{"id":79,"date":"2025-02-13T22:43:31","date_gmt":"2025-02-13T22:43:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions\/"},"modified":"2026-03-09T19:53:01","modified_gmt":"2026-03-09T19:53:01","slug":"graphs-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 1","rendered":"Graphs of Polynomial Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\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>Identifying Local Behavior of Polynomial Functions<\/h2>\r\n<img class=\"wp-image-1911 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/24201855\/Screenshot-2024-06-24-at-1.18.49%E2%80%AFPM.png\" alt=\"\" width=\"400\" height=\"337\" \/>I<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">n addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. <\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">In particular, we are interested in locations where graph behavior changes. A <strong>turning point<\/strong> is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/span>\r\n\r\nWe are also interested in the <strong>intercepts<\/strong>. As with all functions, the [latex]y[\/latex]-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one [latex]y[\/latex]-intercept [latex]\\left(0,{a}_{0}\\right)[\/latex]. The [latex]x[\/latex]<em>-<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one [latex]x[\/latex]<em>-<\/em>intercept.<span style=\"background-color: #ffffff;\">\u00a0<\/span>\r\n\r\n<section class=\"textbox example\">Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the [latex]y[\/latex]\u00a0and [latex]x[\/latex]-intercepts.[reveal-answer q=\"701514\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"701514\"]The <em>y-<\/em>intercept occurs when the input is zero, so substitute [latex]0[\/latex] for [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(0\\right)=\\left(0 - 2\\right)\\left(0+1\\right)\\left(0 - 4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=\\left(-2\\right)\\left(1\\right)\\left(-4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=8\\hfill \\end{array}[\/latex]<\/p>\r\nThe [latex]y[\/latex]<em>-<\/em>intercept is [latex](0, 8)[\/latex].\r\n\r\nThe [latex]x[\/latex]-intercepts occur when the output [latex]f(x)[\/latex] is zero.\r\n<p style=\"text-align: center;\">[latex]0=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllllllll}x - 2=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x+1=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x - 4=0\\hfill \\\\ \\text{}x=2\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{ }x=-1\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x=4 \\end{array}[\/latex]<\/p>\r\nThe\u00a0[latex]x[\/latex]-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].\r\n\r\nWe can see these intercepts on the graph of the function shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194527\/CNX_Precalc_Figure_03_03_0182.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"350\" height=\"453\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox keyTakeaway\">\r\n<h3>intercepts and turning points of polynomial functions<\/h3>\r\n<ul>\r\n \t<li>A <strong>turning point<\/strong> of a graph is a point where the graph changes from increasing to decreasing or decreasing to increasing.<\/li>\r\n \t<li>The [latex]y[\/latex]<em>-<\/em>intercept is the point where the function has an input value of zero.<\/li>\r\n \t<li>The [latex]x[\/latex]-intercepts are the points where the output value is zero.<\/li>\r\n \t<li>A polynomial of degree [latex]n[\/latex]\u00a0will have, at most, [latex]n[\/latex]\u00a0[latex]x[\/latex]-intercepts and [latex]n \u2013 1[\/latex] turning points.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Why do we use the phrase \"<em>at most<\/em> [latex]n[\/latex]\" when describing the number of real roots (x-intercepts) of the graph of an [latex]n^{\\text{th}}[\/latex] degree polynomial? Can it have fewer?[reveal-answer q=\"232068\"]more[\/reveal-answer]\r\n[hidden-answer a=\"232068\"]Consider the graph of the polynomial function [latex]f(x)=x^2-x+1[\/latex]. The function is a [latex]2^{\\text{nd}}[\/latex] degree polynomial, so it must have <em>at most<\/em> [latex]n[\/latex] roots and [latex]n-1[\/latex] turning points.\r\n[latex]\\\\[\/latex]\r\nWe know this function has non-real roots since the discriminant of the quadratic formula is negative. This means that this\u00a0[latex]2^{\\text{nd}}[\/latex] polynomial has no real roots (apply the quadratic formula to prove this to yourself if needed). That is, it has no x-intercepts. But it does have two distinct complex roots.\r\n[latex]\\\\[\/latex]\r\nCan you picture the graph of a quadratic function with one distinct real root? Two? But you can also see that there will never be more than two [latex]x[\/latex]-intercepts. Since a parabola (the graph of a [latex]2^{\\text{nd}}[\/latex] degree polynomial) has only one turning point, it can't cross the [latex]x[\/latex]-axis more than twice.