{"id":964,"date":"2023-07-13T16:14:10","date_gmt":"2023-07-13T16:14:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=964"},"modified":"2023-08-14T06:01:24","modified_gmt":"2023-08-14T06:01:24","slug":"symmetry-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/symmetry-of-a-function\/","title":{"raw":"\u25aa Symmetry of a Function","rendered":"\u25aa Symmetry of a Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning OUTCOMES<\/h3>\r\n<ul>\r\n \t<li>Recognize symmetry of a function.<\/li>\r\n \t<li>Determine even and odd functions.<\/li>\r\n \t<li>Determine symmetry of an equation (with two variables) graphically.<\/li>\r\n \t<li>Determine symmetry of an equation (with two variables) algebraically.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Symmetry of a Function<\/h2>\r\n<p id=\"fs-id1170572213225\">The graphs of certain functions have symmetrical properties that help us understand the function and the shape of its graph. For example, consider the function [latex]f(x)=x^4-2x^2-3[\/latex]\u00a0shown in Figure 2(a). If we take the part of the curve that lies to the right of the [latex]y[\/latex]-axis and flip it over the [latex]y[\/latex]-axis, it lays exactly on top of the curve to the left of the [latex]y[\/latex]-axis. In this case, we say the function has symmetry about the [latex]y[\/latex]-axis. On the other hand, consider the function [latex] f(x)=x^3-4x [\/latex] shown in Figure 2(b). If we take the graph and rotate it 180\u00b0 about the origin, the new graph will look exactly same. In this case, we say the function is\u00a0symmetric about the origin.<\/p>\r\n<img class=\"aligncenter wp-image-932\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs.jpg\" alt=\"\" width=\"550\" height=\"321\" \/>\r\n<div class=\"textbox shaded\">Figure 2. (a) A graph that is symmetric about the y-axis.\u00a0 \u00a0(b) A graph that is symmetric about the origin.<\/div>\r\nIn other words, if we reflect the graph (a) about the [latex]y[\/latex]-axis,\u00a0we can see that the new graph looks exactly same as the original graph. Also, if we reflect the graph (b) about the origin (or about the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis), the new graph looks exactly same as the original graph. To make it happen,\u00a0[latex](-x, y)[\/latex] should be on the graph (a) whenever [latex](x, y)[\/latex] is on the graph (a) and\u00a0[latex](-x, -y)[\/latex] should be on the graph (b) whenever [latex](x, y)[\/latex] is on the graph (b).\u00a0What does this mean?\r\n\r\nFor example, if [latex](1, -4)[\/latex] is on the graph (a), [latex](-1, -4)[\/latex] should be on the graph (a) because it is symmetric about the [latex]y[\/latex]-axis. Also, if\u00a0[latex](1, -3)[\/latex] is on the graph (b), [latex](-1, 3)[\/latex] should be on the graph (b) because it is symmetric about the origin.\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 485.3px;\"><img class=\"size-medium wp-image-973 alignright\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-300x269.png\" alt=\"\" width=\"300\" height=\"269\" \/><\/td>\r\n<td style=\"width: 485.3px;\"><img class=\"size-medium wp-image-974 alignleft\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-1-287x300.png\" alt=\"\" width=\"287\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>A GENERAL NOTE: SYmmetry of a Function<\/h3>\r\n<ul>\r\n \t<li>A function is <strong>symmetric about the y-axis<\/strong> if\u00a0[latex](-x, y)[\/latex] is on the graph of the function whenever [latex](x, y)[\/latex] is on the graph.<\/li>\r\n \t<li>A function is <strong>symmetric about the origin<\/strong> if\u00a0[latex](-x, -y)[\/latex] is on the graph of the function whenever [latex](x, y)[\/latex] is on the graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Even and Odd Functions<\/h2>\r\n<p id=\"fs-id1170572173116\">If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[\/latex] has symmetry? Looking at Figure 2(a) again, we see that since [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis, if the point [latex](x, y)[\/latex] is on the graph, the point [latex](-x, y)[\/latex] is on the graph. In other words, [latex]f(-x)=f(x)[\/latex]. If a function [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an\u00a0<strong>even function<\/strong>, which has symmetry about the [latex]y[\/latex]-axis. For example, [latex]f(x)=x^2[\/latex] is even because<\/p>\r\n<p style=\"text-align: center;\">[latex]f(-x)=(-x)^2=x^2=f(x)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1170572552270\" class=\"equation unnumbered\">\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">In contrast, looking at Figure 2(b) again, if a function [latex]f[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is symmetric about the origin, then whenever the point [latex](x, y)[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is on the graph, the point [latex](-x, -y)[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is also on the graph. In other words, [latex]f(-x)=-f(x)[\/latex].