{"id":974,"date":"2022-03-06T23:19:04","date_gmt":"2022-03-06T23:19:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=974"},"modified":"2026-01-17T02:11:44","modified_gmt":"2026-01-17T02:11:44","slug":"2-4-transformations-of-a-linear-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-4-transformations-of-a-linear-function\/","title":{"raw":"2.4: Transformations of the Linear Function f(x)=x","rendered":"2.4: Transformations of the Linear Function f(x)=x"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-93\" class=\"standard post-93 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning objectives<\/h3>\r\n<ul>\r\n \t<li>Perform a vertical shift on a linear function<\/li>\r\n \t<li>Perform a vertical stretch or compression on a linear function<\/li>\r\n \t<li>Perform a reflection of a linear function across the [latex]x[\/latex]-axis<\/li>\r\n \t<li>Perform a combination of transformations on a linear function<\/li>\r\n \t<li>Explain the transformations performed on\u00a0[latex]f(x)=x[\/latex] given the transformed function<\/li>\r\n \t<li>Write an algebraic function after completing transformations on the parent function<\/li>\r\n<\/ul>\r\n<\/div>\r\nRecall the introduction to graphical\u00a0<em><strong>transformations<\/strong><\/em> of a function we saw in section 1.3. A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression. The original function [latex]f(x)=x[\/latex] is also known as the <em><strong>parent function<\/strong><\/em> and is the linear function used for transformations in this section. We will apply transformations graphically and consider what these transformations mean algebraically.\r\n<h2>Vertical Shifts<\/h2>\r\nMaking a <em><strong>vertical shift<\/strong><\/em> means to move the graph of the linear function vertically up or down a certain number of units. When we shift a line up or down, the slope (steepness) of the line does not change. What changes is the [latex]y[\/latex]-intercept. For example, the function [latex]f(x)=x[\/latex] has a slope of 1 and a [latex]y[\/latex]-intercept at (0, 0). If we shift the line up two units, the slope does not change, but the [latex]y[\/latex]-intercept moves to (0, 2) (Figure 1).\r\n\r\n[caption id=\"attachment_1355\" align=\"aligncenter\" width=\"434\"]<img class=\"wp-image-1355\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05215249\/Vert-shift-up-2-300x276.png\" alt=\"A blue line with y-intercept (0,0) and slope 1 shifted up 2 units becomes a green line with y-intercept (0,2) and slope 1.\" width=\"434\" height=\"399\" \/> Figure 1. Vertical shift[\/caption]\r\n\r\nIf fact, every point on the graph shifts up by 2 units so for every point on the line [latex]y=x[\/latex], the [latex]y[\/latex]-coordinate of each point increases by 2. This is shown in Table 1.\r\n<table style=\"border-collapse: collapse; width: 21.312821745488055%; height: 222px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 32.2997%;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 32.2997%;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 35.6589%;\">[latex]y+2[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 32.299741602067186%;\">0<\/td>\r\n<td style=\"width: 32.299741602067186%;\">0<\/td>\r\n<td style=\"width: 35.65891472868217%;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 32.299741602067186%;\">1<\/td>\r\n<td style=\"width: 32.299741602067186%;\">1<\/td>\r\n<td style=\"width: 35.65891472868217%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 32.299741602067186%;\">2<\/td>\r\n<td style=\"width: 32.299741602067186%;\">2<\/td>\r\n<td style=\"width: 35.65891472868217%;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 32.299741602067186%;\">3<\/td>\r\n<td style=\"width: 32.299741602067186%;\">3<\/td>\r\n<td style=\"width: 35.65891472868217%;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 32.299741602067186%;\">4<\/td>\r\n<td style=\"width: 32.299741602067186%;\">4<\/td>\r\n<td style=\"width: 35.65891472868217%;\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 32.2997%;\" colspan=\"3\">Table 1. Increase in [latex]y[\/latex]-values<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn other words, the function [latex]f(x)=x[\/latex] transforms to [latex]f(x)=x+2[\/latex] because all of the [latex]y[\/latex]-values are increased by 2.\r\n\r\nIf the function [latex]f(x)=x[\/latex] is shifted down two units, it means all the [latex]y[\/latex]-coordinates of the points on the line are decreased by 2 (increased by \u20132) (Table 2).\u00a0In other words, the function [latex]f(x)=x[\/latex] becomes [latex]f(x)=x-2[\/latex] because all of the [latex]y[\/latex]-values are decreased by 2. From\u00a0another perspective, this transformation results in the original [latex]y[\/latex]-intercept (0, 0) moving to (0, \u20132). Therefore, the function becomes [latex]f(x)=x-2[\/latex].\r\n<table style=\"border-collapse: collapse; width: 38.937803723907656%; height: 230px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 111.691px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 111.691px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 87.3162px;\">[latex]y\u20132[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 111.69116973876953px;\">0<\/td>\r\n<td style=\"width: 111.69116973876953px;\">0<\/td>\r\n<td style=\"width: 87.31617736816406px;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 111.69116973876953px;\">1<\/td>\r\n<td style=\"width: 111.69116973876953px;\">1<\/td>\r\n<td style=\"width: 87.31617736816406px;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 111.69116973876953px;\">2<\/td>\r\n<td style=\"width: 111.69116973876953px;\">2<\/td>\r\n<td style=\"width: 87.31617736816406px;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 111.69116973876953px;\">3<\/td>\r\n<td style=\"width: 111.69116973876953px;\">3<\/td>\r\n<td style=\"width: 87.31617736816406px;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 111.69116973876953px;\">4<\/td>\r\n<td style=\"width: 111.69116973876953px;\">4<\/td>\r\n<td style=\"width: 87.31617736816406px;\">6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 111.691px;\" colspan=\"3\">Table 2. Decrease in [latex]y[\/latex]-values<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div><\/div>\r\n<div>Figure 2 shows different vertical transformations of the function [latex]f(x)=x[\/latex] to [latex]f(x)=x+b[\/latex]. If [latex]b&gt;0[\/latex] the line moves up, and if [latex]b&lt;0[\/latex] the line moves down.