{"id":2948,"date":"2019-09-25T19:21:15","date_gmt":"2019-09-25T19:21:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=2948"},"modified":"2025-03-07T22:23:59","modified_gmt":"2025-03-07T22:23:59","slug":"1-7-transformations-stretches-and-compressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/1-7-transformations-stretches-and-compressions\/","title":{"raw":"1.7 Transformations:  Stretches and Compressions","rendered":"1.7 Transformations:  Stretches and Compressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Graph functions using stretches and compressions.<\/li>\r\n \t<li>Combine transformations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Figure_01_05_001\" class=\"medium\"><\/div>\r\n<div id=\"fs-id1165137827988\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137535664\" class=\"bc-section section\"><\/div>\r\n<div id=\"fs-id1165137654768\" class=\"bc-section section\">\r\n<h3>Graphing Functions Using Stretches and Compressions<\/h3>\r\n<p id=\"fs-id1165137654773\">Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\r\n<p id=\"fs-id1165137675403\">We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\r\n\r\n<div id=\"fs-id1165137793506\" class=\"bc-section section\">\r\n<h4>Vertical Stretches and Compressions<\/h4>\r\n<p id=\"fs-id1165137461225\">When we multiply a function by a positive constant, we get a function whose graph is stretched vertically away from or compressed vertically toward the x-axis in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a <strong>vertical compression<\/strong>. <a class=\"autogenerated-content\" href=\"#Figure_01_05_025\">Figure 1<\/a> shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\r\n\r\n<div id=\"Figure_01_05_025\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"441\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205727\/CNX_Precalc_Figure_01_05_024.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"441\" height=\"295\" \/> Figure 1 Vertical stretch and compression[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137472530\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137444261\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=af\\left(x\\right),[\/latex] where [latex]a[\/latex] is a constant, is a <strong>vertical stretch<\/strong> or <strong>vertical compression<\/strong> of the function [latex]f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<ul id=\"fs-id1165135553621\">\r\n \t<li>If [latex]|a|&gt;1,[\/latex] then the graph will be stretched away from the x-axis.<\/li>\r\n \t<li>If [latex]0&lt;|a|&lt;1,[\/latex] then the graph will be compressed toward the x-axis.<\/li>\r\n \t<li>If [latex]a&lt;0,[\/latex] then there will be combination of a vertical stretch or compression with a vertical reflection.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135173107\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165132939216\"><strong>Given a function, graph its vertical stretch or compression.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134190735\" type=\"1\">\r\n \t<li>Identify the value of [latex]a.[\/latex]<\/li>\r\n \t<li>Multiply all range values by [latex]a.[\/latex]<\/li>\r\n \t<li>\r\n<p id=\"eip-id1165133107002\">If [latex]|a|&gt;1,[\/latex] the graph is stretched by a factor of [latex]a.[\/latex]<\/p>\r\n<p id=\"eip-id1165135191979\">If [latex]0&lt;|a|&lt;1,[\/latex] the graph is compressed by a factor of [latex]a.[\/latex]<\/p>\r\n<p id=\"eip-id1165134061939\">If [latex]a&lt;0,[\/latex] the graph is either stretched or compressed and also reflected about the <em>x<\/em>-axis.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_13\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137619296\">\r\n<div id=\"fs-id1165137619298\">\r\n<h3>Example 1:\u00a0 Graphing a Vertical Stretch<\/h3>\r\n<p id=\"fs-id1165137552971\">A function [latex]P\\left(t\\right)[\/latex] models the population of fruit flies. The graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_026\">Figure 2<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_026\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"366\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205730\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"366\" height=\"276\" \/> Figure 2[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135349865\">A scientist is comparing this population to another population, [latex]Q,[\/latex] whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137482306\">[reveal-answer q=\"fs-id1165137482306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137482306\"]\r\n<p id=\"fs-id1165137482308\">Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values. Graphically, this is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_027\">Figure 3<\/a>.<\/p>\r\n<p id=\"fs-id1165137482312\">If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.<\/p>\r\n<p id=\"fs-id1165137705889\">The following shows where the new points for the new graph will be located.<\/p>\r\n\r\n<div id=\"fs-id1165133281393\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{array}[\/latex]<\/div>\r\n<div id=\"Figure_01_05_027\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"365\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205733\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"365\" height=\"275\" \/> Figure 3[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165132939230\">Symbolically, the relationship is written as<\/p>\r\n\r\n<div id=\"fs-id1165135524747\" class=\"unnumbered\" style=\"text-align: center;\">[latex]Q\\left(t\\right)=2P\\left(t\\right).[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135305830\">This means that for any input [latex]t,[\/latex] the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P.[\/latex] Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t,[\/latex] stay the same while the output values are twice as large as before.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137647092\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137442797\"><strong>Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for the transformation. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137812146\" type=\"1\">\r\n \t<li>Determine the value of [latex]a.[\/latex]<\/li>\r\n \t<li>Multiply all of the output values by [latex]a.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_14\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135436655\">\r\n<div id=\"fs-id1165135436657\">\r\n<h3>Example 2:\u00a0 Finding a Vertical Compression of a Tabular Function<\/h3>\r\n<p id=\"fs-id1165134237296\">A function [latex]f[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_09\">Table 1<\/a>. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_05_09\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=1, f(4)=3, f(6)=7, and f(8)=11.\"><caption>Table 1<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25.656px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"width: 35.6563px; text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 25.656px;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"width: 35.6563px; text-align: center;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137780720\">[reveal-answer q=\"fs-id1165137780720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137780720\"]\r\n<p id=\"fs-id1165137889748\">The formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3.[\/latex] Then<\/p>\r\n\r\n<div id=\"fs-id1165134350257\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(4\\right)=\\frac{1}{2}f\\left(4\\right)=\\frac{1}{2}\\left(3\\right)=\\frac{3}{2}.[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\nWe do the same for the other values to produce <a class=\"autogenerated-content\" href=\"#Table_01_05_10\">Table 2<\/a>.\r\n<table id=\"Table_01_05_10\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cg(x)\u201d. The values of x are 2, 4, 6, and 8. So for g(2)=1\/2, g(4)=3\/2, g(6)=7\/2, and g(8)=11\/2.\"><caption>Table 2<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]6[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{3}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{7}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Analysis<\/h3>\r\nThe result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by a factor of [latex]\\frac{1}{2}.[\/latex] Each output value is divided in half, so the graph is half the original height.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137755750\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"fs-id1165137551062\">\r\n<div id=\"fs-id1165137551064\">\r\n<p id=\"fs-id1165137551066\">A function [latex]f[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_011\">Table 3<\/a>. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_05_011\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=12, f(4)=16, f(6)=20, and f(8)=0.\"><caption>Table 3<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">[latex]x[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">16<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165134261677\">[reveal-answer q=\"fs-id1165134261677\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134261677\"]\r\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cg(x)\u201d. The values of x are 2, 4, 6, and 8. So for g(2)=9, g(4)=12, g(6)=15, and g(8)=0.\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">[latex]x[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]g\\left(x\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">9<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_15\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135530673\">\r\n<div id=\"fs-id1165135530675\">\r\n<h3>Example 3:\u00a0 Recognizing a Vertical Stretch<\/h3>\r\n<p id=\"fs-id1165135519281\">The graph in <a class=\"autogenerated-content\" href=\"#Figure_01_05_028\">Figure 4<\/a> is a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{3}.