{"id":127,"date":"2019-02-08T20:59:14","date_gmt":"2019-02-08T20:59:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/transformation-of-functions\/"},"modified":"2025-03-31T20:36:02","modified_gmt":"2025-03-31T20:36:02","slug":"transformation-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/transformation-of-functions\/","title":{"raw":"1.6 Transformation of Functions","rendered":"1.6 Transformation of Functions"},"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>Describe and apply vertical and horizontal shifts and reflections of graphs, tables and function formulas.<\/li>\r\n \t<li>Use function notation to express horizontal and vertical shifts and reflections of functions.<\/li>\r\n \t<li>Determine whether a function is even, odd or neither from its algebraic formula or graph.<\/li>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Figure_01_05_001\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205611\/CNX_Precalc_Figure_01_05_038n.jpg\" alt=\"Figure_01_05_038\" width=\"488\" height=\"325\" \/> Figure 1 (credit: \"Misko\"\/Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137742090\">We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.<\/p>\r\n\r\n<div id=\"fs-id1165137827988\" class=\"bc-section section\">\r\n<h3>Graphing Functions Using Vertical and Horizontal Shifts<\/h3>\r\n<p id=\"fs-id1165137654715\">Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.<\/p>\r\n\r\n<div id=\"fs-id1165137535664\" class=\"bc-section section\">\r\n<h4>Identifying Vertical Shifts<\/h4>\r\n<p id=\"fs-id1165135503932\">One simple kind of <span class=\"no-emphasis\">transformation<\/span> involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a <strong>vertical shift<\/strong>, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function [latex]g\\left(x\\right)=f\\left(x\\right)+k,[\/latex] the function [latex]f\\left(x\\right)[\/latex] is shifted vertically [latex]k[\/latex] units. See <a class=\"autogenerated-content\" href=\"#Figure_01_05_002\">Figure 2<\/a> for an example.<\/p>\r\n\r\n<div id=\"Figure_01_05_002\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205614\/CNX_Precalc_Figure_01_05_001.jpg\" alt=\"Figure_01_05_001\" width=\"487\" height=\"292\" \/> Figure 2 Vertical shift by [latex]k=1[\/latex] of the cube root function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}.[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165137439125\">To help you visualize the concept of a vertical shift, consider that [latex]y=f\\left(x\\right).[\/latex] Therefore, [latex]f\\left(x\\right)+k[\/latex] is equivalent to [latex]y+k.[\/latex] Every unit of [latex]y[\/latex] is replaced by [latex]y+k,[\/latex] so the [latex]y\\text{-}[\/latex]value increases or decreases depending on the value of [latex]k.[\/latex] The result is a shift upward or downward.<\/p>\r\n\r\n<div id=\"fs-id1165137737394\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\nGiven a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=f\\left(x\\right)+k,[\/latex] where\u00a0 [latex]k[\/latex] is a constant, is a <strong>vertical shift<\/strong> of the function [latex]f\\left(x\\right).[\/latex] All the output values change by [latex]k[\/latex] units. If [latex]k[\/latex] is positive, the graph will shift up. If [latex]k[\/latex] is negative, the graph will shift down.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137701104\">\r\n<div id=\"fs-id1165135547346\">\r\n<h3>Example 1:\u00a0 Adding a Constant to a Function<\/h3>\r\n<p id=\"fs-id1165137676255\">To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. <a class=\"autogenerated-content\" href=\"#Figure_01_05_003\">Figure 3<\/a> shows the area of open vents [latex]V[\/latex] (in square feet) throughout the day in hours after midnight, [latex]t.[\/latex] During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.<\/p>\r\n\r\n<div id=\"Figure_01_05_003\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"442\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205616\/CNX_Precalc_Figure_01_05_002.jpg\" alt=\"Figure_01_05_002\" width=\"442\" height=\"296\" \/> Figure 3[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135504997\">[reveal-answer q=\"fs-id1165135504997\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135504997\"]\r\n<p id=\"fs-id1165137667304\">We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_004\">Figure 4<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_004\" class=\"wp-caption aligncenter\" style=\"width: 500px;\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"442\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205619\/CNX_Precalc_Figure_01_05_003a.jpg\" alt=\"Figure_01_05_003a\" width=\"442\" height=\"299\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137662763\">Notice that in <a class=\"autogenerated-content\" href=\"#Figure_01_05_004\">Figure 4<\/a>, for each input value, the output value has increased by 20, so if we call the new function [latex]S\\left(t\\right),[\/latex] we could write<\/p>\r\n\r\n<div id=\"fs-id1165137772392\" class=\"unnumbered\" style=\"text-align: center;\">[latex]S\\left(t\\right)=V\\left(t\\right)+20[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165134164955\">This notation tells us that, any value of [latex]S\\left(t\\right)[\/latex] can be found by evaluating the function [latex]V[\/latex] at the same input and then adding 20 to the result. This defines [latex]S[\/latex] as a transformation of the function [latex]V,[\/latex] in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See <a class=\"autogenerated-content\" href=\"#Table_01_05_018\">Table 1<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_018\" summary=\"Three rows and seven columns. The first row is labeled, \u201ct\u201d, the second is labeled, \u201cV(t)\u201d, and the third is labeled, \u201cS(t)\u201d. The values of t are 0, 8, 10, 17, 19, and 24. So for V(0)=0, V(8)=0, V(10)=220, V(17)=220, V(19)=0, and V(24)=0. For S(0)=20, S(8)=20, S(10)=240, S(17)=240, S(19)=20, and S(24)=20.\"><caption>Table 1<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">17<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">19<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]V\\left(t\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">220<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">220<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]S\\left(t\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">240<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">240<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/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=\"fs-id1165135369225\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165134371214\"><strong>Given a tabular function, create a new row to represent a vertical shift.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135530387\" type=\"1\">\r\n \t<li>Identify the output row or column.<\/li>\r\n \t<li>Determine the <span class=\"no-emphasis\">magnitude<\/span> of the shift.<\/li>\r\n \t<li>Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135581153\">\r\n<div id=\"fs-id1165135443766\">\r\n<h3>Example 2:\u00a0 Shifting a Tabular Function Vertically<\/h3>\r\n<p id=\"fs-id1165137692074\">A function [latex]f\\left(x\\right)[\/latex] is given in <a class=\"autogenerated-content\" href=\"#Table_01_05_01\">Table 2<\/a>. Create a table for the function [latex]g\\left(x\\right)=f\\left(x\\right)-3.[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_05_01\" style=\"height: 24px;\" 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 2<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 30.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 30.6563px; text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 30.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 30.6563px; text-align: center;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137574326\">[reveal-answer q=\"fs-id1165137574326\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137574326\"]\r\n<p id=\"fs-id1165135241392\">The formula [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex] tells us that we can find the output values of [latex]g[\/latex] by subtracting 3 from the output values of [latex]f.[\/latex] For example:<\/p>\r\n\r\n<div id=\"fs-id1165137662784\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(2\\right)&amp;=1\\hfill &amp;&amp; \\text{Given}\\hfill \\\\ g\\left(x\\right)&amp;=f\\left(x\\right)-3\\hfill &amp;&amp; \\text{Given transformation}\\hfill \\\\ g\\left(2\\right)&amp;=f\\left(2\\right)-3\\hfill &amp;&amp; \\hfill \\\\ \\text{ }&amp;=1-3\\hfill &amp;&amp; \\hfill \\\\ \\text{ }&amp;=-2\\hfill &amp;&amp; \\hfill \\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\nSubtracting 3 from each [latex]f\\left(x\\right)[\/latex] value, we can complete a table of values for [latex]g\\left(x\\right)[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Table_01_05_02\">Table 3<\/a>.\r\n<table id=\"Table_01_05_02\" summary=\"Three rows and five columns. The first row is labeled, \u201cx\u201d, the second is labeled, \u201cf(x)\u201d, and the third is labeled, \u201cg(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. For g(2)=-2, g(4)=0, g(6)=4, and g(8)=8.\"><caption>Table 3<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><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\" style=\"text-align: center;\"><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<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u22122<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Analysis<\/h3>\r\nAs with the earlier vertical shift, notice the input values stay the same and only the output values change.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137692134\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_01_05_01\">\r\n<div id=\"fs-id1165137834794\">\r\n<p id=\"fs-id1165137767439\">The function [latex]h\\left(t\\right)=-4.9{t}^{2}+30t[\/latex] gives the height [latex]h[\/latex] of a ball (in meters) thrown upward from the ground after [latex]t[\/latex] seconds. Suppose the ball was instead thrown from the top of a 10 meter building. Relate this new height function [latex]b\\left(t\\right)[\/latex] to [latex]h\\left(t\\right),[\/latex] and then find a formula for [latex]b\\left(t\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137481965\">[reveal-answer q=\"fs-id1165137481965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137481965\"]\r\n<div id=\"fs-id1165137680332\">[latex]b\\left(t\\right)=h\\left(t\\right)+10=-4.9{t}^{2}+30t+10[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137597159\" class=\"bc-section section\">\r\n<h4>Identifying Horizontal Shifts<\/h4>\r\n<p id=\"fs-id1165137404493\">We just saw that the vertical shift is a change to the output or outside of the function. We will now look at how changes to input or the inside of the function change its graph and meaning.<\/p>\r\nA change to the <strong>input<\/strong> results in a movement of the graph of an original function left or right in what is known as a <strong>horizontal shift<\/strong>. We will be creating a new function [latex]g\\left(x\\right)[\/latex] which is based on an original function [latex]f\\left(x\\right)[\/latex] using the following function notation:\u00a0 [latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex] where <strong>[latex]h[\/latex]<\/strong> is a constant.\r\n<div id=\"Figure_01_05_005\" class=\"small\"><\/div>\r\n<p id=\"eip-884\">For example, if [latex]f\\left(x\\right)={x}^{2},[\/latex] then we can create a function in terms of [latex]f[\/latex] by writing [latex]g\\left(x\\right)=f\\left(x-2\\right),[\/latex] which is equivalent to [latex]g\\left(x\\right)={\\left(x-2\\right)}^{2}.[\/latex]\u00a0 Think about what happens carefully.<\/p>\r\nWe can read the statement [latex]g\\left(x\\right)=f\\left(x-2\\right)[\/latex] as saying that the output for [latex]g[\/latex] at [latex]x[\/latex] will be the same as the output we get for the original function [latex]f[\/latex] evaluated two units earlier.\u00a0 Perhaps an easier way to see this is to recognize that if [latex]x[\/latex] is 5, then [latex]g\\left(5\\right)=f\\left(5-2\\right)=f\\left(3\\right).[\/latex]\u00a0 We get the same output for [latex]g[\/latex] at the input of 5 as we did for the function [latex]f[\/latex] for an input two untis earlier.\u00a0 Therefore, in order to produce the graph of [latex]g[\/latex], we will shift our original function [latex]f\\left(x\\right)[\/latex] to the right by two units.\r\n\r\nWhat if [latex]h[\/latex] is negative?\u00a0 Let's consider the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}.[\/latex]\u00a0 If we let [latex]h=-1,[\/latex] then we can consider a new function [latex]m\\left(x\\right)=f\\left(x-\\left(-1\\right)\\right)=f\\left(x+1\\right).[\/latex]\u00a0 This is equivalent to [latex]m\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x+1}.[\/latex] Notice again that it is our input which has changed.\u00a0 We can read this statement as saying that the output for [latex]m[\/latex] evaluated at [latex]x[\/latex] will be the same as the output we get for the original function [latex]f[\/latex] evaluated one unit later.\u00a0 Therefore, [latex]m\\left(5\\right)=f\\left(5+1\\right)=f\\left(6\\right).[\/latex]\u00a0 In order to produce the graph of [latex]m,[\/latex] we will shift the original function [latex]f\\left(x\\right)[\/latex] to the left by one unit.\u00a0 Consider the picture shown in\u00a0<a class=\"autogenerated-content\" href=\"#Figure_01_05_005\">Figure 5<\/a>.\r\n<div id=\"Figure_01_05_005\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205623\/CNX_Precalc_Figure_01_05_004.jpg\" alt=\"Figure_01_05_004\" width=\"487\" height=\"288\" \/> Figure 5 Horizontal shift of the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}.[\/latex] Note that [latex]f\\left(x+1\\right)[\/latex] shifts the graph to the left by one unit.