{"id":1294,"date":"2025-07-24T04:06:20","date_gmt":"2025-07-24T04:06:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1294"},"modified":"2026-03-18T02:53:33","modified_gmt":"2026-03-18T02:53:33","slug":"transformation-of-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/transformation-of-functions-fresh-take\/","title":{"raw":"Transformation of Functions: Fresh Take","rendered":"Transformation of Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph functions using horizontal and vertical stretch and reflection.<\/li>\r\n \t<li>Graph functions using horizontal and vertical shifts.<\/li>\r\n \t<li>Graph functions using a combination of transformations.<\/li>\r\n \t<li>Describe transformations based on a function formula and write the function given its parent and transformations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identifying Vertical Shifts<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A vertical shift moves a function's graph up or down without changing its shape<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Achieved by adding or subtracting a constant to the function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Upward Shift:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x) + c[\/latex], where [latex]c &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Moves the graph up by [latex]c[\/latex] units<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Downward Shift:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x) - c[\/latex], where [latex]c &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Moves the graph down by [latex]c[\/latex] units<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Function Values:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Each [latex]y[\/latex]-coordinate is increased or decreased by [latex]c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-coordinates remain unchanged<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Key Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">y-intercept shifts vertically by [latex]c[\/latex] units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zeros of the function shift vertically (may change number of zeros)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f(x) = x^2[\/latex], describe and graph [latex]g(x) = f(x) - 3[\/latex] and [latex]h(x) = f(x) + 1[\/latex].[reveal-answer q=\"74159\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"74159\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Original function [latex]f(x) = x^2[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Parabola with vertex at (0,0)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">y-intercept: (0,0)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Opens upward<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x) - 3 = x^2 - 3[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] down by 3 units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New vertex: (0,-3)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New y-intercept: (0,-3)<\/li>\r\n \t<li>Below is a graph with the original function [latex]f(x) = x^2[\/latex] graphed in red and the shifter function [latex]f(x) = x^2-3[\/latex] graphed in blue.\r\n\r\n[caption id=\"attachment_3876\" align=\"alignnone\" width=\"619\"]<img class=\"wp-image-3876 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12160718\/Screenshot-2024-09-12-120605.png\" alt=\"Graph showing two parabolas on a coordinate plane. The red parabola represents the function f(x) = x^2, and the blue parabola represents the function f(x) = x^2 - 3. The red parabola has its vertex at the origin (0, 0), while the blue parabola is shifted downward, with its vertex at (0, -3). \" width=\"619\" height=\"585\" \/> Graph showing two parabolas on a coordinate plane[\/caption]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x) + 1 = x^2 + 1[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] up by 1 unit<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New vertex: (0,1)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New y-intercept: (0,1)<\/li>\r\n \t<li>Below is a graph with the original function [latex]f(x) = x^2[\/latex] graphed in red and the shifter function [latex]f(x) = x^2+1[\/latex] graphed in blue.\r\n\r\n[caption id=\"attachment_3877\" align=\"alignnone\" width=\"586\"]<img class=\"wp-image-3877 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12160957\/Screenshot-2024-09-12-120931.png\" alt=\"Graph showing two parabolas on a coordinate plane. The red parabola represents the function f(x) = x^2, and the blue parabola represents the function f(x) = x^2 + 1. The red parabola has its vertex at the origin (0, 0), while the blue parabola is shifted upward, with its vertex at (0, 1).\" width=\"586\" height=\"448\" \/> Graph showing two parabolas on a coordinate plane[\/caption]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Identifying Horizontal Shifts<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A horizontal shift moves a function's graph left or right without changing its shape<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Achieved by adding or subtracting a constant inside the function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rightward Shift:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x - c)[\/latex], where [latex]c &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Moves the graph right by [latex]c[\/latex] units<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Leftward Shift:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x + c)[\/latex], where [latex]c &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Moves the graph left by [latex]c[\/latex] units<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Function Values:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Each [latex]x[\/latex]-coordinate is increased or decreased by [latex]c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-coordinates remain unchanged<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Key Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts shift horizontally by [latex]c[\/latex] units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept may change or disappear<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f(x) = |x|[\/latex], describe and graph [latex]g(x) = f(x - 2)[\/latex] and [latex]h(x) = f(x + 1)[\/latex].