[\/latex] Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right),[\/latex] and then find a formula for [latex]g\\left(x\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_01_05_028\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"331\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205735\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"331\" height=\"300\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135173356\">[reveal-answer q=\"fs-id1165135173356\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135173356\"]\r\n<p id=\"fs-id1165137424163\">When trying to determine a vertical stretch or compression, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that [latex]g\\left(2\\right)=2.[\/latex] With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8.[\/latex] Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]g\\left(2\\right)=\\frac{1}{4}f\\left(2\\right).[\/latex] From this we can fairly safely conclude that [latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135154389\">We can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137634248\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135499881\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_01_05_06\">\r\n<div id=\"fs-id1165137415361\">\r\n<p id=\"fs-id1165137643555\">Write the formula for the function that we get when we vertically stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137834210\">[reveal-answer q=\"fs-id1165137834210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137834210\"]\r\n<p id=\"fs-id1165134380936\">[latex]g\\left(x\\right)=3x-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135344103\" class=\"bc-section section\">\r\n<h4>Horizontal Stretches and Compressions<\/h4>\r\nNow we consider the changes that occur to a function if we multiply the input of an original function [latex]f\\left(x\\right)[\/latex] by some constant.\u00a0 Notice that we are changing the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched horizontally away from or compressed horizontally toward the vertical axis in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.\u00a0 Let\u2019s consider an example.\r\n\r\nSuppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. Let\u2019s let our original population be [latex]P[\/latex] and our new population be [latex]R[\/latex].\u00a0 Our new population, [latex]R,[\/latex] will progress in 1 hour the same amount as the original population [latex]P[\/latex] does in 2 hours, and in 2 hours, the new population [latex]R[\/latex] will progress as much as the original population [latex]P[\/latex] does in 4 hours. Sketch a graph of this population.\r\n\r\nSymbolically, we could write\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}R\\left(1\\right)=P\\left(2\\right),\\hfill \\\\ R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,}\\hfill \\\\ \\text{ }R\\left(t\\right)=P\\left(2t\\right).\\hfill \\end{array}[\/latex]<\/p>\r\n[latex]\\\\[\/latex]\r\n<p id=\"fs-id1165134380331\">See <a class=\"autogenerated-content\" href=\"#Figure_01_05_030\">Figure 5<\/a> for a graphical comparison of the original population and the compressed population.<\/p>\r\n\r\n<div id=\"Figure_01_05_030\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"974\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205741\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).\" width=\"974\" height=\"400\" \/> Figure 5 (a) Original population graph (b) Compressed population graph[\/caption]<\/div>\r\n<h3><\/h3>\r\nNotice that the effect on the graph is a horizontal compression towards the vertical axis where all input values for our new function [latex]R[\/latex] are half of the original input value for [latex]P.[\/latex]\u00a0 You can clearly see that [latex]R\\left(3\\right) = P\\left(6\\right).[\/latex]\u00a0 We have therefore compressed the original graph of [latex]P\\left(t\\right)[\/latex] towards the vertical axis by a factor of \u00bd in order to create the graph of our new function [latex]R\\left(t\\right).[\/latex]\r\n\r\nAnother way to think about this is that if the point (6, 2) is on the graph of [latex]P,[\/latex] then the point (3, 2) will be a point on the graph of [latex]R.[\/latex]\u00a0 We get the same outputs, but the inputs for [latex]R[\/latex] are half as large as the corresponding input for [latex]P.[\/latex]\u00a0 This results in a horizontal compression towards the vertical axis.\r\n\r\nYou should spend some time convincing yourself that if we multiplied our original input by a value between 0 and 1 that we would get a horizontal stretch away from the vertical axis.\u00a0 Think about what would happen if we considered another population of fruit flies who progress through their life span half as fast as those represented by [latex]P\\left(t\\right).[\/latex]\u00a0 \u00a0We can consider [latex]S\\left(t\\right) = P\\left(\\frac{1}{2} t\\right).[\/latex]\u00a0 This means that [latex]S\\left(4\\right) = P\\left(2\\right)[\/latex] and that [latex]S\\left(6\\right) = P\\left(3\\right).[\/latex]\u00a0 Can you see in order to get the same outputs that our inputs for the new function [latex]S[\/latex] are twice as large as the inputs for the original function [latex]P?[\/latex]\u00a0 This means there would be a horizontal stretch by a factor of 2 away from the vertical axis.\u00a0 Our outputs for [latex]S[\/latex] are the same as the outputs for [latex]P[\/latex] when the inputs for [latex]S[\/latex] are double the inputs for [latex]P.[\/latex]\u00a0 If [latex]P[\/latex] has the point (3, 3) on the graph, then [latex]S[\/latex] will have the point (6, 3) on the graph.\r\n<div id=\"fs-id1165137732896\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165134573810\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=f\\left(bx\\right),[\/latex] where [latex]b[\/latex] is a constant, is a <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<ul id=\"eip-456\">\r\n \t<li>If [latex]|b|&gt;1,[\/latex] then the graph will be compressed by a factor of [latex]\\frac{1}{b}[\/latex] toward the y-axis.<\/li>\r\n \t<li>If [latex]0&lt;|b|&lt;1,[\/latex] then the graph will be stretched by a factor [latex]\\frac{1}{b}[\/latex] away from the y-axis.<\/li>\r\n \t<li>If [latex]b&lt;0,[\/latex] then there will be combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137832347\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137784900\"><strong>Given a description of a function, sketch a horizontal compression or stretch. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137784904\" type=\"1\">\r\n \t<li>Write a formula to represent the function.<\/li>\r\n \t<li>Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b&gt;1[\/latex] for a compression or [latex] 0\\lt b&lt;1[\/latex] for a stretch.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_17\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134071619\">\r\n<div id=\"fs-id1165134071621\">\r\n<h3>Example 4:\u00a0 Finding a Horizontal Stretch for a Tabular Function<\/h3>\r\n<p id=\"fs-id1165134071627\">A function [latex]f\\left(x\\right)[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_12\">Table 4<\/a>. Create a table for the function [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right).[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_05_12\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=1, f(4)=3, f(6)=7, and f(8)=11.\"><caption>Table 4<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137401572\">[reveal-answer q=\"fs-id1165137401572\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137401572\"]\r\n<p id=\"fs-id1165137401575\">The formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right),[\/latex] and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135531534\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165137827502\">We do the same for the other values to produce <a class=\"autogenerated-content\" href=\"#Table_01_05_13\">Table 5<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_13\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=1, f(4)=3, f(6)=7, and f(8)=11.\"><caption>Table 5<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<a class=\"autogenerated-content\" href=\"#Figure_01_05_032\">Figure 6<\/a> shows the graphs of both of these sets of points.\r\n<div id=\"Figure_01_05_032\" class=\"wp-caption aligncenter\" style=\"width: 960px;\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"721\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205745\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"721\" height=\"246\" \/> Figure 6[\/caption]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Analysis<\/h3>\r\nBecause each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_18\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137783980\">\r\n<div id=\"fs-id1165137783983\">\r\n<h3>Example 5:\u00a0 Recognizing a Horizontal Compression on a Graph<\/h3>\r\n<p id=\"fs-id1165135545991\">Relate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_05_033\">Figure 7<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_033\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"418\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205748\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"418\" height=\"250\" \/> Figure 7[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137737597\">[reveal-answer q=\"fs-id1165137737597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137737597\"]\r\n<p id=\"fs-id1165137895242\">The graph of [latex]g\\left(x\\right)[\/latex] looks like the graph of [latex]f\\left(x\\right)[\/latex] horizontally compressed. Because [latex]f\\left(x\\right)[\/latex] ends at [latex]\\left(6,4\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] ends at [latex]\\left(2,4\\right),[\/latex] we can see that the [latex]x\\text{-}[\/latex] values have been compressed by a factor of [latex]\\frac{1}{3},[\/latex] because [latex]6\\left(\\frac{1}{3}\\right)=2.