinates of the points on the graph are halved, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are divided by 2, or multiplied by [latex]\\dfrac{1}{2}[\/latex]. The equation of the function after being compressed is [latex]f(x)=\\dfrac{1}{2}\\times\\log_2{x}[\/latex]. The reason for multiplying [latex]\\log_2{x}[\/latex] by [latex]\\dfrac{1}{2}[\/latex] is that each [latex]y[\/latex]-coordinate becomes half of the original value when it is divided by 2. Table 6 shows this change and the graph is shown in figure 6.\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;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 10px;\"><strong>[latex]f(x)=\\dfrac{1}{2}\\log_2{x}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 16.6667%;\" rowspan=\"9\">\r\n<div id=\"attachment_1866\" class=\"wp-caption aligncenter\" style=\"width: 455px;\">\r\n\r\n<img class=\"aligncenter wp-image-3003\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/27031453\/6-2-CompressNew2-300x300.png\" alt=\"A green curve is half as far above or below the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,y\/2) on the green curve. \" width=\"455\" height=\"455\" \/>\r\n<p class=\"wp-caption-text\">Figure 6. Compressing the graph vertically.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-3<\/td>\r\n<td style=\"width: 16.6667%; height: 34px;\">[latex]-\\dfrac{3}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-2<\/td>\r\n<td style=\"width: 16.6667%; height: 34px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-1<\/td>\r\n<td style=\"width: 16.6667%; height: 34px;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\r\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{3}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 42.1941%;\" colspan=\"3\">Table 6.\u00a0Compressing the graph vertically by a factor of [latex]\\dfrac{1}{2}[\/latex] transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\dfrac{1}{2}\\log_2{x}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that vertical stretching and compressing do not change the vertical asymptote.\r\n\r\nMove the red dots in manipulation 4 to change the values of [latex]b[\/latex] and [latex]a[\/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]a[\/latex] and the transformed function.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/sdxxkrpf6o?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><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\nManipulation 4. Vertical stretching and compressing<\/p>\r\n\r\n<div class=\"entry-content\">\r\n<div class=\"textbox shaded\">\r\n<h3>vertical stretching and compressing<\/h3>\r\n<p id=\"fs-id1165137770279\">A stretch or compression of the graph of [latex]f(x)=\\log_2{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\\log_2{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 up by a factor of [latex]a.[\/latex] If [latex]0&lt;a&lt;1[\/latex], the graph is compressed down by a factor of [latex]a[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 7, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-third its height, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{6}\\log_5x[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=5\\log_2x[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWith stretching and compression, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=a\\log_bx[\/latex].\r\n<ol>\r\n \t<li>[latex]a=7[\/latex] so the transformed function is\u00a0[latex]f(x)=7\\log_3x[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{3}[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{3}\\log_7x[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{6}[\/latex] so the transformation was a compression to one-sixth its height<\/li>\r\n \t<li>[latex]a=5[\/latex] so the transformation was a vertical stretch by a factor of 5<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 2, what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-eighth its height, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{4}\\log_5x[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=7\\log_2x[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm442\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm442\"]\r\n<ol>\r\n \t<li>[latex]a=2[\/latex] so the transformed function is\u00a0[latex]f(x)=2\\log_3x[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{8}[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{8}\\log_7x[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{4}[\/latex] so the transformation was a compression to one-quarter its height<\/li>\r\n \t<li>[latex]a=7[\/latex] so the transformation was a vertical stretch by a factor of 7<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we can combine vertical stretches and compressions with horizontal and vertical shifts.\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 7, and moved down by 5 units what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-third its height, moved left by 4 units and moved up by 2 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{5}\\log_5{(x-9)}-6[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=3\\log_2{(x+5)}+7[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWith stretching and compression, combined with shifting the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=a\\log_b{(x-h)}+k[\/latex].\r\n<ol>\r\n \t<li>[latex]a=7,\\; k=-5[\/latex] so the transformed function is\u00a0[latex]f(x)=7\\log_3x-5[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{3},\\;h=-4,\\;k=2[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{3}\\log_7{(x+4)}+2[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{5},\\;h=9,\\;k=-6[\/latex] so the transformation was a compression to one-fifth its height, a shift right by 9 units, and a shift down by 6 units<\/li>\r\n \t<li>[latex]a=3,\\;-5,\\;k=7[\/latex] so the transformation was a vertical stretch by a factor of 3,\u00a0a shift left by 5 units, and a shift up by 7 units<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 5, and moved down by 2 units what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-half its height, moved left by 7 units and moved up by 3 units, what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{3}\\log_5{(x-2)}-3[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=9\\log_2{(x+1)}+5[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm356\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm356\"]\r\n<ol>\r\n \t<li>[latex]a=5,\\; k=-2[\/latex] so the transformed function is\u00a0[latex]f(x)=5\\log_3x-2[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{2},\\;h=-7,\\;k=3[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{2}\\log_7{(x+7)}+3[\/latex]<\/li>\r\n \t<li>[latex]a=\\frac{1}{3},\\;h=2,\\;k=-3[\/latex] so the transformation was a compression to one-third its height, a shift right by 2 units, and a shift down by 3 units<\/li>\r\n \t<li>[latex]a=9,\\;h=-1,\\;k=5[\/latex] so the transformation was a vertical stretch by a factor of 9,\u00a0a shift left by 1 unit, and a shift up by 5 units<\/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)=\\log_2{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)=\\log_2{x}[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-\\log_2{x}[\/latex]. The graph changes from increasing upwards to decreasing downwards. