{"id":982,"date":"2023-07-13T21:15:39","date_gmt":"2023-07-13T21:15:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=982"},"modified":"2023-08-10T00:21:59","modified_gmt":"2023-08-10T00:21:59","slug":"summary-symmetry-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-symmetry-of-a-function\/","title":{"raw":"Summary: Symmetry of a Function","rendered":"Summary: Symmetry of a Function"},"content":{"raw":"<h2>\u00a0Key Concepts<\/h2>\r\n<section id=\"fs-id1165135264626\" class=\"key-concepts\">\r\n<ul id=\"fs-id1165135264630\">\r\n \t<li>The point that is symmetric to [latex](a, b)[\/latex] about the [latex]x[\/latex]-axis is [latex](a, -b)[\/latex].<\/li>\r\n \t<li><span style=\"font-size: 1rem; text-align: initial;\">The point that is symmetric to [latex](a, b)[\/latex] about the [latex]y[\/latex]-axis is [latex](-a, b)[\/latex]<\/span>.<\/li>\r\n \t<li>The point that is symmetric to [latex](a, b)[\/latex] about the origin is [latex](-a, -b)[\/latex].<\/li>\r\n \t<li>A function is symmetric about the y-axis 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 symmetric about the origin 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><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>even function<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>odd function<span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">\u00a0is symmetric about the origin.<\/span><\/li>\r\n \t<li>To test whether an equation with two variables is symmetric about the [latex]x[\/latex]-axis, substitute [latex]-y[\/latex] for [latex]y[\/latex].<\/li>\r\n \t<li>To test whether an equation with two variables is symmetric about the [latex]y[\/latex]-axis, substitute [latex]-x[\/latex] for [latex]x[\/latex].<\/li>\r\n \t<li>To test whether an equation with two variables is symmetric about the origin, substitute\u00a0[latex]-x[\/latex] for [latex]x[\/latex] and [latex]-y[\/latex] for [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137448239\" class=\"definition\">\r\n \t<dt>even function<\/dt>\r\n \t<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex], and is symmetric about the [latex]y\\text{-}[\/latex] axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134259240\" class=\"definition\">\r\n \t<dt>odd function<\/dt>\r\n \t<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex], and is symmetric about the origin<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<h2>\u00a0Key Concepts<\/h2>\n<section id=\"fs-id1165135264626\" class=\"key-concepts\">\n<ul id=\"fs-id1165135264630\">\n<li>The point that is symmetric to [latex](a, b)[\/latex] about the [latex]x[\/latex]-axis is [latex](a, -b)[\/latex].<\/li>\n<li><span style=\"font-size: 1rem; text-align: initial;\">The point that is symmetric to [latex](a, b)[\/latex] about the [latex]y[\/latex]-axis is [latex](-a, b)[\/latex]<\/span>.<\/li>\n<li>The point that is symmetric to [latex](a, b)[\/latex] about the origin is [latex](-a, -b)[\/latex].<\/li>\n<li>A function is symmetric about the y-axis 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 symmetric about the origin 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><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>even function<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>odd function<span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial;\">\u00a0is symmetric about the origin.<\/span><\/li>\n<li>To test whether an equation with two variables is symmetric about the [latex]x[\/latex]-axis, substitute [latex]-y[\/latex] for [latex]y[\/latex].<\/li>\n<li>To test whether an equation with two variables is symmetric about the [latex]y[\/latex]-axis, substitute [latex]-x[\/latex] for [latex]x[\/latex].<\/li>\n<li>To test whether an equation with two variables is symmetric about the origin, substitute\u00a0[latex]-x[\/latex] for [latex]x[\/latex] and [latex]-y[\/latex] for [latex]y[\/latex].<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137448239\" class=\"definition\">\n<dt>even function<\/dt>\n<dd id=\"fs-id1165137448244\">a function whose graph is unchanged by horizontal reflection, [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex], and is symmetric about the [latex]y\\text{-}[\/latex] axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134259240\" class=\"definition\">\n<dt>odd function<\/dt>\n<dd id=\"fs-id1165134259246\">a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex], and is symmetric about the origin<\/dd>\n<\/dl>\n<\/div>\n<\/section>\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-982\">\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>Summary: 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":35,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Summary: 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-982","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":115,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/982","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":3,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/982\/revisions"}],"predecessor-version":[{"id":987,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/982\/revisions\/987"}],"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\/982\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=982"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=982"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=982"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}