[reveal-answer q=\"277792\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"277792\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Original function [latex]f(x) = |x|[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">V-shaped graph with vertex at (0,0)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x - 2) = |x - 2|[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] right by 2 units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New vertex: (2,0)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Below is a graph with the original function [latex]f(x) = |x|[\/latex] graphed in red and the shifter function [latex]f(x) = |x-2|[\/latex] graphed in blue.\r\n\r\n[caption id=\"attachment_3883\" align=\"alignnone\" width=\"617\"]<img class=\"wp-image-3883 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12161824\/Screenshot-2024-09-12-121712.png\" alt=\"Graph showing two absolute value functions on a coordinate plane. The red graph represents the function f(x) = |x|, which forms a V-shape with its vertex at the origin (0, 0). The blue graph represents the function f(x) = |x - 2|, which forms a V-shape with its vertex shifted to (2, 0). \" width=\"617\" height=\"480\" \/> Graph showing two absolute value functions on a coordinate plane[\/caption]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x + 1) = |x + 1|[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] left by 1 unit<\/li>\r\n \t<li class=\"whitespace-normal break-words\">New vertex: (-1,0)<\/li>\r\n \t<li>Below is a graph with the original function [latex]f(x) = |x|[\/latex] graphed in red and the shifter function [latex]f(x) = |x+1|[\/latex] graphed in blue.\r\n\r\n[caption id=\"attachment_3884\" align=\"alignnone\" width=\"580\"]<img class=\"wp-image-3884 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12161942\/Screenshot-2024-09-12-121936.png\" alt=\"Graph showing two absolute value functions on a coordinate plane. The red graph represents the function f(x) = |x|, forming a V-shape with its vertex at the origin (0, 0). The blue graph represents the function f(x) = |x + 1|, forming a V-shape with its vertex shifted to (-1, 0). \" width=\"580\" height=\"399\" \/> Graph showing two absolute value functions on a coordinate plane[\/caption]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=\\sqrt{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?[reveal-answer q=\"193388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"193388\"]A horizontal shift results when a constant is added to or subtracted from the input. A vertical shift results when a constant is added to or subtracted from the output.[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Online graphing calculators can graph transformations using function notation. Use an online graphing calculator to graph the toolkit function [latex]f(x) = x^2[\/latex]\r\nNow, enter [latex]f(x+5)[\/latex], and [latex]f(x)+5[\/latex] in the next two lines.Now have the online graphing calculator make a table of values for the original function. Include integer values on the interval [latex][-5,5][\/latex]. Replace the column labeled [latex]y_{1}[\/latex] with [latex]f(x_{1})[\/latex].Now replace [latex]f(x_{1})[\/latex] with\u00a0[latex]f(x_{1}+3)[\/latex], and\u00a0[latex]f(x_{1})+3[\/latex].What are the corresponding functions associated with the transformations you have graphed?\r\n[reveal-answer q=\"593472\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"593472\"]You have graphed the following transformations:[latex]f(x+3)=(x+3)^2[\/latex][latex]f(x)+3=(x)^2+3[\/latex][\/hidden-answer]<\/section>\r\n<h2>Graphing Functions Using Reflections about the Axes<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Reflection:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reflects graph across the [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Changes sign of output: [latex]g(x) = -f(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Reflection:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reflects graph across the [latex]y[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Changes sign of input: [latex]g(x) = f(-x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Graph:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical reflection: Mirror image about [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal reflection: Mirror image about [latex]y[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain and Range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Reflections can affect the domain and range of functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical reflection may change the range<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal reflection may change the domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Composition of Reflections:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Can be combined with other transformations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Order of operations matters<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to reflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.