<\/div>\r\n<div class=\"wp-nocaption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"566\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201052\/CNX_Precalc_Figure_02_02_0062.jpg\" alt=\"graph showing y = x , y = x+2, y = x+4, y = x-2, y = x-4. All are parallel lines with each having an intercept equal to the value added to x.\" width=\"566\" height=\"477\" \/> Figure 2. Vertical shifts.[\/caption]\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">This can be investigated in Desmos by changing the value of [latex]b[\/latex]. Click on the desmos icon on the bottom right corner of figure 3 and move the slider to change the value of [latex]b[\/latex].<\/p>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/nxc0ola2td?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 3. Transformation of [latex]f(x)=x[\/latex] to [latex]f(x)=x+b[\/latex].<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nThe graph shows the function [latex]f(x)=x[\/latex]. Use this graph to shift the function down by 4 units. Then write the equation of the new function.\r\n\r\n<img class=\"aligncenter wp-image-1360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05235256\/fxx-300x300.png\" alt=\"Graph of f(x)=x. y-intercept (0,0), slope of 1.\" width=\"167\" height=\"167\" \/>\r\n<h4>Solution<\/h4>\r\nWhen the function is shifted down by 4 units, it is the [latex]y[\/latex]-coordinates that are decreased by 4 while the [latex]x[\/latex]-coordinates stay the same.\r\n\r\n<img class=\"aligncenter wp-image-1363 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-274x300.png\" alt=\"A vertical transformation showing each point on f(x) has moved straight down 4 units forming a new line with y-intercept (0,-4) with a slope of 1.\" width=\"274\" height=\"300\" \/>\r\n\r\nThe transformed function takes the form [latex]f(x)=x+b[\/latex] where [latex]b=-4[\/latex]: [latex]f(x)=x-4[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nThe graph shows the function [latex]f(x)=x[\/latex]. Use this graph to shift the function up by 3 units . Then write the equation of the new function.\r\n\r\n<img class=\"aligncenter wp-image-1360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05235256\/fxx-300x300.png\" alt=\"Graph of f(x)=x. y-intercept (0,0), slope of 1.\" width=\"167\" height=\"167\" \/>\r\n\r\n[reveal-answer q=\"hjm152\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm152\"]\r\n<img class=\"aligncenter wp-image-1364 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-300x273.png\" alt=\"A vertical transformation showing each point on f(x) has moved straight down 4 units forming a new line with y-intercept (0,-4) with a slope of 1.\" width=\"300\" height=\"273\" \/>\r\n\r\n[latex]f(x)=x+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Vertical Stretching or Compressing<\/h2>\r\n<h3>Vertically stretching<\/h3>\r\nIf we could grab both ends of the line and pull vertically in opposite directions (e.g., the end goes up if it is above the x-axis, and the end goes down if it is below the x-axis) we would stretch the line causing the slope of the line to increase. This is <em><strong>vertical stretching<\/strong><\/em> of a linear function. Another perspective of viewing vertically stretching a line is that every y coordinate of points on the line is multiplied by a number that is greater than 1. Therefore, if stretching a line vertically twice, the point (1, 2) becomes (1, 4) (e.g., the y-coordinate is twice larger) and the point (-2, -1) becomes (-2, -2).\r\n\r\nTo increase the steepness of a function the slope must increase from 1 to a number that is greater than 1. For example, stretching the function [latex]f(x)=x[\/latex] to be twice as steep means multiplying the original slope of 1 by 2. Therefore, the stretched function becomes [latex]f(x)=2x[\/latex]. It is important to note that the position of the [latex]y[\/latex]-intercept does not change because there is no vertical shift.\r\n<h3>Vertically compressing down<\/h3>\r\nIf instead of being able to stretch the ends of the line in opposite directions we were able to push the ends of the line, causing the slope of the line to decrease, we would witness vertical compression. The result would be to decrease the steepness (slope) of the line.\u00a0Another perspective of viewing vertically compressing a line is that every y-coordinate of points on the line is multiplied by a number that is less than 1. Therefore, if compressing a line vertically to half of its height, the point (1, 2) becomes (1, 1) (e.g., the y-coordinate is half of its height) and the point (-2, -1) becomes (-2, -1\/2).\r\n\r\nTo decrease the steepness of a line means to multiply the original slope of 1 by a number that is between 0 and 1. For example, compressing the function [latex]f(x)=x[\/latex] to one-half of its height means multiplying the slope by [latex]\\dfrac{1}{2}[\/latex]. Therefore, the compressed function is [latex]f(x)=\\dfrac{1}{2}x[\/latex]. It is important to note that the position of the [latex]y[\/latex]-intercept does not change because there is no vertical shift.\r\n\r\nFigure 4 illustrates the graphs of various functions that are the result of stretching ([latex]m&gt;1[\/latex] or compressing ([latex]0\u2264m&lt;1[\/latex] the function [latex]f(x)=x[\/latex] to [latex]f(x)=mx[\/latex].\r\n\r\n[caption id=\"attachment_1928\" align=\"aligncenter\" width=\"409\"]<img class=\"wp-image-1928\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/30001147\/desmos-graph-98-300x300.png\" alt=\"Figure 4. Stretching and compressing a linear function. As the slope increases the line is steeper.\" width=\"409\" height=\"409\" \/> Figure 4. Stretching and compressing a linear function.[\/caption]\r\n\r\n<div class=\"wp-nocaption aligncenter\"><\/div>\r\nThis can be investigated in Desmos by changing the value of [latex]m[\/latex]. Click on the desmos icon on the bottom right corner of figure 5 and move the slider to change the value of [latex]m[\/latex]. Notice what happens to the coordinate (1, 1) on the original graph as the transformation plays out.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/nvqssi6jcx?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 5. Transformation of [latex]f(x)=x[\/latex] to [latex]f(x)=mx[\/latex].<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nDescribe the\u00a0transformation of the function [latex]f(x)=x[\/latex] to the function [latex]f(x)=\\dfrac{2}{3}x[\/latex] and\u00a0draw a graph.\r\n<h4>Solution<\/h4>\r\nThe transformed function has a slope of [latex]m=\\dfrac{2}{3}[\/latex], so this is a vertical compression since [latex]0&lt;m&lt;1[\/latex].