{"id":151,"date":"2023-06-21T13:22:38","date_gmt":"2023-06-21T13:22:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/reflections\/"},"modified":"2023-08-24T05:57:31","modified_gmt":"2023-08-24T05:57:31","slug":"reflections","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/reflections\/","title":{"raw":"\u25aa   Reflections","rendered":"\u25aa   Reflections"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Graph functions using reflections about the [latex]x[\/latex] -axis and the [latex]y[\/latex] -axis.<\/li>\r\n \t<li>Determine whether a function is even, odd, or neither from its graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAnother transformation that can be applied to a function is a reflection over the [latex]x[\/latex]- or [latex]y[\/latex]-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis. The reflections are shown in Figure 9.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203556\/CNX_Precalc_Figure_01_05_0122.jpg\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"487\" height=\"442\" \/> Vertical and horizontal reflections of a function.[\/caption]\r\n\r\nNotice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the [latex]x[\/latex]-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the [latex]y[\/latex]-axis.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Reflections<\/h3>\r\nGiven 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 [latex]x[\/latex]-axis.\r\n\r\nGiven 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 [latex]y[\/latex]-axis.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function, reflect the graph both vertically and horizontally.<\/h3>\r\n<ol>\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 [latex]x[\/latex]-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 [latex]y[\/latex]-axis.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reflecting a Graph Horizontally and Vertically<\/h3>\r\nReflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex]\u00a0 (a) vertically and (b) horizontally.\r\n\r\n[reveal-answer q=\"211400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211400\"]\r\n\r\na. Reflecting the graph vertically means that each output value will be reflected over the horizontal [latex]t[\/latex]<em>-<\/em>axis as shown below.\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\/896\/2016\/10\/18203559\/CNX_Precalc_Figure_01_05_0132.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"975\" height=\"442\" \/> Vertical reflection of the square root function[\/caption]\r\n\r\nBecause each output value is the opposite of the original output value, we can write\r\n<p style=\"text-align: center;\">[latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt{t}[\/latex]<\/p>\r\nNotice that this is an outside change, or vertical shift, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.\r\n\r\nb. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.\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\/896\/2016\/10\/18203602\/CNX_Precalc_Figure_01_05_0142.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"975\" height=\"442\" \/> Horizontal reflection of the square root function[\/caption]\r\n\r\nBecause each input value is the opposite of the original input value, we can write\r\n<p style=\"text-align: center;\">[latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/p>\r\nNotice 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.\r\n\r\nNote 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 ,0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,0\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse an online graphing calculator to reflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.\r\n\r\n[reveal-answer q=\"362828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"362828\"]\r\n\r\na)\r\n\r\n<img class=\"alignnone 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\" \/>\r\n\r\nb)\r\n\r\n<img class=\"alignnone 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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Reflecting a Tabular Function Horizontally and Vertically<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.\r\n<ol>\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\" 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><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"608272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"608272\"]\r\n<ol>\r\n \t<li>For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the [latex]x[\/latex]-values stay the same and each output value will be the opposite of the original output value.\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><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20137<\/td>\r\n<td>\u201311<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>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.\r\n<table id=\"Table_01_05_07\" 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><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>\u22122<\/td>\r\n<td>\u22124<\/td>\r\n<td>\u22126<\/td>\r\n<td>\u22128<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n<td>1<\/td>\r\n<td>3<\/td>\r\n<td>7<\/td>\r\n<td>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 class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\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[ohm_question height=\"210\"]60650[\/ohm_question]\r\n\r\n[ohm_question height=\"340\"]113454[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\r\nSome 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 [latex]y[\/latex]-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong>\r\n\r\nIf the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph.\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\/896\/2016\/10\/18203605\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/> (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\nWe 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>.\r\n\r\nNote: 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].\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=VvUI6E78cN4\r\n<div class=\"textbox\">\r\n<h3>A General Note: Even and Odd Functions<\/h3>\r\nA function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]\r\n\r\n[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]\r\n\r\nThe graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.\r\n\r\nA function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]\r\n\r\n[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]\r\n\r\nThe graph of an odd function is symmetric about the origin.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the formula for a function, determine if the function is even, odd, or neither.<strong>\r\n<\/strong><\/h3>\r\n<ol>\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.<\/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 class=\"textbox examples\">\r\n<h3>recall evaluating functions<\/h3>\r\nWhen evaluating functions for specific input, we can wrap the variable in parentheses first, then drop in the value we want to use to evaluate it. This works whether the value is a constant or a variable.\r\n\r\nBe extra careful when evaluating a function for a negative input. The negative sign goes in the parentheses.\r\n\r\nEx. Evaluate [latex]f(x) = x^2 \\ \\text{ for} -1[\/latex]\r\n\r\n[latex]\\begin{align} f(-1) &amp;= (-1)^2 \\\\ &amp;= 1 \\end{align}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining whether a Function Is Even, Odd, or Neither<\/h3>\r\nIs the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?\r\n\r\n[reveal-answer q=\"936347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"936347\"]\r\n\r\nWithout 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.\r\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\r\nThis does not return us to the original function, so this function is not even. We can now test the rule for odd functions.\r\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\r\nBecause [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.\r\n<h4>Analysis of the Solution<\/h4>\r\nConsider the graph of [latex]f[\/latex]. 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203607\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIs the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?\r\n\r\n[reveal-answer q=\"630369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"630369\"]\r\n\r\nEven\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"250\"]112703[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Graph functions using reflections about the [latex]x[\/latex] -axis and the [latex]y[\/latex] -axis.<\/li>\n<li>Determine whether a function is even, odd, or neither from its graph.