{"id":1328,"date":"2025-07-24T04:33:26","date_gmt":"2025-07-24T04:33:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1328"},"modified":"2026-03-18T03:57:37","modified_gmt":"2026-03-18T03:57:37","slug":"grapsh-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/grapsh-of-polynomial-functions\/","title":{"raw":"Graphs of Polynomial Functions: Fresh Take","rendered":"Graphs of Polynomial Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize characteristics of graphs of polynomial functions.<\/li>\r\n \t<li>Graph polynomial functions.<\/li>\r\n \t<li>Use the Intermediate Value Theorem.<\/li>\r\n \t<li>Write the formula for a polynomial function given its graph.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Recognizing Characteristics of Graphs of Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Smoothness<\/strong>: Polynomial functions of degree [latex]2[\/latex] or higher have graphs with no sharp corners.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Continuity<\/strong>: Polynomial graphs have no breaks; they are continuous over their entire domain.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Domain<\/strong>: All real numbers are valid inputs for polynomial functions.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Visual Distinction<\/strong>: Polynomial graphs can be distinguished from non-polynomial graphs by their smooth and continuous nature.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Degree Impact<\/strong>: The degree of the polynomial influences the overall shape and behavior of the graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbhgdbcg-cJIzQ2L3Pxg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/cJIzQ2L3Pxg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbhgdbcg-cJIzQ2L3Pxg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780698&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dbhgdbcg-cJIzQ2L3Pxg&amp;vembed=0&amp;video_id=cJIzQ2L3Pxg&amp;video_target=tpm-plugin-dbhgdbcg-cJIzQ2L3Pxg\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Determining+if+a+graph+is+a+polynomial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining if a graph is a polynomial\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Using Factoring to Find Zeros of Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Zeros Definition<\/strong>: Values of [latex]x[\/latex] where [latex]f(x) = 0[\/latex] for a polynomial function [latex]f[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zeros correspond to [latex]x[\/latex]-intercepts of the polynomial's graph.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factoring Methods:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Using known techniques (e.g., greatest common factor, trinomial factoring)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Working with pre-factored polynomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Utilizing technology for complex cases<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set each factor to zero and solve for [latex]x[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each linear factor corresponds to a zero of the polynomial.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the [latex]x[\/latex]-intercepts of [latex]f(x) = x^3 - 5x^2 - x + 5[\/latex].[reveal-answer q=\"212412\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"212412\"]Find solutions for [latex]f(x) = 0[\/latex] by factoring.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} x^3 - 5x^2 - x + 5 &amp; = &amp; 0 \\quad \\text{Factor by grouping.} \\\\ x^2(x - 5) - (x - 5) &amp; = &amp; 0 \\quad \\text{Factor out the common factor.} \\\\ (x^2 - 1)(x - 5) &amp; = &amp; 0 \\quad \\text{Factor the difference of squares.} \\\\ (x + 1)(x - 1)(x - 5) &amp; = &amp; 0 \\quad \\text{Set each factor equal to zero.} \\end{array} [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] x + 1 = 0 \\quad \\text{or} \\quad x - 1 = 0 \\quad \\text{or} \\quad x - 5 = 0 [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] x = -1 \\quad \\quad x = 1 \\quad \\quad x = 5 [\/latex]<\/p>\r\nThere are three [latex]x[\/latex]-intercepts: [latex](-1, 0)[\/latex], [latex](1, 0)[\/latex], and [latex](5, 0)[\/latex].\r\n\r\n[caption id=\"attachment_4293\" align=\"aligncenter\" width=\"560\"]<img class=\"wp-image-4293 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30144411\/fe61d4a2221ab3c9b75cbea781f608c93f5c26b4.jpg\" alt=\"Graph of f(x)=x^3-5x^2-x+5 with its three intercepts (-1, 0), (1, 0), and (5, 0).\" width=\"560\" height=\"372\" \/> Graph of f(x)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hegfhhdd-vIGSlHznWdg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vIGSlHznWdg?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hegfhhdd-vIGSlHznWdg\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780699&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hegfhhdd-vIGSlHznWdg&amp;vembed=0&amp;video_id=vIGSlHznWdg&amp;video_target=tpm-plugin-hegfhhdd-vIGSlHznWdg\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+x+and+y+intercepts+given+a+polynomial+function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding x and y intercepts given a polynomial function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Identifying Zeros and Their Behavior<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Zero Definition<\/strong>: Points where a polynomial function crosses or touches the [latex]x[\/latex]-axis.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Multiplicity<\/strong>: The number of times a factor appears in the polynomial's factored form.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Behavior at Zeros<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Odd multiplicity: Graph crosses the [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Even multiplicity: Graph touches or is tangent to the [latex]x[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Degree Relationship<\/strong>: The sum of all zero multiplicities equals the polynomial's degree.