\r\n\r\nThe graph of\u00a0[latex]f(x)=\\dfrac{2}{3}x[\/latex]\u00a0is shown with the original function [latex]f(x)=x[\/latex]:\r\n<div class=\"wp-nocaption size-medium wp-image-2245 aligncenter\"><img class=\"aligncenter wp-image-2245 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07171247\/Screen-Shot-2016-07-07-at-10.12.10-AM-300x289.png\" alt=\"A graph with f(x)=x, with y-intercept (0,0) and slope 1, and a second line y=two thirds x, with y-intercept (0,0) and slope 2\/3.\" width=\"300\" height=\"289\" \/><\/div>\r\nNote how the original function is compressed because the rate of change is \u201cslowed\u201d.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nWrite the equation of the transformation when [latex]f(x)=x[\/latex] is stretched vertically by a factor of 3.\r\n<h4>Solution<\/h4>\r\nAll the function values must be multiplied by 3, while the [latex]x[\/latex]-values stay the same.\r\n\r\n[latex]f(x)=3x[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>try it 2<\/h3>\r\nWrite the equation of the transformation when [latex]f(x)=x[\/latex] is stretched or compressed vertically by the given factor:\r\n<ol>\r\n \t<li>Stretched by a factor of 7<\/li>\r\n \t<li>Compressed to one-third its height<\/li>\r\n \t<li>Compressed to one-fourth its height<\/li>\r\n \t<li>Stretched by a factor of 5<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm906\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm906\"]\r\n<ol>\r\n \t<li>[latex]f(x)=7x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\frac{1}{3}x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\frac{1}{4}x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=5x[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Reflection Across the x-axis<\/h2>\r\nReflecting a linear function across the x-axis means to flip the line representing the function across the x-axis. This results in\u00a0 a mirror image of the original function with respect to the x-axis. To reflect a linear function across the [latex]x[\/latex]-axis means all points on the line will see their [latex]y[\/latex]-coordinates change sign; positive [latex]x[\/latex]-coordinates will become negative (e.g, (2, 2) becomes (2, -2)), and negative [latex]y[\/latex]-coordinates will become positive (e.g., (1, \u20135) becomes (1, 5)). From an algebraic point of view, vertically flipping the graph of the function [latex]f(x)=x[\/latex] is equivalent to multiplying the [latex]x[\/latex]-values by \u20131. That is, the [latex]y[\/latex] value in [latex]y=f(x)=x[\/latex] becomes [latex]y=f(x)=-x[\/latex] (e.g., positive [latex]y[\/latex] values become negative, or negative [latex]y[\/latex] values become positive). \u00a0So [latex]f(x)=x[\/latex] becomes [latex]f(x)=-x[\/latex]. Table 3 shows the transformation.\r\n<table style=\"border-collapse: collapse; width: 0%; height: 162px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 18px;\">\r\n<th style=\"width: 33.3333%; height: 18px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 33.3333%; height: 18px;\">[latex]f(x)=x[\/latex]<\/th>\r\n<th style=\"width: 33.3333%; height: 18px;\">[latex]f(x) = -x[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">-3<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">-3<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">-2<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">-2<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">-1<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">-1<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">0<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">0<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">1<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">1<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">-1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">2<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">2<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">-2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">3<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">3<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">-3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\" colspan=\"3\">Table 3.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div><\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nThe graph shows the function [latex]f(x)=x[\/latex]. Use this graph to reflect the function across the [latex]x[\/latex]-axis. Then write the equation of the new function.\r\n\r\n<img class=\"aligncenter wp-image-1360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05235256\/fxx-300x300.png\" alt=\"Graph of f(x)=x\" width=\"167\" height=\"167\" \/>\r\n<h4>Solution<\/h4>\r\nAll of the [latex]y[\/latex]-values change sign with the function values staying the same. So (3, 3) moves to (3, \u20133), (\u20131, \u20131) moves to (\u20131, 1) etc. Once the new points have been plotted the new line can be drawn.\r\n\r\n<img class=\"aligncenter wp-image-1930 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-300x300.png\" alt=\"reflected lines. The reflected line described above.\" width=\"300\" height=\"300\" \/>\r\n\r\nThe new function is [latex]f(x)=-x[\/latex].\r\n\r\n<\/div>\r\n<h2>Combining Transformations<\/h2>\r\nWe can combine different transformations, one after the other.\u00a0Suppose we want to transform the function [latex]f(x)=x[\/latex] by reflecting this function across the [latex]x[\/latex]-axis, shifting it up 3 units. The reflection across the [latex]x[\/latex]-axis results in the new function [latex]f(x)=-x[\/latex]. The vertical shift of 3 units up results in the function [latex]f(x)=-x+3[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nGraph [latex]f\\left(x\\right)=\\dfrac{1}{2}x - 3[\/latex] using transformations of [latex]f(x)=x[\/latex].\r\n<h4>Solution<\/h4>\r\nThe equation for the function shows that [latex]m=\\dfrac{1}{2}[\/latex], so the original function is vertically compressed to one-half its height. The equation for the function also shows that [latex]b=\u20133[\/latex], so the original function is vertically shifted down by 3 units. First, graph the function [latex]f(x)=x[\/latex] and show the vertical compression (the [latex]y[\/latex]-values are all divided by 2).\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201054\/CNX_Precalc_Figure_02_02_0072.jpg\" alt=\"graph showing the lines y = x and y = (1\/2)x. For each x value the y value has been halved.