{"id":152,"date":"2023-09-20T22:47:54","date_gmt":"2023-09-20T22:47:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/polynomial-functions\/"},"modified":"2025-08-17T15:40:55","modified_gmt":"2025-08-17T15:40:55","slug":"basic-classes-of-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-classes-of-functions-learn-it-1\/","title":{"raw":"Basic Classes of Functions: Learn It 1","rendered":"Basic Classes of Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify polynomial degrees and solutions, and graph basic odd and even polynomials<\/li>\r\n\t<li>Graph a piecewise-defined function<\/li>\r\n\t<li>Describe how algebraic functions, like polynomials, differ from transcendental functions, like sine and exponential functions<\/li>\r\n\t<li>Draw the graph of a function after it has been moved up or down, stretched or shrunk, or flipped across an axis<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h1>Polynomial Functions<\/h1>\r\n<p id=\"fs-id1170573363307\">\u00a0A <strong>polynomial function<\/strong> is any function that can be written in the form<\/p>\r\n<div id=\"fs-id1170573359094\" class=\"equation\" style=\"text-align: center;\">[latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<p>Polynomials are defined by their <strong>degree<\/strong>, which is the highest exponent of the variable x with a non-zero coefficient. The <strong>leading coefficient<\/strong> is the coefficient of the term with the highest power.<\/p>\r\n<p>The simplest polynomial, the zero function [latex]f(x)=0[\/latex], has a degree of [latex]0[\/latex]. A polynomial of degree [latex]1[\/latex] is known as a linear function and can be written as [latex]f(x)=mx+b[\/latex], where [latex]m[\/latex] is non-zero. If a polynomial's highest degree term is [latex]2[\/latex], it's called a quadratic function, such as [latex]f(x)=ax^2+bx+c[\/latex], with [latex]a[\/latex] being non-zero. A polynomial with a degree of 3 is termed cubic, and so forth.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>terminology of polynomial functions<\/h3>\r\n<center>\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\/11\/02194507\/CNX_Precalc_Figure_03_03_010n2.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"487\" height=\"147\" \/> Diagram to show what the components of the leading term in a function are[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Given a Polynomial Function, Identify the Degree and Leading Coefficient<\/strong><\/p>\r\n<ol>\r\n\t<li>Find the highest power of [latex]x[\/latex] to determine the <strong>degree of the function<\/strong>.<\/li>\r\n\t<li>Identify the term containing the highest power of [latex]x[\/latex]<em>\u00a0<\/em>to find the <strong>leading term<\/strong>.<\/li>\r\n\t<li>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"632394\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"632394\"]<\/p>\r\n<p>For the function [latex]f\\left(x\\right)[\/latex], the highest power of [latex]x[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].<\/p>\r\n<p>For the function [latex]g\\left(t\\right)[\/latex], the highest power of [latex]t[\/latex] is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/p>\r\n<p>For the function [latex]h\\left(p\\right)[\/latex], the highest power of [latex]p[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, [latex]\u20131[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]284045[\/ohm_question]<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify polynomial degrees and solutions, and graph basic odd and even polynomials<\/li>\n<li>Graph a piecewise-defined function<\/li>\n<li>Describe how algebraic functions, like polynomials, differ from transcendental functions, like sine and exponential functions<\/li>\n<li>Draw the graph of a function after it has been moved up or down, stretched or shrunk, or flipped across an axis<\/li>\n<\/ul>\n<\/section>\n<h1>Polynomial Functions<\/h1>\n<p id=\"fs-id1170573363307\">\u00a0A <strong>polynomial function<\/strong> is any function that can be written in the form<\/p>\n<div id=\"fs-id1170573359094\" class=\"equation\" style=\"text-align: center;\">[latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Polynomials are defined by their <strong>degree<\/strong>, which is the highest exponent of the variable x with a non-zero coefficient. The <strong>leading coefficient<\/strong> is the coefficient of the term with the highest power.<\/p>\n<p>The simplest polynomial, the zero function [latex]f(x)=0[\/latex], has a degree of [latex]0[\/latex]. A polynomial of degree [latex]1[\/latex] is known as a linear function and can be written as [latex]f(x)=mx+b[\/latex], where [latex]m[\/latex] is non-zero. If a polynomial&#8217;s highest degree term is [latex]2[\/latex], it&#8217;s called a quadratic function, such as [latex]f(x)=ax^2+bx+c[\/latex], with [latex]a[\/latex] being non-zero. A polynomial with a degree of 3 is termed cubic, and so forth.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>terminology of polynomial functions<\/h3>\n<div style=\"text-align: center;\">\n<figure style=\"width: 487px\" 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\/11\/02194507\/CNX_Precalc_Figure_03_03_010n2.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"487\" height=\"147\" \/><figcaption class=\"wp-caption-text\">Diagram to show what the components of the leading term in a function are<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.<\/p>\n<p>&nbsp;<\/p>\n<p>The <strong>leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/p>\n<p>&nbsp;<\/p>\n<p>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Given a Polynomial Function, Identify the Degree and Leading Coefficient<\/strong><\/p>\n<ol>\n<li>Find the highest power of [latex]x[\/latex] to determine the <strong>degree of the function<\/strong>.<\/li>\n<li>Identify the term containing the highest power of [latex]x[\/latex]<em>\u00a0<\/em>to find the <strong>leading term<\/strong>.<\/li>\n<li>The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q632394\">Show Solution<\/button><\/p>\n<div id=\"q632394\" class=\"hidden-answer\" style=\"display: none\">\n<p>For the function [latex]f\\left(x\\right)[\/latex], the highest power of [latex]x[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].<\/p>\n<p>For the function [latex]g\\left(t\\right)[\/latex], the highest power of [latex]t[\/latex] is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/p>\n<p>For the function [latex]h\\left(p\\right)[\/latex], the highest power of [latex]p[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, [latex]\u20131[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm284045\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=284045&theme=lumen&iframe_resize_id=ohm284045&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":6,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.2 Basic Classes of Functions\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"1.2 Basic Classes of Functions","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/152"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/152\/revisions"}],"predecessor-version":[{"id":4727,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/152\/revisions\/4727"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/152\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=152"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=152"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=152"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=152"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}