<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0If [latex]f[\/latex] h<\/span><span style=\"font-size: 1rem; text-align: initial;\">as this property, we say [latex]f[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is an\u00a0<\/span><strong style=\"font-size: 1rem; text-align: initial;\">odd function<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, which has symmetry about the origin. For example, [latex]f(x)=x^3[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is odd because<\/span>\r\n<p style=\"text-align: center;\">[latex]f(-x)=(-x)^3=-x^3=-f(x)[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>A GENERAL NOTE: Even and Odd Functions<\/h3>\r\n<ul>\r\n \t<li><span style=\"background-color: initial; font-size: 1rem; orphans: 1; text-align: initial;\">If [latex]f(-x)=f(x)[\/latex]\u00a0for all [latex]x[\/latex]\u00a0in the domain of [latex]f[\/latex], then [latex]f[\/latex]\u00a0is an even function. An\u00a0<\/span><strong style=\"background-color: initial; font-size: 1rem; orphans: 1; text-align: initial;\">even function<\/strong><span style=\"background-color: initial; font-size: 1rem; orphans: 1; text-align: initial;\">\u00a0is symmetric about the [latex]y[\/latex]-axis.<\/span><\/li>\r\n \t<li><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">If [latex]f(-x)=-f(x)[\/latex]\u00a0for all [latex]x[\/latex]\u00a0in the domain of [latex]f[\/latex], then [latex]f[\/latex]\u00a0is an odd function. An\u00a0<\/span><strong style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">odd function<\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">\u00a0is symmetric about the origin.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572169671\" class=\"textbox exercises\">\r\n<h3>EXAMPLE: EVEN AND ODD FUNCTIONS<\/h3>\r\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\r\n\r\n<ol>\r\n \t<li style=\"list-style-type: none;\">\r\n<ol>\r\n \t<li>[latex]f(x)=-5x^4+7x^2-2[\/latex]<\/li>\r\n \t<li>[latex]g(x)=2x^5-4x+5[\/latex]<\/li>\r\n \t<li>[latex]h(x)=\\frac{3x}{x^2+1}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"676427\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"676427\"]To determine whether a function is even or odd, we evaluate [latex]f(-x)[\/latex] and compare it to [latex]f(x)[\/latex] and [latex]-f(x)[\/latex].\r\n\r\n1. [latex] f(-x)=-5(-x)^4+7(-x)^2-2=-5x^4+7x^2-2=f(x)[\/latex]. Therefore, [latex]f[\/latex] is even.\r\n\r\n2. [latex] g(-x)=2(-x)^5-4(-x)+5=-2x^5+4x+5[\/latex]. Now, [latex]g(-x)\\ne g(x)[\/latex].\u00a0Furthermore, noting that\u00a0[latex]-g(x)=-(2x^5-4x+5)=-2x^5+4x-5[\/latex], we see that [latex]g(-x)\\ne -g(x)[\/latex]. Therefore, [latex]g[\/latex] is neither even nor odd.\r\n\r\n3. [latex]h(-x)= \\frac{3(-x)}{(-x)^2+1}=\\frac{-3x}{x^2+1}=-\\frac{3x}{x^2+1}=-h(x)[\/latex]. Therefore, [latex]h[\/latex] is odd.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine whether [latex]f(x)=4x^3-5x[\/latex] is even, odd, or neither.\r\n\r\n[reveal-answer q=\"82355\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"82355\"][latex]f(x)[\/latex] is odd.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Determine Symmetry of an Equation (with two variables) Graphically<\/h2>\r\n<div class=\"textbox\">\r\n<h3>How to: Determine Symmetry of an Equation (with two variables) Graphically<\/h3>\r\n<ul>\r\n \t<li>An equation with two variables is <strong>symmetric about the\u00a0x-axis<\/strong>\u00a0if we reflect its graph about the [latex]x[\/latex]-axis and the new graph looks exactly same as its original graph.<\/li>\r\n \t<li>An equation with two variables is <strong>symmetric about the y-axis<\/strong>\u00a0if\u00a0we reflect the graph about the [latex]y[\/latex]-axis and the new graph looks exactly same as its original graph.<\/li>\r\n \t<li>An equation is <strong>symmetric about the origin<\/strong>\u00a0if\u00a0we\u00a0rotate the graph 180\u00b0 about the origin (or reflect the graph about the [latex]x[\/latex]-axis and [latex]y[\/latex]-axis) and the\u00a0new graph looks exactly same as its original graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determine Symmetry of an Equation (with two variables) Graphically<\/h3>\r\nUsing the given graph of an equation, determine whether the graph is symmetric about [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and\/or the origin.