\" width=\"487\" height=\"378\" \/><\/div>\r\nNow show the vertical shift.\u00a0The function [latex]y=\\dfrac{1}{2}x[\/latex] is shifted down\u00a0by 3 units.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201055\/CNX_Precalc_Figure_02_02_0082.jpg\" alt=\"Graph showing the lines y = (1\/2)x, and y = (1\/2) + 3. For each x value on y=(1\/2)x the y value has been dropped by 3.\" width=\"487\" height=\"377\" \/><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video describes a linear transformation of the function [latex]f(x)=x[\/latex] and its corresponding graph.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.4-Video-1.odt\">Transcript 2.4 Video 1<\/a>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nUse transformations of [latex]f(x)=x[\/latex] to graph [latex]f(x)=-2x+5[\/latex].\r\n\r\n[reveal-answer q=\"hjm756\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm756\"]\r\n\r\n[latex]m=-2[\/latex] is a vertical stretch by a factor of 2 (the dashed blue line) and a reflection across the [latex]x[\/latex]-axis (the dashed green line). [latex]b=5[\/latex] is a vertical shift up by 5 units (the solid purple line).\r\n\r\n<img class=\"aligncenter wp-image-4410 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-300x300.png\" alt=\"A series of transformations. f(x)=x is first stretched by a factor of 2, then reflected in x, then raised by 5 units.\" width=\"300\" height=\"300\" \/>\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nCombinations of transformations can be investigated using the Desmos graph in figure 8. Move the green \"Change b\" point up or down to change the value of [latex]b[\/latex] and move the \"Change m\" point left or right to change the value of [latex]m[\/latex].\r\n\r\n<\/div>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/zzxnnfj6zz?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 8. Transformation of [latex]f(x)=x[\/latex] to [latex]f(x)=mx+b[\/latex].<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nExplain the transformations performed to the function [latex]f(x)=x[\/latex] to end up with the function:\r\n<ol>\r\n \t<li>[latex]f(x)=x-9[\/latex]<\/li>\r\n \t<li>[latex]f(x)=4x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=3x-5[\/latex]<\/li>\r\n \t<li>[latex]f(x)=-x+6[\/latex]<\/li>\r\n \t<li>[latex]f(x)=-\\frac{1}{3}x+4[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]f(x)=x[\/latex] is shifted down by 9 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is stretched by a factor of 4.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is stretched by a factor of 3 and then shifted down 5 units. It is NOT in the order of \"shifted down 5 unites\" and then \"stretched by a factor of 3\". This is because the equation would be [latex]f(x)=x-5[\/latex], and then [latex]f(x)=3(x-5)[\/latex]. By simplifying the equation, it is [latex]f(x)=3x-15[\/latex]. In other words, the transformations of shifting up\/down should be the last step when explaining the transformations of a function.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis and then shifted up 6 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis, compressed by a factor of [latex]\\dfrac{1}{3}[\/latex], and then shifted up by 4 units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nExplain the transformations performed to the function [latex]f(x)=x[\/latex] to end up with the function:\r\n<ol>\r\n \t<li>[latex]f(x)=x+4[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{1}{5}x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=6x+1[\/latex]<\/li>\r\n \t<li>[latex]f(x)=-x+4[\/latex]<\/li>\r\n \t<li>[latex]f(x)=-5x+9[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm069\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm069\"]\r\n<ol>\r\n \t<li>[latex]f(x)=x[\/latex] is shifted up by 4 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex]\u00a0 is compressed to one-fifth its height.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is stretched by a factor of 6 and shifted up by 1 unit.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis and shifted up by 4 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis, stretched by a factor of 5, and shifted up by 9 units.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nWrite the function that is the result of the transformations performed on\u00a0[latex]f(x)=x[\/latex]:\r\n<ol>\r\n \t<li>[latex]f(x)=x[\/latex] is shifted up by 3 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis and shifted down by 2 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is stretch by a factor of 4 and shifted up by 3 units.<\/li>\r\n \t<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis, compressed to one-third its height, and shifted down by 7 units.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"hjm632\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm632\"]\r\n<ol>\r\n \t<li>[latex]f(x)=x+3[\/latex]<\/li>\r\n \t<li>[latex]f(x)=-x-2[\/latex]<\/li>\r\n \t<li>[latex]f(x)=4x+3[\/latex]<\/li>\r\n \t<li>[latex]f(x)=-\\dfrac{1}{3}x-7[\/latex]<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-93\" class=\"standard post-93 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h3>Learning objectives<\/h3>\n<ul>\n<li>Perform a vertical shift on a linear function<\/li>\n<li>Perform a vertical stretch or compression on a linear function<\/li>\n<li>Perform a reflection of a linear function across the [latex]x[\/latex]-axis<\/li>\n<li>Perform a combination of transformations on a linear function<\/li>\n<li>Explain the transformations performed on\u00a0[latex]f(x)=x[\/latex] given the transformed function<\/li>\n<li>Write an algebraic function after completing transformations on the parent function<\/li>\n<\/ul>\n<\/div>\n<p>Recall the introduction to graphical\u00a0<em><strong>transformations<\/strong><\/em> of a function we saw in section 1.3. A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression. The original function [latex]f(x)=x[\/latex] is also known as the <em><strong>parent function<\/strong><\/em> and is the linear function used for transformations in this section. We will apply transformations graphically and consider what these transformations mean algebraically.<\/p>\n<h2>Vertical Shifts<\/h2>\n<p>Making a <em><strong>vertical shift<\/strong><\/em> means to move the graph of the linear function vertically up or down a certain number of units. When we shift a line up or down, the slope (steepness) of the line does not change. What changes is the [latex]y[\/latex]-intercept. For example, the function [latex]f(x)=x[\/latex] has a slope of 1 and a [latex]y[\/latex]-intercept at (0, 0). If we shift the line up two units, the slope does not change, but the [latex]y[\/latex]-intercept moves to (0, 2) (Figure 1).