[\/caption]<\/div>\r\n<div id=\"fs-id1165137436470\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\nGiven a function [latex]f,[\/latex] a new function [latex]g\\left(x\\right)=f\\left(x-h\\right),[\/latex] where [latex]h[\/latex] is a constant, is a <strong>horizontal shift<\/strong> of the function [latex]f.[\/latex] If [latex]h[\/latex] is positive, the graph will shift right. If [latex]h[\/latex] is negative, the graph will shift left.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135241148\">\r\n<div id=\"fs-id1165134148356\">\r\n<h3>Example 3:\u00a0 Adding a Constant to an Input<\/h3>\r\n<p id=\"fs-id1165137658363\">Returning to our building airflow example from <a class=\"autogenerated-content\" href=\"#Figure_01_05_003\">Figure 3<\/a>, suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137423751\">[reveal-answer q=\"fs-id1165137423751\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137423751\"]\r\n<p id=\"fs-id1165137423753\">We can set [latex]V\\left(t\\right)[\/latex] to be the original program and [latex]F\\left(t\\right)[\/latex] to be the revised program.<\/p>\r\n\r\n<div id=\"fs-id1165135195278\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\\\[\/latex][latex]\\begin{align*}V\\left(t\\right)&amp;=\\text{ the original venting plan}\\\\ \\text{F}\\left(t\\right)&amp;=\\text{ starting 2 hrs sooner}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165137714274\">In the new graph, at each time, the airflow is the same as the original function [latex]V[\/latex] was 2 hours later. For example, in the original function [latex]V,[\/latex] the airflow starts to change at 8 a.m., whereas for the function [latex]F,[\/latex] the airflow starts to change at 6 a.m. The comparable function values are [latex]F\\left(6\\right)=V\\left(8\\right).[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_05_006\">Figure 6<\/a>. Notice also that the vents first opened to [latex]220{\\text{ ft}}^{2}[\/latex] at 10 a.m. under the original plan, while under the new plan the vents reach [latex]220{\\text{ ft}}^{\\text{2}}[\/latex] at 8 a.m., so [latex]F\\left(8\\right)=V\\left(10\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165137411313\">In both cases, we see that, because [latex]F\\left(t\\right)[\/latex] starts 2 hours sooner, [latex]h=-2.[\/latex] That means that the same output values are reached when [latex]F\\left(t\\right)=V\\left(t-\\left(-2\\right)\\right)=V\\left(t+2\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_01_05_006\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"444\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205625\/CNX_Precalc_Figure_01_05_005a.jpg\" alt=\"Figure_01_05_005a\" width=\"444\" height=\"300\" \/> Figure 6[\/caption]\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165137757835\">Note that [latex]V\\left(t+2\\right)[\/latex] has the effect of shifting the graph to the <em>left<\/em>.<\/p>\r\n<p id=\"fs-id1165137473462\">Horizontal changes or \u201cinside changes\u201d affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function [latex]F\\left(t\\right)[\/latex] uses the same outputs as [latex]V\\left(t\\right),[\/latex] but matches those outputs to inputs 2 hours earlier than those of [latex]V\\left(t\\right).[\/latex] Said another way, we must add 2 hours to the input of [latex]V[\/latex] to find the corresponding output for[latex]F:F\\left(t\\right)=V\\left(t+2\\right).[\/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-id1165137644724\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137667199\"><strong>Given a tabular function, create a new row to represent a horizontal shift.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137652957\" type=\"1\">\r\n \t<li>Identify the input row or column.<\/li>\r\n \t<li>Determine the magnitude of the shift.<\/li>\r\n \t<li>Add the shift to the value in each input cell.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137831912\">\r\n<div id=\"fs-id1165137831914\">\r\n<h3>Example 4:\u00a0 Shifting a Tabular Function Horizontally<\/h3>\r\n<p id=\"fs-id1165137761563\">A function [latex]f\\left(x\\right)[\/latex] is given in <a class=\"autogenerated-content\" href=\"#Table_01_05_03\">Table 4<\/a>. Create a table for the function [latex]g\\left(x\\right)=f\\left(x-3\\right).[\/latex]<\/p>\r\n\r\n<table id=\"Table_01_05_03\" style=\"width: 596px;\" 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\" style=\"text-align: center; width: 221px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 375px;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center; width: 221px;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 375px;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center; width: 221px;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 375px;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center; width: 221px;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 375px;\">7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center; width: 221px;\">8<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 375px;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165134401703\">[reveal-answer q=\"fs-id1165134401703\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134401703\"]\r\n<p id=\"fs-id1165137735632\">The formula [latex]g\\left(x\\right)=f\\left(x-3\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are the same as the output value of [latex]f[\/latex] when the input value is 3 less. For example, we know that [latex]f\\left(2\\right)=1.[\/latex] To get the same output from the function [latex]g,[\/latex] we will need an input value that is 3 <em>larger<\/em>. We input a value that is 3 larger for [latex]g\\left(x\\right)[\/latex] because the function takes 3 away before evaluating the function [latex]f.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137645517\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}g\\left(5\\right)&amp;=f\\left(5-3\\right)\\hfill \\\\ \\text{ }&amp;=f\\left(2\\right)\\hfill \\\\ \\text{ }&amp;=1\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135531624\">We continue with the other values to create <a class=\"autogenerated-content\" href=\"#Table_01_05_04\">Table 5<\/a>. In our table for [latex]g\\left(x\\right)[\/latex], we need to increase each input value for [latex]f[\/latex] by 3.<\/p>\r\n\r\n<table id=\"Table_01_05_04\" style=\"height: 48px; width: 493px;\" summary=\"Three rows and five columns. The first row is labeled, \u201cx\u201d, the second is labeled, \u201cf(x)\u201d, and the third is labeled, \u201cg(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. For g(2)=1, g(4)=3, g(6)=7, and g(8)=11.\"><caption>Table 5<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\"><strong>[latex]g\\left(x\\right)=f\\left(x-3\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\">[latex]g\\left(5\\right)=f\\left(5-3\\right)=f\\left(2\\right)=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\">[latex]g\\left(7\\right)=f\\left(7-3\\right)=f\\left(4\\right)=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\">9<\/td>\r\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\">[latex]g\\left(9\\right)=f\\left(9-3\\right)=f\\left(6\\right)=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 193px; text-align: center;\">11<\/td>\r\n<td class=\"border\" style=\"width: 300px; text-align: center;\">[latex]g\\left(11\\right)=f\\left(11-3\\right)=f\\left(8\\right)=11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137678273\">The result is that the function [latex]g\\left(x\\right)[\/latex] has been shifted to the right by 3. Notice the output values for [latex]g\\left(x\\right)[\/latex] remain the same as the output values for [latex]f\\left(x\\right),[\/latex] but the corresponding input values, [latex]x,[\/latex] have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165135438856\"><a class=\"autogenerated-content\" href=\"#Figure_01_05_007\">Figure 7<\/a> represents both of the functions. We can see the horizontal shift in each point.<\/p>\r\n\r\n<div id=\"Figure_01_05_007\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"282\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205629\/CNX_Precalc_Figure_01_05_006.jpg\" alt=\"Graph of the points from the previous table for f(x) and g(x)=f(x-3).\" width=\"282\" height=\"318\" \/> Figure 7[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135196829\">\r\n<div id=\"Figure_01_05_007\" class=\"small\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135528978\">\r\n<div id=\"fs-id1165135528980\">\r\n<h3>Example 5:\u00a0 Identifying a Horizontal Shift of a Toolkit Function<\/h3>\r\n<p id=\"fs-id1165137582443\"><a class=\"autogenerated-content\" href=\"#Figure_01_05_008\">Figure 8<\/a> represents a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{2}.[\/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_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"436\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205633\/CNX_Precalc_Figure_01_05_007.jpg\" alt=\"Graph of a parabola.\" width=\"436\" height=\"294\" \/> Figure 8[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135173899\">[reveal-answer q=\"fs-id1165135173899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135173899\"]\r\n<p id=\"fs-id1165135173902\">Notice that the graph is identical in shape to the [latex]f\\left(x\\right)={x}^{2}[\/latex] function, but the <em>x-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so<\/p>\r\n\r\n<div id=\"fs-id1165133349274\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(x\\right)=f\\left(x-2\\right)[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165137561935\">Notice how we must input the value [latex]x=2[\/latex] to get the output value [latex]y=0;[\/latex] the <em>x<\/em>-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the [latex]f\\left(x\\right)[\/latex] function to write a formula for [latex]g\\left(x\\right)[\/latex] by evaluating [latex]f\\left(x-2\\right).[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137444147\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(x\\right)&amp;={x}^{2}\\hfill \\\\ g\\left(x\\right)&amp;=f\\left(x-2\\right)\\hfill \\\\ g\\left(x\\right)&amp;=f\\left(x-2\\right)={\\left(x-2\\right)}^{2}\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<h3>Analysis<\/h3>\r\n<div>To determine whether the shift is [latex]+2[\/latex] or [latex]-2[\/latex], consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function, [latex]f\\left(0\\right)=0.[\/latex] In our shifted function, [latex]g\\left(2\\right)=0.[\/latex] To obtain the output value of 0 from the function [latex]f,[\/latex] we need to decide whether a plus or a minus sign will work to satisfy [latex]g\\left(2\\right)=f\\left(x-2\\right)=f\\left(0\\right)=0.[\/latex] For this to work, we will need to <em>subtract<\/em> 2 units from our input values.<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137437869\">\r\n<div id=\"fs-id1165137786325\">\r\n<h3>Example 6:\u00a0 Interpreting Horizontal versus Vertical Shifts<\/h3>\r\n<p id=\"fs-id1165137574172\">The function [latex]G\\left(m\\right)[\/latex] gives the number of gallons of gas required to drive [latex]m[\/latex] miles. Interpret [latex]G\\left(m\\right)+10[\/latex] and [latex]G\\left(m+10\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135690669\">[reveal-answer q=\"fs-id1165135690669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135690669\"]\r\n<p id=\"fs-id1165137921579\">[latex]G\\left(m\\right)+10[\/latex] can be interpreted as adding 10 to the output, gallons. This is the gas required to drive [latex]m[\/latex] miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.<\/p>\r\n<p id=\"fs-id1165137434382\">[latex]G\\left(m+10\\right)[\/latex] can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than [latex]m[\/latex] miles. The graph would indicate a horizontal shift.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137781562\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_01_05_02\">\r\n<div id=\"fs-id1165137470341\">\r\n<p id=\"fs-id1165137557596\">Given the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x},[\/latex] graph the original function [latex]f\\left(x\\right)[\/latex] and the transformation [latex]g\\left(x\\right)=f\\left(x+2\\right)[\/latex] on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135548998\">[reveal-answer q=\"fs-id1165135548998\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135548998\"]\r\n<p id=\"fs-id1165135548999\">The graphs of [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.<\/p>\r\n<span id=\"fs-id1165137761611\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205637\/CNX_Precalc_Figure_01_05_008.jpg\" alt=\"Graph of a square root function and a horizontally shift square foot function.\" width=\"431\" height=\"255\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135250592\" class=\"bc-section section\">\r\n<h4>Combining Vertical and Horizontal Shifts<\/h4>\r\n<p id=\"fs-id1165137676099\">Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output, [latex]y\\text{-}[\/latex] axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input, [latex]x\\text{-}[\/latex] axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down <em>and<\/em> right or left.<\/p>\r\n\r\n<div id=\"fs-id1165137628099\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137668111\"><strong>Given a function and both a vertical and a horizontal shift, sketch the graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137558226\" type=\"1\">\r\n \t<li>Identify the vertical and horizontal shifts from the formula.<\/li>\r\n \t<li>The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.<\/li>\r\n \t<li>The horizontal shift results from a constant subtracted from the input. Move the graph right for a positive constant and left for a negative constant.<\/li>\r\n \t<li>Apply the shifts to the graph in either order.