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 640px;\" border=\"none\">\r\n<tbody>\r\n<tr style=\"height: 303px;\">\r\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 303px;\">\r\n<div class=\"textbox shaded\">1.\u00a0<img class=\"aligncenter wp-image-959 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7.png\" alt=\"A graph of hyperbola\" width=\"244\" height=\"234\" \/><\/div><\/td>\r\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 303px;\">\r\n<div class=\"textbox shaded\">2.\u00a0<img class=\"aligncenter wp-image-954 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4.png\" alt=\"A graph of sine function\" width=\"362\" height=\"234\" \/><\/div><\/td>\r\n<\/tr>\r\n<tr style=\"height: 337px;\">\r\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 337px;\">\r\n<div class=\"textbox shaded\">3.\u00a0<img class=\"aligncenter wp-image-953 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-3.png\" alt=\"A graph of exponential function\" width=\"260\" height=\"265\" \/><\/div><\/td>\r\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 337px;\">\r\n<div class=\"textbox shaded\">4.\u00a0<img class=\"aligncenter wp-image-955 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5.png\" alt=\"A graph of quadratic function\" width=\"276\" height=\"264\" \/><\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center; vertical-align: top;\">\r\n<div class=\"textbox shaded\">5.\u00a0<img class=\"aligncenter wp-image-960 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8.png\" alt=\"A graph of x+y^2=4\" width=\"298\" height=\"264\" \/><\/div><\/td>\r\n<td style=\"width: 50%; text-align: center; vertical-align: top;\">\r\n<div class=\"textbox shaded\">6.\u00a0<img class=\"aligncenter wp-image-958 \" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6.png\" alt=\"A graph of ellipse\" width=\"321\" height=\"263\" \/><\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"843042\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"843042\"]\r\n\r\n1. Symmetric about the [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and the origin\r\n\r\n2. Symmetric about the origin\r\n\r\n3. No symmetry\r\n\r\n4. Symmetric about the [latex]y[\/latex]-axis\r\n\r\n5. Symmetric about the [latex]x[\/latex]-axis\r\n\r\n6. Symmetric about the\u00a0[latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and the origin [\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<h2>Determine Symmetry of an Equation (with two variables) Algebraically<\/h2>\r\n<div class=\"textbox\">\r\n<h3>How to: Determine Symmetry of an Equation (with two variables) Algebraically<\/h3>\r\n<div class=\"textbox shaded\"><strong>Symmetry about the x-axis<\/strong><\/div>\r\n<ol>\r\n \t<li>Substitute [latex]-y[\/latex] for all the [latex]y[\/latex]'s in the equation.<\/li>\r\n \t<li>Simplify the equation.<\/li>\r\n \t<li>If the simplified equation is exactly same as the original equation, the equation is symmetric about the [latex]x[\/latex]-axis.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n<div class=\"textbox shaded\"><strong>Symmetry about the y-axis<\/strong><\/div>\r\n<ol>\r\n \t<li>Substitute [latex]-x[\/latex] for all the [latex]x[\/latex]'s in the equation.<\/li>\r\n \t<li>Simplify the equation.<\/li>\r\n \t<li>If the simplified equation is exactly same as the original equation, the equation is symmetric about the [latex]y[\/latex]-axis.<\/li>\r\n<\/ol>\r\n&nbsp;\r\n<div class=\"textbox shaded\"><strong>Symmetry about the origin<\/strong><\/div>\r\n<ol>\r\n \t<li>Substitute [latex]-x[\/latex] for all the [latex]x[\/latex]'s and [latex]-y[\/latex] for all the [latex]y[\/latex]'s in the equation.<\/li>\r\n \t<li>Simplify the equation.<\/li>\r\n \t<li>If the simplified equation is exactly same as the original equation, the equation is symmetric about the origin.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determine Symmetry of a Function Algebraically<\/h3>\r\nDetermine whether each equation is symmetric about [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and\/or the origin.\r\n<ol>\r\n \t<li>[latex]x^2+y^2=9[\/latex]<\/li>\r\n \t<li>[latex]y=x^2-5[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"760361\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"760361\"]\r\n\r\n1. (a) About x-axis: If we substitute [latex]-y[\/latex] for [latex]y[\/latex], [latex]x^2+(-y)^2=9[\/latex]. So, [latex]x^2+y^2=9[\/latex], which is same as the original equation.