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Graphical Interpretation<\/strong>: The shape of the graph near a zero indicates its multiplicity.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201600\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/> Graph of a polynomial function[\/caption]\r\n\r\n[reveal-answer q=\"874458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"874458\"]The graph has a zero of \u20135 with multiplicity 1, a zero of \u20131 with multiplicity 2, and a zero of 3 with multiplicity 2.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fffbbdca-RbCZ0gvynxs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RbCZ0gvynxs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fffbbdca-RbCZ0gvynxs\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846611&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fffbbdca-RbCZ0gvynxs&amp;vembed=0&amp;video_id=RbCZ0gvynxs&amp;video_target=tpm-plugin-fffbbdca-RbCZ0gvynxs\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Determine+the+Zeros%3ARoots+and+Multiplicity+From+a+Graph+of+a+Polynomial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine the Zeros\/Roots and Multiplicity From a Graph of a Polynomial\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Graphing Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Key Elements for Graphing:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Zeros and their behavior<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercepts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">End behavior<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Local behavior (turning points and concavity)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Zeros and Their Multiplicities:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Even multiplicity: Graph touches [latex]x[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Odd multiplicity: Graph crosses [latex]x[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>End Behavior:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determined by the degree (odd or even) and sign of the leading coefficient<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Turning Points:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Maximum number is one less than the polynomial's degree<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Symmetry:<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Even functions: Symmetric about [latex]y[\/latex]-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Odd functions: Symmetric about origin<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a possible graph for [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].Check yourself with an online graphing calculator when you are done.[reveal-answer q=\"812296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812296\"]\r\n\r\n[caption id=\"attachment_2911\" align=\"aligncenter\" width=\"487\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\"><img class=\"wp-image-2911 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" width=\"487\" height=\"366\" \/><\/a> Graph of a polynomial function[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-agadcgcg-PWLvZZfpC5I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/PWLvZZfpC5I?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-agadcgcg-PWLvZZfpC5I\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328600&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-agadcgcg-PWLvZZfpC5I&amp;vembed=0&amp;video_id=PWLvZZfpC5I&amp;video_target=tpm-plugin-agadcgcg-PWLvZZfpC5I\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+Sketch+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Sketch a Polynomial Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Using the Intermediate Value Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Intermediate Value Theorem (IVT):<\/strong> For a continuous function [latex]f(x)[\/latex] on [latex][a,b][\/latex], if [latex]y[\/latex] is between [latex]f(a)[\/latex] and [latex]f(b)[\/latex], then there exists a [latex]c[\/latex] in [latex][a,b][\/latex] where [latex]f(c) = y[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Application to Zeros:<\/strong> If [latex]f(a)[\/latex] and [latex]f(b)[\/latex] have opposite signs, there's at least one zero between [latex]a[\/latex] and [latex]b[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Helps prove existence of zeros without directly solving equations.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Continuity:<\/strong> The theorem relies on the function being continuous.<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Show that the function [latex]f\\left(x\\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[\/latex] has at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].[reveal-answer q=\"778313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"778313\"]Because <em>f<\/em>\u00a0is a polynomial function and since [latex]f\\left(1\\right)[\/latex] is negative and [latex]f\\left(2\\right)[\/latex] is positive, there is at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbfdgagc-5hK6RWeg0yU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5hK6RWeg0yU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbfdgagc-5hK6RWeg0yU\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846612&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbfdgagc-5hK6RWeg0yU&amp;vembed=0&amp;video_id=5hK6RWeg0yU&amp;video_target=tpm-plugin-fbfdgagc-5hK6RWeg0yU\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Intermediate+Value+Theorem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntermediate Value Theorem\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Writing Formulas for Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Factor Form<\/strong>: [latex]f(x) = a(x-x_1)^{p_1}(x-x_2)^{p_2}...