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137874554\">\r\n<div id=\"fs-id1165137874556\">\r\n<h3>Example 7:\u00a0 Graphing Combined Vertical and Horizontal Shifts<\/h3>\r\n<p id=\"fs-id1165135168426\">Given [latex]f\\left(x\\right)=|x|,[\/latex] sketch a graph of [latex]h\\left(x\\right)=f\\left(x+1\\right)-3.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137841697\">[reveal-answer q=\"fs-id1165137841697\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137841697\"]\r\n<p id=\"fs-id1165137841699\">The function [latex]f[\/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of [latex]h[\/latex] has transformed [latex]f[\/latex] in two ways: [latex]f\\left(x+1\\right)[\/latex] is a change on the inside of the function, giving a horizontal shift left by 1 since [latex]h=-1[\/latex], and the subtraction by 3 in [latex]f\\left(x+1\\right)-3[\/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in <a class=\"autogenerated-content\" href=\"#Figure_01_05_010a\">Figure 9<\/a>.<\/p>\r\n<p id=\"fs-id1165133276230\">Let us follow one point of the graph of [latex]f\\left(x\\right)=|x|.[\/latex]<\/p>\r\n\r\n<ul id=\"eip-id1165135605357\">\r\n \t<li>The point [latex]\\left(0,0\\right)[\/latex]is transformed first by shifting left 1 unit: [latex]\\left(0,0\\right)\\to \\left(-1,0\\right)[\/latex]<\/li>\r\n \t<li>The point [latex]\\left(-1,0\\right)[\/latex]is transformed next by shifting down 3 units: [latex]\\left(-1,0\\right)\\to \\left(-1,-3\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<div id=\"Figure_01_05_010a\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"364\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205640\/CNX_Precalc_Figure_01_05_009a.jpg\" alt=\"Graph of an absolute function, y=|x|, and how it was transformed to y=|x+1|-3.\" width=\"364\" height=\"300\" \/> Figure 9[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137705975\"><a class=\"autogenerated-content\" href=\"#Figure_01_05_010b\">Figure 10<\/a> shows the graph of [latex]h.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_01_05_010b\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"363\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205643\/CNX_Precalc_Figure_01_05_009b.jpg\" alt=\"The final function y=|x+1|-3.\" width=\"363\" height=\"299\" \/> Figure 10[\/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-id1165135192765\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"fs-id1165137400636\">\r\n<div id=\"fs-id1165137400639\">\r\n<p id=\"fs-id1165135479104\">Given[latex]f\\left(x\\right)=|x|,[\/latex] sketch a graph of [latex]h\\left(x\\right)=f\\left(x-2\\right)+4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137422205\">[reveal-answer q=\"fs-id1165137422205\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137422205\"]<span id=\"fs-id1165137557535\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205646\/CNX_Precalc_Figure_01_05_010.jpg\" alt=\"Graph of h(x)=|x-2|+4.\" width=\"362\" height=\"299\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_08\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137452719\">\r\n<div id=\"fs-id1165137452721\">\r\n<h3>Example 8:\u00a0 Identifying Combined Vertical and Horizontal Shifts<\/h3>\r\n<p id=\"fs-id1165137731917\">Write a formula for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_012\">Figure 11<\/a>, which is a transformation of the toolkit square root function.<\/p>\r\n\r\n<div id=\"Figure_01_05_012\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"470\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205648\/CNX_Precalc_Figure_01_05_011.jpg\" alt=\"Graph of a square root function transposed right one unit and up 2.\" width=\"470\" height=\"282\" \/> Figure 11[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137549427\">[reveal-answer q=\"fs-id1165137549427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137549427\"]\r\n<p id=\"fs-id1165137549429\">The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as<\/p>\r\n\r\n<div id=\"fs-id1165137692779\" class=\"unnumbered\" style=\"text-align: center;\">[latex]h\\left(x\\right)=f\\left(x-1\\right)+2[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135175226\">Using the formula for the square root function, we can write<\/p>\r\n\r\n<div id=\"fs-id1165135176449\" class=\"unnumbered\" style=\"text-align: center;\">[latex]h\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x-1}+2[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<h3>Analysis<\/h3>\r\n<div>Note that this transformation has changed the domain and range of the function. This new graph has domain [latex]\\left[1,\\infty \\right)[\/latex] and range [latex]\\left[2,\\infty \\right).[\/latex]<\/div>\r\n<div class=\"unnumbered\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135401680\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_01_05_03\">\r\n<div id=\"fs-id1165134187253\">\r\n<p id=\"fs-id1165137653909\">Write a formula for a transformation of the toolkit reciprocal function [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] that shifts the function\u2019s graph one unit to the right and one unit up.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137727791\">[reveal-answer q=\"fs-id1165137727791\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137727791\"]\r\n<p id=\"fs-id1165137727793\">[latex]g\\left(x\\right)=\\frac{1}{x-1}+1[\/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-id1165137600415\" class=\"bc-section section\">\r\n<h3>Graphing Functions Using Reflections about the Axes<\/h3>\r\n<p id=\"fs-id1165137772409\">Another transformation that can be applied to a function is a reflection over the <em>x<\/em>- or <em>y<\/em>-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the <em>x<\/em>-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the <em>y<\/em>-axis. The reflections are shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_013\">Figure 12<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_013\" 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\/08205651\/CNX_Precalc_Figure_01_05_012.jpg\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"325\" height=\"295\" \/> Figure 12 Vertical and horizontal reflections of a function.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137642152\">Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the <em>x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the <em>y<\/em>-axis.<\/p>\r\n\r\n<div id=\"fs-id1165137432318\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definitions<\/h3>\r\n<p id=\"fs-id1165134040633\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] is a <strong>vertical reflection<\/strong> of the function [latex]f\\left(x\\right),[\/latex] sometimes called a reflection about (or over, or through) the <em>x<\/em>-axis.<\/p>\r\n<p id=\"fs-id1165135203741\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] is a <strong>horizontal reflection<\/strong> of the function [latex]f\\left(x\\right),[\/latex] sometimes called a reflection about the <em>y<\/em>-axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137557940\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135187109\"><strong>Given a function, reflect the graph both vertically and horizontally. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137920678\" type=\"1\">\r\n \t<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the <em>x<\/em>-axis.<\/li>\r\n \t<li>Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the <em>y<\/em>-axis.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_09\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135195785\">\r\n<div id=\"fs-id1165137838801\">\r\n<h3>Example 9:\u00a0 Reflecting a Graph Horizontally and Vertically<\/h3>\r\n<p id=\"fs-id1165134351127\">Reflect the graph of [latex]s\\left(t\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{t}[\/latex] (a) vertically and (b) horizontally.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135696191\">[reveal-answer q=\"fs-id1165135696191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135696191\"]\r\n<ol id=\"fs-id1165135386517\" type=\"a\">\r\n \t<li>\r\n<p id=\"fs-id1165137455471\">Reflecting the graph vertically means that each output value will be reflected over the horizontal <em>t-<\/em>axis as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_014\">Figure 13<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_014\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"588\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205654\/CNX_Precalc_Figure_01_05_013.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"588\" height=\"267\" \/> Figure 13 Vertical reflection of the square root function[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137431214\">Because each output value is the opposite of the original output value, we can write<\/p>\r\n\r\n<div id=\"fs-id1165137425686\" class=\"unnumbered\" style=\"text-align: center;\">[latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{t}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div id=\"fs-id1165137425686\" class=\"unnumbered\" style=\"text-align: center;\"><\/div>\r\nNotice that this is an outside change, or vertical reflection, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.<\/li>\r\n \t<li>\r\n<p id=\"fs-id1165134393050\">Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_015\">Figure 14<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_015\" class=\"wp-caption alignnone\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205657\/CNX_Precalc_Figure_01_05_014.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"975\" height=\"442\" \/> Figure 14 Horizontal reflection of the square root function[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165133408855\">Because each input value is the opposite of the original input value, we can write<\/p>\r\n\r\n<div id=\"fs-id1165137470692\" class=\"unnumbered\" style=\"text-align: center;\">[latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-t}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137742575\">Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.<\/p>\r\n<p id=\"fs-id1165137664617\">Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\\left[0,\\infty \\right)[\/latex] and range [latex]\\left[0,\\infty \\right),[\/latex] the vertical reflection gives the [latex]V\\left(t\\right)[\/latex] function the range [latex]\\left(-\\infty ,\\text{ }0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,\\text{ }0\\right].[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135330596\" class=\"precalculus tryit\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"fs-id1165137437703\">\r\n<div id=\"fs-id1165137726983\">\r\n<p id=\"fs-id1165137726985\">Reflect the graph of [latex]f\\left(x\\right)=|x-1|[\/latex] (a) vertically and (b) horizontally.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134234186\">[reveal-answer q=\"fs-id1165134234186\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134234186\"]\r\n<ol id=\"fs-id1165137444428\" type=\"a\">\r\n \t<li><span id=\"fs-id1165137693977\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205700\/CNX_Precalc_Figure_01_05_015a.jpg\" alt=\"Graph of a vertically reflected absolute function.\" \/><\/span><\/li>\r\n \t<li><span id=\"fs-id1165135436637\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205703\/CNX_Precalc_Figure_01_05_015b.jpg\" alt=\"Graph of an absolute function translated one unit left.\" \/><\/span><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137696924\">\r\n<div id=\"fs-id1165137405053\">\r\n<h3>Example 10:\u00a0 Reflecting a Tabular Function Horizontally and Vertically<\/h3>\r\n<p id=\"fs-id1165137564278\">A function [latex]f\\left(x\\right)[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_05\">Table 6<\/a>. Create a table for the functions below.<\/p>\r\n\r\n<ol id=\"fs-id1165135485824\" type=\"a\">\r\n \t<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\r\n \t<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<table id=\"Table_01_05_05\" style=\"width: 355px;\" 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 6<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 27.217px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"width: 13px; text-align: center;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 27.217px;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">7<\/td>\r\n<td class=\"border\" style=\"width: 13px; text-align: center;\">11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165134199548\">[reveal-answer q=\"fs-id1165134199548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134199548\"]\r\n<ol id=\"fs-id1165134199550\" type=\"a\">\r\n \t<li>\r\n<p id=\"fs-id1165137411052\">For [latex]g\\left(x\\right),[\/latex] the negative sign outside the function indicates a vertical reflection, so the <em>x<\/em>-values stay the same and each output value will be the opposite of the original output value. See <a class=\"autogenerated-content\" href=\"#Table_01_05_06\">Table 7<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_06\" 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, g(4)=-3, g(6)=-7, and g(8)=-11.\"><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]g\\left(x\\right)[\/latex]<\/strong><\/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;\">\u20137<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u201311<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<p id=\"fs-id1165137749533\">For [latex]h\\left(x\\right),[\/latex] the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\\left(x\\right)[\/latex] values stay the same as the [latex]f\\left(x\\right)[\/latex] values. See <a class=\"autogenerated-content\" href=\"#Table_01_05_07\">Table 8<\/a>.<\/p>\r\n\r\n<table id=\"Table_01_05_07\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201ch(x)\u201d. The values of x are -2, -4, -6, and -8. So for h(-2)=1, h(-4)=3, h(-6)=7, and h(-8)=11.