{"id":217,"date":"2023-06-21T13:22:45","date_gmt":"2023-06-21T13:22:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-polynomial-functions\/"},"modified":"2023-06-21T13:22:45","modified_gmt":"2023-06-21T13:22:45","slug":"summary-graphs-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-polynomial-functions\/","title":{"raw":"Summary: Graphs of Polynomial Functions","rendered":"Summary: Graphs of Polynomial Functions"},"content":{"raw":"\n\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\n \t<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\n \t<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the <em>x<\/em>-axis.<\/li>\n \t<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercept.<\/li>\n \t<li>The graph of a polynomial will cross the x-axis at a zero with odd multiplicity.<\/li>\n \t<li>The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity.<\/li>\n \t<li>The end behavior of a polynomial function depends on the leading term.<\/li>\n \t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n \t<li>A polynomial function of degree <em>n<\/em>&nbsp;has at most&nbsp;<em>n <\/em>\u2013&nbsp;1 turning points.<\/li>\n \t<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>&nbsp;n <\/em>\u2013&nbsp;1 turning points.<\/li>\n \t<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\n \t<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]&nbsp;have opposite signs, then there exists at least one value <em>c<\/em>&nbsp;between <em>a<\/em>&nbsp;and <em>b<\/em>&nbsp;for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n \t<dt><strong>global maximum<\/strong><\/dt>\n \t<dd id=\"fs-id1165132943525\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]&nbsp;where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]&nbsp;for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n \t<dt><b>global minimum<\/b><\/dt>\n \t<dd id=\"fs-id1165132943525\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]&nbsp;where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n \t<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\n \t<dd id=\"fs-id1165132943525\">for two numbers <em>a<\/em>&nbsp;and <em>b<\/em>&nbsp;in the domain of <em>f<\/em>,&nbsp;if [latex]a&lt;b[\/latex]&nbsp;and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],&nbsp;then the function <em>f<\/em>&nbsp;takes on every value between [latex]f\\left(a\\right)[\/latex]&nbsp;and [latex]f\\left(b\\right)[\/latex];&nbsp;specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n \t<dt><strong>multiplicity<\/strong><\/dt>\n \t<dd id=\"fs-id1165132943525\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]&nbsp;is a zero of multiplicity <em>p<\/em>.<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\n<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\n<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the <em>x<\/em>-axis.<\/li>\n<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercept.<\/li>\n<li>The graph of a polynomial will cross the x-axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity.<\/li>\n<li>The end behavior of a polynomial function depends on the leading term.<\/li>\n<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>A polynomial function of degree <em>n<\/em>&nbsp;has at most&nbsp;<em>n <\/em>\u2013&nbsp;1 turning points.<\/li>\n<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>&nbsp;n <\/em>\u2013&nbsp;1 turning points.<\/li>\n<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\n<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]&nbsp;have opposite signs, then there exists at least one value <em>c<\/em>&nbsp;between <em>a<\/em>&nbsp;and <em>b<\/em>&nbsp;for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>global maximum<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]&nbsp;where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]&nbsp;for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><b>global minimum<\/b><\/dt>\n<dd id=\"fs-id1165132943525\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]&nbsp;where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">for two numbers <em>a<\/em>&nbsp;and <em>b<\/em>&nbsp;in the domain of <em>f<\/em>,&nbsp;if [latex]a<b[\/latex]&nbsp;and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],&nbsp;then the function <em>f<\/em>&nbsp;takes on every value between [latex]f\\left(a\\right)[\/latex]&nbsp;and [latex]f\\left(b\\right)[\/latex];&nbsp;specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>multiplicity<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]&nbsp;is a zero of multiplicity <em>p<\/em>.<\/dd>\n<\/dl>\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-217\">\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><\/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":14,"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\"}]","CANDELA_OUTCOMES_GUID":"93854fb6-2394-411d-80cb-326ecb8224fd","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-217","chapter","type-chapter","status-publish","hentry"],"part":202,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/217","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\/395986"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/217\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/202"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/217\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=217"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=217"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=217"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=217"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}