<\/li>\n<\/ul>\n<\/div>\n<p>Another transformation that can be applied to a function is a reflection over the [latex]x[\/latex]&#8211; or [latex]y[\/latex]-axis. A <strong>vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis, while a <strong>horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis. The reflections are shown in Figure 9.<\/p>\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\/896\/2016\/10\/18203556\/CNX_Precalc_Figure_01_05_0122.jpg\" alt=\"Graph of the vertical and horizontal reflection of a function.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical and horizontal reflections of a function.<\/p>\n<\/div>\n<p>Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the [latex]x[\/latex]-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the [latex]y[\/latex]-axis.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Reflections<\/h3>\n<p>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 [latex]x[\/latex]-axis.<\/p>\n<p>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 [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function, reflect the graph both vertically and horizontally.<\/h3>\n<ol>\n<li>Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the [latex]x[\/latex]-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 [latex]y[\/latex]-axis.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Reflecting a Graph Horizontally and Vertically<\/h3>\n<p>Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex]\u00a0 (a) vertically and (b) horizontally.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211400\">Show Solution<\/span><\/p>\n<div id=\"q211400\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. Reflecting the graph vertically means that each output value will be reflected over the horizontal [latex]t[\/latex]<em>&#8211;<\/em>axis as shown below.<\/p>\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\/896\/2016\/10\/18203559\/CNX_Precalc_Figure_01_05_0132.jpg\" alt=\"Graph of the vertical reflection of the square root function.\" width=\"975\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Vertical reflection of the square root function<\/p>\n<\/div>\n<p>Because each output value is the opposite of the original output value, we can write<\/p>\n<p style=\"text-align: center;\">[latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt{t}[\/latex]<\/p>\n<p>Notice that this is an outside change, or vertical shift, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.<\/p>\n<p>b. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.<\/p>\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\/896\/2016\/10\/18203602\/CNX_Precalc_Figure_01_05_0142.jpg\" alt=\"Graph of the horizontal reflection of the square root function.\" width=\"975\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Horizontal reflection of the square root function<\/p>\n<\/div>\n<p>Because each input value is the opposite of the original input value, we can write<\/p>\n<p style=\"text-align: center;\">[latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/p>\n<p>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>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 ,0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,0\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q362828\">Show Solution<\/span><\/p>\n<div id=\"q362828\" class=\"hidden-answer\" style=\"display: none\">\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone 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\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone 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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Reflecting a Tabular Function Horizontally and Vertically<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.<\/p>\n<ol>\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\" 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><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q608272\">Show Solution<\/span><\/p>\n<div id=\"q608272\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the [latex]x[\/latex]-values stay the same and each output value will be the opposite of the original output value.<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><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>2<\/td>\n<td>4<\/td>\n<td>6<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>\u20131<\/td>\n<td>\u20133<\/td>\n<td>\u20137<\/td>\n<td>\u201311<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>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.<br \/>\n<table id=\"Table_01_05_07\" 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><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>\u22122<\/td>\n<td>\u22124<\/td>\n<td>\u22126<\/td>\n<td>\u22128<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\n<td>1<\/td>\n<td>3<\/td>\n<td>7<\/td>\n<td>11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q230301\">Show Solution<\/span><\/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<p><iframe loading=\"lazy\" id=\"ohm60650\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=60650&theme=oea&iframe_resize_id=ohm60650&show_question_numbers\" width=\"100%\" height=\"210\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm113454\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=113454&theme=oea&iframe_resize_id=ohm113454&show_question_numbers\" width=\"100%\" height=\"340\"><\/iframe><\/p>\n<\/div>\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\n<p>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 [latex]y[\/latex]-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong><\/p>\n<p>If the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph.<\/p>\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\/896\/2016\/10\/18203605\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\">(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<p>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>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<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Introduction to Odd and Even Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VvUI6E78cN4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Even and Odd Functions<\/h3>\n<p>A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<p>[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<p>The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.<\/p>\n<p>A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<p>[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/p>\n<p>The graph of an odd function is symmetric about the origin.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the formula for a function, determine if the function is even, odd, or neither.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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.<\/li>\n<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>recall evaluating functions<\/h3>\n<p>When evaluating functions for specific input, we can wrap the variable in parentheses first, then drop in the value we want to use to evaluate it. This works whether the value is a constant or a variable.<\/p>\n<p>Be extra careful when evaluating a function for a negative input. The negative sign goes in the parentheses.<\/p>\n<p>Ex. Evaluate [latex]f(x) = x^2 \\ \\text{ for} -1[\/latex]<\/p>\n<p>[latex]\\begin{align} f(-1) &= (-1)^2 \\\\ &= 1 \\end{align}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining whether a Function Is Even, Odd, or Neither<\/h3>\n<p>Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q936347\">Show Solution<\/span><\/p>\n<div id=\"q936347\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\n<p>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<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\n<p>Because [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Consider the graph of [latex]f[\/latex]. 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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203607\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q630369\">Show Solution<\/span><\/p>\n<div id=\"q630369\" class=\"hidden-answer\" style=\"display: none\">\n<p>Even<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm112703\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=112703&theme=oea&iframe_resize_id=ohm112703&show_question_numbers\" width=\"100%\" height=\"250\"><\/iframe><\/p>\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-151\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 112701, 60650, 113454,112703. <strong>Authored by<\/strong>: Lumen Learning. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":28,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"original\",\"description\":\"Question ID 112701, 60650, 113454,112703\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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