[reveal-answer q=\"362828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"362828\"]a)\r\n\r\n[caption id=\"attachment_6733\" align=\"alignnone\" width=\"351\"]<img class=\"wp-image-6733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08195713\/Screen-Shot-2019-07-08-at-12.55.48-PM.png\" alt=\"Graph of f(x)=|x-1| and f(x)=-|x-1|\" width=\"351\" height=\"347\" \/> Graph of f(x)=|x-1| and f(x)=-|x-1|[\/caption]\r\n\r\nb)\r\n\r\n[caption id=\"attachment_6741\" align=\"alignnone\" width=\"351\"]<img class=\"wp-image-6741\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08200623\/Screen-Shot-2019-07-08-at-1.05.11-PM.png\" alt=\"Graph of f(x)=|x-1| and f(x)=|(-x)-1|\" width=\"351\" height=\"347\" \/> Graph of f(x)=|x-1| and f(x)=|(-x)-1|[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<table id=\"Table_01_05_08\" summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>5<\/td>\r\n<td>10<\/td>\r\n<td>15<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing the function [latex]f\\left(x\\right)[\/latex] given in the table above, create a table for the functions below.\r\n\r\na. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]\r\n\r\nb. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]\r\n\r\n[reveal-answer q=\"230301\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"230301\"]\r\n<ol>\r\n \t<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]\r\n<table summary=\"Two rows and five columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>-2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]-10[\/latex]<\/td>\r\n<td>[latex]-15[\/latex]<\/td>\r\n<td>[latex]-20[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]\r\n<table summary=\"Two rows and five columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>-2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]h\\left(x\\right)[\/latex]<\/td>\r\n<td>15<\/td>\r\n<td>10<\/td>\r\n<td>5<\/td>\r\n<td>unknown<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 30px;\">[latex]x=4[\/latex] is unknown in the last problem because you are looking for what [latex]f(x)[\/latex] was when the [latex]x[\/latex]-value equaled [latex]-x[\/latex], or in this case, [latex]-4[\/latex]. There is no [latex]f(x)[\/latex] value give for [latex]x=-4[\/latex] in the original function table, so the [latex]h(x)[\/latex] value is <em>unknown<\/em>.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Graphing Functions Using Stretches and Compressions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Stretches and Compressions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Affect output values: [latex]g(x) = a \\cdot f(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a &gt; 1[\/latex]: Vertical stretch<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0 &lt; a &lt; 1[\/latex]: Vertical compression<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Stretches and Compressions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Affect input values: [latex]g(x) = f(b \\cdot x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0 &lt; b &lt; 1[\/latex]: Horizontal stretch by factor [latex]\\frac{1}{b}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b &gt; 1[\/latex]: Horizontal compression by factor [latex]\\frac{1}{b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Effect on Graph:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical: Changes height of graph<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal: Changes width of graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative Values:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a &lt; 0[\/latex]: Combine vertical stretch\/compression with reflection over x-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]b &lt; 0[\/latex]: Combine horizontal stretch\/compression with reflection over y-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain and Range:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Vertical transformations may affect the range<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal transformations may affect the domain<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A function [latex]f[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right)[\/latex].\r\n<table summary=\"Two rows and five columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<td>12<\/td>\r\n<td>16<\/td>\r\n<td>20<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"805921\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"805921\"]\r\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled, \"><colgroup> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<td>6<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\r\n<td>9<\/td>\r\n<td>12<\/td>\r\n<td>15<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the formula for the function that we get when we vertically stretch (or scale) the identity toolkit function by a factor of 3, and then shift it down by 2 units.\r\nCheck your work with an online graphing calculator.[reveal-answer q=\"473017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473017\"][latex]g(x)=3x-2[\/latex]\r\n\r\n[caption id=\"attachment_6746\" align=\"alignnone\" width=\"452\"]<img class=\"wp-image-6746\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08201018\/Screen-Shot-2019-07-08-at-1.09.59-PM.png\" alt=\"Graph of f(x)=x and f(x)=3x-2\" width=\"452\" height=\"442\" \/> Graph of two lines[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write a formula for the toolkit square root function horizontally stretched by a factor of 3.\r\nUse an online graphing calculator to check your work.