<\/p>\n<div id=\"attachment_1355\" style=\"width: 444px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1355\" class=\"wp-image-1355\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05215249\/Vert-shift-up-2-300x276.png\" alt=\"A blue line with y-intercept (0,0) and slope 1 shifted up 2 units becomes a green line with y-intercept (0,2) and slope 1.\" width=\"434\" height=\"399\" \/><\/p>\n<p id=\"caption-attachment-1355\" class=\"wp-caption-text\">Figure 1. Vertical shift<\/p>\n<\/div>\n<p>If fact, every point on the graph shifts up by 2 units so for every point on the line [latex]y=x[\/latex], the [latex]y[\/latex]-coordinate of each point increases by 2. This is shown in Table 1.<\/p>\n<table style=\"border-collapse: collapse; width: 21.312821745488055%; height: 222px;\">\n<tbody>\n<tr>\n<th style=\"width: 32.2997%;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 32.2997%;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 35.6589%;\">[latex]y+2[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 32.299741602067186%;\">0<\/td>\n<td style=\"width: 32.299741602067186%;\">0<\/td>\n<td style=\"width: 35.65891472868217%;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 32.299741602067186%;\">1<\/td>\n<td style=\"width: 32.299741602067186%;\">1<\/td>\n<td style=\"width: 35.65891472868217%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 32.299741602067186%;\">2<\/td>\n<td style=\"width: 32.299741602067186%;\">2<\/td>\n<td style=\"width: 35.65891472868217%;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 32.299741602067186%;\">3<\/td>\n<td style=\"width: 32.299741602067186%;\">3<\/td>\n<td style=\"width: 35.65891472868217%;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 32.299741602067186%;\">4<\/td>\n<td style=\"width: 32.299741602067186%;\">4<\/td>\n<td style=\"width: 35.65891472868217%;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 32.2997%;\" colspan=\"3\">Table 1. Increase in [latex]y[\/latex]-values<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In other words, the function [latex]f(x)=x[\/latex] transforms to [latex]f(x)=x+2[\/latex] because all of the [latex]y[\/latex]-values are increased by 2.<\/p>\n<p>If the function [latex]f(x)=x[\/latex] is shifted down two units, it means all the [latex]y[\/latex]-coordinates of the points on the line are decreased by 2 (increased by \u20132) (Table 2).\u00a0In other words, the function [latex]f(x)=x[\/latex] becomes [latex]f(x)=x-2[\/latex] because all of the [latex]y[\/latex]-values are decreased by 2. From\u00a0another perspective, this transformation results in the original [latex]y[\/latex]-intercept (0, 0) moving to (0, \u20132). Therefore, the function becomes [latex]f(x)=x-2[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 38.937803723907656%; height: 230px;\">\n<tbody>\n<tr>\n<th style=\"width: 111.691px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 111.691px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 87.3162px;\">[latex]y\u20132[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 111.69116973876953px;\">0<\/td>\n<td style=\"width: 111.69116973876953px;\">0<\/td>\n<td style=\"width: 87.31617736816406px;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 111.69116973876953px;\">1<\/td>\n<td style=\"width: 111.69116973876953px;\">1<\/td>\n<td style=\"width: 87.31617736816406px;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 111.69116973876953px;\">2<\/td>\n<td style=\"width: 111.69116973876953px;\">2<\/td>\n<td style=\"width: 87.31617736816406px;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 111.69116973876953px;\">3<\/td>\n<td style=\"width: 111.69116973876953px;\">3<\/td>\n<td style=\"width: 87.31617736816406px;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 111.69116973876953px;\">4<\/td>\n<td style=\"width: 111.69116973876953px;\">4<\/td>\n<td style=\"width: 87.31617736816406px;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 111.691px;\" colspan=\"3\">Table 2. Decrease in [latex]y[\/latex]-values<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div><\/div>\n<div>Figure 2 shows different vertical transformations of the function [latex]f(x)=x[\/latex] to [latex]f(x)=x+b[\/latex]. If [latex]b>0[\/latex] the line moves up, and if [latex]b<0[\/latex] the line moves down.<\/div>\n<div class=\"wp-nocaption aligncenter\">\n<div style=\"width: 576px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201052\/CNX_Precalc_Figure_02_02_0062.jpg\" alt=\"graph showing y = x , y = x+2, y = x+4, y = x-2, y = x-4. All are parallel lines with each having an intercept equal to the value added to x.\" width=\"566\" height=\"477\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Vertical shifts.<\/p>\n<\/div>\n<\/div>\n<p style=\"text-align: left;\">This can be investigated in Desmos by changing the value of [latex]b[\/latex]. Click on the desmos icon on the bottom right corner of figure 3 and move the slider to change the value of [latex]b[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/nxc0ola2td?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 3. Transformation of [latex]f(x)=x[\/latex] to [latex]f(x)=x+b[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>The graph shows the function [latex]f(x)=x[\/latex]. Use this graph to shift the function down by 4 units. Then write the equation of the new function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05235256\/fxx-300x300.png\" alt=\"Graph of f(x)=x. y-intercept (0,0), slope of 1.\" width=\"167\" height=\"167\" \/><\/p>\n<h4>Solution<\/h4>\n<p>When the function is shifted down by 4 units, it is the [latex]y[\/latex]-coordinates that are decreased by 4 while the [latex]x[\/latex]-coordinates stay the same.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1363 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-274x300.png\" alt=\"A vertical transformation showing each point on f(x) has moved straight down 4 units forming a new line with y-intercept (0,-4) with a slope of 1.\" width=\"274\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-274x300.png 274w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-768x841.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-935x1024.png 935w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-65x71.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-225x246.