[\/hidden-answer]<\/section><section class=\"textbox example\">Without graphing the function, determine the local behavior of the function by finding the maximum number of [latex]x[\/latex]-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].[reveal-answer q=\"96529\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96529\"]The polynomial has a degree of [latex]10[\/latex], so there are at most [latex]10[\/latex] [latex]x[\/latex]-intercepts and at most [latex]10 \u2013 1 = 9[\/latex] turning points.[\/hidden-answer]<\/section>Now we can bring the two concepts of turning points and intercepts together to get a general picture of the behavior of polynomial functions. \u00a0These types of analyses on polynomials developed before the advent of mass computing as a way to quickly understand the general behavior of a polynomial function. We now have a quick way, with computers, to graph and calculate important characteristics of polynomials that once took a lot of algebra.\r\n\r\n<section class=\"textbox example\">Given the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194531\/CNX_Precalc_Figure_03_03_0202.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/>[reveal-answer q=\"200904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200904\"]\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194532\/CNX_Precalc_Figure_03_03_0212.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/>The end behavior of the graph tells us this is the graph of an even-degree polynomial. The graph has [latex]2[\/latex] [latex]x[\/latex]-intercepts, suggesting a degree of [latex]2[\/latex] or greater, and [latex]3[\/latex] turning points, suggesting a degree of [latex]4[\/latex] or greater. Based on this, it would be reasonable to conclude that the degree is even and at least [latex]4[\/latex].[\/hidden-answer]<\/section>Now you try to determine the least possible degree of a polynomial given its graph.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Given the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194534\/CNX_Precalc_Figure_03_03_0224.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/>[reveal-answer q=\"492375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"492375\"]The end behavior indicates an odd-degree polynomial function; there are [latex]3[\/latex] [latex]x[\/latex]-intercepts and [latex]2[\/latex] turning points, so the degree is odd and at least [latex]3[\/latex]. Because of the end behavior, we know that the leading coefficient must be negative.[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318822[\/ohm_question]<\/section><section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318823[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318824[\/ohm_question]<\/section><\/section>\r\n<div id=\"Example_03_04_07\" class=\"example\">\r\n<div id=\"fs-id1165134374690\" class=\"exercise\"><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1165134060425\">Find the maximum number of turning points of each polynomial function.<\/p>\r\n\r\n<ol id=\"fs-id1165134060428\">\r\n \t<li>[latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}+1[\/latex]<\/li>\r\n \t<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"157524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157524\"]\r\n<ol id=\"fs-id1165137784430\">\r\n \t<li>[latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\r\n<p id=\"fs-id1165135335895\">First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\r\n<p id=\"fs-id1165135453844\">Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\r\n<p id=\"fs-id1165135341233\">The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\r\n<\/li>\r\n \t<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\"><img class=\"aligncenter wp-image-15117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\" alt=\"Graphic of f(x) showing to multiply the first term of (x-1)^2 and 2x^2 to determine the leading term.\" width=\"487\" height=\"67\" \/><\/a>\r\n<p style=\"text-align: center;\">[latex]a_{n}=-\\left(x^2\\right)\\left(2x^2\\right)=-2x^4[\/latex]<\/p>\r\n<p id=\"fs-id1165133104532\">First, identify the leading term of the polynomial function if the function were expanded.