\"><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;\">\u22122<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u22124<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u22126<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u22128<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]h\\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<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165132937220\" class=\"precalculus tryit\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"fs-id1165137757772\">\r\n<div id=\"fs-id1165137757774\">\r\n<p id=\"fs-id1165135397310\">A function [latex]f\\left(x\\right)[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_08\">Table 9<\/a>. Create a table for the functions below.<\/p>\r\n\r\n<ol id=\"fs-id1165137401553\" type=\"a\">\r\n \t<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\r\n \t<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<table id=\"Table_01_05_08\" 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, 0, 2, and 4. So for f(-2)=5, f(0)=10, f(2)=15, and f(4)=20.\"><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;\">\u22122<\/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<td class=\"border\" style=\"text-align: center;\">4<\/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;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137553074\">[reveal-answer q=\"fs-id1165137553074\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137553074\"]\r\n<ol id=\"fs-id1165137642579\" type=\"a\">\r\n \t<li>\r\n<p id=\"fs-id1165137637578\">[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<table id=\"fs-id1165137431044\" class=\"unnumbered\" 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, 0, 2, and 4. So for f(-2)=-5, f(0)=-10, f(2)=-15, and f(4)=-20.\"><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;\">0<\/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<\/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;\">[latex]-5[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]-10[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]-15[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]-20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>\r\n<p id=\"fs-id1165137871009\">[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<table id=\"fs-id1165134042357\" class=\"unnumbered\" 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, 0, -2, and -4. So for f(-2)=5, f(0)=10, f(-2)=15, and f(-4)=-20.\"><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;\">0<\/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<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]h\\left(x\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">unknown<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_05_11\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137657438\">\r\n<div id=\"fs-id1165137432328\">\r\n<h3>Example 11:\u00a0 Applying a Learning Model Equation<\/h3>\r\n<p id=\"fs-id1165137938642\">A common model for learning has an equation similar to [latex]k\\left(t\\right)=-{2}^{-t}+1,[\/latex] where [latex]k[\/latex] is the percentage of mastery that can be achieved after [latex]t[\/latex] practice sessions. This is a transformation of the function [latex]f\\left(t\\right)={2}^{t}[\/latex] shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_017\">Figure 15<\/a>. Sketch a graph of [latex]k\\left(t\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_01_05_017\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"333\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205707\/CNX_Precalc_Figure_01_05_016.jpg\" alt=\"Graph of k(t)\" width=\"333\" height=\"302\" \/> Figure 15[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137731123\">[reveal-answer q=\"fs-id1165137731123\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137731123\"]\r\n<p id=\"fs-id1165137731125\">This equation combines three transformations into one equation.<\/p>\r\n\r\n<ul id=\"fs-id1165137442908\">\r\n \t<li>A horizontal reflection: [latex]f\\left(-t\\right)={2}^{-t}[\/latex]<\/li>\r\n \t<li>followed by a vertical reflection: [latex]-f\\left(-t\\right)=-{2}^{-t}[\/latex]<\/li>\r\n \t<li>and finally vertical shift: [latex]-f\\left(-t\\right)+1=-{2}^{-t}+1[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137698491\">We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).<\/p>\r\n\r\n<ol id=\"fs-id1165135319410\" type=\"1\">\r\n \t<li>First, we apply a horizontal reflection to (0,1) and (1,2) by negating the input value to get (0, 1) and (-1, 2) respectively.<\/li>\r\n \t<li>Then, we apply a vertical reflection by negating the second coordinate to get (0, \u22121) and (-1, -2) respectively.<\/li>\r\n \t<li>Finally, we apply a vertical shift by adding 1 giving the points (0, 0) and (-1, -1) on the function [latex]k\\left(t\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165135176725\">This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.<\/p>\r\n<p id=\"fs-id1165137723144\">In <a class=\"autogenerated-content\" href=\"#Figure_01_05_018\">Figure 16<\/a>, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.<\/p>\r\n\r\n<div id=\"Figure_01_05_018\" class=\"wp-caption alignnone\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"707\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205710\/CNX_Precalc_Figure_01_05_017abc.jpg\" alt=\"Graphs of all the transformations.\" width=\"707\" height=\"316\" \/> Figure 16[\/caption]\r\n\r\n<\/div>\r\n<h3>Analysis<\/h3>\r\nAs a model for learning, this function would be limited to a domain of [latex]t\\ge 0,[\/latex] with corresponding range [latex]\\left[0,1\\right).[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137472858\" class=\"precalculus tryit\">\r\n<h3>Try it #7<\/h3>\r\n<div id=\"ti_01_05_04\">\r\n<div id=\"fs-id1165135548260\">\r\n<p id=\"fs-id1165137827376\">Given the toolkit function [latex]f\\left(x\\right)={x}^{2},[\/latex] graph [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)=f\\left(-x\\right).[\/latex] Take note of any surprising behavior for these functions.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137894550\">[reveal-answer q=\"fs-id1165137894550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137894550\"]<span id=\"fs-id1165137778960\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205715\/CNX_Precalc_Figure_01_05_020.jpg\" alt=\"Graph of x^2 and its reflections.\" width=\"336\" height=\"302\" \/><\/span>\r\n<p id=\"fs-id1165135255910\">Notice: [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] looks the same as [latex]f\\left(x\\right)[\/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-id1165135182974\" class=\"bc-section section\">\r\n<h3>Determining Even and Odd Functions<\/h3>\r\n<p id=\"fs-id1165135532474\">Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\\left(x\\right)={x}^{2}[\/latex] or [latex]f\\left(x\\right)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the <em>y<\/em>-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong><\/p>\r\n<p id=\"fs-id1165137939530\">If the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph, as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_022\">Figure 17<\/a>.<\/p>\r\n\r\n<div id=\"Figure_01_05_022\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205720\/CNX_Precalc_Figure_01_05_021abc.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/> Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137406881\">We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\r\n<p id=\"fs-id1165134573214\">Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137619398\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137407995\">A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135424702\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135552902\">The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex]axis.[latex]\\\\[\/latex]<\/p>\r\n<p id=\"fs-id1165137501973\">A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137762060\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex] or equivalently [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135503845\">The graph of an odd function is symmetric about the origin.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135503849\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165133353947\"><strong>Given the formula for a function, determine if the function is even, odd, or neither. <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137552979\" type=\"1\">\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex] If it does, it is even.<\/li>\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right).[\/latex] If it does, it is odd.\u00a0 Note that you can also show the equivalent statement [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\r\n \t<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_05_12\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137415536\">\r\n<div id=\"fs-id1165137415539\">\r\n<h3>Example 12:\u00a0 Determining whether a Function Is Even, Odd, or Neither<\/h3>\r\n<p id=\"fs-id1165135252115\">Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137784966\">[reveal-answer q=\"fs-id1165137784966\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137784966\"]\r\n<p id=\"fs-id1165137784968\">Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\r\n\r\n<div id=\"fs-id1165137401549\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165137771042\">This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\r\n\r\n<div id=\"fs-id1165137740781\" class=\"unnumbered\" style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135667851\">Because [latex]-f\\left(-x\\right)=f\\left(x\\right),[\/latex] this is an odd function.<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1165133050510\">Consider the graph of [latex]f[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_05_039\">Figure 18<\/a>. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f,[\/latex] and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.<\/p>\r\n\r\n<div id=\"Figure_01_05_039\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"383\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205724\/CNX_Precalc_Figure_01_05_039.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"383\" height=\"256\" \/> Figure 18[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137480929\">\r\n<div id=\"Figure_01_05_039\" class=\"medium\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135400987\" class=\"precalculus tryit\">\r\n<h3>Try it #8<\/h3>\r\n<div id=\"ti_01_05_05\">\r\n<div id=\"fs-id1165137897939\">\r\n<p id=\"fs-id1165137897941\">Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137757764\">[reveal-answer q=\"fs-id1165137757764\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137757764\"]\r\n<p id=\"fs-id1165137757766\">even<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137654768\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137793506\" class=\"bc-section section\">\r\n<div id=\"Figure_01_05_025\" class=\"small\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137676302\" class=\"bc-section section\">\r\n<div id=\"fs-id1165135497140\" class=\"precalculus media\">\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]\\left(0\\lt a\\lt 1\\right)[\/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]\\left(0\\lt b\\lt 1\\right)[\/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 shifted vertically by adding a constant to the output.<\/li>\r\n \t<li>A function can be shifted horizontally by adding a constant to the input.<\/li>\r\n \t<li>Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.<\/li>\r\n \t<li>Vertical and horizontal shifts are often combined.<\/li>\r\n \t<li>A vertical reflection reflects a graph about the [latex]x\\text{-}[\/latex]axis. A graph can be reflected vertically by multiplying the output by \u20131.<\/li>\r\n \t<li>A horizontal reflection reflects a graph about the [latex]y\\text{-}[\/latex]axis. A graph can be reflected horizontally by multiplying the input by \u20131.<\/li>\r\n \t<li>A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.<\/li>\r\n \t<li>A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.<\/li>\r\n \t<li>A function presented as an equation can be reflected by applying transformations one at a time.<\/li>\r\n \t<li>Even functions are symmetric about the [latex]y\\text{-}[\/latex]axis, whereas odd functions are symmetric about the origin.<\/li>\r\n \t<li>Even functions satisfy the condition [latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex]<\/li>\r\n \t<li>Odd functions satisfy the condition [latex]f\\left(x\\right)=-f\\left(-x\\right).[\/latex]<\/li>\r\n \t<li>A function can be odd, even, or neither.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137448239\">\r\n \t<dt>even function<\/dt>\r\n \t<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right),[\/latex] and is symmetric about the [latex]y\\text{-}[\/latex]axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133242964\"><\/dl>\r\n<dl id=\"fs-id1165135440170\">\r\n \t<dt>horizontal reflection<\/dt>\r\n \t<dd id=\"fs-id1165137602051\">a transformation that reflects a function\u2019s graph across the <em>y<\/em>-axis by multiplying the input by [latex]-1[\/latex]<\/dd>\r\n<\/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<dl id=\"fs-id1165137862450\">\r\n \t<dd id=\"fs-id1165132971698\"><\/dd>\r\n<\/dl>\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>Describe and apply vertical and horizontal shifts and reflections of graphs, tables and function formulas.<\/li>\n<li>Use function notation to express horizontal and vertical shifts and reflections of functions.<\/li>\n<li>Determine whether a function is even, odd or neither from its algebraic formula or graph.<\/li>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_01_05_001\" class=\"medium\">\n<div style=\"width: 498px\" 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\/08205611\/CNX_Precalc_Figure_01_05_038n.jpg\" alt=\"Figure_01_05_038\" width=\"488\" height=\"325\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1 (credit: &#8220;Misko&#8221;\/Flickr)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137742090\">We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.