[\/latex] We might also notice that [latex]g\\left(2\\right)=f\\left(6\\right)[\/latex] and [latex]g\\left(1\\right)=f\\left(3\\right).[\/latex] Either way, we can describe this relationship as [latex]g\\left(x\\right)=f\\left(3x\\right).[\/latex] This is a horizontal compression by a factor of [latex]\\frac{1}{3}.[\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNotice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\\frac{1}{4}[\/latex] in our function: [latex]f\\left(\\frac{1}{4}x\\right).[\/latex] This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135613633\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_01_05_07\">\r\n<div id=\"fs-id1165137451792\">\r\n<p id=\"fs-id1165137451793\">Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132945546\">[reveal-answer q=\"fs-id1165132945546\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132945546\"]\r\n<p id=\"fs-id1165132945547\">[latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex] so using the square root function we get [latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{1}{3}x}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137676302\" class=\"bc-section section\">\r\n<h3>Performing a Sequence of Transformations<\/h3>\r\n<p id=\"fs-id1165137387533\">When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by a factor of 2 does not create the same graph as vertically stretching by a factor of 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.<\/p>\r\n<p id=\"fs-id1165137387540\">When we see an expression such as [latex]2f\\left(x\\right)+3,[\/latex] which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of [latex]f\\left(x\\right),[\/latex] we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.<\/p>\r\n<p id=\"fs-id1165137483273\">Horizontal transformations are a little trickier to think about. When we write [latex]g\\left(x\\right)=f\\left(2x+3\\right),[\/latex] for example, we have to think about how the inputs to the function [latex]g[\/latex] relate to the inputs to the function [latex]f.[\/latex] Suppose we know [latex]f\\left(7\\right)=12.[\/latex] What input to [latex]g[\/latex] would produce that output? In other words, what value of [latex]x[\/latex] will allow [latex]g\\left(x\\right)=f\\left(2x+3\\right)=12?[\/latex] We would need [latex]2x+3=7.[\/latex] To solve for [latex]x,[\/latex] we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.<\/p>\r\n<p id=\"fs-id1165137580228\">This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch or compress a graph before shifting. We can work around this by factoring inside the function.<\/p>\r\n\r\n<div id=\"fs-id1165135241282\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(bx+p\\right)=f\\left(b\\left(x+\\frac{p}{b}\\right)\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165137427526\">Let\u2019s work through an example.<\/p>\r\n\r\n<div id=\"fs-id1165137705217\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2x+4\\right)}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165137438427\">We can factor out a 2.<\/p>\r\n\r\n<div id=\"fs-id1165137694985\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2\\left(x+2\\right)\\right)}^{2}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135349189\">Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.<\/p>\r\n\r\n<div id=\"fs-id1165137715290\">\r\n<div class=\"textbox examples\">\r\n<h3>How To<\/h3>\r\n<strong>Given a transformation, determine the order in which they should be preformed.<\/strong>\r\n<ol>\r\n \t<li id=\"fs-id1165137469882\">When combining vertical transformations written in the form [latex]af\\left(x\\right)+k,[\/latex] first vertically stretch or compress by a factor of [latex]a[\/latex] and then vertically shift by [latex]k.[\/latex]<\/li>\r\n \t<li id=\"fs-id1165137423095\">When combining horizontal transformations written in the form [latex]f\\left(bx-h\\right),[\/latex] first horizontally shift by [latex]h[\/latex] and then horizontally stretch or compress by a factor of [latex]\\frac{1}{b}.[\/latex]<\/li>\r\n \t<li id=\"fs-id1165135186003\">When combining horizontal transformations written in the form [latex]f\\left(b\\left(x-h\\right)\\right),[\/latex] first horizontally stretch or compress by a factor of [latex]\\frac{1}{b}[\/latex] and then horizontally shift by [latex]h.[\/latex]<\/li>\r\n \t<li id=\"fs-id1165135191611\">Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_19\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135309915\">\r\n<div id=\"fs-id1165135309917\">\r\n<h3>Example 6:\u00a0 Finding a Triple Transformation of a Tabular Function<\/h3>\r\n<p id=\"fs-id1165134188798\">Given <a class=\"autogenerated-content\" href=\"#Table_01_05_14\">Table 6<\/a> for the function [latex]f\\left(x\\right),[\/latex] create a table of values for the function [latex]g\\left(x\\right)=2f\\left(3x\\right)+1.[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_05_14\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 6, 12, 18, and 24. So for f(6)=10, f(12)=14, f(18)=15, and f(24)=17.\"><caption>Table 6<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">18<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">14<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135151321\">[reveal-answer q=\"fs-id1165135151321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135151321\"]\r\n<p id=\"fs-id1165135151323\">There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, [latex]f\\left(3x\\right)[\/latex] is a horizontal compression by a factor of [latex]\\frac{1}{3},[\/latex] which means we multiply each [latex]x\\text{-}[\/latex]value by [latex]\\frac{1}{3}.[\/latex] See <a class=\"autogenerated-content\" href=\"#Table_01_05_15\">Table 7<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_15\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(3x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=10, f(4)=14, f(6)=15, and f(8)=17.\"><caption>Table 7<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]f\\left(3x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">14<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">17<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137460249\">Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See <a class=\"autogenerated-content\" href=\"#Table_01_05_016\">Table 8<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_016\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201c2f(3x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=20, f(4)=28, f(6)=30, and f(8)=34.\"><caption>Table 8<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]2f\\left(3x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">28<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">30<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">34<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137661581\">Finally, we can apply the vertical shift, which will add 1 to all the output values. See <a class=\"autogenerated-content\" href=\"#Table_01_05_17\">Table 9<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_17\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cg(x)=2f(3x)+1\u201d. The values of x are 2, 4, 6, and 8. So for g(2)=21, g(4)=29, g(6)=31, and g(8)=35.\"><caption>Table 9<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]g\\left(x\\right)=2f\\left(3x\\right)+1[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">21<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">29<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">31<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">35<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_20\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137642734\">\r\n<div id=\"fs-id1165137571821\">\r\n<h3>Example 7:\u00a0 Finding a Triple Transformation of a Graph<\/h3>\r\n<p id=\"fs-id1165137571826\">Use the graph of [latex]f\\left(x\\right)[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_05_035\">Figure 8<\/a> to sketch a graph of [latex]k\\left(x\\right)=f\\left(\\frac{1}{2}x+1\\right)-3.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_01_05_035\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205750\/CNX_Precalc_Figure_01_05_034.jpg\" alt=\"Graph of a half-circle.\" width=\"325\" height=\"295\" \/> Figure 8[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134148412\">[reveal-answer q=\"fs-id1165134148412\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134148412\"]\r\n<p id=\"fs-id1165134148414\">To simplify, let\u2019s start by factoring out the inside of the function.<\/p>\r\n\r\n<div id=\"fs-id1165137831276\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}x+1\\right)-3=f\\left(\\frac{1}{2}\\left(x+2\\right)\\right)-3[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135532397\">By factoring the inside, we can first horizontally stretch by a factor of 2, as indicated by the [latex]\\frac{1}{2}[\/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See <a class=\"autogenerated-content\" href=\"#Figure_01_05_036\">Figure 9<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_036\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"285\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205753\/CNX_Precalc_Figure_01_05_035.jpg\" alt=\"Graph of a vertically stretch half-circle.\" width=\"285\" height=\"294\" \/> Figure 9[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137736781\">Next, we horizontally shift left by 2 units, as indicated by [latex]x+2.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_05_037\">Figure 10<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_037\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"292\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205756\/CNX_Precalc_Figure_01_05_036.