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4-350x383.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/yx-to-yx-4.png 1136w\" sizes=\"auto, (max-width: 274px) 100vw, 274px\" \/><\/p>\n<p>The transformed function takes the form [latex]f(x)=x+b[\/latex] where [latex]b=-4[\/latex]: [latex]f(x)=x-4[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>The graph shows the function [latex]f(x)=x[\/latex]. Use this graph to shift the function up by 3 units . Then write the equation of the new function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05235256\/fxx-300x300.png\" alt=\"Graph of f(x)=x. y-intercept (0,0), slope of 1.\" width=\"167\" height=\"167\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm152\">Show Answer<\/span><\/p>\n<div id=\"qhjm152\" class=\"hidden-answer\" style=\"display: none\">\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1364 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-300x273.png\" alt=\"A vertical transformation showing each point on f(x) has moved straight down 4 units forming a new line with y-intercept (0,-4) with a slope of 1.\" width=\"300\" height=\"273\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-300x273.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-768x698.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-1024x931.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-65x59.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-225x205.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up-350x318.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Vertical-shift-up.png 1142w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>[latex]f(x)=x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Vertical Stretching or Compressing<\/h2>\n<h3>Vertically stretching<\/h3>\n<p>If we could grab both ends of the line and pull vertically in opposite directions (e.g., the end goes up if it is above the x-axis, and the end goes down if it is below the x-axis) we would stretch the line causing the slope of the line to increase. This is <em><strong>vertical stretching<\/strong><\/em> of a linear function. Another perspective of viewing vertically stretching a line is that every y coordinate of points on the line is multiplied by a number that is greater than 1. Therefore, if stretching a line vertically twice, the point (1, 2) becomes (1, 4) (e.g., the y-coordinate is twice larger) and the point (-2, -1) becomes (-2, -2).<\/p>\n<p>To increase the steepness of a function the slope must increase from 1 to a number that is greater than 1. For example, stretching the function [latex]f(x)=x[\/latex] to be twice as steep means multiplying the original slope of 1 by 2. Therefore, the stretched function becomes [latex]f(x)=2x[\/latex]. It is important to note that the position of the [latex]y[\/latex]-intercept does not change because there is no vertical shift.<\/p>\n<h3>Vertically compressing down<\/h3>\n<p>If instead of being able to stretch the ends of the line in opposite directions we were able to push the ends of the line, causing the slope of the line to decrease, we would witness vertical compression. The result would be to decrease the steepness (slope) of the line.\u00a0Another perspective of viewing vertically compressing a line is that every y-coordinate of points on the line is multiplied by a number that is less than 1. Therefore, if compressing a line vertically to half of its height, the point (1, 2) becomes (1, 1) (e.g., the y-coordinate is half of its height) and the point (-2, -1) becomes (-2, -1\/2).<\/p>\n<p>To decrease the steepness of a line means to multiply the original slope of 1 by a number that is between 0 and 1. For example, compressing the function [latex]f(x)=x[\/latex] to one-half of its height means multiplying the slope by [latex]\\dfrac{1}{2}[\/latex]. Therefore, the compressed function is [latex]f(x)=\\dfrac{1}{2}x[\/latex]. It is important to note that the position of the [latex]y[\/latex]-intercept does not change because there is no vertical shift.<\/p>\n<p>Figure 4 illustrates the graphs of various functions that are the result of stretching ([latex]m>1[\/latex] or compressing ([latex]0\u2264m<1[\/latex] the function [latex]f(x)=x[\/latex] to [latex]f(x)=mx[\/latex].\n\n\n\n<div id=\"attachment_1928\" style=\"width: 419px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1928\" class=\"wp-image-1928\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/30001147\/desmos-graph-98-300x300.png\" alt=\"Figure 4. Stretching and compressing a linear function. As the slope increases the line is steeper.\" width=\"409\" height=\"409\" \/><\/p>\n<p id=\"caption-attachment-1928\" class=\"wp-caption-text\">Figure 4. Stretching and compressing a linear function.<\/p>\n<\/div>\n<div class=\"wp-nocaption aligncenter\"><\/div>\n<p>This can be investigated in Desmos by changing the value of [latex]m[\/latex]. Click on the desmos icon on the bottom right corner of figure 5 and move the slider to change the value of [latex]m[\/latex]. Notice what happens to the coordinate (1, 1) on the original graph as the transformation plays out.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/nvqssi6jcx?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 5. Transformation of [latex]f(x)=x[\/latex] to [latex]f(x)=mx[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Describe the\u00a0transformation of the function [latex]f(x)=x[\/latex] to the function [latex]f(x)=\\dfrac{2}{3}x[\/latex] and\u00a0draw a graph.<\/p>\n<h4>Solution<\/h4>\n<p>The transformed function has a slope of [latex]m=\\dfrac{2}{3}[\/latex], so this is a vertical compression since [latex]0<m<1[\/latex].\n\nThe graph of\u00a0[latex]f(x)=\\dfrac{2}{3}x[\/latex]\u00a0is shown with the original function [latex]f(x)=x[\/latex]:\n\n\n<div class=\"wp-nocaption size-medium wp-image-2245 aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2245 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/07171247\/Screen-Shot-2016-07-07-at-10.12.10-AM-300x289.png\" alt=\"A graph with f(x)=x, with y-intercept (0,0) and slope 1, and a second line y=two thirds x, with y-intercept (0,0) and slope 2\/3.\" width=\"300\" height=\"289\" \/><\/div>\n<p>Note how the original function is compressed because the rate of change is \u201cslowed\u201d.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Write the equation of the transformation when [latex]f(x)=x[\/latex] is stretched vertically by a factor of 3.