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 7.\r\n<table style=\"border-collapse: collapse; width: 54.9578%; height: 306px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]-\\log_2{x}[\/latex]<\/th>\r\n<td style=\"width: 16.6667%; height: 306px;\" rowspan=\"9\">\r\n<div id=\"attachment_1868\" class=\"wp-caption aligncenter\" style=\"width: 436px;\">\r\n\r\n<img class=\"aligncenter wp-image-2768 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-300x300.png\" alt=\"A green curve is the reflection of the blue curve for all x-values across the x-axis: each point (x,y) on the blue curve corresponds to a point (x,-y) on the green curve. \" width=\"300\" height=\"300\" \/>\r\n<p class=\"wp-caption-text\">Figure 7. Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across the [latex]x[\/latex]-axis.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-3<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-2<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">-1<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">-3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 154px;\">\r\n<td style=\"width: 38.2911%; height: 154px;\" colspan=\"3\">Table 7.\u00a0Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=-\\log_2{x}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that reflecting across the [latex]x[\/latex]-axis does not change the vertical asymptote.\r\n<h3>Across the [latex]y[\/latex]-axis<\/h3>\r\nWhen the graph of the function [latex]f(x)=\\log_2{x}[\/latex] is 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, while the [latex]y[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\\log_2{x}[\/latex] is reflected across the [latex]y[\/latex]-axis is [latex]f(x)=\\log_2{(-x)}[\/latex]. The graph changes from increasing from the left to decreasing from the left. Table 8 shows the effect of such a reflection on the functions values and the graph is shown in figure 8.\r\n<table style=\"border-collapse: collapse; width: 58.75077591776798%; height: 683px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 10px;\">\r\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]-x[\/latex]<\/th>\r\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{(-x)}[\/latex]<\/th>\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-2770 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-300x300.png\" alt=\"A green curve is the reflection of the blue curve for all y-values across the y-axis: each point (x,y) on the blue curve corresponds to a point (-x,y) on the green curve. \" width=\"300\" height=\"300\" \/>\r\n<p class=\"wp-caption-text\">Figure 8. Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across the [latex]y[\/latex]-axis.<\/p>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">\u20133<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">\u20132<\/td>\r\n<\/tr>\r\n<tr style=\"height: 34px;\">\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 12.7637%; height: 34px;\">\u20131<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20131<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20132<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20134<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"width: 12.7637%; height: 19px;\">\u20138<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\r\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 154px;\">\r\n<td style=\"width: 38.2911%; height: 154px;\" colspan=\"3\">Table 8.\u00a0Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across the [latex]y[\/latex]-axis transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\log_2{(-x)}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div>Notice that reflecting across the [latex]x[\/latex]-axis does not change the vertical asymptote.<\/div>\r\n<\/div>\r\n<\/div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div><\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is reflected across the [latex]x[\/latex]-axis what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is reflected across the [latex]y[\/latex]-axis what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=-\\log_5{x}-6[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function [latex]f(x)=\\log_2x[\/latex] if the transformed function is [latex]f(x)=-3\\log_2{(-x)}[\/latex]?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>The [latex]y[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=-\\log_3x[\/latex]<\/li>\r\n \t<li>\u00a0The [latex]x[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=\\log_7{(-x)}[\/latex]<\/li>\r\n \t<li>[latex]a=-1,\\;k=-6[\/latex] so the transformation was a reflection across the [latex]x[\/latex]-axis and a shift down by 6 units<\/li>\r\n \t<li>[latex]a=-3[\/latex] and [latex]x[\/latex] is\u00a0[latex]-x[\/latex] so the transformation was a vertical stretch by a factor of 3,\u00a0a reflection across the [latex]x[\/latex], and\u00a0a reflection across the [latex]y[\/latex]-axis<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_9x[\/latex] is reflected across the [latex]x[\/latex]-axis what is the equation of the transformed function?<\/li>\r\n \t<li>If [latex]f(x)=\\log_4x[\/latex] is reflected across the [latex]y[\/latex]-axis what is the equation of the transformed function?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(-x)}+4[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=2\\log_2{(-x)}[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm098\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm098\"]\r\n<ol>\r\n \t<li>The [latex]y[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=-\\log_9x[\/latex]<\/li>\r\n \t<li>\u00a0The [latex]x[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=\\log_4{(-x)}[\/latex]<\/li>\r\n \t<li>The transformation was a reflection across the [latex]y[\/latex]-axis and a shift up by 4 units<\/li>\r\n \t<li>The transformation was a vertical stretch by a factor of 2, and a reflection across the [latex]y[\/latex]-axis<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Combining Transformations<\/h2>\r\nAfter learning all the transformations for the function [latex]f(x)=\\log_b{x}[\/latex], we should be able to write a transformed function given specific transformations, and also determine what transformations have been performed on the function [latex]f(x)=\\log_b{x}[\/latex], given an arbitrary transformed function [latex]f(x)=a\\log_b{(x-h)}+k[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nWhat transformations have been done to the parent function\u00a0[latex]f(x)=\\log_2{x}[\/latex] to get the transformed function\u00a0[latex]f(x)=-3\\log_2{(x+3)}-6[\/latex]?\r\n<h4>Solution<\/h4>\r\nWe need to identify [latex]a,\\;h,\\;k[\/latex] and whether or not [latex]x[\/latex] has a negative sign in front of it. To do this we line up the transformed function\u00a0[latex]f(x)=-3\\log_2{(x+3)}-6[\/latex] with the standard function\u00a0[latex]f(x)=a\\log_2{(x-h)}+k[\/latex]:\r\n\r\n[latex]a=-3[\/latex] means it has been stretched by a factor of 3 and reflected across the [latex]x[\/latex]-axis.