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"292\" height=\"302\" \/> Figure 10[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137696198\">Last, we vertically shift down by 3 to complete our sketch, as indicated by the [latex]-3[\/latex] on the outside of the function. See <a class=\"autogenerated-content\" href=\"#Figure_01_05_038\">Figure 11<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_038\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"296\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205759\/CNX_Precalc_Figure_01_05_037.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"296\" height=\"306\" \/> Figure 11[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135497140\" class=\"precalculus media\">\r\n<div id=\"Example_03_02_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135460939\">\r\n<div id=\"fs-id1165135460941\">\r\n<h3>Example 8:\u00a0 Writing the Equation of a Quadratic Function from the Graph<\/h3>\r\nWrite an equation for the quadratic function [latex]g[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_03_02_007\">Figure 12<\/a> as a transformation of [latex]f\\left(x\\right)={x}^{2},[\/latex].\u00a0 First, write your solutions in the <strong>vertex form of a quadratic function<\/strong> [latex]g\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.\u00a0 Then expand the formula, and simplify terms to write the equation in general form.\r\n<div id=\"Figure_03_02_007\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"334\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07151928\/CNX_Precalc_Figure_03_02_007.jpg\" alt=\"Graph of a parabola with its vertex at (-2, -3).\" width=\"334\" height=\"304\" \/> Figure 12[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134211341\">[reveal-answer q=\"fs-id1165134211341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134211341\"]\r\n<p id=\"fs-id1165137742565\">We can see the graph of <em>g <\/em>is the graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] shifted to the left 2 and down 3, giving a formula in the form [latex]g\\left(x\\right)=a{\\left(x+2\\right)}^{2}\u20133.[\/latex]<\/p>\r\n<p id=\"fs-id1165134064001\">Substituting the coordinates of a point on the curve, such as [latex]\\left(0,-1\\right),[\/latex] we can solve for the vertical stretch or compression factor.<\/p>\r\n\r\n<div id=\"eip-id1165134221671\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}-1&amp;=a{\\left(0+2\\right)}^{2}-3\\hfill \\\\ \\text{ }2&amp;=4a\\hfill \\\\ \\text{ }a&amp;=\\frac{1}{2}\\hfill \\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165137895371\">In vertex form, the algebraic model for this graph is [latex]g\\left(x\\right)=\\frac{1}{2}{\\left(x+2\\right)}^{2}\u20133.[\/latex]<\/p>\r\n<p id=\"fs-id1165137844164\">To write this in general polynomial form, we can expand the formula and simplify terms.<\/p>\r\n\r\n<div id=\"eip-id1165137463836\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}g\\left(x\\right)&amp;=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3\\hfill \\\\ \\text{ }&amp;=\\frac{1}{2}\\left(x+2\\right)\\left(x+2\\right)-3\\hfill \\\\ \\text{ }&amp;=\\frac{1}{2}\\left({x}^{2}+4x+4\\right)-3\\hfill \\\\ \\text{ }&amp;=\\frac{1}{2}{x}^{2}+2x+2-3\\hfill \\\\ \\text{ }&amp;=\\frac{1}{2}{x}^{2}+2x-1\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137914060\">Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165137803212\">We can check our work using the table feature on a graphing utility. First enter [latex]\\text{Y1}=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3.[\/latex] Next, select [latex]\\text{TBLSET,}[\/latex] then use [latex]\\text{TblStart}=\u20136[\/latex] and [latex]\\Delta \\text{Tbl = 2,}[\/latex] and select [latex]\\text{TABLE}\\text{.}[\/latex] See <a class=\"autogenerated-content\" href=\"#Table_03_02_01\">Table 10<\/a>.<\/p>\r\n\r\n<table id=\"Table_03_02_01\" summary=\"..\"><caption>Table 10<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20136<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20134<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20132<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20131<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20133<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20131<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135570238\">The ordered pairs in the table correspond to points on the graph.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137527658\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_03_02_01\">\r\n<div id=\"fs-id1165137933940\">\r\n<p id=\"fs-id1165137933941\">A coordinate grid has been superimposed over the quadratic path of a basketball in <a class=\"autogenerated-content\" href=\"#Figure_03_02_008\">Figure 13<\/a>. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\r\n\r\n<div id=\"Figure_03_02_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07151931\/CNX_Precalc_Figure_03_02_008.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" \/> Figure 13.\u00a0(credit: modification of work by Dan Meyer)[\/caption]\r\n\r\n<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135414238\">[reveal-answer q=\"fs-id1165135414238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135414238\"]\r\n<p id=\"fs-id1165135414239\">The path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right),[\/latex] so [latex]\\left(h\\right)x=\u2013\\frac{7}{16}{\\left(x+4\\right)}^{2}+7.[\/latex] To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(\u20137.5\\right)\\approx 1.64;[\/latex] he doesn\u2019t make it.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137410202\">Access this online resource for additional instruction and practice with transformation of functions.<\/p>\r\n\r\n<ul id=\"fs-id1165137410206\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/functrans\">Function Transformations<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135499979\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165134474082\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Vertical shift<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex] (up for [latex]k&gt;0[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Horizontal shift<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex] (right for [latex]h&gt;0[\/latex])<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Vertical reflection<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Horizontal reflection<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Vertical stretch<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] ([latex]a&gt;1[\/latex] )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Vertical compression<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] ([latex] 0\\lt a&lt;1[\/latex] )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Horizontal stretch<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] ([latex] 0\\lt b \\lt\u00a0 1[\/latex] )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Horizontal compression<\/td>\r\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] ([latex]b&gt;1[\/latex])<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135264626\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165135264630\">\r\n \t<li>A function can be compressed or stretched vertically by multiplying the output by a constant.<\/li>\r\n \t<li>A function can be compressed or stretched horizontally by multiplying the input by a constant.<\/li>\r\n \t<li>The order in which different transformations are applied does affect the final function. Shifts and stretches must be applied in the order given. However,\u00a0 vertical transformations may be done before or after horizontal transformations.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137448239\"><\/dl>\r\n<dl id=\"fs-id1165133242964\">\r\n \t<dt>horizontal compression<\/dt>\r\n \t<dd id=\"fs-id1165137833874\">a transformation that compresses a function\u2019s graph horizontally, by multiplying the input by a constant [latex]b&gt;1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135440170\"><\/dl>\r\n<dl id=\"fs-id1165137922367\">\r\n \t<dt>horizontal shift<\/dt>\r\n \t<dd id=\"fs-id1165137922373\">a transformation that shifts a function\u2019s graph left or right by adding a positive or negative constant to the input<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137922379\"><\/dl>\r\n<dl id=\"fs-id1165134259240\">\r\n \t<dt>odd function<\/dt>\r\n \t<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right),[\/latex] and is symmetric about the origin<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137676545\"><\/dl>\r\n<dl id=\"fs-id1165137662611\">\r\n \t<dt>vertical reflection<\/dt>\r\n \t<dd id=\"fs-id1165137834403\">a transformation that reflects a function\u2019s graph across the <em>x<\/em>-axis by multiplying the output by [latex]-1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135580354\">\r\n \t<dt>vertical shift<\/dt>\r\n \t<dd id=\"fs-id1165137862443\">a transformation that shifts a function\u2019s graph up or down by adding a positive or negative constant to the output<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Graph functions using stretches and compressions.<\/li>\n<li>Combine transformations.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_01_05_001\" class=\"medium\"><\/div>\n<div id=\"fs-id1165137827988\" class=\"bc-section section\">\n<div id=\"fs-id1165137535664\" class=\"bc-section section\"><\/div>\n<div id=\"fs-id1165137654768\" class=\"bc-section section\">\n<h3>Graphing Functions Using Stretches and Compressions<\/h3>\n<p id=\"fs-id1165137654773\">Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\n<p id=\"fs-id1165137675403\">We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\n<div id=\"fs-id1165137793506\" class=\"bc-section section\">\n<h4>Vertical Stretches and Compressions<\/h4>\n<p id=\"fs-id1165137461225\">When we multiply a function by a positive constant, we get a function whose graph is stretched vertically away from or compressed vertically toward the x-axis in relation to the graph of the original function. If the constant is greater than 1, we get a <strong>vertical stretch<\/strong>; if the constant is between 0 and 1, we get a <strong>vertical compression<\/strong>. <a class=\"autogenerated-content\" href=\"#Figure_01_05_025\">Figure 1<\/a> shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\n<div id=\"Figure_01_05_025\" class=\"small\">\n<div style=\"width: 451px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205727\/CNX_Precalc_Figure_01_05_024.jpg\" alt=\"Graph of a function that shows vertical stretching and compression.