<span id=\"fs-id1165134130071\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135551181\">Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\r\n<p id=\"fs-id1165135551185\">The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\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>Identifying Local Behavior of Polynomial Functions<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1911 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/24201855\/Screenshot-2024-06-24-at-1.18.49%E2%80%AFPM.png\" alt=\"\" width=\"400\" height=\"337\" \/>I<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">n addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. <\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">In particular, we are interested in locations where graph behavior changes. A <strong>turning point<\/strong> is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/span><\/p>\n<p>We are also interested in the <strong>intercepts<\/strong>. As with all functions, the [latex]y[\/latex]-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one [latex]y[\/latex]-intercept [latex]\\left(0,{a}_{0}\\right)[\/latex]. The [latex]x[\/latex]<em>&#8211;<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one [latex]x[\/latex]<em>&#8211;<\/em>intercept.<span style=\"background-color: #ffffff;\">\u00a0<\/span><\/p>\n<section class=\"textbox example\">Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the [latex]y[\/latex]\u00a0and [latex]x[\/latex]-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q701514\">Show Solution<\/button><\/p>\n<div id=\"q701514\" class=\"hidden-answer\" style=\"display: none\">The <em>y-<\/em>intercept occurs when the input is zero, so substitute [latex]0[\/latex] for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(0\\right)=\\left(0 - 2\\right)\\left(0+1\\right)\\left(0 - 4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=\\left(-2\\right)\\left(1\\right)\\left(-4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=8\\hfill \\end{array}[\/latex]<\/p>\n<p>The [latex]y[\/latex]<em>&#8211;<\/em>intercept is [latex](0, 8)[\/latex].<\/p>\n<p>The [latex]x[\/latex]-intercepts occur when the output [latex]f(x)[\/latex] is zero.<\/p>\n<p style=\"text-align: center;\">[latex]0=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllllllll}x - 2=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x+1=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x - 4=0\\hfill \\\\ \\text{}x=2\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{ }x=-1\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x=4 \\end{array}[\/latex]<\/p>\n<p>The\u00a0[latex]x[\/latex]-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\n<p>We can see these intercepts on the graph of the function shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194527\/CNX_Precalc_Figure_03_03_0182.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"350\" height=\"453\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>intercepts and turning points of polynomial functions<\/h3>\n<ul>\n<li>A <strong>turning point<\/strong> of a graph is a point where the graph changes from increasing to decreasing or decreasing to increasing.<\/li>\n<li>The [latex]y[\/latex]<em>&#8211;<\/em>intercept is the point where the function has an input value of zero.<\/li>\n<li>The [latex]x[\/latex]-intercepts are the points where the output value is zero.<\/li>\n<li>A polynomial of degree [latex]n[\/latex]\u00a0will have, at most, [latex]n[\/latex]\u00a0[latex]x[\/latex]-intercepts and [latex]n \u2013 1[\/latex] turning points.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Why do we use the phrase &#8220;<em>at most<\/em> [latex]n[\/latex]&#8221; when describing the number of real roots (x-intercepts) of the graph of an [latex]n^{\\text{th}}[\/latex] degree polynomial? Can it have fewer?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q232068\">more<\/button><\/p>\n<div id=\"q232068\" class=\"hidden-answer\" style=\"display: none\">Consider the graph of the polynomial function [latex]f(x)=x^2-x+1[\/latex]. The function is a [latex]2^{\\text{nd}}[\/latex] degree polynomial, so it must have <em>at most<\/em> [latex]n[\/latex] roots and [latex]n-1[\/latex] turning points.<br \/>\n[latex]\\\\[\/latex]<br \/>\nWe know this function has non-real roots since the discriminant of the quadratic formula is negative. This means that this\u00a0[latex]2^{\\text{nd}}[\/latex] polynomial has no real roots (apply the quadratic formula to prove this to yourself if needed). That is, it has no x-intercepts. But it does have two distinct complex roots.<br \/>\n[latex]\\\\[\/latex]<br \/>\nCan you picture the graph of a quadratic function with one distinct real root? Two? But you can also see that there will never be more than two [latex]x[\/latex]-intercepts. Since a parabola (the graph of a [latex]2^{\\text{nd}}[\/latex] degree polynomial) has only one turning point, it can&#8217;t cross the [latex]x[\/latex]-axis more than twice.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Without graphing the function, determine the local behavior of the function by finding the maximum number of [latex]x[\/latex]-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q96529\">Show Solution<\/button><\/p>\n<div id=\"q96529\" class=\"hidden-answer\" style=\"display: none\">The polynomial has a degree of [latex]10[\/latex], so there are at most [latex]10[\/latex] [latex]x[\/latex]-intercepts and at most [latex]10 \u2013 1 = 9[\/latex] turning points.