[reveal-answer q=\"35233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35233\"][latex]g\\left(x\\right)=\\sqrt{\\frac{1}{3}x}[\/latex]\r\n\r\n[caption id=\"attachment_6743\" align=\"alignnone\" width=\"419\"]<img class=\"wp-image-6743\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08200821\/Screen-Shot-2019-07-08-at-1.08.04-PM.png\" alt=\"Graph of f(x)=sqrt(x) and f(x)=sqrt(1\/3 x)\" width=\"419\" height=\"274\" \/> Graph of two functions[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Performing a Sequence of Transformations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Order of Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]y = a \\cdot f(b(x-c))+d[\/latex], the order is: a. Horizontal shift b. Horizontal stretch\/compression c. Reflections d. Vertical stretch\/compression e. Vertical shift<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Horizontal Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Shift: [latex]f(x-c)[\/latex] shifts right by [latex]c[\/latex] units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stretch\/Compression: [latex]f(bx)[\/latex] stretches by factor [latex]\\frac{1}{b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Shift: [latex]f(x)+d[\/latex] shifts up by [latex]d[\/latex] units<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Stretch\/Compression: [latex]a \\cdot f(x)[\/latex] stretches by factor [latex]a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reflections:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal: [latex]f(-x)[\/latex] reflects across y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical: [latex]-f(x)[\/latex] reflects across x-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Composite Transformations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Apply transformations from inside to outside<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pay attention to the order of operations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f\\left(x\\right)=|x|[\/latex], sketch a graph of [latex]h\\left(x\\right)=f\\left(x - 2\\right)+4[\/latex].\r\nCheck your work with an online graphing calculator.[reveal-answer q=\"807890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807890\"]\r\n\r\n[caption id=\"attachment_2752\" align=\"alignnone\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\"><img class=\"wp-image-2752 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\" alt=\"cnx_precalc_figure_01_05_010\" width=\"487\" height=\"402\" \/><\/a> Graph of absolute value function[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write a formula for a transformation of the toolkit reciprocal function [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] that shifts the function\u2019s graph three units to the left and one unit down.[reveal-answer q=\"126023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"126023\"][latex]g(x)=\\dfrac{1}{x+3}-1[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">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.[reveal-answer q=\"386010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"386010\"]\r\n\r\n[caption id=\"attachment_2755\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\"><img class=\"wp-image-2755 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\" alt=\"cnx_precalc_figure_01_05_020\" width=\"487\" height=\"438\" \/><\/a> Graph of f(x) and g(x)[\/caption]\r\n\r\nNotice: [latex]g(x)=f(\u2212x)[\/latex]\u2009looks the same as [latex]f(x)[\/latex].[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gbhbabff-An29CALYjAA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/An29CALYjAA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gbhbabff-An29CALYjAA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844429&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gbhbabff-An29CALYjAA&amp;vembed=0&amp;video_id=An29CALYjAA&amp;video_target=tpm-plugin-gbhbabff-An29CALYjAA\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Functions+Transformations+-++A+Summary_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFunctions Transformations: A Summary\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Even Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Symmetrical about the y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = f(-x)[\/latex] for all x in the domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Odd Functions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Symmetrical about the origin<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = -f(-x)[\/latex] for all x in the domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^3[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Neither Even nor Odd:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Functions that don't satisfy either condition<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = 2^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Special Cases:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = 0[\/latex] is both even and odd<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Polynomial functions with only even powers are even<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Polynomial functions with only odd powers are odd<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Even: Unchanged when reflected over y-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Odd: Unchanged when rotated 180\u00b0 about the origin<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?[reveal-answer q=\"630369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"630369\"]Even[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dfhdbbeg-VvUI6E78cN4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VvUI6E78cN4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dfhdbbeg-VvUI6E78cN4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=6454976&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dfhdbbeg-VvUI6E78cN4&amp;vembed=0&amp;video_id=VvUI6E78cN4&amp;video_target=tpm-plugin-dfhdbbeg-VvUI6E78cN4\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Odd+and+Even+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Odd and Even Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph functions using horizontal and vertical stretch and reflection.<\/li>\n<li>Graph functions using horizontal and vertical shifts.<\/li>\n<li>Graph functions using a combination of transformations.<\/li>\n<li>Describe transformations based on a function formula and write the function given its parent and transformations.