\" width=\"441\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1 Vertical stretch and compression<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472530\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137444261\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=af\\left(x\\right),[\/latex] where [latex]a[\/latex] is a constant, is a <strong>vertical stretch<\/strong> or <strong>vertical compression<\/strong> of the function [latex]f\\left(x\\right).[\/latex]<\/p>\n<ul id=\"fs-id1165135553621\">\n<li>If [latex]|a|>1,[\/latex] then the graph will be stretched away from the x-axis.<\/li>\n<li>If [latex]0<|a|<1,[\/latex] then the graph will be compressed toward the x-axis.<\/li>\n<li>If [latex]a<0,[\/latex] then there will be combination of a vertical stretch or compression with a vertical reflection.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173107\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165132939216\"><strong>Given a function, graph its vertical stretch or compression.<\/strong><\/p>\n<ol id=\"fs-id1165134190735\" type=\"1\">\n<li>Identify the value of [latex]a.[\/latex]<\/li>\n<li>Multiply all range values by [latex]a.[\/latex]<\/li>\n<li>\n<p id=\"eip-id1165133107002\">If [latex]|a|>1,[\/latex] the graph is stretched by a factor of [latex]a.[\/latex]<\/p>\n<p id=\"eip-id1165135191979\">If [latex]0<|a|<1,[\/latex] the graph is compressed by a factor of [latex]a.[\/latex]<\/p>\n<p id=\"eip-id1165134061939\">If [latex]a<0,[\/latex] the graph is either stretched or compressed and also reflected about the <em>x<\/em>-axis.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_13\" class=\"textbox examples\">\n<div id=\"fs-id1165137619296\">\n<div id=\"fs-id1165137619298\">\n<h3>Example 1:\u00a0 Graphing a Vertical Stretch<\/h3>\n<p id=\"fs-id1165137552971\">A function [latex]P\\left(t\\right)[\/latex] models the population of fruit flies. The graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_026\">Figure 2<\/a>.<\/p>\n<div id=\"Figure_01_05_026\" class=\"small\">\n<div style=\"width: 376px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205730\/CNX_Precalc_Figure_01_05_025.jpg\" alt=\"Graph to represent the growth of the population of fruit flies.\" width=\"366\" height=\"276\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135349865\">A scientist is comparing this population to another population, [latex]Q,[\/latex] whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.<\/p>\n<\/div>\n<div id=\"fs-id1165137482306\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137482306\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137482306\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137482308\">Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values. Graphically, this is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_027\">Figure 3<\/a>.<\/p>\n<p id=\"fs-id1165137482312\">If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.<\/p>\n<p id=\"fs-id1165137705889\">The following shows where the new points for the new graph will be located.<\/p>\n<div id=\"fs-id1165133281393\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_01_05_027\" class=\"small\">\n<div style=\"width: 375px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205733\/CNX_Precalc_Figure_01_05_026.jpg\" alt=\"Graph of the population function doubled.\" width=\"365\" height=\"275\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165132939230\">Symbolically, the relationship is written as<\/p>\n<div id=\"fs-id1165135524747\" class=\"unnumbered\" style=\"text-align: center;\">[latex]Q\\left(t\\right)=2P\\left(t\\right).[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135305830\">This means that for any input [latex]t,[\/latex] the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P.[\/latex] Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t,[\/latex] stay the same while the output values are twice as large as before.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137647092\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137442797\"><strong>Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for the transformation. <\/strong><\/p>\n<ol id=\"fs-id1165137812146\" type=\"1\">\n<li>Determine the value of [latex]a.[\/latex]<\/li>\n<li>Multiply all of the output values by [latex]a.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_14\" class=\"textbox examples\">\n<div id=\"fs-id1165135436655\">\n<div id=\"fs-id1165135436657\">\n<h3>Example 2:\u00a0 Finding a Vertical Compression of a Tabular Function<\/h3>\n<p id=\"fs-id1165134237296\">A function [latex]f[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_09\">Table 1<\/a>. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right).[\/latex]<\/p>\n<table id=\"Table_01_05_09\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=1, f(4)=3, f(6)=7, and f(8)=11.\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 25.656px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"width: 35.6563px; text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 25.656px;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"width: 25.6563px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"width: 35.6563px; text-align: center;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137780720\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137780720\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137780720\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137889748\">The formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3.[\/latex] Then<\/p>\n<div id=\"fs-id1165134350257\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(4\\right)=\\frac{1}{2}f\\left(4\\right)=\\frac{1}{2}\\left(3\\right)=\\frac{3}{2}.[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p>We do the same for the other values to produce <a class=\"autogenerated-content\" href=\"#Table_01_05_10\">Table 2<\/a>.<\/p>\n<table id=\"Table_01_05_10\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cg(x)\u201d. The values of x are 2, 4, 6, and 8. So for g(2)=1\/2, g(4)=3\/2, g(6)=7\/2, and g(8)=11\/2.\">\n<caption>Table 2<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]6[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{3}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{7}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]\\frac{11}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Analysis<\/h3>\n<p>The result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by a factor of [latex]\\frac{1}{2}.[\/latex] Each output value is divided in half, so the graph is half the original height.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755750\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"fs-id1165137551062\">\n<div id=\"fs-id1165137551064\">\n<p id=\"fs-id1165137551066\">A function [latex]f[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_011\">Table 3<\/a>. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right).[\/latex]<\/p>\n<table id=\"Table_01_05_011\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=12, f(4)=16, f(6)=20, and f(8)=0.\">\n<caption>Table 3<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\">[latex]x[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\n<td class=\"border\" style=\"text-align: center;\">16<\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165134261677\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134261677\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134261677\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cg(x)\u201d. The values of x are 2, 4, 6, and 8. So for g(2)=9, g(4)=12, g(6)=15, and g(8)=0.\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\">[latex]x[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]g\\left(x\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">9<\/td>\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_15\" class=\"textbox examples\">\n<div id=\"fs-id1165135530673\">\n<div id=\"fs-id1165135530675\">\n<h3>Example 3:\u00a0 Recognizing a Vertical Stretch<\/h3>\n<p id=\"fs-id1165135519281\">The graph in <a class=\"autogenerated-content\" href=\"#Figure_01_05_028\">Figure 4<\/a> is a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{3}.[\/latex] Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right),[\/latex] and then find a formula for [latex]g\\left(x\\right).[\/latex]<\/p>\n<div id=\"Figure_01_05_028\" class=\"small\">\n<div style=\"width: 341px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205735\/CNX_Precalc_Figure_01_05_027.jpg\" alt=\"Graph of a transformation of f(x)=x^3.\" width=\"331\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173356\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135173356\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135173356\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137424163\">When trying to determine a vertical stretch or compression, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that [latex]g\\left(2\\right)=2.[\/latex] With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8.[\/latex] Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]g\\left(2\\right)=\\frac{1}{4}f\\left(2\\right).[\/latex] From this we can fairly safely conclude that [latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135154389\">We can write a formula for [latex]g[\/latex] by using the definition of the function [latex]f.