<\/p>\n<div id=\"fs-id1165137827988\" class=\"bc-section section\">\n<h3>Graphing Functions Using Vertical and Horizontal Shifts<\/h3>\n<p id=\"fs-id1165137654715\">Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.<\/p>\n<div id=\"fs-id1165137535664\" class=\"bc-section section\">\n<h4>Identifying Vertical Shifts<\/h4>\n<p id=\"fs-id1165135503932\">One simple kind of <span class=\"no-emphasis\">transformation<\/span> involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a <strong>vertical shift<\/strong>, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function [latex]g\\left(x\\right)=f\\left(x\\right)+k,[\/latex] the function [latex]f\\left(x\\right)[\/latex] is shifted vertically [latex]k[\/latex] units. See <a class=\"autogenerated-content\" href=\"#Figure_01_05_002\">Figure 2<\/a> for an example.<\/p>\n<div id=\"Figure_01_05_002\" 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\/02\/08205614\/CNX_Precalc_Figure_01_05_001.jpg\" alt=\"Figure_01_05_001\" width=\"487\" height=\"292\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2 Vertical shift by [latex]k=1[\/latex] of the cube root function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137439125\">To help you visualize the concept of a vertical shift, consider that [latex]y=f\\left(x\\right).[\/latex] Therefore, [latex]f\\left(x\\right)+k[\/latex] is equivalent to [latex]y+k.[\/latex] Every unit of [latex]y[\/latex] is replaced by [latex]y+k,[\/latex] so the [latex]y\\text{-}[\/latex]value increases or decreases depending on the value of [latex]k.[\/latex] The result is a shift upward or downward.<\/p>\n<div id=\"fs-id1165137737394\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p>Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=f\\left(x\\right)+k,[\/latex] where\u00a0 [latex]k[\/latex] is a constant, is a <strong>vertical shift<\/strong> of the function [latex]f\\left(x\\right).[\/latex] All the output values change by [latex]k[\/latex] units. If [latex]k[\/latex] is positive, the graph will shift up. If [latex]k[\/latex] is negative, the graph will shift down.<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137701104\">\n<div id=\"fs-id1165135547346\">\n<h3>Example 1:\u00a0 Adding a Constant to a Function<\/h3>\n<p id=\"fs-id1165137676255\">To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. <a class=\"autogenerated-content\" href=\"#Figure_01_05_003\">Figure 3<\/a> shows the area of open vents [latex]V[\/latex] (in square feet) throughout the day in hours after midnight, [latex]t.[\/latex] During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.<\/p>\n<div id=\"Figure_01_05_003\" class=\"small\">\n<div style=\"width: 452px\" 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\/08205616\/CNX_Precalc_Figure_01_05_002.jpg\" alt=\"Figure_01_05_002\" width=\"442\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135504997\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135504997\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135504997\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137667304\">We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_004\">Figure 4<\/a>.<\/p>\n<div id=\"Figure_01_05_004\" class=\"wp-caption aligncenter\" style=\"width: 500px;\">\n<div style=\"width: 452px\" 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\/08205619\/CNX_Precalc_Figure_01_05_003a.jpg\" alt=\"Figure_01_05_003a\" width=\"442\" height=\"299\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137662763\">Notice that in <a class=\"autogenerated-content\" href=\"#Figure_01_05_004\">Figure 4<\/a>, for each input value, the output value has increased by 20, so if we call the new function [latex]S\\left(t\\right),[\/latex] we could write<\/p>\n<div id=\"fs-id1165137772392\" class=\"unnumbered\" style=\"text-align: center;\">[latex]S\\left(t\\right)=V\\left(t\\right)+20[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165134164955\">This notation tells us that, any value of [latex]S\\left(t\\right)[\/latex] can be found by evaluating the function [latex]V[\/latex] at the same input and then adding 20 to the result. This defines [latex]S[\/latex] as a transformation of the function [latex]V,[\/latex] in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See <a class=\"autogenerated-content\" href=\"#Table_01_05_018\">Table 1<\/a>.<\/p>\n<table id=\"Table_01_05_018\" summary=\"Three rows and seven columns. The first row is labeled, \u201ct\u201d, the second is labeled, \u201cV(t)\u201d, and the third is labeled, \u201cS(t)\u201d. The values of t are 0, 8, 10, 17, 19, and 24. So for V(0)=0, V(8)=0, V(10)=220, V(17)=220, V(19)=0, and V(24)=0. For S(0)=20, S(8)=20, S(10)=240, S(17)=240, S(19)=20, and S(24)=20.\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]t[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">17<\/td>\n<td class=\"border\" style=\"text-align: center;\">19<\/td>\n<td class=\"border\" style=\"text-align: center;\">24<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]V\\left(t\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">220<\/td>\n<td class=\"border\" style=\"text-align: center;\">220<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]S\\left(t\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<td class=\"border\" style=\"text-align: center;\">240<\/td>\n<td class=\"border\" style=\"text-align: center;\">240<\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135369225\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134371214\"><strong>Given a tabular function, create a new row to represent a vertical shift.<\/strong><\/p>\n<ol id=\"fs-id1165135530387\" type=\"1\">\n<li>Identify the output row or column.<\/li>\n<li>Determine the <span class=\"no-emphasis\">magnitude<\/span> of the shift.<\/li>\n<li>Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135581153\">\n<div id=\"fs-id1165135443766\">\n<h3>Example 2:\u00a0 Shifting a Tabular Function Vertically<\/h3>\n<p id=\"fs-id1165137692074\">A function [latex]f\\left(x\\right)[\/latex] is given in <a class=\"autogenerated-content\" href=\"#Table_01_05_01\">Table 2<\/a>. Create a table for the function [latex]g\\left(x\\right)=f\\left(x\\right)-3.[\/latex]<\/p>\n<table id=\"Table_01_05_01\" style=\"height: 24px;\" 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 2<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 30.656px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"height: 12px; width: 30.6563px; text-align: center;\">8<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 30.656px; text-align: center;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"height: 12px; width: 18.6563px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"height: 12px; width: 30.6563px; text-align: center;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137574326\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137574326\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137574326\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135241392\">The formula [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex] tells us that we can find the output values of [latex]g[\/latex] by subtracting 3 from the output values of [latex]f.[\/latex] For example:<\/p>\n<div id=\"fs-id1165137662784\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(2\\right)&=1\\hfill && \\text{Given}\\hfill \\\\ g\\left(x\\right)&=f\\left(x\\right)-3\\hfill && \\text{Given transformation}\\hfill \\\\ g\\left(2\\right)&=f\\left(2\\right)-3\\hfill && \\hfill \\\\ \\text{ }&=1-3\\hfill && \\hfill \\\\ \\text{ }&=-2\\hfill && \\hfill \\end{align*}[\/latex]<\/div>\n<div><\/div>\n<p>Subtracting 3 from each [latex]f\\left(x\\right)[\/latex] value, we can complete a table of values for [latex]g\\left(x\\right)[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#Table_01_05_02\">Table 3<\/a>.<\/p>\n<table id=\"Table_01_05_02\" summary=\"Three rows and five columns. The first row is labeled, \u201cx\u201d, the second is labeled, \u201cf(x)\u201d, and the third is labeled, \u201cg(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. For g(2)=-2, g(4)=0, g(6)=4, and g(8)=8.\">\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\" style=\"text-align: center;\"><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\" style=\"text-align: center;\"><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<tr>\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">\u22122<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Analysis<\/h3>\n<p>As with the earlier vertical shift, notice the input values stay the same and only the output values change.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692134\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_01_05_01\">\n<div id=\"fs-id1165137834794\">\n<p id=\"fs-id1165137767439\">The function [latex]h\\left(t\\right)=-4.9{t}^{2}+30t[\/latex] gives the height [latex]h[\/latex] of a ball (in meters) thrown upward from the ground after [latex]t[\/latex] seconds. Suppose the ball was instead thrown from the top of a 10 meter building. Relate this new height function [latex]b\\left(t\\right)[\/latex] to [latex]h\\left(t\\right),[\/latex] and then find a formula for [latex]b\\left(t\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137481965\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137481965\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137481965\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165137680332\">[latex]b\\left(t\\right)=h\\left(t\\right)+10=-4.9{t}^{2}+30t+10[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137597159\" class=\"bc-section section\">\n<h4>Identifying Horizontal Shifts<\/h4>\n<p id=\"fs-id1165137404493\">We just saw that the vertical shift is a change to the output or outside of the function. We will now look at how changes to input or the inside of the function change its graph and meaning.<\/p>\n<p>A change to the <strong>input<\/strong> results in a movement of the graph of an original function left or right in what is known as a <strong>horizontal shift<\/strong>. We will be creating a new function [latex]g\\left(x\\right)[\/latex] which is based on an original function [latex]f\\left(x\\right)[\/latex] using the following function notation:\u00a0 [latex]g\\left(x\\right)=f\\left(x-h\\right)[\/latex] where <strong>[latex]h[\/latex]<\/strong> is a constant.<\/p>\n<div id=\"Figure_01_05_005\" class=\"small\"><\/div>\n<p id=\"eip-884\">For example, if [latex]f\\left(x\\right)={x}^{2},[\/latex] then we can create a function in terms of [latex]f[\/latex] by writing [latex]g\\left(x\\right)=f\\left(x-2\\right),[\/latex] which is equivalent to [latex]g\\left(x\\right)={\\left(x-2\\right)}^{2}.[\/latex]\u00a0 Think about what happens carefully.<\/p>\n<p>We can read the statement [latex]g\\left(x\\right)=f\\left(x-2\\right)[\/latex] as saying that the output for [latex]g[\/latex] at [latex]x[\/latex] will be the same as the output we get for the original function [latex]f[\/latex] evaluated two units earlier.\u00a0 Perhaps an easier way to see this is to recognize that if [latex]x[\/latex] is 5, then [latex]g\\left(5\\right)=f\\left(5-2\\right)=f\\left(3\\right).[\/latex]\u00a0 We get the same output for [latex]g[\/latex] at the input of 5 as we did for the function [latex]f[\/latex] for an input two untis earlier.\u00a0 Therefore, in order to produce the graph of [latex]g[\/latex], we will shift our original function [latex]f\\left(x\\right)[\/latex] to the right by two units.<\/p>\n<p>What if [latex]h[\/latex] is negative?\u00a0 Let&#8217;s consider the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}.[\/latex]\u00a0 If we let [latex]h=-1,[\/latex] then we can consider a new function [latex]m\\left(x\\right)=f\\left(x-\\left(-1\\right)\\right)=f\\left(x+1\\right).[\/latex]\u00a0 This is equivalent to [latex]m\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x+1}.[\/latex] Notice again that it is our input which has changed.\u00a0 We can read this statement as saying that the output for [latex]m[\/latex] evaluated at [latex]x[\/latex] will be the same as the output we get for the original function [latex]f[\/latex] evaluated one unit later.\u00a0 Therefore, [latex]m\\left(5\\right)=f\\left(5+1\\right)=f\\left(6\\right).[\/latex]\u00a0 In order to produce the graph of [latex]m,[\/latex] we will shift the original function [latex]f\\left(x\\right)[\/latex] to the left by one unit.\u00a0 Consider the picture shown in\u00a0<a class=\"autogenerated-content\" href=\"#Figure_01_05_005\">Figure 5<\/a>.<\/p>\n<div id=\"Figure_01_05_005\" 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\/02\/08205623\/CNX_Precalc_Figure_01_05_004.jpg\" alt=\"Figure_01_05_004\" width=\"487\" height=\"288\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5 Horizontal shift of the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}.[\/latex] Note that [latex]f\\left(x+1\\right)[\/latex] shifts the graph to the left by one unit.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436470\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p>Given a function [latex]f,[\/latex] a new function [latex]g\\left(x\\right)=f\\left(x-h\\right),[\/latex] where [latex]h[\/latex] is a constant, is a <strong>horizontal shift<\/strong> of the function [latex]f.[\/latex] If [latex]h[\/latex] is positive, the graph will shift right. If [latex]h[\/latex] is negative, the graph will shift left.<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135241148\">\n<div id=\"fs-id1165134148356\">\n<h3>Example 3:\u00a0 Adding a Constant to an Input<\/h3>\n<p id=\"fs-id1165137658363\">Returning to our building airflow example from <a class=\"autogenerated-content\" href=\"#Figure_01_05_003\">Figure 3<\/a>, suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.