<\/li>\n<\/ul>\n<\/section>\n<h2>Identifying Vertical Shifts<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A vertical shift moves a function&#8217;s graph up or down without changing its shape<\/li>\n<li class=\"whitespace-normal break-words\">Achieved by adding or subtracting a constant to the function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Upward Shift:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x) + c[\/latex], where [latex]c > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Moves the graph up by [latex]c[\/latex] units<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Downward Shift:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x) - c[\/latex], where [latex]c > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Moves the graph down by [latex]c[\/latex] units<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Function Values:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each [latex]y[\/latex]-coordinate is increased or decreased by [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-coordinates remain unchanged<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Key Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">y-intercept shifts vertically by [latex]c[\/latex] units<\/li>\n<li class=\"whitespace-normal break-words\">Zeros of the function shift vertically (may change number of zeros)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f(x) = x^2[\/latex], describe and graph [latex]g(x) = f(x) - 3[\/latex] and [latex]h(x) = f(x) + 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q74159\">Show Answer<\/button><\/p>\n<div id=\"q74159\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Original function [latex]f(x) = x^2[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Parabola with vertex at (0,0)<\/li>\n<li class=\"whitespace-normal break-words\">y-intercept: (0,0)<\/li>\n<li class=\"whitespace-normal break-words\">Opens upward<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x) - 3 = x^2 - 3[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] down by 3 units<\/li>\n<li class=\"whitespace-normal break-words\">New vertex: (0,-3)<\/li>\n<li class=\"whitespace-normal break-words\">New y-intercept: (0,-3)<\/li>\n<li>Below is a graph with the original function [latex]f(x) = x^2[\/latex] graphed in red and the shifter function [latex]f(x) = x^2-3[\/latex] graphed in blue.<br \/>\n<figure id=\"attachment_3876\" aria-describedby=\"caption-attachment-3876\" style=\"width: 619px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3876 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12160718\/Screenshot-2024-09-12-120605.png\" alt=\"Graph showing two parabolas on a coordinate plane. The red parabola represents the function f(x) = x^2, and the blue parabola represents the function f(x) = x^2 - 3. The red parabola has its vertex at the origin (0, 0), while the blue parabola is shifted downward, with its vertex at (0, -3).\" width=\"619\" height=\"585\" \/><figcaption id=\"caption-attachment-3876\" class=\"wp-caption-text\">Graph showing two parabolas on a coordinate plane<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x) + 1 = x^2 + 1[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] up by 1 unit<\/li>\n<li class=\"whitespace-normal break-words\">New vertex: (0,1)<\/li>\n<li class=\"whitespace-normal break-words\">New y-intercept: (0,1)<\/li>\n<li>Below is a graph with the original function [latex]f(x) = x^2[\/latex] graphed in red and the shifter function [latex]f(x) = x^2+1[\/latex] graphed in blue.<br \/>\n<figure id=\"attachment_3877\" aria-describedby=\"caption-attachment-3877\" style=\"width: 586px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3877 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12160957\/Screenshot-2024-09-12-120931.png\" alt=\"Graph showing two parabolas on a coordinate plane. The red parabola represents the function f(x) = x^2, and the blue parabola represents the function f(x) = x^2 + 1. The red parabola has its vertex at the origin (0, 0), while the blue parabola is shifted upward, with its vertex at (0, 1).\" width=\"586\" height=\"448\" \/><figcaption id=\"caption-attachment-3877\" class=\"wp-caption-text\">Graph showing two parabolas on a coordinate plane<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<h2>Identifying Horizontal Shifts<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A horizontal shift moves a function&#8217;s graph left or right without changing its shape<\/li>\n<li class=\"whitespace-normal break-words\">Achieved by adding or subtracting a constant inside the function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rightward Shift:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x - c)[\/latex], where [latex]c > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Moves the graph right by [latex]c[\/latex] units<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Leftward Shift:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x + c)[\/latex], where [latex]c > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Moves the graph left by [latex]c[\/latex] units<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Function Values:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each [latex]x[\/latex]-coordinate is increased or decreased by [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-coordinates remain unchanged<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Key Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts shift horizontally by [latex]c[\/latex] units<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept may change or disappear<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f(x) = |x|[\/latex], describe and graph [latex]g(x) = f(x - 2)[\/latex] and [latex]h(x) = f(x + 1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q277792\">Show Answer<\/button><\/p>\n<div id=\"q277792\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Original function [latex]f(x) = |x|[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">V-shaped graph with vertex at (0,0)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">[latex]g(x) = f(x - 2) = |x - 2|[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] right by 2 units<\/li>\n<li class=\"whitespace-normal break-words\">New vertex: (2,0)<\/li>\n<li class=\"whitespace-normal break-words\">Below is a graph with the original function [latex]f(x) = |x|[\/latex] graphed in red and the shifter function [latex]f(x) = |x-2|[\/latex] graphed in blue.