\r\n\r\n[latex]h=-3[\/latex] means it has been shifted left by 3 units.\r\n\r\n[latex]k=-6[\/latex] means it has been shifted down by 6 units.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\n<ol>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{-x}+7[\/latex]?<\/li>\r\n \t<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=-2\\log_2{(x-5)}-4[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm529\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm529\"]\r\n<ol>\r\n \t<li>The transformation was a reflection across the [latex]y[\/latex]-axis and a shift up by 7 units<\/li>\r\n \t<li>The transformation was a vertical stretch by a factor of 2, a reflection across the [latex]x[\/latex]-axis, a shift right by 5 units, and a shift down by 4 units.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is reflected across the [latex]x[\/latex]-axis, stretched by a factor of 3, and shifted left by 2 units, what is the equation of the transformed function? What happens to the vertical asymptote?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is reflected across the [latex]y[\/latex]-axis, compressed to half its height, and shifted up by 7 units, what is the equation of the transformed function?\u00a0What happens to the vertical asymptote?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]a=-3,\\;h=-2[\/latex] so the transformed function is [latex]-3\\log_3{(x+2)}[\/latex].\u00a0The vertical asymptote is shifted left by 2 units from [latex]x=0[\/latex] to [latex]x=-2[\/latex].<\/li>\r\n \t<li>[latex]a=\\frac{1}{2},\\;k=7[\/latex] and [latex]x[\/latex] has a negative coefficient so the transformed function is [latex]f(x)=\\frac{1}{2}\\log_7{(-x)}+7[\/latex]. Since there are no horizontal shifts, nothing happens to the vertical asymptote.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 8<\/h3>\r\n<ol>\r\n \t<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 7, shifted right by 4 units and shifted down by 4 units, what is the equation of the transformed function?\u00a0What happens to the vertical asymptote?<\/li>\r\n \t<li>If [latex]f(x)=\\log_7x[\/latex] is reflected across the [latex]x[\/latex]-axis, compressed to one-sixth its height, shifted left by 1 unit, and shifted up by 7 units, what is the equation of the transformed function?\u00a0What happens to the vertical asymptote?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm831\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm831\"]\r\n<ol>\r\n \t<li>[latex]a=7,\\;h=4,\\;k=-4[\/latex] so the transformed function is [latex]f(x)=7\\log_3{(x-4)}-4[\/latex].\u00a0The vertical asymptote is shifted right by 4 units from [latex]x=0[\/latex] to [latex]x=4[\/latex].<\/li>\r\n \t<li>[latex]a=-\\frac{1}{6},\\;h=-1,\\;k=7[\/latex] so the transformed function is [latex]f(x)=-\\frac{1}{6}\\log_7{(x+1)}+7[\/latex].\u00a0The vertical asymptote is shifted left by 1 unit from [latex]x=0[\/latex] to [latex]x=-1[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nMove the red dots in manipulation 5 to change the values of [latex]a, h, k[\/latex] and [latex]b[\/latex] or to reflect the graph across the [latex]y[\/latex]-axis. Pay attention to what happens to the graph and the relationship between the values of [latex]a,\\;h,\\;k[\/latex] and the transformed function.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/olgugm4jyu?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><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\nManipulation 5. Transformations on [latex]f(x)=\\log_b{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div><\/div>\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<p>For the logarithmic function [latex]f(x)=\\log_b{x}[\/latex],<\/p>\n<ul>\n<li>Perform vertical and horizontal shifts<\/li>\n<li>Perform vertical compressions and\u00a0stretches<\/li>\n<li>Perform reflections across the [latex]x[\/latex]-axis<\/li>\n<li>Perform reflections across the [latex]y[\/latex]-axis<\/li>\n<li>Determine the transformations of the logarithmic function\u00a0[latex]f(x)=a\\log_b{(x-h)}+k[\/latex]<\/li>\n<li>Determine the equation of a function given the transformations<\/li>\n<li>Determine what happens to the vertical asymptote as transformations are made<\/li>\n<\/ul>\n<\/div>\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<h2>Vertical Shifts<\/h2>\n<p id=\"fs-id1165137770279\">If we shift the graph of the logarithmic function [latex]f(x)=\\log_2{x}[\/latex] up 5 units, all of the points on the graph increase their [latex]y[\/latex]-coordinates by 5, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\log_2{x}[\/latex] after it has been shifted up 5 units transforms to [latex]f(x)=\\log_2{x}+5[\/latex]. The vertical asymptote at [latex]x=0[\/latex] remains the same. 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;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\n<th style=\"width: 29.277286135693217%; height: 10px;\" scope=\"row\">[latex]\\log_2{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: 389px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2752 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-300x300.png\" alt=\"A green curve is 5 units above the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,y+5) on the green curve.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftUpNew.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Shifting the graph of [latex]f(x)=\\log_2{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;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 54.80468613569322%;\" colspan=\"3\">Table 1. [latex]f(x)=\\log_2{x}[\/latex] is transformed to [latex]f(x)=\\log_2{x}+5[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137770279\">If we shift the graph of the function [latex]f(x)=\\log_2{x}[\/latex] down 6 units, all of the points on the graph decrease their [latex]y[\/latex]-coordinates by 6, but their [latex]x[\/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\\log_2{x}[\/latex] after it has been shifted down 6 units transforms to [latex]f(x)=\\log_2{x}-6[\/latex].The vertical asymptote at [latex]x=0[\/latex] remains the same. 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: 71.4714%; height: 222px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\n<th style=\"width: 29.277286135693217%; height: 10px;\" scope=\"row\">[latex]\\log_2{x}-6[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 222px;\" rowspan=\"9\">\n<div id=\"attachment_1825\" class=\"wp-caption aligncenter\" style=\"width: 370px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2755 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-300x300.png\" alt=\"A green curve is 6 units below the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,y-6) on the green curve.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ShiftDownNew1.