(x-x_n)^{p_n}[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x_i[\/latex] are zeros (x-intercepts)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]p_i[\/latex] are multiplicities (behavior at zeros)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is the stretch factor<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>From Graph to Formula<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]x[\/latex]-intercepts (zeros)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine multiplicity at each zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find stretch factor using another point<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Alternative Approaches<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Reverse Engineering<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Start with simple polynomials and modify them to create desired features.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Steps:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">a. Begin with a basic polynomial, e.g., [latex]f(x) = x^3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">b. Add\/subtract a constant to shift vertically: [latex]f(x) = x^3 + 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">c. Add\/subtract [latex]x[\/latex] to shift horizontally: [latex]f(x) = (x-1)^3 + 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">d. Multiply by a constant to stretch\/compress: [latex]f(x) = 2(x-1)^3 + 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">e. Add lower-degree terms to adjust shape: [latex]f(x) = 2(x-1)^3 - 3x + 2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Practice: Start with [latex]f(x) = x^2[\/latex] and modify it to match a given graph.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Technology-Aided Discovery<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use graphing calculators or software (e.g., Desmos, GeoGebra) to visualize functions.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exploration process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">a. Input a general polynomial form: [latex]f(x) = ax^3 + bx^2 + cx + d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">b. Create sliders for coefficients [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex], and [latex]d[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">c. Adjust sliders to match the given graph<\/li>\r\n \t<li class=\"whitespace-normal break-words\">d. Observe how each coefficient affects the graph's shape<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Advanced: Use parametric equations to visualize how zeros \"move\" as coefficients change.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given the graph below, write a formula for the function shown.\r\n\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\/02201629\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/> Graph of a polynomial function[\/caption]\r\n\r\n[reveal-answer q=\"427364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"427364\"][latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to help you write the equation of a degree 5 polynomial function with roots at [latex](-1,0),(0,2),\\text{and },(0,3)[\/latex] with multiplicities 3, 1, and 1 respectively, that passes through the point [latex](1,-32)[\/latex].[reveal-answer q=\"135031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"135031\"][latex]f(x)=-2(x-2)(x+1)^3(x-2)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ecefdfea-Ly8mygYGwLo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ly8mygYGwLo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ecefdfea-Ly8mygYGwLo\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846613&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ecefdfea-Ly8mygYGwLo&amp;vembed=0&amp;video_id=Ly8mygYGwLo&amp;video_target=tpm-plugin-ecefdfea-Ly8mygYGwLo\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex2+-+Find+an+Equation+of+a+Degree+5+Polynomial+Function+From+the+Graph+of+the+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx2: Find an Equation of a Degree 5 Polynomial Function From the Graph of the Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Local and Global Extrema<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Local Extrema<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Highest\/lowest points in an open interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can be multiple local extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Global Extrema<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Highest\/lowest points on entire function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Only even-degree polynomials have both<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Turning Points<\/strong>:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Where function changes from increasing to decreasing or vice versa<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Maximum number: degree of polynomial minus [latex]1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbhggaba-lRVCKYfSsCc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/lRVCKYfSsCc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbhggaba-lRVCKYfSsCc\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780726&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbhggaba-lRVCKYfSsCc&amp;vembed=0&amp;video_id=lRVCKYfSsCc&amp;video_target=tpm-plugin-fbhggaba-lRVCKYfSsCc\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Local+and+Global+Extrema_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLocal and Global Extrema\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize characteristics of graphs of polynomial functions.<\/li>\n<li>Graph polynomial functions.<\/li>\n<li>Use the Intermediate Value Theorem.<\/li>\n<li>Write the formula for a polynomial function given its graph.<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Recognizing Characteristics of Graphs of Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Smoothness<\/strong>: Polynomial functions of degree [latex]2[\/latex] or higher have graphs with no sharp corners.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Continuity<\/strong>: Polynomial graphs have no breaks; they are continuous over their entire domain.