<br \/>\n<figure id=\"attachment_3883\" aria-describedby=\"caption-attachment-3883\" style=\"width: 617px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3883 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12161824\/Screenshot-2024-09-12-121712.png\" alt=\"Graph showing two absolute value functions on a coordinate plane. The red graph represents the function f(x) = |x|, which forms a V-shape with its vertex at the origin (0, 0). The blue graph represents the function f(x) = |x - 2|, which forms a V-shape with its vertex shifted to (2, 0).\" width=\"617\" height=\"480\" \/><figcaption id=\"caption-attachment-3883\" class=\"wp-caption-text\">Graph showing two absolute value functions on a coordinate plane<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">[latex]h(x) = f(x + 1) = |x + 1|[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shifts [latex]f(x)[\/latex] left by 1 unit<\/li>\n<li class=\"whitespace-normal break-words\">New vertex: (-1,0)<\/li>\n<li>Below is a graph with the original function [latex]f(x) = |x|[\/latex] graphed in red and the shifter function [latex]f(x) = |x+1|[\/latex] graphed in blue.<br \/>\n<figure id=\"attachment_3884\" aria-describedby=\"caption-attachment-3884\" style=\"width: 580px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3884 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/12161942\/Screenshot-2024-09-12-121936.png\" alt=\"Graph showing two absolute value functions on a coordinate plane. The red graph represents the function f(x) = |x|, forming a V-shape with its vertex at the origin (0, 0). The blue graph represents the function f(x) = |x + 1|, forming a V-shape with its vertex shifted to (-1, 0).\" width=\"580\" height=\"399\" \/><figcaption id=\"caption-attachment-3884\" class=\"wp-caption-text\">Graph showing two absolute value functions on a coordinate plane<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=\\sqrt{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 class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q193388\">Show Solution<\/button><\/p>\n<div id=\"q193388\" class=\"hidden-answer\" style=\"display: none\">A horizontal shift results when a constant is added to or subtracted from the input. A vertical shift results when a constant is added to or subtracted from the output.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Online graphing calculators can graph transformations using function notation. Use an online graphing calculator to graph the toolkit function [latex]f(x) = x^2[\/latex]<br \/>\nNow, enter [latex]f(x+5)[\/latex], and [latex]f(x)+5[\/latex] in the next two lines.Now have the online graphing calculator make a table of values for the original function. Include integer values on the interval [latex][-5,5][\/latex]. Replace the column labeled [latex]y_{1}[\/latex] with [latex]f(x_{1})[\/latex].Now replace [latex]f(x_{1})[\/latex] with\u00a0[latex]f(x_{1}+3)[\/latex], and\u00a0[latex]f(x_{1})+3[\/latex].What are the corresponding functions associated with the transformations you have graphed?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q593472\">Show Solution<\/button><\/p>\n<div id=\"q593472\" class=\"hidden-answer\" style=\"display: none\">You have graphed the following transformations:[latex]f(x+3)=(x+3)^2[\/latex][latex]f(x)+3=(x)^2+3[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Graphing Functions Using Reflections about the Axes<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Vertical Reflection:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reflects graph across the [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Changes sign of output: [latex]g(x) = -f(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Reflection:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reflects graph across the [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Changes sign of input: [latex]g(x) = f(-x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Graph:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical reflection: Mirror image about [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal reflection: Mirror image about [latex]y[\/latex]-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reflections can affect the domain and range of functions<\/li>\n<li class=\"whitespace-normal break-words\">Vertical reflection may change the range<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal reflection may change the domain<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Composition of Reflections:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can be combined with other transformations<\/li>\n<li class=\"whitespace-normal break-words\">Order of operations matters<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to reflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q362828\">Show Solution<\/button><\/p>\n<div id=\"q362828\" class=\"hidden-answer\" style=\"display: none\">a)<\/p>\n<figure id=\"attachment_6733\" aria-describedby=\"caption-attachment-6733\" style=\"width: 351px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6733\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08195713\/Screen-Shot-2019-07-08-at-12.55.48-PM.png\" alt=\"Graph of f(x)=|x-1| and f(x)=-|x-1|\" width=\"351\" height=\"347\" \/><figcaption id=\"caption-attachment-6733\" class=\"wp-caption-text\">Graph of f(x)=|x-1| and f(x)=-|x-1|<\/figcaption><\/figure>\n<p>b)<\/p>\n<figure id=\"attachment_6741\" aria-describedby=\"caption-attachment-6741\" style=\"width: 351px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6741\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08200623\/Screen-Shot-2019-07-08-at-1.05.11-PM.png\" alt=\"Graph of f(x)=|x-1| and f(x)=|(-x)-1|\" width=\"351\" height=\"347\" \/><figcaption id=\"caption-attachment-6741\" class=\"wp-caption-text\">Graph of f(x)=|x-1| and f(x)=|(-x)-1|<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<table id=\"Table_01_05_08\" summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>5<\/td>\n<td>10<\/td>\n<td>15<\/td>\n<td>20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using the function [latex]f\\left(x\\right)[\/latex] given in the table above, create a table for the functions below.