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Shifting the graph of [latex]f(x)=\\log_2{x}[\/latex] down 6 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]-9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]-8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 34px;\">[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 29.277286135693217%; height: 19px;\">[latex]-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)=\\log_2{x}[\/latex] is transformed to [latex]f(x)=\\log_2{x}-6[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that vertical shifts up or down do not change the vertical asymptote.<\/p>\n<p>Move the red dots in manipulation 1 to change the values of [latex]b[\/latex] and [latex]k[\/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]k[\/latex] and the transformed 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\/n6lahppwjk?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><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;\">Manipulation 1. Vertical shifts<\/p>\n<div class=\"textbox shaded\">\n<h3>Vertical shifts<\/h3>\n<p>We can represent a vertical shift of the graph of [latex]f(x)=\\log_2{x}[\/latex] by adding or subtracting a constant, [latex]k[\/latex], to the function:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\log_2{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<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is shifted vertically up by 7 units, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is shifted vertically down by 4 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5x+9[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2x-3[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>With vertical shifts, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=\\log_bx+k[\/latex].<\/p>\n<ol>\n<li>[latex]k=7[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3x+7[\/latex]<\/li>\n<li>[latex]k=-4[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7x-4[\/latex]<\/li>\n<li>[latex]k=9[\/latex] so the transformation was a vertical shift up by 9 units.<\/li>\n<li>[latex]k=-3[\/latex] so the transformation was a vertical shift down by 3 units.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is shifted vertically up by 2 units, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is shifted vertically down by 9 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5x+3[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2x-8[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm624\">Show Answer<\/span><\/p>\n<div id=\"qhjm624\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]k=2[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3x+2[\/latex]<\/li>\n<li>[latex]k=-9[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7x-9[\/latex]<\/li>\n<li>[latex]k=3[\/latex] so the transformation was a vertical shift up by 3 units.<\/li>\n<li>[latex]k=-8[\/latex] so the transformation was a vertical shift down by 8 units.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Horizontal Shifts<\/h2>\n<\/div>\n<p>If we shift the graph of the function [latex]f(x)=\\log_2{x}[\/latex] right 8 units, all of the points on the graph increase their [latex]x[\/latex]-coordinates by 8, but their [latex]y[\/latex]-coordinates remain the same. The [latex]x[\/latex]-intercept (1, 0) in the original graph is moved to (9, 0) (figure 3). The vertical asymptote at [latex]x=0[\/latex] shifts right by 8 units to [latex]x=8[\/latex]. Any point [latex](x, y)[\/latex] on the original graph is moved to [latex](x+8, y)[\/latex].<\/p>\n<p>But what happens to the original function [latex]f(x)=\\log_2{x}[\/latex]? An automatic assumption may be that since [latex]x[\/latex] moves to [latex]x+8[\/latex] that the function will become [latex]f(x)=\\log_2{(x+8)}[\/latex]. But that is NOT the case. Remember that the [latex]x[\/latex]-intercept is moved to (9, 0) and if we substitute [latex]x=9[\/latex] into the function\u00a0[latex]f(x)=\\log_2{(x+8)}[\/latex] we get\u00a0[latex]f(9)=\\log_2{(9+8)}=4.0875 \\neq 1[\/latex]!! The way to get a function value of 0 is for the transformed function to be [latex]f(x)=\\log_2{(x-8)}[\/latex]. Then [latex]f(9)=\\log_2{(9-8)}=0[\/latex]. So the function [latex]f(x)=\\log_2{x}[\/latex] transforms to [latex]f(x)=\\log_2{(x-8)}[\/latex] after being shifted 8 units to the right. The reason is that when we move the function 8 units to the right, the [latex]x[\/latex]-value increases by 8 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 8 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 3.<\/p>\n<table style=\"border-collapse: collapse; width: 73.9452%; height: 337px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]x-8[\/latex]<\/th>\n<td style=\"width: 31.7511%; height: 10px;\"><strong>[latex]\\log_2{(x-8)}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 337px;\" rowspan=\"9\">\n<div id=\"attachment_1828\" class=\"wp-caption aligncenter\" style=\"width: 425px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/27152325\/6-2-ShiftRightNew3-300x300.png\" alt=\"A green curve is 8 units to the right of the blue curve for all y-values where each point (x,y) on the blue curve corresponds to a point (x+8,y) on the green curve.\" width=\"425\" height=\"425\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Shifting the graph right 8 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{65}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 34px;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{33}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 34px;\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{17}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 34px;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 31.7511%; height: 19px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 149px;\">\n<td style=\"width: 57.2785%; height: 149px;\" colspan=\"3\">Table 3. Shifting the graph right by 8 units transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\log_2{(x-8)}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the vertical asymptote also shifts from [latex]x=0[\/latex] to [latex]x=8[\/latex].<\/p>\n<p>On the other hand, if we shift the graph of the function [latex]f(x)=\\log_2{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)=\\log_2{(x+11)}[\/latex]. Table 4 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: 74.1562%; height: 296px;\">\n<tbody>\n<tr style=\"height: 19px;\">\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]x+11[\/latex]<\/th>\n<td style=\"width: 31.9621%; height: 10px;\"><strong>[latex]f(x)=\\log_2{(x+11)}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%; height: 296px;\" rowspan=\"9\">\n<div id=\"attachment_1830\" class=\"wp-caption aligncenter\" style=\"width: 439px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3031\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/27152356\/6-2-ShiftLeftNew3-300x300.png\" alt=\"A green curve is 11 units to the left of the blue curve for all y-values where each point (x,y) on the blue curve corresponds to a point (x-11,y) on the green curve.\" width=\"439\" height=\"439\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. Shifting the graph left 11 units.