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Domain<\/strong>: All real numbers are valid inputs for polynomial functions.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Visual Distinction<\/strong>: Polynomial graphs can be distinguished from non-polynomial graphs by their smooth and continuous nature.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Degree Impact<\/strong>: The degree of the polynomial influences the overall shape and behavior of the graph.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbhgdbcg-cJIzQ2L3Pxg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/cJIzQ2L3Pxg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbhgdbcg-cJIzQ2L3Pxg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780698&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dbhgdbcg-cJIzQ2L3Pxg&amp;vembed=0&amp;video_id=cJIzQ2L3Pxg&amp;video_target=tpm-plugin-dbhgdbcg-cJIzQ2L3Pxg\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Determining+if+a+graph+is+a+polynomial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining if a graph is a polynomial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Using Factoring to Find Zeros of Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Zeros Definition<\/strong>: Values of [latex]x[\/latex] where [latex]f(x) = 0[\/latex] for a polynomial function [latex]f[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Zeros correspond to [latex]x[\/latex]-intercepts of the polynomial&#8217;s graph.<\/li>\n<li class=\"whitespace-normal break-words\">Factoring Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Using known techniques (e.g., greatest common factor, trinomial factoring)<\/li>\n<li class=\"whitespace-normal break-words\">Working with pre-factored polynomials<\/li>\n<li class=\"whitespace-normal break-words\">Utilizing technology for complex cases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Set each factor to zero and solve for [latex]x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Each linear factor corresponds to a zero of the polynomial.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the [latex]x[\/latex]-intercepts of [latex]f(x) = x^3 - 5x^2 - x + 5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q212412\">Show Answer<\/button><\/p>\n<div id=\"q212412\" class=\"hidden-answer\" style=\"display: none\">Find solutions for [latex]f(x) = 0[\/latex] by factoring.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} x^3 - 5x^2 - x + 5 & = & 0 \\quad \\text{Factor by grouping.} \\\\ x^2(x - 5) - (x - 5) & = & 0 \\quad \\text{Factor out the common factor.} \\\\ (x^2 - 1)(x - 5) & = & 0 \\quad \\text{Factor the difference of squares.} \\\\ (x + 1)(x - 1)(x - 5) & = & 0 \\quad \\text{Set each factor equal to zero.} \\end{array}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x + 1 = 0 \\quad \\text{or} \\quad x - 1 = 0 \\quad \\text{or} \\quad x - 5 = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x = -1 \\quad \\quad x = 1 \\quad \\quad x = 5[\/latex]<\/p>\n<p>There are three [latex]x[\/latex]-intercepts: [latex](-1, 0)[\/latex], [latex](1, 0)[\/latex], and [latex](5, 0)[\/latex].<\/p>\n<figure id=\"attachment_4293\" aria-describedby=\"caption-attachment-4293\" style=\"width: 560px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4293 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/30144411\/fe61d4a2221ab3c9b75cbea781f608c93f5c26b4.jpg\" alt=\"Graph of f(x)=x^3-5x^2-x+5 with its three intercepts (-1, 0), (1, 0), and (5, 0).\" width=\"560\" height=\"372\" \/><figcaption id=\"caption-attachment-4293\" class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hegfhhdd-vIGSlHznWdg\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/vIGSlHznWdg?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hegfhhdd-vIGSlHznWdg\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780699&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-hegfhhdd-vIGSlHznWdg&amp;vembed=0&amp;video_id=vIGSlHznWdg&amp;video_target=tpm-plugin-hegfhhdd-vIGSlHznWdg\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Finding+x+and+y+intercepts+given+a+polynomial+function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding x and y intercepts given a polynomial function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Identifying Zeros and Their Behavior<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Zero Definition<\/strong>: Points where a polynomial function crosses or touches the [latex]x[\/latex]-axis.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Multiplicity<\/strong>: The number of times a factor appears in the polynomial&#8217;s factored form.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Behavior at Zeros<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Odd multiplicity: Graph crosses the [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Even multiplicity: Graph touches or is tangent to the [latex]x[\/latex]-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Degree Relationship<\/strong>: The sum of all zero multiplicities equals the polynomial&#8217;s degree.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Graphical Interpretation<\/strong>: The shape of the graph near a zero indicates its multiplicity.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201600\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/><figcaption class=\"wp-caption-text\">Graph of a polynomial function<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q874458\">Show Solution<\/button><\/p>\n<div id=\"q874458\" class=\"hidden-answer\" style=\"display: none\">The graph has a zero of \u20135 with multiplicity 1, a zero of \u20131 with multiplicity 2, and a zero of 3 with multiplicity 2.