[\/latex]<\/p>\n<div id=\"fs-id1165137634248\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(x\\right)=\\frac{1}{4}f\\left(x\\right)=\\frac{1}{4}{x}^{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135499881\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_01_05_06\">\n<div id=\"fs-id1165137415361\">\n<p id=\"fs-id1165137643555\">Write the formula for the function that we get when we vertically stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.<\/p>\n<\/div>\n<div id=\"fs-id1165137834210\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137834210\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137834210\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134380936\">[latex]g\\left(x\\right)=3x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135344103\" class=\"bc-section section\">\n<h4>Horizontal Stretches and Compressions<\/h4>\n<p>Now we consider the changes that occur to a function if we multiply the input of an original function [latex]f\\left(x\\right)[\/latex] by some constant.\u00a0 Notice that we are changing the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched horizontally away from or compressed horizontally toward the vertical axis in relation to the graph of the original function. If the constant is between 0 and 1, we get a <strong>horizontal stretch<\/strong>; if the constant is greater than 1, we get a <strong>horizontal compression<\/strong> of the function.\u00a0 Let\u2019s consider an example.<\/p>\n<p>Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. Let\u2019s let our original population be [latex]P[\/latex] and our new population be [latex]R[\/latex].\u00a0 Our new population, [latex]R,[\/latex] will progress in 1 hour the same amount as the original population [latex]P[\/latex] does in 2 hours, and in 2 hours, the new population [latex]R[\/latex] will progress as much as the original population [latex]P[\/latex] does in 4 hours. Sketch a graph of this population.<\/p>\n<p>Symbolically, we could write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}R\\left(1\\right)=P\\left(2\\right),\\hfill \\\\ R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,}\\hfill \\\\ \\text{ }R\\left(t\\right)=P\\left(2t\\right).\\hfill \\end{array}[\/latex]<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<p id=\"fs-id1165134380331\">See <a class=\"autogenerated-content\" href=\"#Figure_01_05_030\">Figure 5<\/a> for a graphical comparison of the original population and the compressed population.<\/p>\n<div id=\"Figure_01_05_030\" class=\"wp-caption aligncenter\">\n<div style=\"width: 984px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205741\/CNX_Precalc_Figure_01_05_029ab.jpg\" alt=\"Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).\" width=\"974\" height=\"400\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5 (a) Original population graph (b) Compressed population graph<\/p>\n<\/div>\n<\/div>\n<h3><\/h3>\n<p>Notice that the effect on the graph is a horizontal compression towards the vertical axis where all input values for our new function [latex]R[\/latex] are half of the original input value for [latex]P.[\/latex]\u00a0 You can clearly see that [latex]R\\left(3\\right) = P\\left(6\\right).[\/latex]\u00a0 We have therefore compressed the original graph of [latex]P\\left(t\\right)[\/latex] towards the vertical axis by a factor of \u00bd in order to create the graph of our new function [latex]R\\left(t\\right).[\/latex]<\/p>\n<p>Another way to think about this is that if the point (6, 2) is on the graph of [latex]P,[\/latex] then the point (3, 2) will be a point on the graph of [latex]R.[\/latex]\u00a0 We get the same outputs, but the inputs for [latex]R[\/latex] are half as large as the corresponding input for [latex]P.[\/latex]\u00a0 This results in a horizontal compression towards the vertical axis.<\/p>\n<p>You should spend some time convincing yourself that if we multiplied our original input by a value between 0 and 1 that we would get a horizontal stretch away from the vertical axis.\u00a0 Think about what would happen if we considered another population of fruit flies who progress through their life span half as fast as those represented by [latex]P\\left(t\\right).[\/latex]\u00a0 \u00a0We can consider [latex]S\\left(t\\right) = P\\left(\\frac{1}{2} t\\right).[\/latex]\u00a0 This means that [latex]S\\left(4\\right) = P\\left(2\\right)[\/latex] and that [latex]S\\left(6\\right) = P\\left(3\\right).[\/latex]\u00a0 Can you see in order to get the same outputs that our inputs for the new function [latex]S[\/latex] are twice as large as the inputs for the original function [latex]P?[\/latex]\u00a0 This means there would be a horizontal stretch by a factor of 2 away from the vertical axis.\u00a0 Our outputs for [latex]S[\/latex] are the same as the outputs for [latex]P[\/latex] when the inputs for [latex]S[\/latex] are double the inputs for [latex]P.[\/latex]\u00a0 If [latex]P[\/latex] has the point (3, 3) on the graph, then [latex]S[\/latex] will have the point (6, 3) on the graph.<\/p>\n<div id=\"fs-id1165137732896\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165134573810\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=f\\left(bx\\right),[\/latex] where [latex]b[\/latex] is a constant, is a <strong>horizontal stretch<\/strong> or <strong>horizontal compression<\/strong> of the function [latex]f\\left(x\\right).[\/latex]<\/p>\n<ul id=\"eip-456\">\n<li>If [latex]|b|>1,[\/latex] then the graph will be compressed by a factor of [latex]\\frac{1}{b}[\/latex] toward the y-axis.<\/li>\n<li>If [latex]0<|b|<1,[\/latex] then the graph will be stretched by a factor [latex]\\frac{1}{b}[\/latex] away from the y-axis.<\/li>\n<li>If [latex]b<0,[\/latex] then there will be combination of a horizontal stretch or compression with a horizontal reflection.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137832347\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137784900\"><strong>Given a description of a function, sketch a horizontal compression or stretch. <\/strong><\/p>\n<ol id=\"fs-id1165137784904\" type=\"1\">\n<li>Write a formula to represent the function.<\/li>\n<li>Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b>1[\/latex] for a compression or [latex]0\\lt b<1[\/latex] for a stretch.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_17\" class=\"textbox examples\">\n<div id=\"fs-id1165134071619\">\n<div id=\"fs-id1165134071621\">\n<h3>Example 4:\u00a0 Finding a Horizontal Stretch for a Tabular Function<\/h3>\n<p id=\"fs-id1165134071627\">A function [latex]f\\left(x\\right)[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_12\">Table 4<\/a>. Create a table for the function [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right).[\/latex]<\/p>\n<table id=\"Table_01_05_12\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=1, f(4)=3, f(6)=7, and f(8)=11.\">\n<caption>Table 4<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\n<td class=\"border\" style=\"text-align: center;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137401572\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137401572\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137401572\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137401575\">The formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right),[\/latex] and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]<\/p>\n<div id=\"fs-id1165135531534\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165137827502\">We do the same for the other values to produce <a class=\"autogenerated-content\" href=\"#Table_01_05_13\">Table 5<\/a>.<\/p>\n<table id=\"Table_01_05_13\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=1, f(4)=3, f(6)=7, and f(8)=11.\">\n<caption>Table 5<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\n<td class=\"border\" style=\"text-align: center;\">16<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\n<td class=\"border\" style=\"text-align: center;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a class=\"autogenerated-content\" href=\"#Figure_01_05_032\">Figure 6<\/a> shows the graphs of both of these sets of points.<\/p>\n<div id=\"Figure_01_05_032\" class=\"wp-caption aligncenter\" style=\"width: 960px;\">\n<div style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205745\/CNX_Precalc_Figure_01_05_031.jpg\" alt=\"Graph of the previous table.\" width=\"721\" height=\"246\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Analysis<\/h3>\n<p>Because each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_18\" class=\"textbox examples\">\n<div id=\"fs-id1165137783980\">\n<div id=\"fs-id1165137783983\">\n<h3>Example 5:\u00a0 Recognizing a Horizontal Compression on a Graph<\/h3>\n<p id=\"fs-id1165135545991\">Relate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_05_033\">Figure 7<\/a>.<\/p>\n<div id=\"Figure_01_05_033\" class=\"small\">\n<div style=\"width: 428px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205748\/CNX_Precalc_Figure_01_05_032.jpg\" alt=\"Graph of f(x) being vertically compressed to g(x).\" width=\"418\" height=\"250\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737597\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137737597\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137737597\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137895242\">The graph of [latex]g\\left(x\\right)[\/latex] looks like the graph of [latex]f\\left(x\\right)[\/latex] horizontally compressed. Because [latex]f\\left(x\\right)[\/latex] ends at [latex]\\left(6,4\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] ends at [latex]\\left(2,4\\right),[\/latex] we can see that the [latex]x\\text{-}[\/latex] values have been compressed by a factor of [latex]\\frac{1}{3},[\/latex] because [latex]6\\left(\\frac{1}{3}\\right)=2.[\/latex] We might also notice that [latex]g\\left(2\\right)=f\\left(6\\right)[\/latex] and [latex]g\\left(1\\right)=f\\left(3\\right).