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{87}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 31.9621%; height: 34px;\">\u20133<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{43}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 31.9621%; height: 34px;\">\u20132<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{21}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 31.9621%; height: 34px;\">\u20131<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u201310<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20139<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20137<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20133<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\n<td style=\"width: 31.9621%; height: 19px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 108px;\">\n<td style=\"width: 57.4895%; height: 108px;\" colspan=\"3\">Table 4. Shifting the graph left by 11 units transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\log_2{(x+11)}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"entry-content\">\n<p>Notice that the vertical asymptote also shifts from [latex]x=0[\/latex] to [latex]x=-11[\/latex].<\/p>\n<p>Move the red dots in manipulation 2 to change the values of [latex]b[\/latex] and [latex]h[\/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]h[\/latex] and the transformed function.<\/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\/ukwqebsewa?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><br \/>\nManipulation 2. Horizontal shifts<\/p>\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 [latex]f(x)=\\log_2{x}[\/latex] by adding or subtracting a constant, [latex]h[\/latex], to the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=\\log_2{(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\nThe vertical asymptote [latex]x=0[\/latex] shifts to [latex]x=h[\/latex].\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is shifted right by 2 units, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 9 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x+3)}[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x-8)}[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>With horizontal shifts, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=\\log_b{(x-h)}[\/latex].<\/p>\n<ol>\n<li>[latex]h=2[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-2)}[\/latex]<\/li>\n<li>[latex]h=-9[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+9)}[\/latex]<\/li>\n<li>[latex]h=-3[\/latex] so the transformation was a horizontal shift left by 3 units.<\/li>\n<li>[latex]h=8[\/latex] so the transformation was a horizontal shift right by 8 units<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is shifted right by 7 units, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 4 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x-5)}[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x+4)}[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm832\">Show Answer<\/span><\/p>\n<div id=\"qhjm832\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]h=7[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-7)}[\/latex]<\/li>\n<li>[latex]h=-4[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+4)}[\/latex]<\/li>\n<li>[latex]h=5[\/latex] so the transformation was a horizontal shift right by 5 units.<\/li>\n<li>[latex]h=-4[\/latex] so the transformation was a horizontal shift left by 4 units<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>We can combine vertical and horizontal shifts by transforming [latex]f(x)=\\log_b{(x-h)}+k[\/latex].<\/p>\n<p>Move the red dots in manipulation 3 to change the values of [latex]b, \\;h[\/latex] and [latex]k[\/latex]. Pay attention to what happens to the graph and the relationship between the values of [latex]h[\/latex] and[latex]h[\/latex] and the transformed 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\/tq5mj8l9xf?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><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\nManipulation 3. Vertical and horizontal shifts<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is shifted vertically down by 3 units and right by 2 units, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 9 units and up by 6 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x-2)}+7[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x+5)}-4[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>With horizontal and vertical shifts, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=\\log_b{(x-h)}+k[\/latex].<\/p>\n<ol>\n<li>[latex]h=2,\\;k=-3[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-2)}-3[\/latex]<\/li>\n<li>[latex]h=-9,\\;k=6[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+9)}+6[\/latex]<\/li>\n<li>[latex]h=2,\\;k=7[\/latex] so the transformation was a horizontal shift right by 2 units and up by 7 units.<\/li>\n<li>[latex]h=-5,\\;k=-4[\/latex] so the transformation was a horizontal shift left by 5 units and down by 4 units.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is shifted down by 7 units and right by 5 units, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is shifted left by 4 units and up by 3 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(x-2)}-1[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_2{(x+4)}+7[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm302\">Show Answer<\/span><\/p>\n<div id=\"qhjm302\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]h=5,\\;k=-7[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_3{(x-5)}-7[\/latex]<\/li>\n<li>[latex]h=-4,\\;k=3[\/latex] so the transformed function is\u00a0[latex]f(x)=\\log_7{(x+4)}+3[\/latex]<\/li>\n<li>[latex]h=2,\\;k=-1[\/latex] so the transformation was a horizontal shift right by 2 units and down by 1 unit.<\/li>\n<li>[latex]h=-4,\\;k=7[\/latex] so the transformation was a horizontal shift left by 4 units and up by 7 units.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Vertical Stretching and Compressing<\/h2>\n<p>If we vertically stretch the graph of the function [latex]f(x)=\\log_2{x}[\/latex] by a factor of 2, all of the[latex]y[\/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[\/latex]-coordinates remain the same. The equation of the function after the graph is stretched up by a factor of 2 is [latex]f(x)=2\\log_2{x}[\/latex]. The reason for multiplying\u00a0 [latex]\\log_2{x}[\/latex] by 2 is that each [latex]y[\/latex]-coordinate is doubled, and since [latex]y=\\log_2{x}[\/latex], [latex]\\log_2{x}[\/latex] is doubled. Table 5 shows this change and the graph is shown in figure 5.<\/p>\n<table style=\"border-collapse: collapse; width: 58.860800000000005%; height: 365px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 10px;\"><strong>[latex]f(x)=2\\log_2{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: 425px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3002 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-300x300.png\" alt=\"A green curve is twice as far above or below the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,2y) on the green curve.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-StretchNew2.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. Stretching the graph vertically.