<\/p>\n<p>a. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<p>b. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q230301\">Show Solution<\/button><\/p>\n<div id=\"q230301\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<br \/>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>-2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]-10[\/latex]<\/td>\n<td>[latex]-15[\/latex]<\/td>\n<td>[latex]-20[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>[latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<br \/>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>-2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>[latex]h\\left(x\\right)[\/latex]<\/td>\n<td>15<\/td>\n<td>10<\/td>\n<td>5<\/td>\n<td>unknown<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p style=\"padding-left: 30px;\">[latex]x=4[\/latex] is unknown in the last problem because you are looking for what [latex]f(x)[\/latex] was when the [latex]x[\/latex]-value equaled [latex]-x[\/latex], or in this case, [latex]-4[\/latex]. There is no [latex]f(x)[\/latex] value give for [latex]x=-4[\/latex] in the original function table, so the [latex]h(x)[\/latex] value is <em>unknown<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Graphing Functions Using Stretches and Compressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Vertical Stretches and Compressions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Affect output values: [latex]g(x) = a \\cdot f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a > 1[\/latex]: Vertical stretch<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0 < a < 1[\/latex]: Vertical compression<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Stretches and Compressions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Affect input values: [latex]g(x) = f(b \\cdot x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0 < b < 1[\/latex]: Horizontal stretch by factor [latex]\\frac{1}{b}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b > 1[\/latex]: Horizontal compression by factor [latex]\\frac{1}{b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effect on Graph:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical: Changes height of graph<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal: Changes width of graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Negative Values:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a < 0[\/latex]: Combine vertical stretch\/compression with reflection over x-axis<\/li>\n<li class=\"whitespace-normal break-words\">[latex]b < 0[\/latex]: Combine horizontal stretch\/compression with reflection over y-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain and Range:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Vertical transformations may affect the range<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal transformations may affect the domain<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A function [latex]f[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=\\frac{3}{4}f\\left(x\\right)[\/latex].<\/p>\n<table summary=\"Two rows and five columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\n<td>12<\/td>\n<td>16<\/td>\n<td>20<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q805921\">Show Solution<\/button><\/p>\n<div id=\"q805921\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"fs-id1165134261681\" class=\"unnumbered\" summary=\"Two rows and five columns. The first row is labeled,\">\n<colgroup> <\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>9<\/td>\n<td>12<\/td>\n<td>15<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the formula for the function that we get when we vertically stretch (or scale) the identity toolkit function by a factor of 3, and then shift it down by 2 units.<br \/>\nCheck your work with an online graphing calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q473017\">Show Solution<\/button><\/p>\n<div id=\"q473017\" class=\"hidden-answer\" style=\"display: none\">[latex]g(x)=3x-2[\/latex]<\/p>\n<figure id=\"attachment_6746\" aria-describedby=\"caption-attachment-6746\" style=\"width: 452px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6746\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08201018\/Screen-Shot-2019-07-08-at-1.09.59-PM.png\" alt=\"Graph of f(x)=x and f(x)=3x-2\" width=\"452\" height=\"442\" \/><figcaption id=\"caption-attachment-6746\" class=\"wp-caption-text\">Graph of two lines<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write a formula for the toolkit square root function horizontally stretched by a factor of 3.<br \/>\nUse an online graphing calculator to check your work.