\r\n(b) About y-axis: If we substitute [latex]-x[\/latex] for [latex]x[\/latex], [latex](-x)^2+y^2=9[\/latex]. So, [latex]x^2+y^2=9[\/latex], which is same as the original equation.\r\n(c) About origin: If we substitute [latex]-x[\/latex] for [latex]x[\/latex] and [latex]-y[\/latex] for [latex]y[\/latex], [latex](-x)^2+(-y)^2=9[\/latex]. So, [latex]x^2+y^2=9[\/latex], which is same as the original equation.\r\nTherefore, [latex]x^2+y^2=9[\/latex] is <strong>symmetric about the x-axis, the y-axis, and the origin<\/strong>.\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">2.\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">(a) About x-axis: If we substitute [latex]-y[\/latex] for [latex]y[\/latex], [latex](-y)=x^2-5[\/latex]. So, [latex]y=-(x^2-5)[\/latex] and [latex]y=-x^2+5[\/latex], which is not same as the original equation.\r\n<\/span>(b) About y-axis: If we substitute [latex]-x[\/latex] for [latex]x[\/latex], [latex]y=(-x)^2-5[\/latex]. So, [latex]y=x^2-5[\/latex], which is same as the original equation.\r\n(c) About origin: If we substitute [latex]-x[\/latex] for [latex]x[\/latex] and [latex]-y[\/latex] for [latex]y[\/latex], [latex](-y)=(-x)^2-5[\/latex]. So, [latex]-y=x^2-5[\/latex] and [latex]y=-(x^2-5)[\/latex]. Thus, [latex]y=-x^2+5[\/latex], which is not same as the original equation.\r\nTherefore, [latex]y=x^2-5[\/latex] is <strong>symmetric about the y-axis<\/strong>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine whether each equation is symmetric about [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and\/or the origin.\r\n<ol>\r\n \t<li>[latex]y=x^3+1[\/latex]<\/li>\r\n \t<li>[latex]y^2+4x-8=0[\/latex]<\/li>\r\n \t<li>[latex]4x^2+9y^2=36[\/latex]<\/li>\r\n \t<li>[latex]y=\\sqrt[3]{x}[\/latex]<\/li>\r\n \t<li>[latex]y=x^6-7x^2+3[\/latex]<\/li>\r\n \t<li>[latex]y=\\frac{x}{x^2-1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"142186\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"142186\"]\r\n\r\n1. [latex]y=x^3+1[\/latex] is not symmetric.\r\n\r\n2. [latex]y^2+4x-8=0[\/latex] is symmetric about the x-axis.\r\n\r\n3. [latex]4x^2+9y^2=36[\/latex] is symmetric about the x-axis, the y-axis, and the origin.\r\n\r\n4. [latex]y=\\sqrt[4]{x}[\/latex] is not symmetric.\r\n\r\n5. [latex]y=x^6-7x^2+3[\/latex] is symmetric about the y-axis.\r\n\r\n6. [latex]y=\\frac{x}{x^2-1}[\/latex] is symmetric about the x-axis, the y-axis, and the origin.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning OUTCOMES<\/h3>\n<ul>\n<li>Recognize symmetry of a function.<\/li>\n<li>Determine even and odd functions.<\/li>\n<li>Determine symmetry of an equation (with two variables) graphically.<\/li>\n<li>Determine symmetry of an equation (with two variables) algebraically.<\/li>\n<\/ul>\n<\/div>\n<h2>Symmetry of a Function<\/h2>\n<p id=\"fs-id1170572213225\">The graphs of certain functions have symmetrical properties that help us understand the function and the shape of its graph. For example, consider the function [latex]f(x)=x^4-2x^2-3[\/latex]\u00a0shown in Figure 2(a). If we take the part of the curve that lies to the right of the [latex]y[\/latex]-axis and flip it over the [latex]y[\/latex]-axis, it lays exactly on top of the curve to the left of the [latex]y[\/latex]-axis. In this case, we say the function has symmetry about the [latex]y[\/latex]-axis. On the other hand, consider the function [latex]f(x)=x^3-4x[\/latex] shown in Figure 2(b). If we take the graph and rotate it 180\u00b0 about the origin, the new graph will look exactly same. In this case, we say the function is\u00a0symmetric about the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-932\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs.jpg\" alt=\"\" width=\"550\" height=\"321\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs.jpg 731w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs-300x175.jpg 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs-65x38.jpg 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs-225x131.jpg 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/Symmetric-Function-Graphs-350x204.jpg 350w\" sizes=\"auto, (max-width: 550px) 100vw, 550px\" \/><\/p>\n<div class=\"textbox shaded\">Figure 2. (a) A graph that is symmetric about the y-axis.\u00a0 \u00a0(b) A graph that is symmetric about the origin.