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fffbbdca-RbCZ0gvynxs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RbCZ0gvynxs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fffbbdca-RbCZ0gvynxs\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846611&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fffbbdca-RbCZ0gvynxs&amp;vembed=0&amp;video_id=RbCZ0gvynxs&amp;video_target=tpm-plugin-fffbbdca-RbCZ0gvynxs\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Determine+the+Zeros%3ARoots+and+Multiplicity+From+a+Graph+of+a+Polynomial_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine the Zeros\/Roots and Multiplicity From a Graph of a Polynomial\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphing Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Key Elements for Graphing:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Zeros and their behavior<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercepts<\/li>\n<li class=\"whitespace-normal break-words\">End behavior<\/li>\n<li class=\"whitespace-normal break-words\">Local behavior (turning points and concavity)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Zeros and Their Multiplicities:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even multiplicity: Graph touches [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Odd multiplicity: Graph crosses [latex]x[\/latex]-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>End Behavior:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determined by the degree (odd or even) and sign of the leading coefficient<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Turning Points:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Maximum number is one less than the polynomial&#8217;s degree<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Symmetry:<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even functions: Symmetric about [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Odd functions: Symmetric about origin<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a possible graph for [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].Check yourself with an online graphing calculator when you are done.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q812296\">Show Solution<\/button><\/p>\n<div id=\"q812296\" class=\"hidden-answer\" style=\"display: none\">\n<figure id=\"attachment_2911\" aria-describedby=\"caption-attachment-2911\" style=\"width: 487px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2911 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" width=\"487\" height=\"366\" \/><\/a><figcaption id=\"caption-attachment-2911\" class=\"wp-caption-text\">Graph of a polynomial function<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-agadcgcg-PWLvZZfpC5I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/PWLvZZfpC5I?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-agadcgcg-PWLvZZfpC5I\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=11328600&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-agadcgcg-PWLvZZfpC5I&amp;vembed=0&amp;video_id=PWLvZZfpC5I&amp;video_target=tpm-plugin-agadcgcg-PWLvZZfpC5I\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+Sketch+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Sketch a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Using the Intermediate Value Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Intermediate Value Theorem (IVT):<\/strong> For a continuous function [latex]f(x)[\/latex] on [latex][a,b][\/latex], if [latex]y[\/latex] is between [latex]f(a)[\/latex] and [latex]f(b)[\/latex], then there exists a [latex]c[\/latex] in [latex][a,b][\/latex] where [latex]f(c) = y[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Application to Zeros:<\/strong> If [latex]f(a)[\/latex] and [latex]f(b)[\/latex] have opposite signs, there&#8217;s at least one zero between [latex]a[\/latex] and [latex]b[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Helps prove existence of zeros without directly solving equations.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Continuity:<\/strong> The theorem relies on the function being continuous.<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Show that the function [latex]f\\left(x\\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[\/latex] has at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q778313\">Show Solution<\/button><\/p>\n<div id=\"q778313\" class=\"hidden-answer\" style=\"display: none\">Because <em>f<\/em>\u00a0is a polynomial function and since [latex]f\\left(1\\right)[\/latex] is negative and [latex]f\\left(2\\right)[\/latex] is positive, there is at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbfdgagc-5hK6RWeg0yU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/5hK6RWeg0yU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbfdgagc-5hK6RWeg0yU\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846612&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbfdgagc-5hK6RWeg0yU&amp;vembed=0&amp;video_id=5hK6RWeg0yU&amp;video_target=tpm-plugin-fbfdgagc-5hK6RWeg0yU\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Intermediate+Value+Theorem_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntermediate Value Theorem\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Formulas for Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Factor Form<\/strong>: [latex]f(x) = a(x-x_1)^{p_1}(x-x_2)^{p_2}...(x-x_n)^{p_n}[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x_i[\/latex] are zeros (x-intercepts)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]p_i[\/latex] are multiplicities (behavior at zeros)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] is the stretch factor<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>From Graph to Formula<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify [latex]x[\/latex]-intercepts (zeros)<\/li>\n<li class=\"whitespace-normal break-words\">Determine multiplicity at each zero<\/li>\n<li class=\"whitespace-normal break-words\">Find stretch factor using another point<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Alternative Approaches<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Reverse Engineering<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Start with simple polynomials and modify them to create desired features.