<\/p>\n<\/div>\n<div id=\"fs-id1165137423751\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137423751\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137423751\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137423753\">We can set [latex]V\\left(t\\right)[\/latex] to be the original program and [latex]F\\left(t\\right)[\/latex] to be the revised program.<\/p>\n<div id=\"fs-id1165135195278\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\\\[\/latex][latex]\\begin{align*}V\\left(t\\right)&=\\text{ the original venting plan}\\\\ \\text{F}\\left(t\\right)&=\\text{ starting 2 hrs sooner}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165137714274\">In the new graph, at each time, the airflow is the same as the original function [latex]V[\/latex] was 2 hours later. For example, in the original function [latex]V,[\/latex] the airflow starts to change at 8 a.m., whereas for the function [latex]F,[\/latex] the airflow starts to change at 6 a.m. The comparable function values are [latex]F\\left(6\\right)=V\\left(8\\right).[\/latex] See <a class=\"autogenerated-content\" href=\"#Figure_01_05_006\">Figure 6<\/a>. Notice also that the vents first opened to [latex]220{\\text{ ft}}^{2}[\/latex] at 10 a.m. under the original plan, while under the new plan the vents reach [latex]220{\\text{ ft}}^{\\text{2}}[\/latex] at 8 a.m., so [latex]F\\left(8\\right)=V\\left(10\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137411313\">In both cases, we see that, because [latex]F\\left(t\\right)[\/latex] starts 2 hours sooner, [latex]h=-2.[\/latex] That means that the same output values are reached when [latex]F\\left(t\\right)=V\\left(t-\\left(-2\\right)\\right)=V\\left(t+2\\right).[\/latex]<\/p>\n<div id=\"Figure_01_05_006\" class=\"small\">\n<div style=\"width: 454px\" 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\/08205625\/CNX_Precalc_Figure_01_05_005a.jpg\" alt=\"Figure_01_05_005a\" width=\"444\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<h3>Analysis<\/h3>\n<p id=\"fs-id1165137757835\">Note that [latex]V\\left(t+2\\right)[\/latex] has the effect of shifting the graph to the <em>left<\/em>.<\/p>\n<p id=\"fs-id1165137473462\">Horizontal changes or \u201cinside changes\u201d affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function [latex]F\\left(t\\right)[\/latex] uses the same outputs as [latex]V\\left(t\\right),[\/latex] but matches those outputs to inputs 2 hours earlier than those of [latex]V\\left(t\\right).[\/latex] Said another way, we must add 2 hours to the input of [latex]V[\/latex] to find the corresponding output for[latex]F:F\\left(t\\right)=V\\left(t+2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137644724\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137667199\"><strong>Given a tabular function, create a new row to represent a horizontal shift.<\/strong><\/p>\n<ol id=\"fs-id1165137652957\" type=\"1\">\n<li>Identify the input row or column.<\/li>\n<li>Determine the magnitude of the shift.<\/li>\n<li>Add the shift to the value in each input cell.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137831912\">\n<div id=\"fs-id1165137831914\">\n<h3>Example 4:\u00a0 Shifting a Tabular Function Horizontally<\/h3>\n<p id=\"fs-id1165137761563\">A function [latex]f\\left(x\\right)[\/latex] is given in <a class=\"autogenerated-content\" href=\"#Table_01_05_03\">Table 4<\/a>. Create a table for the function [latex]g\\left(x\\right)=f\\left(x-3\\right).[\/latex]<\/p>\n<table id=\"Table_01_05_03\" style=\"width: 596px;\" 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\" style=\"text-align: center; width: 221px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center; width: 375px;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center; width: 221px;\">2<\/td>\n<td class=\"border\" style=\"text-align: center; width: 375px;\">1<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center; width: 221px;\">4<\/td>\n<td class=\"border\" style=\"text-align: center; width: 375px;\">3<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center; width: 221px;\">6<\/td>\n<td class=\"border\" style=\"text-align: center; width: 375px;\">7<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center; width: 221px;\">8<\/td>\n<td class=\"border\" style=\"text-align: center; width: 375px;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165134401703\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134401703\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134401703\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137735632\">The formula [latex]g\\left(x\\right)=f\\left(x-3\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are the same as the output value of [latex]f[\/latex] when the input value is 3 less. For example, we know that [latex]f\\left(2\\right)=1.[\/latex] To get the same output from the function [latex]g,[\/latex] we will need an input value that is 3 <em>larger<\/em>. We input a value that is 3 larger for [latex]g\\left(x\\right)[\/latex] because the function takes 3 away before evaluating the function [latex]f.[\/latex]<\/p>\n<div id=\"fs-id1165137645517\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}g\\left(5\\right)&=f\\left(5-3\\right)\\hfill \\\\ \\text{ }&=f\\left(2\\right)\\hfill \\\\ \\text{ }&=1\\hfill \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135531624\">We continue with the other values to create <a class=\"autogenerated-content\" href=\"#Table_01_05_04\">Table 5<\/a>. In our table for [latex]g\\left(x\\right)[\/latex], we need to increase each input value for [latex]f[\/latex] by 3.<\/p>\n<table id=\"Table_01_05_04\" style=\"height: 48px; width: 493px;\" summary=\"Three rows and five columns. The first row is labeled, \u201cx\u201d, the second is labeled, \u201cf(x)\u201d, and the third is labeled, \u201cg(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. For g(2)=1, g(4)=3, g(6)=7, and g(8)=11.\">\n<caption>Table 5<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\"><strong>[latex]g\\left(x\\right)=f\\left(x-3\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\">5<\/td>\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\">[latex]g\\left(5\\right)=f\\left(5-3\\right)=f\\left(2\\right)=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\">[latex]g\\left(7\\right)=f\\left(7-3\\right)=f\\left(4\\right)=3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td class=\"border\" style=\"height: 12px; width: 193px; text-align: center;\">9<\/td>\n<td class=\"border\" style=\"height: 12px; width: 300px; text-align: center;\">[latex]g\\left(9\\right)=f\\left(9-3\\right)=f\\left(6\\right)=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 193px; text-align: center;\">11<\/td>\n<td class=\"border\" style=\"width: 300px; text-align: center;\">[latex]g\\left(11\\right)=f\\left(11-3\\right)=f\\left(8\\right)=11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137678273\">The result is that the function [latex]g\\left(x\\right)[\/latex] has been shifted to the right by 3. Notice the output values for [latex]g\\left(x\\right)[\/latex] remain the same as the output values for [latex]f\\left(x\\right),[\/latex] but the corresponding input values, [latex]x,[\/latex] have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.<\/p>\n<h3>Analysis<\/h3>\n<p id=\"fs-id1165135438856\"><a class=\"autogenerated-content\" href=\"#Figure_01_05_007\">Figure 7<\/a> represents both of the functions. We can see the horizontal shift in each point.<\/p>\n<div id=\"Figure_01_05_007\" class=\"small\">\n<div style=\"width: 292px\" 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\/08205629\/CNX_Precalc_Figure_01_05_006.jpg\" alt=\"Graph of the points from the previous table for f(x) and g(x)=f(x-3).\" width=\"282\" height=\"318\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135196829\">\n<div id=\"Figure_01_05_007\" class=\"small\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_05\" class=\"textbox examples\">\n<div id=\"fs-id1165135528978\">\n<div id=\"fs-id1165135528980\">\n<h3>Example 5:\u00a0 Identifying a Horizontal Shift of a Toolkit Function<\/h3>\n<p id=\"fs-id1165137582443\"><a class=\"autogenerated-content\" href=\"#Figure_01_05_008\">Figure 8<\/a> represents a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{2}.[\/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_008\" class=\"small\">\n<div style=\"width: 446px\" 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\/08205633\/CNX_Precalc_Figure_01_05_007.jpg\" alt=\"Graph of a parabola.\" width=\"436\" height=\"294\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173899\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135173899\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135173899\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135173902\">Notice that the graph is identical in shape to the [latex]f\\left(x\\right)={x}^{2}[\/latex] function, but the <em>x-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so<\/p>\n<div id=\"fs-id1165133349274\" class=\"unnumbered\" style=\"text-align: center;\">[latex]g\\left(x\\right)=f\\left(x-2\\right)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137561935\">Notice how we must input the value [latex]x=2[\/latex] to get the output value [latex]y=0;[\/latex] the <em>x<\/em>-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the [latex]f\\left(x\\right)[\/latex] function to write a formula for [latex]g\\left(x\\right)[\/latex] by evaluating [latex]f\\left(x-2\\right).[\/latex]<\/p>\n<div id=\"fs-id1165137444147\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(x\\right)&={x}^{2}\\hfill \\\\ g\\left(x\\right)&=f\\left(x-2\\right)\\hfill \\\\ g\\left(x\\right)&=f\\left(x-2\\right)={\\left(x-2\\right)}^{2}\\hfill \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<h3>Analysis<\/h3>\n<div>To determine whether the shift is [latex]+2[\/latex] or [latex]-2[\/latex], consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function, [latex]f\\left(0\\right)=0.[\/latex] In our shifted function, [latex]g\\left(2\\right)=0.[\/latex] To obtain the output value of 0 from the function [latex]f,[\/latex] we need to decide whether a plus or a minus sign will work to satisfy [latex]g\\left(2\\right)=f\\left(x-2\\right)=f\\left(0\\right)=0.[\/latex] For this to work, we will need to <em>subtract<\/em> 2 units from our input values.<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137437869\">\n<div id=\"fs-id1165137786325\">\n<h3>Example 6:\u00a0 Interpreting Horizontal versus Vertical Shifts<\/h3>\n<p id=\"fs-id1165137574172\">The function [latex]G\\left(m\\right)[\/latex] gives the number of gallons of gas required to drive [latex]m[\/latex] miles. Interpret [latex]G\\left(m\\right)+10[\/latex] and [latex]G\\left(m+10\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135690669\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135690669\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135690669\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137921579\">[latex]G\\left(m\\right)+10[\/latex] can be interpreted as adding 10 to the output, gallons. This is the gas required to drive [latex]m[\/latex] miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.<\/p>\n<p id=\"fs-id1165137434382\">[latex]G\\left(m+10\\right)[\/latex] can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than [latex]m[\/latex] miles. The graph would indicate a horizontal shift.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137781562\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_01_05_02\">\n<div id=\"fs-id1165137470341\">\n<p id=\"fs-id1165137557596\">Given the function [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x},[\/latex] graph the original function [latex]f\\left(x\\right)[\/latex] and the transformation [latex]g\\left(x\\right)=f\\left(x+2\\right)[\/latex] on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?<\/p>\n<\/div>\n<div id=\"fs-id1165135548998\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135548998\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135548998\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135548999\">The graphs of [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex] are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.<\/p>\n<p><span id=\"fs-id1165137761611\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205637\/CNX_Precalc_Figure_01_05_008.jpg\" alt=\"Graph of a square root function and a horizontally shift square foot function.\" width=\"431\" height=\"255\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135250592\" class=\"bc-section section\">\n<h4>Combining Vertical and Horizontal Shifts<\/h4>\n<p id=\"fs-id1165137676099\">Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output, [latex]y\\text{-}[\/latex] axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input, [latex]x\\text{-}[\/latex] axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down <em>and<\/em> right or left.<\/p>\n<div id=\"fs-id1165137628099\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137668111\"><strong>Given a function and both a vertical and a horizontal shift, sketch the graph.<\/strong><\/p>\n<ol id=\"fs-id1165137558226\" type=\"1\">\n<li>Identify the vertical and horizontal shifts from the formula.<\/li>\n<li>The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.<\/li>\n<li>The horizontal shift results from a constant subtracted from the input. Move the graph right for a positive constant and left for a negative constant.<\/li>\n<li>Apply the shifts to the graph in either order.