<\/div>\n<p>In other words, if we reflect the graph (a) about the [latex]y[\/latex]-axis,\u00a0we can see that the new graph looks exactly same as the original graph. Also, if we reflect the graph (b) about the origin (or about the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis), the new graph looks exactly same as the original graph. To make it happen,\u00a0[latex](-x, y)[\/latex] should be on the graph (a) whenever [latex](x, y)[\/latex] is on the graph (a) and\u00a0[latex](-x, -y)[\/latex] should be on the graph (b) whenever [latex](x, y)[\/latex] is on the graph (b).\u00a0What does this mean?<\/p>\n<p>For example, if [latex](1, -4)[\/latex] is on the graph (a), [latex](-1, -4)[\/latex] should be on the graph (a) because it is symmetric about the [latex]y[\/latex]-axis. Also, if\u00a0[latex](1, -3)[\/latex] is on the graph (b), [latex](-1, 3)[\/latex] should be on the graph (b) because it is symmetric about the origin.<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 485.3px;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-973 alignright\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-300x269.png\" alt=\"\" width=\"300\" height=\"269\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-300x269.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-65x58.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-225x202.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y.png 345w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 485.3px;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-974 alignleft\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-1-287x300.png\" alt=\"\" width=\"287\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-1-287x300.png 287w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-1-65x68.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-1-225x235.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function_-x-y-1.png 325w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A GENERAL NOTE: SYmmetry of a Function<\/h3>\n<ul>\n<li>A function is <strong>symmetric about the y-axis<\/strong> if\u00a0[latex](-x, y)[\/latex] is on the graph of the function whenever [latex](x, y)[\/latex] is on the graph.<\/li>\n<li>A function is <strong>symmetric about the origin<\/strong> if\u00a0[latex](-x, -y)[\/latex] is on the graph of the function whenever [latex](x, y)[\/latex] is on the graph.<\/li>\n<\/ul>\n<\/div>\n<h2>Even and Odd Functions<\/h2>\n<p id=\"fs-id1170572173116\">If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[\/latex] has symmetry? Looking at Figure 2(a) again, we see that since [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis, if the point [latex](x, y)[\/latex] is on the graph, the point [latex](-x, y)[\/latex] is on the graph. In other words, [latex]f(-x)=f(x)[\/latex]. If a function [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an\u00a0<strong>even function<\/strong>, which has symmetry about the [latex]y[\/latex]-axis. For example, [latex]f(x)=x^2[\/latex] is even because<\/p>\n<p style=\"text-align: center;\">[latex]f(-x)=(-x)^2=x^2=f(x)[\/latex].<\/p>\n<div id=\"fs-id1170572552270\" class=\"equation unnumbered\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">In contrast, looking at Figure 2(b) again, if a function [latex]f[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is symmetric about the origin, then whenever the point [latex](x, y)[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is on the graph, the point [latex](-x, -y)[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is also on the graph. In other words, [latex]f(-x)=-f(x)[\/latex].<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0If [latex]f[\/latex] h<\/span><span style=\"font-size: 1rem; text-align: initial;\">as this property, we say [latex]f[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is an\u00a0<\/span><strong style=\"font-size: 1rem; text-align: initial;\">odd function<\/strong><span style=\"font-size: 1rem; text-align: initial;\">, which has symmetry about the origin. For example, [latex]f(x)=x^3[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is odd because<\/span><\/p>\n<p style=\"text-align: center;\">[latex]f(-x)=(-x)^3=-x^3=-f(x)[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<div class=\"textbox\">\n<h3>A GENERAL NOTE: Even and Odd Functions<\/h3>\n<ul>\n<li><span style=\"background-color: initial; font-size: 1rem; orphans: 1; text-align: initial;\">If [latex]f(-x)=f(x)[\/latex]\u00a0for all [latex]x[\/latex]\u00a0in the domain of [latex]f[\/latex], then [latex]f[\/latex]\u00a0is an even function. An\u00a0<\/span><strong style=\"background-color: initial; font-size: 1rem; orphans: 1; text-align: initial;\">even function<\/strong><span style=\"background-color: initial; font-size: 1rem; orphans: 1; text-align: initial;\">\u00a0is symmetric about the [latex]y[\/latex]-axis.