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-3<\/td>\n<td style=\"width: 16.6667%; height: 34px;\">-6<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-2<\/td>\n<td style=\"width: 16.6667%; height: 34px;\">-4<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-1<\/td>\n<td style=\"width: 16.6667%; height: 34px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">4<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">6<\/td>\n<\/tr>\n<tr style=\"height: 177px;\">\n<td style=\"width: 42.1941%; height: 177px;\" colspan=\"3\">Table 5.\u00a0Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=2\\log_2{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]f(x)=\\log_2{x}[\/latex] to half of its original height, we multiply the function by the factor [latex]\\dfrac{1}{2}[\/latex]. All of the [latex]y[\/latex]-coordinates of the points on the graph are halved, but their [latex]x[\/latex]-coordinates remain the same. This means the [latex]y[\/latex]-coordinates are divided by 2, or multiplied by [latex]\\dfrac{1}{2}[\/latex]. The equation of the function after being compressed is [latex]f(x)=\\dfrac{1}{2}\\times\\log_2{x}[\/latex]. The reason for multiplying [latex]\\log_2{x}[\/latex] by [latex]\\dfrac{1}{2}[\/latex] is that each [latex]y[\/latex]-coordinate becomes half of the original value when it is divided by 2. Table 6 shows this change and the graph is shown in figure 6.<\/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;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 10px;\"><strong>[latex]f(x)=\\dfrac{1}{2}\\log_2{x}[\/latex]<\/strong><\/td>\n<td style=\"width: 16.6667%;\" rowspan=\"9\">\n<div id=\"attachment_1866\" class=\"wp-caption aligncenter\" style=\"width: 455px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3003\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/27031453\/6-2-CompressNew2-300x300.png\" alt=\"A green curve is half as far above or below the blue curve for all x-values where each point (x,y) on the blue curve corresponds to a point (x,y\/2) on the green curve.\" width=\"455\" height=\"455\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. Compressing the graph vertically.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-3<\/td>\n<td style=\"width: 16.6667%; height: 34px;\">[latex]-\\dfrac{3}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-2<\/td>\n<td style=\"width: 16.6667%; height: 34px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-1<\/td>\n<td style=\"width: 16.6667%; height: 34px;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\n<td style=\"width: 16.6667%; height: 19px;\">[latex]\\dfrac{3}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 42.1941%;\" colspan=\"3\">Table 6.\u00a0Compressing the graph vertically by a factor of [latex]\\dfrac{1}{2}[\/latex] transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\dfrac{1}{2}\\log_2{x}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that vertical stretching and compressing do not change the vertical asymptote.<\/p>\n<p>Move the red dots in manipulation 4 to change the values of [latex]b[\/latex] and [latex]a[\/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]a[\/latex] and the transformed 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\/sdxxkrpf6o?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><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\nManipulation 4. Vertical stretching and compressing<\/p>\n<div class=\"entry-content\">\n<div class=\"textbox shaded\">\n<h3>vertical stretching and compressing<\/h3>\n<p id=\"fs-id1165137770279\">A stretch or compression of the graph of [latex]f(x)=\\log_2{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\\log_2{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 up by a factor of [latex]a.[\/latex] If [latex]0<a<1[\/latex], the graph is compressed down by a factor of [latex]a[\/latex].\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 7, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-third its height, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{6}\\log_5x[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=5\\log_2x[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>With stretching and compression, the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=a\\log_bx[\/latex].<\/p>\n<ol>\n<li>[latex]a=7[\/latex] so the transformed function is\u00a0[latex]f(x)=7\\log_3x[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{3}[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{3}\\log_7x[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{6}[\/latex] so the transformation was a compression to one-sixth its height<\/li>\n<li>[latex]a=5[\/latex] so the transformation was a vertical stretch by a factor of 5<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 2, what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-eighth its height, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{4}\\log_5x[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=7\\log_2x[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm442\">Show Answer<\/span><\/p>\n<div id=\"qhjm442\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]a=2[\/latex] so the transformed function is\u00a0[latex]f(x)=2\\log_3x[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{8}[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{8}\\log_7x[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{4}[\/latex] so the transformation was a compression to one-quarter its height<\/li>\n<li>[latex]a=7[\/latex] so the transformation was a vertical stretch by a factor of 7<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we can combine vertical stretches and compressions with horizontal and vertical shifts.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 7, and moved down by 5 units what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-third its height, moved left by 4 units and moved up by 2 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{5}\\log_5{(x-9)}-6[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=3\\log_2{(x+5)}+7[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>With stretching and compression, combined with shifting the parent function\u00a0[latex]f(x)=\\log_bx[\/latex] is transformed to\u00a0[latex]f(x)=a\\log_b{(x-h)}+k[\/latex].