[\/latex] Either way, we can describe this relationship as [latex]g\\left(x\\right)=f\\left(3x\\right).[\/latex] This is a horizontal compression by a factor of [latex]\\frac{1}{3}.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\\frac{1}{4}[\/latex] in our function: [latex]f\\left(\\frac{1}{4}x\\right).[\/latex] This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135613633\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_01_05_07\">\n<div id=\"fs-id1165137451792\">\n<p id=\"fs-id1165137451793\">Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<\/p>\n<\/div>\n<div id=\"fs-id1165132945546\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132945546\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132945546\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165132945547\">[latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex] so using the square root function we get [latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{1}{3}x}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676302\" class=\"bc-section section\">\n<h3>Performing a Sequence of Transformations<\/h3>\n<p id=\"fs-id1165137387533\">When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by a factor of 2 does not create the same graph as vertically stretching by a factor of 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.<\/p>\n<p id=\"fs-id1165137387540\">When we see an expression such as [latex]2f\\left(x\\right)+3,[\/latex] which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of [latex]f\\left(x\\right),[\/latex] we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.<\/p>\n<p id=\"fs-id1165137483273\">Horizontal transformations are a little trickier to think about. When we write [latex]g\\left(x\\right)=f\\left(2x+3\\right),[\/latex] for example, we have to think about how the inputs to the function [latex]g[\/latex] relate to the inputs to the function [latex]f.[\/latex] Suppose we know [latex]f\\left(7\\right)=12.[\/latex] What input to [latex]g[\/latex] would produce that output? In other words, what value of [latex]x[\/latex] will allow [latex]g\\left(x\\right)=f\\left(2x+3\\right)=12?[\/latex] We would need [latex]2x+3=7.[\/latex] To solve for [latex]x,[\/latex] we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.<\/p>\n<p id=\"fs-id1165137580228\">This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch or compress a graph before shifting. We can work around this by factoring inside the function.<\/p>\n<div id=\"fs-id1165135241282\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(bx+p\\right)=f\\left(b\\left(x+\\frac{p}{b}\\right)\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137427526\">Let\u2019s work through an example.<\/p>\n<div id=\"fs-id1165137705217\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2x+4\\right)}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165137438427\">We can factor out a 2.<\/p>\n<div id=\"fs-id1165137694985\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)={\\left(2\\left(x+2\\right)\\right)}^{2}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135349189\">Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.<\/p>\n<div id=\"fs-id1165137715290\">\n<div class=\"textbox examples\">\n<h3>How To<\/h3>\n<p><strong>Given a transformation, determine the order in which they should be preformed.<\/strong><\/p>\n<ol>\n<li id=\"fs-id1165137469882\">When combining vertical transformations written in the form [latex]af\\left(x\\right)+k,[\/latex] first vertically stretch or compress by a factor of [latex]a[\/latex] and then vertically shift by [latex]k.[\/latex]<\/li>\n<li id=\"fs-id1165137423095\">When combining horizontal transformations written in the form [latex]f\\left(bx-h\\right),[\/latex] first horizontally shift by [latex]h[\/latex] and then horizontally stretch or compress by a factor of [latex]\\frac{1}{b}.[\/latex]<\/li>\n<li id=\"fs-id1165135186003\">When combining horizontal transformations written in the form [latex]f\\left(b\\left(x-h\\right)\\right),[\/latex] first horizontally stretch or compress by a factor of [latex]\\frac{1}{b}[\/latex] and then horizontally shift by [latex]h.[\/latex]<\/li>\n<li id=\"fs-id1165135191611\">Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_19\" class=\"textbox examples\">\n<div id=\"fs-id1165135309915\">\n<div id=\"fs-id1165135309917\">\n<h3>Example 6:\u00a0 Finding a Triple Transformation of a Tabular Function<\/h3>\n<p id=\"fs-id1165134188798\">Given <a class=\"autogenerated-content\" href=\"#Table_01_05_14\">Table 6<\/a> for the function [latex]f\\left(x\\right),[\/latex] create a table of values for the function [latex]g\\left(x\\right)=2f\\left(3x\\right)+1.[\/latex]<\/p>\n<table id=\"Table_01_05_14\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(x)\u201d. The values of x are 6, 12, 18, and 24. So for f(6)=10, f(12)=14, f(18)=15, and f(24)=17.\">\n<caption>Table 6<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">12<\/td>\n<td class=\"border\" style=\"text-align: center;\">18<\/td>\n<td class=\"border\" style=\"text-align: center;\">24<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">14<\/td>\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\n<td class=\"border\" style=\"text-align: center;\">17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135151321\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135151321\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135151321\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135151323\">There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, [latex]f\\left(3x\\right)[\/latex] is a horizontal compression by a factor of [latex]\\frac{1}{3},[\/latex] which means we multiply each [latex]x\\text{-}[\/latex]value by [latex]\\frac{1}{3}.[\/latex] See <a class=\"autogenerated-content\" href=\"#Table_01_05_15\">Table 7<\/a>.<\/p>\n<table id=\"Table_01_05_15\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cf(3x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=10, f(4)=14, f(6)=15, and f(8)=17.\">\n<caption>Table 7<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]f\\left(3x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">14<\/td>\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\n<td class=\"border\" style=\"text-align: center;\">17<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137460249\">Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See <a class=\"autogenerated-content\" href=\"#Table_01_05_016\">Table 8<\/a>.<\/p>\n<table id=\"Table_01_05_016\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201c2f(3x)\u201d. The values of x are 2, 4, 6, and 8. So for f(2)=20, f(4)=28, f(6)=30, and f(8)=34.\">\n<caption>Table 8<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]2f\\left(3x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<td class=\"border\" style=\"text-align: center;\">28<\/td>\n<td class=\"border\" style=\"text-align: center;\">30<\/td>\n<td class=\"border\" style=\"text-align: center;\">34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137661581\">Finally, we can apply the vertical shift, which will add 1 to all the output values. See <a class=\"autogenerated-content\" href=\"#Table_01_05_17\">Table 9<\/a>.<\/p>\n<table id=\"Table_01_05_17\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201cg(x)=2f(3x)+1\u201d. The values of x are 2, 4, 6, and 8. So for g(2)=21, g(4)=29, g(6)=31, and g(8)=35.\">\n<caption>Table 9<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]g\\left(x\\right)=2f\\left(3x\\right)+1[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">21<\/td>\n<td class=\"border\" style=\"text-align: center;\">29<\/td>\n<td class=\"border\" style=\"text-align: center;\">31<\/td>\n<td class=\"border\" style=\"text-align: center;\">35<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_20\" class=\"textbox examples\">\n<div id=\"fs-id1165137642734\">\n<div id=\"fs-id1165137571821\">\n<h3>Example 7:\u00a0 Finding a Triple Transformation of a Graph<\/h3>\n<p id=\"fs-id1165137571826\">Use the graph of [latex]f\\left(x\\right)[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_05_035\">Figure 8<\/a> to sketch a graph of [latex]k\\left(x\\right)=f\\left(\\frac{1}{2}x+1\\right)-3.[\/latex]<\/p>\n<div id=\"Figure_01_05_035\" class=\"small\">\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205750\/CNX_Precalc_Figure_01_05_034.jpg\" alt=\"Graph of a half-circle.\" width=\"325\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134148412\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134148412\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134148412\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134148414\">To simplify, let\u2019s start by factoring out the inside of the function.<\/p>\n<div id=\"fs-id1165137831276\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(\\frac{1}{2}x+1\\right)-3=f\\left(\\frac{1}{2}\\left(x+2\\right)\\right)-3[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135532397\">By factoring the inside, we can first horizontally stretch by a factor of 2, as indicated by the [latex]\\frac{1}{2}[\/latex] on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See <a class=\"autogenerated-content\" href=\"#Figure_01_05_036\">Figure 9<\/a>.<\/p>\n<div id=\"Figure_01_05_036\" class=\"small\">\n<div style=\"width: 295px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205753\/CNX_Precalc_Figure_01_05_035.jpg\" alt=\"Graph of a vertically stretch half-circle.\" width=\"285\" height=\"294\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137736781\">Next, we horizontally shift left by 2 units, as indicated by [latex]x+2.[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_05_037\">Figure 10<\/a>.<\/p>\n<div id=\"Figure_01_05_037\" class=\"small\">\n<div style=\"width: 302px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205756\/CNX_Precalc_Figure_01_05_036.