<\/span><\/li>\n<li><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">If [latex]f(-x)=-f(x)[\/latex]\u00a0for all [latex]x[\/latex]\u00a0in the domain of [latex]f[\/latex], then [latex]f[\/latex]\u00a0is an odd function. An\u00a0<\/span><strong style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">odd function<\/strong><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">\u00a0is symmetric about the origin.<\/span><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572169671\" class=\"textbox exercises\">\n<h3>EXAMPLE: EVEN AND ODD FUNCTIONS<\/h3>\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>[latex]f(x)=-5x^4+7x^2-2[\/latex]<\/li>\n<li>[latex]g(x)=2x^5-4x+5[\/latex]<\/li>\n<li>[latex]h(x)=\\frac{3x}{x^2+1}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q676427\">Show Answer<\/span><\/p>\n<div id=\"q676427\" class=\"hidden-answer\" style=\"display: none\">To determine whether a function is even or odd, we evaluate [latex]f(-x)[\/latex] and compare it to [latex]f(x)[\/latex] and [latex]-f(x)[\/latex].<\/p>\n<p>1. [latex]f(-x)=-5(-x)^4+7(-x)^2-2=-5x^4+7x^2-2=f(x)[\/latex]. Therefore, [latex]f[\/latex] is even.<\/p>\n<p>2. [latex]g(-x)=2(-x)^5-4(-x)+5=-2x^5+4x+5[\/latex]. Now, [latex]g(-x)\\ne g(x)[\/latex].\u00a0Furthermore, noting that\u00a0[latex]-g(x)=-(2x^5-4x+5)=-2x^5+4x-5[\/latex], we see that [latex]g(-x)\\ne -g(x)[\/latex]. Therefore, [latex]g[\/latex] is neither even nor odd.<\/p>\n<p>3. [latex]h(-x)= \\frac{3(-x)}{(-x)^2+1}=\\frac{-3x}{x^2+1}=-\\frac{3x}{x^2+1}=-h(x)[\/latex]. Therefore, [latex]h[\/latex] is odd.<\/p><\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine whether [latex]f(x)=4x^3-5x[\/latex] is even, odd, or neither.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q82355\">Show Answer<\/span><\/p>\n<div id=\"q82355\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)[\/latex] is odd.<\/div>\n<\/div>\n<\/div>\n<h2>Determine Symmetry of an Equation (with two variables) Graphically<\/h2>\n<div class=\"textbox\">\n<h3>How to: Determine Symmetry of an Equation (with two variables) Graphically<\/h3>\n<ul>\n<li>An equation with two variables is <strong>symmetric about the\u00a0x-axis<\/strong>\u00a0if we reflect its graph about the [latex]x[\/latex]-axis and the new graph looks exactly same as its original graph.<\/li>\n<li>An equation with two variables is <strong>symmetric about the y-axis<\/strong>\u00a0if\u00a0we reflect the graph about the [latex]y[\/latex]-axis and the new graph looks exactly same as its original graph.<\/li>\n<li>An equation is <strong>symmetric about the origin<\/strong>\u00a0if\u00a0we\u00a0rotate the graph 180\u00b0 about the origin (or reflect the graph about the [latex]x[\/latex]-axis and [latex]y[\/latex]-axis) and the\u00a0new graph looks exactly same as its original graph.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determine Symmetry of an Equation (with two variables) Graphically<\/h3>\n<p>Using the given graph of an equation, determine whether the graph is symmetric about [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and\/or the origin.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 640px;\">\n<tbody>\n<tr style=\"height: 303px;\">\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 303px;\">\n<div class=\"textbox shaded\">1.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-959\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7.png\" alt=\"A graph of hyperbola\" width=\"244\" height=\"234\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7.png 431w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7-300x288.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7-65x62.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7-225x216.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-7-350x336.png 350w\" sizes=\"auto, (max-width: 244px) 100vw, 244px\" \/><\/div>\n<\/td>\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 303px;\">\n<div class=\"textbox shaded\">2.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-954\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4.png\" alt=\"A graph of sine function\" width=\"362\" height=\"234\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4.png 438w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4-300x194.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4-65x42.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4-225x145.