<\/p>\n<ol>\n<li>[latex]a=7,\\; k=-5[\/latex] so the transformed function is\u00a0[latex]f(x)=7\\log_3x-5[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{3},\\;h=-4,\\;k=2[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{3}\\log_7{(x+4)}+2[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{5},\\;h=9,\\;k=-6[\/latex] so the transformation was a compression to one-fifth its height, a shift right by 9 units, and a shift down by 6 units<\/li>\n<li>[latex]a=3,\\;-5,\\;k=7[\/latex] so the transformation was a vertical stretch by a factor of 3,\u00a0a shift left by 5 units, and a shift up by 7 units<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 5, and moved down by 2 units what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is compressed to one-half its height, moved left by 7 units and moved up by 3 units, what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\frac{1}{3}\\log_5{(x-2)}-3[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=9\\log_2{(x+1)}+5[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm356\">Show Answer<\/span><\/p>\n<div id=\"qhjm356\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]a=5,\\; k=-2[\/latex] so the transformed function is\u00a0[latex]f(x)=5\\log_3x-2[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{2},\\;h=-7,\\;k=3[\/latex] so the transformed function is\u00a0[latex]f(x)=\\frac{1}{2}\\log_7{(x+7)}+3[\/latex]<\/li>\n<li>[latex]a=\\frac{1}{3},\\;h=2,\\;k=-3[\/latex] so the transformation was a compression to one-third its height, a shift right by 2 units, and a shift down by 3 units<\/li>\n<li>[latex]a=9,\\;h=-1,\\;k=5[\/latex] so the transformation was a vertical stretch by a factor of 9,\u00a0a shift left by 1 unit, and a shift up by 5 units<\/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)=\\log_2{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)=\\log_2{x}[\/latex] is reflected across the [latex]x[\/latex]-axis is [latex]f(x)=-\\log_2{x}[\/latex]. The graph changes from increasing upwards to decreasing downwards. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 7.<\/p>\n<table style=\"border-collapse: collapse; width: 54.9578%; height: 306px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{x}[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]-\\log_2{x}[\/latex]<\/th>\n<td style=\"width: 16.6667%; height: 306px;\" rowspan=\"9\">\n<div id=\"attachment_1868\" class=\"wp-caption aligncenter\" style=\"width: 436px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2768 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-300x300.png\" alt=\"A green curve is the reflection of the blue curve for all x-values across the x-axis: each point (x,y) on the blue curve corresponds to a point (x,-y) on the green curve.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossX1.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across 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;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-3<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-2<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">-1<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">-3<\/td>\n<\/tr>\n<tr style=\"height: 154px;\">\n<td style=\"width: 38.2911%; height: 154px;\" colspan=\"3\">Table 7.\u00a0Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across the [latex]x[\/latex]-axis transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=-\\log_2{x}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that reflecting across the [latex]x[\/latex]-axis does not change the vertical asymptote.<\/p>\n<h3>Across the [latex]y[\/latex]-axis<\/h3>\n<p>When the graph of the function [latex]f(x)=\\log_2{x}[\/latex] is 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, while the [latex]y[\/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\\log_2{x}[\/latex] is reflected across the [latex]y[\/latex]-axis is [latex]f(x)=\\log_2{(-x)}[\/latex]. The graph changes from increasing from the left to decreasing from the left. Table 8 shows the effect of such a reflection on the functions values and the graph is shown in figure 8.<\/p>\n<table style=\"border-collapse: collapse; width: 58.75077591776798%; height: 683px;\">\n<tbody>\n<tr style=\"height: 10px;\">\n<th style=\"width: 12.7637%; height: 10px;\" scope=\"row\">[latex]x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]-x[\/latex]<\/th>\n<th style=\"width: 12.7637%; height: 10px;\">[latex]\\log_2{(-x)}[\/latex]<\/th>\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-2770 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-300x300.png\" alt=\"A green curve is the reflection of the blue curve for all y-values across the y-axis: each point (x,y) on the blue curve corresponds to a point (-x,y) on the green curve.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/6-2-ReflectionAcrossY1.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across 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;\">[latex]-\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">\u20133<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">\u20132<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 12.7637%; height: 34px;\">[latex]-\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 12.7637%; height: 34px;\">\u20131<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20131<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20132<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20134<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">4<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"width: 12.7637%; height: 19px;\">\u20138<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">8<\/td>\n<td style=\"width: 12.7637%; height: 19px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 154px;\">\n<td style=\"width: 38.2911%; height: 154px;\" colspan=\"3\">Table 8.\u00a0Reflecting the graph of [latex]f(x)=\\log_2{x}[\/latex] across the [latex]y[\/latex]-axis transforms [latex]f(x)=\\log_2{x}[\/latex] into [latex]f(x)=\\log_2{(-x)}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div>Notice that reflecting across the [latex]x[\/latex]-axis does not change the vertical asymptote.<\/div>\n<\/div>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<div id=\"post-403\" class=\"standard post-403 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div><\/div>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is reflected across the [latex]x[\/latex]-axis what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is reflected across the [latex]y[\/latex]-axis what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=-\\log_5{x}-6[\/latex]?<\/li>\n<li>What transformation was made to the parent function [latex]f(x)=\\log_2x[\/latex] if the transformed function is [latex]f(x)=-3\\log_2{(-x)}[\/latex]?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>The [latex]y[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=-\\log_3x[\/latex]<\/li>\n<li>\u00a0The [latex]x[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=\\log_7{(-x)}[\/latex]<\/li>\n<li>[latex]a=-1,\\;k=-6[\/latex] so the transformation was a reflection across the [latex]x[\/latex]-axis and a shift down by 6 units<\/li>\n<li>[latex]a=-3[\/latex] and [latex]x[\/latex] is\u00a0[latex]-x[\/latex] so the transformation was a vertical stretch by a factor of 3,\u00a0a reflection across the [latex]x[\/latex], and\u00a0a reflection across the [latex]y[\/latex]-axis<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_9x[\/latex] is reflected across the [latex]x[\/latex]-axis what is the equation of the transformed function?<\/li>\n<li>If [latex]f(x)=\\log_4x[\/latex] is reflected across the [latex]y[\/latex]-axis what is the equation of the transformed function?