<\/li>\n<li class=\"whitespace-normal break-words\">Steps:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">a. Begin with a basic polynomial, e.g., [latex]f(x) = x^3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">b. Add\/subtract a constant to shift vertically: [latex]f(x) = x^3 + 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">c. Add\/subtract [latex]x[\/latex] to shift horizontally: [latex]f(x) = (x-1)^3 + 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">d. Multiply by a constant to stretch\/compress: [latex]f(x) = 2(x-1)^3 + 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">e. Add lower-degree terms to adjust shape: [latex]f(x) = 2(x-1)^3 - 3x + 2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Practice: Start with [latex]f(x) = x^2[\/latex] and modify it to match a given graph.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Technology-Aided Discovery<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use graphing calculators or software (e.g., Desmos, GeoGebra) to visualize functions.<\/li>\n<li class=\"whitespace-normal break-words\">Exploration process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">a. Input a general polynomial form: [latex]f(x) = ax^3 + bx^2 + cx + d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">b. Create sliders for coefficients [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex], and [latex]d[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">c. Adjust sliders to match the given graph<\/li>\n<li class=\"whitespace-normal break-words\">d. Observe how each coefficient affects the graph&#8217;s shape<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advanced: Use parametric equations to visualize how zeros &#8220;move&#8221; as coefficients change.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given the graph below, write a formula for the function shown.<\/p>\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\/02201629\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/><figcaption class=\"wp-caption-text\">Graph of a polynomial function<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q427364\">Show Solution<\/button><\/p>\n<div id=\"q427364\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use an online graphing calculator to help you write the equation of a degree 5 polynomial function with roots at [latex](-1,0),(0,2),\\text{and },(0,3)[\/latex] with multiplicities 3, 1, and 1 respectively, that passes through the point [latex](1,-32)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q135031\">Show Solution<\/button><\/p>\n<div id=\"q135031\" class=\"hidden-answer\" style=\"display: none\">[latex]f(x)=-2(x-2)(x+1)^3(x-2)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ecefdfea-Ly8mygYGwLo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Ly8mygYGwLo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ecefdfea-Ly8mygYGwLo\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12846613&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ecefdfea-Ly8mygYGwLo&amp;vembed=0&amp;video_id=Ly8mygYGwLo&amp;video_target=tpm-plugin-ecefdfea-Ly8mygYGwLo\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex2+-+Find+an+Equation+of+a+Degree+5+Polynomial+Function+From+the+Graph+of+the+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx2: Find an Equation of a Degree 5 Polynomial Function From the Graph of the Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Local and Global Extrema<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Local Extrema<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Highest\/lowest points in an open interval<\/li>\n<li class=\"whitespace-normal break-words\">Can be multiple local extrema<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Global Extrema<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Highest\/lowest points on entire function<\/li>\n<li class=\"whitespace-normal break-words\">Only even-degree polynomials have both<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Turning Points<\/strong>:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Where function changes from increasing to decreasing or vice versa<\/li>\n<li class=\"whitespace-normal break-words\">Maximum number: degree of polynomial minus [latex]1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbhggaba-lRVCKYfSsCc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/lRVCKYfSsCc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbhggaba-lRVCKYfSsCc\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780726&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbhggaba-lRVCKYfSsCc&amp;vembed=0&amp;video_id=lRVCKYfSsCc&amp;video_target=tpm-plugin-fbhggaba-lRVCKYfSsCc\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Local+and+Global+Extrema_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLocal and Global Extrema\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Determining if a graph is a polynomial\",\"author\":\"\",\"organization\":\"UHDCMS\",\"url\":\"https:\/\/youtu.be\/cJIzQ2L3Pxg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Finding x and y intercepts given a polynomial function\",\"author\":\"\",\"organization\":\"KSpinMATH\",\"url\":\"https:\/\/youtu.be\/vIGSlHznWdg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determine the Zeros\/Roots and Multiplicity From a Graph of a Polynomial\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/RbCZ0gvynxs\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How to Sketch a Polynomial Function\",\"author\":\"\",\"organization\":\"mroldridge\",\"url\":\"https:\/\/youtu.be\/PWLvZZfpC5I\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Intermediate Value Theorem\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/5hK6RWeg0yU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex2: Find an Equation of a Degree 5 Polynomial Function From the Graph of the Function\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Ly8mygYGwLo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Local and Global Extrema\",\"author\":\"\",\"organization\":\"Professor Heather 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