{"id":1729,"date":"2016-11-02T20:14:26","date_gmt":"2016-11-02T20:14:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1729"},"modified":"2017-04-26T16:50:49","modified_gmt":"2017-04-26T16:50:49","slug":"summary-characteristics-of-power-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/summary-characteristics-of-power-and-polynomial-functions\/","title":{"raw":"Summary: Characteristics of Power and Polynomial Functions","rendered":"Summary: Characteristics of Power and Polynomial Functions"},"content":{"raw":"<section id=\"fs-id1165137724050\" class=\"key-equations\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Key Equations<\/h2>\r\n<table id=\"eip-id1165134063974\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"middle\" data-align=\"left\">general form of a polynomial function<\/td>\r\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137731646\" class=\"key-concepts\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135438864\">\r\n \t<li>A power function is a variable base raised to a number power.<\/li>\r\n \t<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\r\n \t<li>The end behavior depends on whether the power is even or odd.<\/li>\r\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\r\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\r\n \t<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\r\n \t<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em data-effect=\"italics\">x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\r\n<\/ul>\r\n<div data-type=\"glossary\">\r\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\r\n<dl id=\"fs-id1165137668266\" class=\"definition\">\r\n \t<dt><strong>coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135194918\" class=\"definition\">\r\n \t<dt><strong>continuous function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832108\" class=\"definition\">\r\n \t<dt><strong>degree<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832115\" class=\"definition\">\r\n \t<dt><strong>end behavior<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>leading coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>leading term<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong>power function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=k{x}^{p}[\/latex]\u00a0where <em>k\u00a0<\/em>is a constant, the base is a variable, and the exponent, <em>p<\/em>,\u00a0is a constant\u00a0smooth curve\u00a0a graph with no sharp corners<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>term of a polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133085661\" class=\"definition\">\r\n \t<dt><strong>turning point<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<section id=\"fs-id1165137724050\" class=\"key-equations\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Equations<\/h2>\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">general form of a polynomial function<\/td>\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137731646\" class=\"key-concepts\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Concepts<\/h2>\n<ul id=\"fs-id1165135438864\">\n<li>A power function is a variable base raised to a number power.<\/li>\n<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\n<li>The end behavior depends on whether the power is even or odd.<\/li>\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\n<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\n<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em data-effect=\"italics\">x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\n<\/ul>\n<div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165137668266\" class=\"definition\">\n<dt><strong>coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\" class=\"definition\">\n<dt><strong>continuous function<\/strong><\/dt>\n<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\" class=\"definition\">\n<dt><strong>degree<\/strong><\/dt>\n<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\" class=\"definition\">\n<dt><strong>end behavior<\/strong><\/dt>\n<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>leading coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>leading term<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>power function<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=k{x}^{p}[\/latex]\u00a0where <em>k\u00a0<\/em>is a constant, the base is a variable, and the exponent, <em>p<\/em>,\u00a0is a constant\u00a0smooth curve\u00a0a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>term of a polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>turning point<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/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-1729\">\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":21,"menu_order":6,"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":"73ac2cf5-35b4-4cde-962f-a8024f812f13","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1729","chapter","type-chapter","status-publish","hentry"],"part":1700,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1729","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1729\/revisions"}],"predecessor-version":[{"id":4315,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1729\/revisions\/4315"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1700"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1729\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=1729"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1729"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1729"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=1729"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}