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{(-x)}+4[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=2\\log_2{(-x)}[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm098\">Show Answer<\/span><\/p>\n<div id=\"qhjm098\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The [latex]y[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=-\\log_9x[\/latex]<\/li>\n<li>\u00a0The [latex]x[\/latex]-values change sign so the transformed function is\u00a0[latex]f(x)=\\log_4{(-x)}[\/latex]<\/li>\n<li>The transformation was a reflection across the [latex]y[\/latex]-axis and a shift up by 4 units<\/li>\n<li>The transformation was a vertical stretch by a factor of 2, and a reflection across the [latex]y[\/latex]-axis<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Combining Transformations<\/h2>\n<p>After learning all the transformations for the function [latex]f(x)=\\log_b{x}[\/latex], we should be able to write a transformed function given specific transformations, and also determine what transformations have been performed on the function [latex]f(x)=\\log_b{x}[\/latex], given an arbitrary transformed function [latex]f(x)=a\\log_b{(x-h)}+k[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>What transformations have been done to the parent function\u00a0[latex]f(x)=\\log_2{x}[\/latex] to get the transformed function\u00a0[latex]f(x)=-3\\log_2{(x+3)}-6[\/latex]?<\/p>\n<h4>Solution<\/h4>\n<p>We need to identify [latex]a,\\;h,\\;k[\/latex] and whether or not [latex]x[\/latex] has a negative sign in front of it. To do this we line up the transformed function\u00a0[latex]f(x)=-3\\log_2{(x+3)}-6[\/latex] with the standard function\u00a0[latex]f(x)=a\\log_2{(x-h)}+k[\/latex]:<\/p>\n<p>[latex]a=-3[\/latex] means it has been stretched by a factor of 3 and reflected across the [latex]x[\/latex]-axis.<\/p>\n<p>[latex]h=-3[\/latex] means it has been shifted left by 3 units.<\/p>\n<p>[latex]k=-6[\/latex] means it has been shifted down by 6 units.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<ol>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_5x[\/latex] if the transformed function is\u00a0[latex]f(x)=\\log_5{-x}+7[\/latex]?<\/li>\n<li>What transformation was made to the parent function\u00a0[latex]f(x)=\\log_2x[\/latex] if the transformed function is\u00a0[latex]f(x)=-2\\log_2{(x-5)}-4[\/latex]?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm529\">Show Answer<\/span><\/p>\n<div id=\"qhjm529\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The transformation was a reflection across the [latex]y[\/latex]-axis and a shift up by 7 units<\/li>\n<li>The transformation was a vertical stretch by a factor of 2, a reflection across the [latex]x[\/latex]-axis, a shift right by 5 units, and a shift down by 4 units.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is reflected across the [latex]x[\/latex]-axis, stretched by a factor of 3, and shifted left by 2 units, what is the equation of the transformed function? What happens to the vertical asymptote?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is reflected across the [latex]y[\/latex]-axis, compressed to half its height, and shifted up by 7 units, what is the equation of the transformed function?\u00a0What happens to the vertical asymptote?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]a=-3,\\;h=-2[\/latex] so the transformed function is [latex]-3\\log_3{(x+2)}[\/latex].\u00a0The vertical asymptote is shifted left by 2 units from [latex]x=0[\/latex] to [latex]x=-2[\/latex].<\/li>\n<li>[latex]a=\\frac{1}{2},\\;k=7[\/latex] and [latex]x[\/latex] has a negative coefficient so the transformed function is [latex]f(x)=\\frac{1}{2}\\log_7{(-x)}+7[\/latex]. Since there are no horizontal shifts, nothing happens to the vertical asymptote.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 8<\/h3>\n<ol>\n<li>If [latex]f(x)=\\log_3x[\/latex] is stretched by a factor of 7, shifted right by 4 units and shifted down by 4 units, what is the equation of the transformed function?\u00a0What happens to the vertical asymptote?<\/li>\n<li>If [latex]f(x)=\\log_7x[\/latex] is reflected across the [latex]x[\/latex]-axis, compressed to one-sixth its height, shifted left by 1 unit, and shifted up by 7 units, what is the equation of the transformed function?\u00a0What happens to the vertical asymptote?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm831\">Show Answer<\/span><\/p>\n<div id=\"qhjm831\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]a=7,\\;h=4,\\;k=-4[\/latex] so the transformed function is [latex]f(x)=7\\log_3{(x-4)}-4[\/latex].\u00a0The vertical asymptote is shifted right by 4 units from [latex]x=0[\/latex] to [latex]x=4[\/latex].<\/li>\n<li>[latex]a=-\\frac{1}{6},\\;h=-1,\\;k=7[\/latex] so the transformed function is [latex]f(x)=-\\frac{1}{6}\\log_7{(x+1)}+7[\/latex].\u00a0The vertical asymptote is shifted left by 1 unit from [latex]x=0[\/latex] to [latex]x=-1[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Move the red dots in manipulation 5 to change the values of [latex]a, h, k[\/latex] and [latex]b[\/latex] or to reflect the graph across the [latex]y[\/latex]-axis. Pay attention to what happens to the graph and the relationship between the values of [latex]a,\\;h,\\;k[\/latex] and the transformed 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\/olgugm4jyu?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><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\nManipulation 5. Transformations on [latex]f(x)=\\log_b{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div><\/div>\n<\/div>\n<\/div>\n<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-2358\">\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 logarithmic function f(x)=log_b{x}. <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. <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>All Examples and Try Its: hjm624; hjm832; hjm302; 442; hjm356; hjm098; hjm529; hjm831. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All manipulations created using Desmos. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/desmos.com\">http:\/\/desmos.com<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Transformations of the logarithmic function f(x)=log_b{x}\",\"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\",\"organization\":\"Utah Valley University\",\"url\":\"www.desmos.com\/calculator\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All Examples and Try Its: hjm624; hjm832; hjm302; 442; hjm356; hjm098; hjm529; hjm831\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All manipulations created using Desmos\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"desmos.com\",\"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-2358","chapter","type-chapter","status-publish","hentry"],"part":2310,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2358","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":81,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2358\/revisions"}],"predecessor-version":[{"id":4824,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2358\/revisions\/4824"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2310"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2358\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2358"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2358"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2358"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}