<\/div>\n<\/div>\n<\/section>\n<p>Now we can bring the two concepts of turning points and intercepts together to get a general picture of the behavior of polynomial functions. \u00a0These types of analyses on polynomials developed before the advent of mass computing as a way to quickly understand the general behavior of a polynomial function. We now have a quick way, with computers, to graph and calculate important characteristics of polynomials that once took a lot of algebra.<\/p>\n<section class=\"textbox example\">Given the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194531\/CNX_Precalc_Figure_03_03_0202.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q200904\">Show Solution<\/button><\/p>\n<div id=\"q200904\" class=\"hidden-answer\" style=\"display: none\">\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194532\/CNX_Precalc_Figure_03_03_0212.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/>The end behavior of the graph tells us this is the graph of an even-degree polynomial. The graph has [latex]2[\/latex] [latex]x[\/latex]-intercepts, suggesting a degree of [latex]2[\/latex] or greater, and [latex]3[\/latex] turning points, suggesting a degree of [latex]4[\/latex] or greater. Based on this, it would be reasonable to conclude that the degree is even and at least [latex]4[\/latex].<\/div>\n<\/div>\n<\/section>\n<p>Now you try to determine the least possible degree of a polynomial given its graph.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Given the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194534\/CNX_Precalc_Figure_03_03_0224.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q492375\">Show Solution<\/button><\/p>\n<div id=\"q492375\" class=\"hidden-answer\" style=\"display: none\">The end behavior indicates an odd-degree polynomial function; there are [latex]3[\/latex] [latex]x[\/latex]-intercepts and [latex]2[\/latex] turning points, so the degree is odd and at least [latex]3[\/latex]. Because of the end behavior, we know that the leading coefficient must be negative.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318822\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318822&theme=lumen&iframe_resize_id=ohm318822&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318823\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318823&theme=lumen&iframe_resize_id=ohm318823&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318824\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318824&theme=lumen&iframe_resize_id=ohm318824&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<div id=\"Example_03_04_07\" class=\"example\">\n<div id=\"fs-id1165134374690\" class=\"exercise\">\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1165134060425\">Find the maximum number of turning points of each polynomial function.<\/p>\n<ol id=\"fs-id1165134060428\">\n<li>[latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}+1[\/latex]<\/li>\n<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q157524\">Show Solution<\/button><\/p>\n<div id=\"q157524\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137784430\">\n<li>[latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\n<p id=\"fs-id1165135335895\">First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\n<p id=\"fs-id1165135453844\">Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\n<p id=\"fs-id1165135341233\">The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\n<\/li>\n<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\" alt=\"Graphic of f(x) showing to multiply the first term of (x-1)^2 and 2x^2 to determine the leading term.\" width=\"487\" height=\"67\" \/><\/a><\/p>\n<p style=\"text-align: center;\">[latex]a_{n}=-\\left(x^2\\right)\\left(2x^2\\right)=-2x^4[\/latex]<\/p>\n<p id=\"fs-id1165133104532\">First, identify the leading term of the polynomial function if the function were expanded.<span id=\"fs-id1165134130071\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135551181\">Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\n<p id=\"fs-id1165135551185\">The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":6,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"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":74,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/79"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/79\/revisions"}],"predecessor-version":[{"id":5845,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/79\/revisions\/5845"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/79\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=79"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=79"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=79"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=79"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}