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137874554\">\n<div id=\"fs-id1165137874556\">\n<h3>Example 7:\u00a0 Graphing Combined Vertical and Horizontal Shifts<\/h3>\n<p id=\"fs-id1165135168426\">Given [latex]f\\left(x\\right)=|x|,[\/latex] sketch a graph of [latex]h\\left(x\\right)=f\\left(x+1\\right)-3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137841697\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137841697\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137841697\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137841699\">The function [latex]f[\/latex] is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of [latex]h[\/latex] has transformed [latex]f[\/latex] in two ways: [latex]f\\left(x+1\\right)[\/latex] is a change on the inside of the function, giving a horizontal shift left by 1 since [latex]h=-1[\/latex], and the subtraction by 3 in [latex]f\\left(x+1\\right)-3[\/latex] is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in <a class=\"autogenerated-content\" href=\"#Figure_01_05_010a\">Figure 9<\/a>.<\/p>\n<p id=\"fs-id1165133276230\">Let us follow one point of the graph of [latex]f\\left(x\\right)=|x|.[\/latex]<\/p>\n<ul id=\"eip-id1165135605357\">\n<li>The point [latex]\\left(0,0\\right)[\/latex]is transformed first by shifting left 1 unit: [latex]\\left(0,0\\right)\\to \\left(-1,0\\right)[\/latex]<\/li>\n<li>The point [latex]\\left(-1,0\\right)[\/latex]is transformed next by shifting down 3 units: [latex]\\left(-1,0\\right)\\to \\left(-1,-3\\right)[\/latex]<\/li>\n<\/ul>\n<div id=\"Figure_01_05_010a\" class=\"small\">\n<div style=\"width: 374px\" 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\/08205640\/CNX_Precalc_Figure_01_05_009a.jpg\" alt=\"Graph of an absolute function, y=|x|, and how it was transformed to y=|x+1|-3.\" width=\"364\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137705975\"><a class=\"autogenerated-content\" href=\"#Figure_01_05_010b\">Figure 10<\/a> shows the graph of [latex]h.[\/latex]<\/p>\n<div id=\"Figure_01_05_010b\" class=\"small\">\n<div style=\"width: 373px\" 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\/08205643\/CNX_Precalc_Figure_01_05_009b.jpg\" alt=\"The final function y=|x+1|-3.\" width=\"363\" height=\"299\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192765\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"fs-id1165137400636\">\n<div id=\"fs-id1165137400639\">\n<p id=\"fs-id1165135479104\">Given[latex]f\\left(x\\right)=|x|,[\/latex] sketch a graph of [latex]h\\left(x\\right)=f\\left(x-2\\right)+4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137422205\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137422205\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137422205\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165137557535\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205646\/CNX_Precalc_Figure_01_05_010.jpg\" alt=\"Graph of h(x)=|x-2|+4.\" width=\"362\" height=\"299\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137452719\">\n<div id=\"fs-id1165137452721\">\n<h3>Example 8:\u00a0 Identifying Combined Vertical and Horizontal Shifts<\/h3>\n<p id=\"fs-id1165137731917\">Write a formula for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_012\">Figure 11<\/a>, which is a transformation of the toolkit square root function.<\/p>\n<div id=\"Figure_01_05_012\" class=\"small\">\n<div style=\"width: 480px\" 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\/08205648\/CNX_Precalc_Figure_01_05_011.jpg\" alt=\"Graph of a square root function transposed right one unit and up 2.\" width=\"470\" height=\"282\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 11<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137549427\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137549427\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137549427\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137549429\">The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as<\/p>\n<div id=\"fs-id1165137692779\" class=\"unnumbered\" style=\"text-align: center;\">[latex]h\\left(x\\right)=f\\left(x-1\\right)+2[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135175226\">Using the formula for the square root function, we can write<\/p>\n<div id=\"fs-id1165135176449\" class=\"unnumbered\" style=\"text-align: center;\">[latex]h\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x-1}+2[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<h3>Analysis<\/h3>\n<div>Note that this transformation has changed the domain and range of the function. This new graph has domain [latex]\\left[1,\\infty \\right)[\/latex] and range [latex]\\left[2,\\infty \\right).[\/latex]<\/div>\n<div class=\"unnumbered\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135401680\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_01_05_03\">\n<div id=\"fs-id1165134187253\">\n<p id=\"fs-id1165137653909\">Write a formula for a transformation of the toolkit reciprocal function [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] that shifts the function\u2019s graph one unit to the right and one unit up.<\/p>\n<\/div>\n<div id=\"fs-id1165137727791\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137727791\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137727791\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137727793\">[latex]g\\left(x\\right)=\\frac{1}{x-1}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600415\" class=\"bc-section section\">\n<h3>Graphing Functions Using Reflections about the Axes<\/h3>\n<p id=\"fs-id1165137772409\">Another transformation that can be applied to a function is a reflection over the <em>x<\/em>&#8211; or <em>y<\/em>-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the <em>x<\/em>-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the <em>y<\/em>-axis. The reflections are shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_013\">Figure 12<\/a>.<\/p>\n<div id=\"Figure_01_05_013\" 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\/08205651\/CNX_Precalc_Figure_01_05_012.jpg\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"325\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 12 Vertical and horizontal reflections of a function.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137642152\">Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the <em>x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the <em>y<\/em>-axis.<\/p>\n<div id=\"fs-id1165137432318\">\n<div class=\"textbox definitions\">\n<h3>Definitions<\/h3>\n<p id=\"fs-id1165134040633\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] is a <strong>vertical reflection<\/strong> of the function [latex]f\\left(x\\right),[\/latex] sometimes called a reflection about (or over, or through) the <em>x<\/em>-axis.<\/p>\n<p id=\"fs-id1165135203741\">Given a function [latex]f\\left(x\\right),[\/latex] a new function [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] is a <strong>horizontal reflection<\/strong> of the function [latex]f\\left(x\\right),[\/latex] sometimes called a reflection about the <em>y<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137557940\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135187109\"><strong>Given a function, reflect the graph both vertically and horizontally. <\/strong><\/p>\n<ol id=\"fs-id1165137920678\" type=\"1\">\n<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the <em>x<\/em>-axis.<\/li>\n<li>Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the <em>y<\/em>-axis.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135195785\">\n<div id=\"fs-id1165137838801\">\n<h3>Example 9:\u00a0 Reflecting a Graph Horizontally and Vertically<\/h3>\n<p id=\"fs-id1165134351127\">Reflect the graph of [latex]s\\left(t\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{t}[\/latex] (a) vertically and (b) horizontally.<\/p>\n<\/div>\n<div id=\"fs-id1165135696191\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135696191\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135696191\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165135386517\" type=\"a\">\n<li>\n<p id=\"fs-id1165137455471\">Reflecting the graph vertically means that each output value will be reflected over the horizontal <em>t-<\/em>axis as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_014\">Figure 13<\/a>.<\/p>\n<div id=\"Figure_01_05_014\" class=\"wp-caption aligncenter\">\n<div style=\"width: 598px\" 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\/08205654\/CNX_Precalc_Figure_01_05_013.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"588\" height=\"267\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 13 Vertical reflection of the square root function<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137431214\">Because each output value is the opposite of the original output value, we can write<\/p>\n<div id=\"fs-id1165137425686\" class=\"unnumbered\" style=\"text-align: center;\">[latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{t}[\/latex][latex]\\\\[\/latex]<\/div>\n<div id=\"fs-id1165137425686\" class=\"unnumbered\" style=\"text-align: center;\"><\/div>\n<p>Notice that this is an outside change, or vertical reflection, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.<\/li>\n<li>\n<p id=\"fs-id1165134393050\">Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_015\">Figure 14<\/a>.<\/p>\n<div id=\"Figure_01_05_015\" class=\"wp-caption alignnone\">\n<div style=\"width: 985px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205657\/CNX_Precalc_Figure_01_05_014.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"975\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 14 Horizontal reflection of the square root function<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165133408855\">Because each input value is the opposite of the original input value, we can write<\/p>\n<div id=\"fs-id1165137470692\" class=\"unnumbered\" style=\"text-align: center;\">[latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-t}.[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137742575\">Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.<\/p>\n<p id=\"fs-id1165137664617\">Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\\left[0,\\infty \\right)[\/latex] and range [latex]\\left[0,\\infty \\right),[\/latex] the vertical reflection gives the [latex]V\\left(t\\right)[\/latex] function the range [latex]\\left(-\\infty ,\\text{ }0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,\\text{ }0\\right].[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135330596\" class=\"precalculus tryit\">\n<h3>Try it #5<\/h3>\n<div id=\"fs-id1165137437703\">\n<div id=\"fs-id1165137726983\">\n<p id=\"fs-id1165137726985\">Reflect the graph of [latex]f\\left(x\\right)=|x-1|[\/latex] (a) vertically and (b) horizontally.<\/p>\n<\/div>\n<div id=\"fs-id1165134234186\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134234186\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134234186\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137444428\" type=\"a\">\n<li><span id=\"fs-id1165137693977\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205700\/CNX_Precalc_Figure_01_05_015a.jpg\" alt=\"Graph of a vertically reflected absolute function.\" \/><\/span><\/li>\n<li><span id=\"fs-id1165135436637\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205703\/CNX_Precalc_Figure_01_05_015b.jpg\" alt=\"Graph of an absolute function translated one unit left.\" \/><\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137696924\">\n<div id=\"fs-id1165137405053\">\n<h3>Example 10:\u00a0 Reflecting a Tabular Function Horizontally and Vertically<\/h3>\n<p id=\"fs-id1165137564278\">A function [latex]f\\left(x\\right)[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_05\">Table 6<\/a>. Create a table for the functions below.<\/p>\n<ol id=\"fs-id1165135485824\" type=\"a\">\n<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\n<\/ol>\n<table id=\"Table_01_05_05\" style=\"width: 355px;\" 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 6<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 27.217px;\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">2<\/td>\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">4<\/td>\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">6<\/td>\n<td class=\"border\" style=\"width: 13px; text-align: center;\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 27.217px;\"><strong>[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">1<\/td>\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">3<\/td>\n<td class=\"border\" style=\"width: 27.2333px; text-align: center;\">7<\/td>\n<td class=\"border\" style=\"width: 13px; text-align: center;\">11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165134199548\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134199548\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134199548\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165134199550\" type=\"a\">\n<li>\n<p id=\"fs-id1165137411052\">For [latex]g\\left(x\\right),[\/latex] the negative sign outside the function indicates a vertical reflection, so the <em>x<\/em>-values stay the same and each output value will be the opposite of the original output value. See <a class=\"autogenerated-content\" href=\"#Table_01_05_06\">Table 7<\/a>.<\/p>\n<table id=\"Table_01_05_06\" 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, g(4)=-3, g(6)=-7, and g(8)=-11.\">\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]g\\left(x\\right)[\/latex]<\/strong><\/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;\">\u20137<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u201311<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<p id=\"fs-id1165137749533\">For [latex]h\\left(x\\right),[\/latex] the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\\left(x\\right)[\/latex] values stay the same as the [latex]f\\left(x\\right)[\/latex] values. See <a class=\"autogenerated-content\" href=\"#Table_01_05_07\">Table 8<\/a>.<\/p>\n<table id=\"Table_01_05_07\" summary=\"Two rows and five columns. The first row is labeled, \u201cx\u201d, and the second is labeled, \u201ch(x)\u201d. The values of x are -2, -4, -6, and -8. So for h(-2)=1, h(-4)=3, h(-6)=7, and h(-8)=11.