<\/p>\n<h4>Solution<\/h4>\n<p>All the function values must be multiplied by 3, while the [latex]x[\/latex]-values stay the same.<\/p>\n<p>[latex]f(x)=3x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>try it 2<\/h3>\n<p>Write the equation of the transformation when [latex]f(x)=x[\/latex] is stretched or compressed vertically by the given factor:<\/p>\n<ol>\n<li>Stretched by a factor of 7<\/li>\n<li>Compressed to one-third its height<\/li>\n<li>Compressed to one-fourth its height<\/li>\n<li>Stretched by a factor of 5<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm906\">Show Answer<\/span><\/p>\n<div id=\"qhjm906\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=7x[\/latex]<\/li>\n<li>[latex]f(x)=\\frac{1}{3}x[\/latex]<\/li>\n<li>[latex]f(x)=\\frac{1}{4}x[\/latex]<\/li>\n<li>[latex]f(x)=5x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Reflection Across the x-axis<\/h2>\n<p>Reflecting a linear function across the x-axis means to flip the line representing the function across the x-axis. This results in\u00a0 a mirror image of the original function with respect to the x-axis. To reflect a linear function across the [latex]x[\/latex]-axis means all points on the line will see their [latex]y[\/latex]-coordinates change sign; positive [latex]x[\/latex]-coordinates will become negative (e.g, (2, 2) becomes (2, -2)), and negative [latex]y[\/latex]-coordinates will become positive (e.g., (1, \u20135) becomes (1, 5)). From an algebraic point of view, vertically flipping the graph of the function [latex]f(x)=x[\/latex] is equivalent to multiplying the [latex]x[\/latex]-values by \u20131. That is, the [latex]y[\/latex] value in [latex]y=f(x)=x[\/latex] becomes [latex]y=f(x)=-x[\/latex] (e.g., positive [latex]y[\/latex] values become negative, or negative [latex]y[\/latex] values become positive). \u00a0So [latex]f(x)=x[\/latex] becomes [latex]f(x)=-x[\/latex]. Table 3 shows the transformation.<\/p>\n<table style=\"border-collapse: collapse; width: 0%; height: 162px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<th style=\"width: 33.3333%; height: 18px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 33.3333%; height: 18px;\">[latex]f(x)=x[\/latex]<\/th>\n<th style=\"width: 33.3333%; height: 18px;\">[latex]f(x) = -x[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">-3<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">-3<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">-2<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">-2<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">-1<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">-1<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">0<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">0<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">0<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">1<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">1<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">-1<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">2<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">2<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">-2<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">3<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">3<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">-3<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\" colspan=\"3\">Table 3.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div><\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>The graph shows the function [latex]f(x)=x[\/latex]. Use this graph to reflect the function across the [latex]x[\/latex]-axis. Then write the equation of the new function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1360\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/05235256\/fxx-300x300.png\" alt=\"Graph of f(x)=x\" width=\"167\" height=\"167\" \/><\/p>\n<h4>Solution<\/h4>\n<p>All of the [latex]y[\/latex]-values change sign with the function values staying the same. So (3, 3) moves to (3, \u20133), (\u20131, \u20131) moves to (\u20131, 1) etc. Once the new points have been plotted the new line can be drawn.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1930 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-300x300.png\" alt=\"reflected lines. The reflected line described above.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-100.png 2000w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The new function is [latex]f(x)=-x[\/latex].<\/p>\n<\/div>\n<h2>Combining Transformations<\/h2>\n<p>We can combine different transformations, one after the other.\u00a0Suppose we want to transform the function [latex]f(x)=x[\/latex] by reflecting this function across the [latex]x[\/latex]-axis, shifting it up 3 units. The reflection across the [latex]x[\/latex]-axis results in the new function [latex]f(x)=-x[\/latex]. The vertical shift of 3 units up results in the function [latex]f(x)=-x+3[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Graph [latex]f\\left(x\\right)=\\dfrac{1}{2}x - 3[\/latex] using transformations of [latex]f(x)=x[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The equation for the function shows that [latex]m=\\dfrac{1}{2}[\/latex], so the original function is vertically compressed to one-half its height. The equation for the function also shows that [latex]b=\u20133[\/latex], so the original function is vertically shifted down by 3 units. First, graph the function [latex]f(x)=x[\/latex] and show the vertical compression (the [latex]y[\/latex]-values are all divided by 2).<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201054\/CNX_Precalc_Figure_02_02_0072.jpg\" alt=\"graph showing the lines y = x and y = (1\/2)x. For each x value the y value has been halved.\" width=\"487\" height=\"378\" \/><\/div>\n<p>Now show the vertical shift.\u00a0The function [latex]y=\\dfrac{1}{2}x[\/latex] is shifted down\u00a0by 3 units.<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201055\/CNX_Precalc_Figure_02_02_0082.jpg\" alt=\"Graph showing the lines y = (1\/2)x, and y = (1\/2) + 3. For each x value on y=(1\/2)x the y value has been dropped by 3.\" width=\"487\" height=\"377\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video describes a linear transformation of the function [latex]f(x)=x[\/latex] and its corresponding graph.