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q35233\">Show Solution<\/button><\/p>\n<div id=\"q35233\" class=\"hidden-answer\" style=\"display: none\">[latex]g\\left(x\\right)=\\sqrt{\\frac{1}{3}x}[\/latex]<\/p>\n<figure id=\"attachment_6743\" aria-describedby=\"caption-attachment-6743\" style=\"width: 419px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6743\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08200821\/Screen-Shot-2019-07-08-at-1.08.04-PM.png\" alt=\"Graph of f(x)=sqrt(x) and f(x)=sqrt(1\/3 x)\" width=\"419\" height=\"274\" \/><figcaption id=\"caption-attachment-6743\" class=\"wp-caption-text\">Graph of two functions<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Performing a Sequence of Transformations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Order of Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]y = a \\cdot f(b(x-c))+d[\/latex], the order is: a. Horizontal shift b. Horizontal stretch\/compression c. Reflections d. Vertical stretch\/compression e. Vertical shift<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shift: [latex]f(x-c)[\/latex] shifts right by [latex]c[\/latex] units<\/li>\n<li class=\"whitespace-normal break-words\">Stretch\/Compression: [latex]f(bx)[\/latex] stretches by factor [latex]\\frac{1}{b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shift: [latex]f(x)+d[\/latex] shifts up by [latex]d[\/latex] units<\/li>\n<li class=\"whitespace-normal break-words\">Stretch\/Compression: [latex]a \\cdot f(x)[\/latex] stretches by factor [latex]a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflections:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal: [latex]f(-x)[\/latex] reflects across y-axis<\/li>\n<li class=\"whitespace-normal break-words\">Vertical: [latex]-f(x)[\/latex] reflects across x-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Composite Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Apply transformations from inside to outside<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the order of operations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f\\left(x\\right)=|x|[\/latex], sketch a graph of [latex]h\\left(x\\right)=f\\left(x - 2\\right)+4[\/latex].<br \/>\nCheck your work with an online graphing calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q807890\">Show Solution<\/button><\/p>\n<div id=\"q807890\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_2752\" aria-describedby=\"caption-attachment-2752\" style=\"width: 487px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2752 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14224048\/CNX_Precalc_Figure_01_05_010.jpg\" alt=\"cnx_precalc_figure_01_05_010\" width=\"487\" height=\"402\" \/><\/a><figcaption id=\"caption-attachment-2752\" class=\"wp-caption-text\">Graph of absolute value function<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write a formula for a transformation of the toolkit reciprocal function [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] that shifts the function\u2019s graph three units to the left and one unit down.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q126023\">Show Solution<\/button><\/p>\n<div id=\"q126023\" class=\"hidden-answer\" style=\"display: none\">[latex]g(x)=\\dfrac{1}{x+3}-1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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 class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q386010\">Show Solution<\/button><\/p>\n<div id=\"q386010\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_2755\" aria-describedby=\"caption-attachment-2755\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2755 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/14225805\/CNX_Precalc_Figure_01_05_020.jpg\" alt=\"cnx_precalc_figure_01_05_020\" width=\"487\" height=\"438\" \/><\/a><figcaption id=\"caption-attachment-2755\" class=\"wp-caption-text\">Graph of f(x) and g(x)<\/figcaption><\/figure>\n<p>Notice: [latex]g(x)=f(\u2212x)[\/latex]\u2009looks the same as [latex]f(x)[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gbhbabff-An29CALYjAA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/An29CALYjAA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gbhbabff-An29CALYjAA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844429&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gbhbabff-An29CALYjAA&amp;vembed=0&amp;video_id=An29CALYjAA&amp;video_target=tpm-plugin-gbhbabff-An29CALYjAA\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Functions+Transformations+-++A+Summary_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFunctions Transformations: A Summary\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Even Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Symmetrical about the y-axis<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x) = f(-x)[\/latex] for all x in the domain<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Odd Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Symmetrical about the origin<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x) = -f(-x)[\/latex] for all x in the domain<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = x^3[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Neither Even nor Odd:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Functions that don&#8217;t satisfy either condition<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]f(x) = 2^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Special Cases:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = 0[\/latex] is both even and odd<\/li>\n<li class=\"whitespace-normal break-words\">Polynomial functions with only even powers are even<\/li>\n<li class=\"whitespace-normal break-words\">Polynomial functions with only odd powers are odd<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even: Unchanged when reflected over y-axis<\/li>\n<li class=\"whitespace-normal break-words\">Odd: Unchanged when rotated 180\u00b0 about the origin<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q630369\">Show Solution<\/button><\/p>\n<div id=\"q630369\" class=\"hidden-answer\" style=\"display: none\">Even<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dfhdbbeg-VvUI6E78cN4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/VvUI6E78cN4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dfhdbbeg-VvUI6E78cN4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=6454976&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dfhdbbeg-VvUI6E78cN4&amp;vembed=0&amp;video_id=VvUI6E78cN4&amp;video_target=tpm-plugin-dfhdbbeg-VvUI6E78cN4\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Odd+and+Even+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Odd and Even Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Functions Transformations: A 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