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-4-350x226.png 350w\" sizes=\"auto, (max-width: 362px) 100vw, 362px\" \/><\/div>\n<\/td>\n<\/tr>\n<tr style=\"height: 337px;\">\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 337px;\">\n<div class=\"textbox shaded\">3.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-953\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-3.png\" alt=\"A graph of exponential function\" width=\"260\" height=\"265\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-3.png 279w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-3-65x66.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-3-225x229.png 225w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/div>\n<\/td>\n<td style=\"width: 50%; text-align: center; vertical-align: top; height: 337px;\">\n<div class=\"textbox shaded\">4.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-955\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5.png\" alt=\"A graph of quadratic function\" width=\"276\" height=\"264\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5.png 372w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5-300x287.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5-65x62.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5-225x215.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-5-350x335.png 350w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center; vertical-align: top;\">\n<div class=\"textbox shaded\">5.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-960\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8.png\" alt=\"A graph of x+y^2=4\" width=\"298\" height=\"264\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8.png 404w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8-300x266.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8-65x58.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8-225x199.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-8-350x310.png 350w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/div>\n<\/td>\n<td style=\"width: 50%; text-align: center; vertical-align: top;\">\n<div class=\"textbox shaded\">6.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-958\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6.png\" alt=\"A graph of ellipse\" width=\"321\" height=\"263\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6.png 434w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6-300x246.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6-65x53.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6-225x185.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.6-Symmetry-of-a-Function-6-350x287.png 350w\" sizes=\"auto, (max-width: 321px) 100vw, 321px\" \/><\/div>\n<\/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=\"q843042\">Show Answer<\/span><\/p>\n<div id=\"q843042\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. Symmetric about the [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and the origin<\/p>\n<p>2. Symmetric about the origin<\/p>\n<p>3. No symmetry<\/p>\n<p>4. Symmetric about the [latex]y[\/latex]-axis<\/p>\n<p>5. Symmetric about the [latex]x[\/latex]-axis<\/p>\n<p>6. Symmetric about the\u00a0[latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and the origin <\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h2>Determine Symmetry of an Equation (with two variables) Algebraically<\/h2>\n<div class=\"textbox\">\n<h3>How to: Determine Symmetry of an Equation (with two variables) Algebraically<\/h3>\n<div class=\"textbox shaded\"><strong>Symmetry about the x-axis<\/strong><\/div>\n<ol>\n<li>Substitute [latex]-y[\/latex] for all the [latex]y[\/latex]&#8216;s in the equation.<\/li>\n<li>Simplify the equation.<\/li>\n<li>If the simplified equation is exactly same as the original equation, the equation is symmetric about the [latex]x[\/latex]-axis.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\"><strong>Symmetry about the y-axis<\/strong><\/div>\n<ol>\n<li>Substitute [latex]-x[\/latex] for all the [latex]x[\/latex]&#8216;s in the equation.<\/li>\n<li>Simplify the equation.<\/li>\n<li>If the simplified equation is exactly same as the original equation, the equation is symmetric about the [latex]y[\/latex]-axis.<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\"><strong>Symmetry about the origin<\/strong><\/div>\n<ol>\n<li>Substitute [latex]-x[\/latex] for all the [latex]x[\/latex]&#8216;s and [latex]-y[\/latex] for all the [latex]y[\/latex]&#8216;s in the equation.<\/li>\n<li>Simplify the equation.<\/li>\n<li>If the simplified equation is exactly same as the original equation, the equation is symmetric about the origin.