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/h9zn_ODlgbM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Transcript-2.4-Video-1.odt\">Transcript 2.4 Video 1<\/a><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Use transformations of [latex]f(x)=x[\/latex] to graph [latex]f(x)=-2x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm756\">Show Answer<\/span><\/p>\n<div id=\"qhjm756\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]m=-2[\/latex] is a vertical stretch by a factor of 2 (the dashed blue line) and a reflection across the [latex]x[\/latex]-axis (the dashed green line). [latex]b=5[\/latex] is a vertical shift up by 5 units (the solid purple line).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4410 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-300x300.png\" alt=\"A series of transformations. f(x)=x is first stretched by a factor of 2, then reflected in x, then raised by 5 units.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2.4-reflection.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Combinations of transformations can be investigated using the Desmos graph in figure 8. Move the green &#8220;Change b&#8221; point up or down to change the value of [latex]b[\/latex] and move the &#8220;Change m&#8221; point left or right to change the value of [latex]m[\/latex].<\/p>\n<\/div>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/zzxnnfj6zz?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 8. Transformation of [latex]f(x)=x[\/latex] to [latex]f(x)=mx+b[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Explain the transformations performed to the function [latex]f(x)=x[\/latex] to end up with the function:<\/p>\n<ol>\n<li>[latex]f(x)=x-9[\/latex]<\/li>\n<li>[latex]f(x)=4x[\/latex]<\/li>\n<li>[latex]f(x)=3x-5[\/latex]<\/li>\n<li>[latex]f(x)=-x+6[\/latex]<\/li>\n<li>[latex]f(x)=-\\frac{1}{3}x+4[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]f(x)=x[\/latex] is shifted down by 9 units.<\/li>\n<li>[latex]f(x)=x[\/latex] is stretched by a factor of 4.<\/li>\n<li>[latex]f(x)=x[\/latex] is stretched by a factor of 3 and then shifted down 5 units. It is NOT in the order of &#8220;shifted down 5 unites&#8221; and then &#8220;stretched by a factor of 3&#8221;. This is because the equation would be [latex]f(x)=x-5[\/latex], and then [latex]f(x)=3(x-5)[\/latex]. By simplifying the equation, it is [latex]f(x)=3x-15[\/latex]. In other words, the transformations of shifting up\/down should be the last step when explaining the transformations of a function.<\/li>\n<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis and then shifted up 6 units.<\/li>\n<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis, compressed by a factor of [latex]\\dfrac{1}{3}[\/latex], and then shifted up by 4 units.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Explain the transformations performed to the function [latex]f(x)=x[\/latex] to end up with the function:<\/p>\n<ol>\n<li>[latex]f(x)=x+4[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{1}{5}x[\/latex]<\/li>\n<li>[latex]f(x)=6x+1[\/latex]<\/li>\n<li>[latex]f(x)=-x+4[\/latex]<\/li>\n<li>[latex]f(x)=-5x+9[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm069\">Show Answer<\/span><\/p>\n<div id=\"qhjm069\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=x[\/latex] is shifted up by 4 units.<\/li>\n<li>[latex]f(x)=x[\/latex]\u00a0 is compressed to one-fifth its height.<\/li>\n<li>[latex]f(x)=x[\/latex] is stretched by a factor of 6 and shifted up by 1 unit.<\/li>\n<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis and shifted up by 4 units.<\/li>\n<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis, stretched by a factor of 5, and shifted up by 9 units.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Write the function that is the result of the transformations performed on\u00a0[latex]f(x)=x[\/latex]:<\/p>\n<ol>\n<li>[latex]f(x)=x[\/latex] is shifted up by 3 units.<\/li>\n<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis and shifted down by 2 units.<\/li>\n<li>[latex]f(x)=x[\/latex] is stretch by a factor of 4 and shifted up by 3 units.<\/li>\n<li>[latex]f(x)=x[\/latex] is reflected across the [latex]x[\/latex]-axis, compressed to one-third its height, and shifted down by 7 units.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm632\">Show Answer<\/span><\/p>\n<div id=\"qhjm632\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]f(x)=x+3[\/latex]<\/li>\n<li>[latex]f(x)=-x-2[\/latex]<\/li>\n<li>[latex]f(x)=4x+3[\/latex]<\/li>\n<li>[latex]f(x)=-\\dfrac{1}{3}x-7[\/latex]<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<\/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-974\">\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>2.4: Transformations of Linear Functions. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graph a Linear Function as a Transformation of f(x)=x. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/h9zn_ODlgbM\">https:\/\/youtu.be\/h9zn_ODlgbM<\/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 graphs created by Desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Lumen Learning. <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><li>Examples and Try Its: hjm756, hjm069, hjm632, hjm392, hjm906, hjm152. <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>FIgure 3. Interactive transformation from f(x)=x to f(x)=x+b. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/nxc0ola2td\">https:\/\/www.desmos.com\/calculator\/nxc0ola2td<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 5. Interactive transformation from f(x)=x to f(x)=mx. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/nvqssi6jcx\">https:\/\/www.desmos.com\/calculator\/nvqssi6jcx<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 8. Interactive transformation of f(x)=x to f(x)=mx+b. <strong>Authored by<\/strong>: John Jarvis and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/zzxnnfj6zz\">https:\/\/www.desmos.com\/calculator\/zzxnnfj6zz<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/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":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"2.4: Transformations of Linear Functions\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Graph a Linear Function as a Transformation of f(x)=x\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/h9zn_ODlgbM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created by Desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Lumen Learning\",\"url\":\"desmos.com\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples and Try Its: hjm756, hjm069, hjm632, hjm392, hjm906, hjm152\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"FIgure 3. 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