\">\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;\">\u22122<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u22124<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u22126<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u22128<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]h\\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<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132937220\" class=\"precalculus tryit\">\n<h3>Try it #6<\/h3>\n<div id=\"fs-id1165137757772\">\n<div id=\"fs-id1165137757774\">\n<p id=\"fs-id1165135397310\">A function [latex]f\\left(x\\right)[\/latex] is given as <a class=\"autogenerated-content\" href=\"#Table_01_05_08\">Table 9<\/a>. Create a table for the functions below.<\/p>\n<ol id=\"fs-id1165137401553\" type=\"a\">\n<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\n<\/ol>\n<table id=\"Table_01_05_08\" 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, 0, 2, and 4. So for f(-2)=5, f(0)=10, f(2)=15, and f(4)=20.\">\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;\">\u22122<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/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;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\n<td class=\"border\" style=\"text-align: center;\">20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137553074\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137553074\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137553074\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137642579\" type=\"a\">\n<li>\n<p id=\"fs-id1165137637578\">[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<table id=\"fs-id1165137431044\" class=\"unnumbered\" 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, 0, 2, and 4. So for f(-2)=-5, f(0)=-10, f(2)=-15, and f(4)=-20.\">\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;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]g\\left(x\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]-5[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]-10[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]-15[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]-20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<p id=\"fs-id1165137871009\">[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<table id=\"fs-id1165134042357\" class=\"unnumbered\" 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, 0, -2, and -4. So for f(-2)=5, f(0)=10, f(-2)=15, and f(-4)=-20.\">\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;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]h\\left(x\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">15<\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">unknown<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_05_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137657438\">\n<div id=\"fs-id1165137432328\">\n<h3>Example 11:\u00a0 Applying a Learning Model Equation<\/h3>\n<p id=\"fs-id1165137938642\">A common model for learning has an equation similar to [latex]k\\left(t\\right)=-{2}^{-t}+1,[\/latex] where [latex]k[\/latex] is the percentage of mastery that can be achieved after [latex]t[\/latex] practice sessions. This is a transformation of the function [latex]f\\left(t\\right)={2}^{t}[\/latex] shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_017\">Figure 15<\/a>. Sketch a graph of [latex]k\\left(t\\right).[\/latex]<\/p>\n<div id=\"Figure_01_05_017\" class=\"small\">\n<div style=\"width: 343px\" 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\/08205707\/CNX_Precalc_Figure_01_05_016.jpg\" alt=\"Graph of k(t)\" width=\"333\" height=\"302\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 15<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137731123\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137731123\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137731123\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137731125\">This equation combines three transformations into one equation.<\/p>\n<ul id=\"fs-id1165137442908\">\n<li>A horizontal reflection: [latex]f\\left(-t\\right)={2}^{-t}[\/latex]<\/li>\n<li>followed by a vertical reflection: [latex]-f\\left(-t\\right)=-{2}^{-t}[\/latex]<\/li>\n<li>and finally vertical shift: [latex]-f\\left(-t\\right)+1=-{2}^{-t}+1[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137698491\">We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).<\/p>\n<ol id=\"fs-id1165135319410\" type=\"1\">\n<li>First, we apply a horizontal reflection to (0,1) and (1,2) by negating the input value to get (0, 1) and (-1, 2) respectively.<\/li>\n<li>Then, we apply a vertical reflection by negating the second coordinate to get (0, \u22121) and (-1, -2) respectively.<\/li>\n<li>Finally, we apply a vertical shift by adding 1 giving the points (0, 0) and (-1, -1) on the function [latex]k\\left(t\\right)[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1165135176725\">This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.<\/p>\n<p id=\"fs-id1165137723144\">In <a class=\"autogenerated-content\" href=\"#Figure_01_05_018\">Figure 16<\/a>, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.<\/p>\n<div id=\"Figure_01_05_018\" class=\"wp-caption alignnone\">\n<div style=\"width: 717px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205710\/CNX_Precalc_Figure_01_05_017abc.jpg\" alt=\"Graphs of all the transformations.\" width=\"707\" height=\"316\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 16<\/p>\n<\/div>\n<\/div>\n<h3>Analysis<\/h3>\n<p>As a model for learning, this function would be limited to a domain of [latex]t\\ge 0,[\/latex] with corresponding range [latex]\\left[0,1\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137472858\" class=\"precalculus tryit\">\n<h3>Try it #7<\/h3>\n<div id=\"ti_01_05_04\">\n<div id=\"fs-id1165135548260\">\n<p id=\"fs-id1165137827376\">Given the toolkit function [latex]f\\left(x\\right)={x}^{2},[\/latex] graph [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] and [latex]h\\left(x\\right)=f\\left(-x\\right).[\/latex] Take note of any surprising behavior for these functions.<\/p>\n<\/div>\n<div id=\"fs-id1165137894550\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137894550\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137894550\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165137778960\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/02\/08205715\/CNX_Precalc_Figure_01_05_020.jpg\" alt=\"Graph of x^2 and its reflections.\" width=\"336\" height=\"302\" \/><\/span><\/p>\n<p id=\"fs-id1165135255910\">Notice: [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] looks the same as [latex]f\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135182974\" class=\"bc-section section\">\n<h3>Determining Even and Odd Functions<\/h3>\n<p id=\"fs-id1165135532474\">Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\\left(x\\right)={x}^{2}[\/latex] or [latex]f\\left(x\\right)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the <em>y<\/em>-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong><\/p>\n<p id=\"fs-id1165137939530\">If the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph, as shown in <a class=\"autogenerated-content\" href=\"#Figure_01_05_022\">Figure 17<\/a>.<\/p>\n<div id=\"Figure_01_05_022\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" 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\/08205720\/CNX_Precalc_Figure_01_05_021abc.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137406881\">We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\n<p id=\"fs-id1165134573214\">Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0.[\/latex]<\/p>\n<div id=\"fs-id1165137619398\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137407995\">A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<div id=\"fs-id1165135424702\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135552902\">The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex]axis.[latex]\\\\[\/latex]<\/p>\n<p id=\"fs-id1165137501973\">A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<div id=\"fs-id1165137762060\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex] or equivalently [latex]f\\left(-x\\right)=-f\\left(x\\right)[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135503845\">The graph of an odd function is symmetric about the origin.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135503849\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165133353947\"><strong>Given the formula for a function, determine if the function is even, odd, or neither. <\/strong><\/p>\n<ol id=\"fs-id1165137552979\" type=\"1\">\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex] If it does, it is even.<\/li>\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right).[\/latex] If it does, it is odd.\u00a0 Note that you can also show the equivalent statement [latex]f\\left(-x\\right)=-f\\left(x\\right).[\/latex]<\/li>\n<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_05_12\" class=\"textbox examples\">\n<div id=\"fs-id1165137415536\">\n<div id=\"fs-id1165137415539\">\n<h3>Example 12:\u00a0 Determining whether a Function Is Even, Odd, or Neither<\/h3>\n<p id=\"fs-id1165135252115\">Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<\/div>\n<div id=\"fs-id1165137784966\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137784966\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137784966\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137784968\">Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\n<div id=\"fs-id1165137401549\" class=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165137771042\">This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\n<div id=\"fs-id1165137740781\" class=\"unnumbered\" style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135667851\">Because [latex]-f\\left(-x\\right)=f\\left(x\\right),[\/latex] this is an odd function.<\/p>\n<h3>Analysis<\/h3>\n<p id=\"fs-id1165133050510\">Consider the graph of [latex]f[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_05_039\">Figure 18<\/a>. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f,[\/latex] and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.<\/p>\n<div id=\"Figure_01_05_039\" class=\"medium\">\n<div style=\"width: 393px\" 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\/08205724\/CNX_Precalc_Figure_01_05_039.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"383\" height=\"256\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 18<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137480929\">\n<div id=\"Figure_01_05_039\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135400987\" class=\"precalculus tryit\">\n<h3>Try it #8<\/h3>\n<div id=\"ti_01_05_05\">\n<div id=\"fs-id1165137897939\">\n<p id=\"fs-id1165137897941\">Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\n<\/div>\n<div id=\"fs-id1165137757764\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137757764\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137757764\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137757766\">even<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137654768\" class=\"bc-section section\">\n<div id=\"fs-id1165137793506\" class=\"bc-section section\">\n<div id=\"Figure_01_05_025\" class=\"small\"><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137676302\" class=\"bc-section section\">\n<div id=\"fs-id1165135497140\" class=\"precalculus media\">\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]\\left(0\\lt a\\lt 1\\right)[\/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]\\left(0\\lt b\\lt 1\\right)[\/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 shifted vertically by adding a constant to the output.<\/li>\n<li>A function can be shifted horizontally by adding a constant to the input.<\/li>\n<li>Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.<\/li>\n<li>Vertical and horizontal shifts are often combined.<\/li>\n<li>A vertical reflection reflects a graph about the [latex]x\\text{-}[\/latex]axis. A graph can be reflected vertically by multiplying the output by \u20131.<\/li>\n<li>A horizontal reflection reflects a graph about the [latex]y\\text{-}[\/latex]axis. A graph can be reflected horizontally by multiplying the input by \u20131.<\/li>\n<li>A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph.<\/li>\n<li>A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.<\/li>\n<li>A function presented as an equation can be reflected by applying transformations one at a time.<\/li>\n<li>Even functions are symmetric about the [latex]y\\text{-}[\/latex]axis, whereas odd functions are symmetric about the origin.<\/li>\n<li>Even functions satisfy the condition [latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex]<\/li>\n<li>Odd functions satisfy the condition [latex]f\\left(x\\right)=-f\\left(-x\\right).[\/latex]<\/li>\n<li>A function can be odd, even, or neither.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137448239\">\n<dt>even function<\/dt>\n<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right),[\/latex] and is symmetric about the [latex]y\\text{-}[\/latex]axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133242964\"><\/dl>\n<dl id=\"fs-id1165135440170\">\n<dt>horizontal reflection<\/dt>\n<dd id=\"fs-id1165137602051\">a transformation that reflects a function\u2019s graph across the <em>y<\/em>-axis by multiplying the input by [latex]-1[\/latex]<\/dd>\n<\/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<dl id=\"fs-id1165137862450\">\n<dd id=\"fs-id1165132971698\"><\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-127\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Transformation of Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:HY2ubE-L@11\/Transformation-of-Functions\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:HY2ubE-L@11\/Transformation-of-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/977ffc66-5462-4966-b1a9-ed62f798136f@1.94<\/li><li><strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Transformation of 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