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"292\" height=\"302\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137696198\">Last, we vertically shift down by 3 to complete our sketch, as indicated by the [latex]-3[\/latex] on the outside of the function. See <a class=\"autogenerated-content\" href=\"#Figure_01_05_038\">Figure 11<\/a>.<\/p>\n<div id=\"Figure_01_05_038\" class=\"small\">\n<div style=\"width: 306px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205759\/CNX_Precalc_Figure_01_05_037.jpg\" alt=\"Graph of a vertically stretch and translated half-circle.\" width=\"296\" height=\"306\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 11<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135497140\" class=\"precalculus media\">\n<div id=\"Example_03_02_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135460939\">\n<div id=\"fs-id1165135460941\">\n<h3>Example 8:\u00a0 Writing the Equation of a Quadratic Function from the Graph<\/h3>\n<p>Write an equation for the quadratic function [latex]g[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_03_02_007\">Figure 12<\/a> as a transformation of [latex]f\\left(x\\right)={x}^{2},[\/latex].\u00a0 First, write your solutions in the <strong>vertex form of a quadratic function<\/strong> [latex]g\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.\u00a0 Then expand the formula, and simplify terms to write the equation in general form.<\/p>\n<div id=\"Figure_03_02_007\" class=\"small\">\n<div style=\"width: 344px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07151928\/CNX_Precalc_Figure_03_02_007.jpg\" alt=\"Graph of a parabola with its vertex at (-2, -3).\" width=\"334\" height=\"304\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 12<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134211341\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134211341\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134211341\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137742565\">We can see the graph of <em>g <\/em>is the graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] shifted to the left 2 and down 3, giving a formula in the form [latex]g\\left(x\\right)=a{\\left(x+2\\right)}^{2}\u20133.[\/latex]<\/p>\n<p id=\"fs-id1165134064001\">Substituting the coordinates of a point on the curve, such as [latex]\\left(0,-1\\right),[\/latex] we can solve for the vertical stretch or compression factor.<\/p>\n<div id=\"eip-id1165134221671\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}-1&=a{\\left(0+2\\right)}^{2}-3\\hfill \\\\ \\text{ }2&=4a\\hfill \\\\ \\text{ }a&=\\frac{1}{2}\\hfill \\end{align*}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165137895371\">In vertex form, the algebraic model for this graph is [latex]g\\left(x\\right)=\\frac{1}{2}{\\left(x+2\\right)}^{2}\u20133.[\/latex]<\/p>\n<p id=\"fs-id1165137844164\">To write this in general polynomial form, we can expand the formula and simplify terms.<\/p>\n<div id=\"eip-id1165137463836\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}g\\left(x\\right)&=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3\\hfill \\\\ \\text{ }&=\\frac{1}{2}\\left(x+2\\right)\\left(x+2\\right)-3\\hfill \\\\ \\text{ }&=\\frac{1}{2}\\left({x}^{2}+4x+4\\right)-3\\hfill \\\\ \\text{ }&=\\frac{1}{2}{x}^{2}+2x+2-3\\hfill \\\\ \\text{ }&=\\frac{1}{2}{x}^{2}+2x-1\\hfill \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137914060\">Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.<\/p>\n<h3>Analysis<\/h3>\n<p id=\"fs-id1165137803212\">We can check our work using the table feature on a graphing utility. First enter [latex]\\text{Y1}=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3.[\/latex] Next, select [latex]\\text{TBLSET,}[\/latex] then use [latex]\\text{TblStart}=\u20136[\/latex] and [latex]\\Delta \\text{Tbl = 2,}[\/latex] and select [latex]\\text{TABLE}\\text{.}[\/latex] See <a class=\"autogenerated-content\" href=\"#Table_03_02_01\">Table 10<\/a>.<\/p>\n<table id=\"Table_03_02_01\" summary=\"..\">\n<caption>Table 10<\/caption>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20136<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20134<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20132<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20131<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20133<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20131<\/td>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135570238\">The ordered pairs in the table correspond to points on the graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137527658\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_03_02_01\">\n<div id=\"fs-id1165137933940\">\n<p id=\"fs-id1165137933941\">A coordinate grid has been superimposed over the quadratic path of a basketball in <a class=\"autogenerated-content\" href=\"#Figure_03_02_008\">Figure 13<\/a>. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\n<div id=\"Figure_03_02_008\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07151931\/CNX_Precalc_Figure_03_02_008.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 13.\u00a0(credit: modification of work by Dan Meyer)<\/p>\n<\/div>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<div id=\"fs-id1165135414238\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135414238\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135414238\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135414239\">The path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right),[\/latex] so [latex]\\left(h\\right)x=\u2013\\frac{7}{16}{\\left(x+4\\right)}^{2}+7.[\/latex] To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(\u20137.5\\right)\\approx 1.64;[\/latex] he doesn\u2019t make it.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137410202\">Access this online resource for additional instruction and practice with transformation of functions.<\/p>\n<ul id=\"fs-id1165137410206\">\n<li><a href=\"http:\/\/openstax.org\/l\/functrans\">Function Transformations<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135499979\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134474082\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\">Vertical shift<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(x\\right)+k[\/latex] (up for [latex]k>0[\/latex])<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Horizontal shift<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex] (right for [latex]h>0[\/latex])<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Vertical reflection<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Horizontal reflection<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Vertical stretch<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] ([latex]a>1[\/latex] )<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Vertical compression<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=af\\left(x\\right)[\/latex] ([latex]0\\lt a<1[\/latex] )<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Horizontal stretch<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] ([latex]0\\lt b \\lt\u00a0 1[\/latex] )<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Horizontal compression<\/td>\n<td class=\"border\">[latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] ([latex]b>1[\/latex])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135264626\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135264630\">\n<li>A function can be compressed or stretched vertically by multiplying the output by a constant.<\/li>\n<li>A function can be compressed or stretched horizontally by multiplying the input by a constant.<\/li>\n<li>The order in which different transformations are applied does affect the final function. Shifts and stretches must be applied in the order given. However,\u00a0 vertical transformations may be done before or after horizontal transformations.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137448239\"><\/dl>\n<dl id=\"fs-id1165133242964\">\n<dt>horizontal compression<\/dt>\n<dd id=\"fs-id1165137833874\">a transformation that compresses a function\u2019s graph horizontally, by multiplying the input by a constant [latex]b>1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135440170\"><\/dl>\n<dl id=\"fs-id1165137922367\">\n<dt>horizontal shift<\/dt>\n<dd id=\"fs-id1165137922373\">a transformation that shifts a function\u2019s graph left or right by adding a positive or negative constant to the input<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137922379\"><\/dl>\n<dl id=\"fs-id1165134259240\">\n<dt>odd function<\/dt>\n<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right),[\/latex] and is symmetric about the origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137676545\"><\/dl>\n<dl id=\"fs-id1165137662611\">\n<dt>vertical reflection<\/dt>\n<dd id=\"fs-id1165137834403\">a transformation that reflects a function\u2019s graph across the <em>x<\/em>-axis by multiplying the output by [latex]-1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135580354\">\n<dt>vertical shift<\/dt>\n<dd id=\"fs-id1165137862443\">a transformation that shifts a function\u2019s graph up or down by adding a positive or negative constant to the output<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n","protected":false},"author":158108,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2948","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2948","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158108"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2948\/revisions"}],"predecessor-version":[{"id":3274,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2948\/revisions\/3274"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2948\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2948"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2948"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2948"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}