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determine Symmetry of a Function Algebraically<\/h3>\n<p>Determine whether each equation is symmetric about [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and\/or the origin.<\/p>\n<ol>\n<li>[latex]x^2+y^2=9[\/latex]<\/li>\n<li>[latex]y=x^2-5[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q760361\">Show Answer<\/span><\/p>\n<div id=\"q760361\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. (a) About x-axis: If we substitute [latex]-y[\/latex] for [latex]y[\/latex], [latex]x^2+(-y)^2=9[\/latex]. So, [latex]x^2+y^2=9[\/latex], which is same as the original equation.<br \/>\n(b) About y-axis: If we substitute [latex]-x[\/latex] for [latex]x[\/latex], [latex](-x)^2+y^2=9[\/latex]. So, [latex]x^2+y^2=9[\/latex], which is same as the original equation.<br \/>\n(c) About origin: If we substitute [latex]-x[\/latex] for [latex]x[\/latex] and [latex]-y[\/latex] for [latex]y[\/latex], [latex](-x)^2+(-y)^2=9[\/latex]. So, [latex]x^2+y^2=9[\/latex], which is same as the original equation.<br \/>\nTherefore, [latex]x^2+y^2=9[\/latex] is <strong>symmetric about the x-axis, the y-axis, and the origin<\/strong>.<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">2.\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">(a) About x-axis: If we substitute [latex]-y[\/latex] for [latex]y[\/latex], [latex](-y)=x^2-5[\/latex]. So, [latex]y=-(x^2-5)[\/latex] and [latex]y=-x^2+5[\/latex], which is not same as the original equation.<br \/>\n<\/span>(b) About y-axis: If we substitute [latex]-x[\/latex] for [latex]x[\/latex], [latex]y=(-x)^2-5[\/latex]. So, [latex]y=x^2-5[\/latex], which is same as the original equation.<br \/>\n(c) About origin: If we substitute [latex]-x[\/latex] for [latex]x[\/latex] and [latex]-y[\/latex] for [latex]y[\/latex], [latex](-y)=(-x)^2-5[\/latex]. So, [latex]-y=x^2-5[\/latex] and [latex]y=-(x^2-5)[\/latex]. Thus, [latex]y=-x^2+5[\/latex], which is not same as the original equation.<br \/>\nTherefore, [latex]y=x^2-5[\/latex] is <strong>symmetric about the y-axis<\/strong>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine whether each equation is symmetric about [latex]x[\/latex]-axis, [latex]y[\/latex]-axis, and\/or the origin.<\/p>\n<ol>\n<li>[latex]y=x^3+1[\/latex]<\/li>\n<li>[latex]y^2+4x-8=0[\/latex]<\/li>\n<li>[latex]4x^2+9y^2=36[\/latex]<\/li>\n<li>[latex]y=\\sqrt[3]{x}[\/latex]<\/li>\n<li>[latex]y=x^6-7x^2+3[\/latex]<\/li>\n<li>[latex]y=\\frac{x}{x^2-1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q142186\">Show Answer<\/span><\/p>\n<div id=\"q142186\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]y=x^3+1[\/latex] is not symmetric.<\/p>\n<p>2. [latex]y^2+4x-8=0[\/latex] is symmetric about the x-axis.<\/p>\n<p>3. [latex]4x^2+9y^2=36[\/latex] is symmetric about the x-axis, the y-axis, and the origin.<\/p>\n<p>4. [latex]y=\\sqrt[4]{x}[\/latex] is not symmetric.<\/p>\n<p>5. [latex]y=x^6-7x^2+3[\/latex] is symmetric about the y-axis.<\/p>\n<p>6. [latex]y=\\frac{x}{x^2-1}[\/latex] is symmetric about the x-axis, the y-axis, and the origin.<\/p>\n<\/div>\n<\/div>\n<\/div>\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-964\">\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>Symmetry of a Function. <strong>Authored by<\/strong>: Michelle Eunhee Chung. <strong>Provided by<\/strong>: Georgia State University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":705214,"menu_order":34,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Symmetry of a Function\",\"author\":\"Michelle Eunhee Chung\",\"organization\":\"Georgia State University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mchung12"],"pb_section_license":""},"chapter-type":[],"contributor":[62],"license":[],"class_list":["post-964","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":115,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/964","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/705214"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/964\/revisions"}],"predecessor-version":[{"id":1352,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/964\/revisions\/1352"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/115"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/964\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=964"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=964"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=964"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=964"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}