{"id":1324,"date":"2025-07-24T04:32:13","date_gmt":"2025-07-24T04:32:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1324"},"modified":"2026-03-18T03:48:50","modified_gmt":"2026-03-18T03:48:50","slug":"polynomial-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-functions-fresh-take\/","title":{"raw":"Polynomial Functions: Fresh Take","rendered":"Polynomial Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\r\n \t<li>Identify end behavior of polynomial functions.<\/li>\r\n \t<li>Identify intercepts of factored polynomial functions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Power 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\">A power function is of the form [latex]f(x) = ax^n[\/latex], where [latex]a[\/latex] and [latex]n[\/latex] are real numbers.\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">The '[latex]a[\/latex]' in the power function is called the coefficient.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The '[latex]n[\/latex]' in the power function can be any real number, not just positive integers.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Power functions include constant, identity, quadratic, cubic, reciprocal, and root functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Which functions are power functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"105254\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"105254\"]\r\n\r\n[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Identifying End Behavior 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\">End behavior describes how a function behaves as [latex]x[\/latex] approaches positive or negative infinity.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Even-power functions ([latex]f(x) = ax^n[\/latex], [latex]n[\/latex] is even):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to \\infty[\/latex] if [latex]a &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex] if [latex]a &lt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Odd-power functions ([latex]f(x) = ax^n[\/latex], [latex]n[\/latex] is odd):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex] if [latex]a &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x \\to \\infty[\/latex], [latex]f(x) \\to \\infty[\/latex] if [latex]a &gt; 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Behavior is reversed if [latex]a &lt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Impact of exponent: Higher powers lead to flatter graphs near the origin and steeper graphs away from it.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Symmetry: Even-power functions are symmetric about the [latex]y[\/latex]-axis; odd-power functions are symmetric about the origin.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Describe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].[reveal-answer q=\"631242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"631242\"]The exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex] and as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex].\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194505\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/> Graph of f(x)=-x^9[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can check our work by using the table feature on an online graphing calculator.\r\n<ol>\r\n \t<li>Enter the function\u00a0[latex]f\\left(x\\right)=-{x}^{9}[\/latex] into an online graphing calculator<\/li>\r\n \t<li>Create a table with the following x values, and observe the sign of the outputs. [latex]-10,-5,0,5,10[\/latex]<\/li>\r\n \t<li>Now, enter the function\u00a0[latex]g\\left(x\\right)={x}^{9}[\/latex], and create a similar table. Compare the signs of the outputs for both functions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Describe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].[reveal-answer q=\"582534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"582534\"]As [latex]x[\/latex]\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.[\/hidden-answer]<\/section>\r\n<h2>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\">A polynomial function is of the form [latex]f(x) = a_nx^n + ... + a_2x^2 + a_1x + a_0[\/latex], where [latex]n[\/latex] is a non-negative integer and [latex]a_i[\/latex] are real number coefficients.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Polynomial functions can be created by combining simpler functions, including power functions.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Terms: Each [latex]a_ix^i[\/latex] is a term of the polynomial function.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Degree: The highest power of [latex]x[\/latex] in the polynomial is its degree.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Degree and Leading Coefficient\u00a0of a Polynomial Function<\/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\">General Form: Polynomials are typically written in descending order of variable powers.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Degree: The highest power of the variable in the polynomial.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Leading Term: The term with the highest degree.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Leading Coefficient: The coefficient of the leading term.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Importance: These concepts help classify and analyze polynomial behavior.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].[reveal-answer q=\"435637\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"435637\"]The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbgfbhbf-F_G_w82s0QA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbgfbhbf-F_G_w82s0QA\" 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=12843132&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbgfbhbf-F_G_w82s0QA&amp;vembed=0&amp;video_id=F_G_w82s0QA&amp;video_target=tpm-plugin-bbgfbhbf-F_G_w82s0QA\" 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\/Degree%2C+Leading+Term%2C+and+Leading+Coefficient+of+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDegree, Leading Term, and Leading Coefficient of a Polynomial Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Identifying End Behavior 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\">The end behavior of a polynomial is determined by its leading term.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The degree (even or odd) and sign of the leading coefficient determine the specific end behavior pattern.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">End behavior is often described using limit notation, such as [latex]\\text{as } x \\to \\infty, f(x) \\to \\infty[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">To identify end behavior, polynomials should be in expanded (general) form.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The graph of a polynomial function reflects its end behavior.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Describe the end behavior of the polynomial function in the graph below.\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\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/> Graph of a polynomial[\/caption]\r\n\r\n[reveal-answer q=\"304329\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304329\"]As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]. It has the shape of an even degree power function with a negative coefficient.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you'll see more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbbeeccc-y78Dpr9LLN0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/y78Dpr9LLN0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbbeeccc-y78Dpr9LLN0\" 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=12845030&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbbeeccc-y78Dpr9LLN0&amp;vembed=0&amp;video_id=y78Dpr9LLN0&amp;video_target=tpm-plugin-fbbeeccc-y78Dpr9LLN0\" 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\/Summary+of+End+Behavior+or+Long+Run+Behavior+of+Polynomial+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSummary of End Behavior or Long Run Behavior of Polynomial Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.[reveal-answer q=\"657153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"657153\"]The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.[\/hidden-answer]<\/section>\r\n<h2>Identifying Local Behavior 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\">Turning Points: Locations where the function changes from increasing to decreasing or vice versa.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Intercepts:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: Point where the graph crosses the [latex]y[\/latex]-axis [latex](0, a_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts: Points where the graph crosses the [latex]x[\/latex]-axis (roots of the polynomial)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Continuity and Smoothness: Polynomial functions are both continuous and smooth.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Degree-Behavior Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Maximum number of [latex]x[\/latex]-intercepts = degree of polynomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Maximum number of turning points = degree of polynomial - 1<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">End Behavior: Determined by the degree (odd or even) and the sign of the leading coefficient.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex][reveal-answer q=\"304362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304362\"]There are at most 12 <em>x<\/em>-intercepts and at most 11 turning points.[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given its degree.\r\n<script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bagddhhg-9WW0EetLD4Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/9WW0EetLD4Q?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bagddhhg-9WW0EetLD4Q\" 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=12846610&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bagddhhg-9WW0EetLD4Q&amp;vembed=0&amp;video_id=9WW0EetLD4Q&amp;video_target=tpm-plugin-bagddhhg-9WW0EetLD4Q\" 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\/Turning+Points+and+X+Intercepts+of+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTurning Points and X Intercepts of a Polynomial Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex],\u00a0determine the local behavior.[reveal-answer q=\"978752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"978752\"]The <em>y<\/em>-intercept is found by evaluating [latex]f\\left(0\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(0\\right)=-4\\left(0\\right)\\left(0+3\\right)\\left(0 - 4\\right)\\hfill \\hfill \\\\ \\text{}f\\left(0\\right)=0\\hfill \\end{array}[\/latex]<\/p>\r\nThe <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].\r\n\r\nThe <em>x<\/em>-intercepts are found by setting the function equal to 0.\r\n<p style=\"text-align: center;\">[latex]0=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllllllllll}-4x=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x+3=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x - 4=0\\hfill \\\\ x=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{}x=-3\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{}x=4\\end{array}[\/latex]<\/p>\r\nThe <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(-3,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].\r\n\r\nThe degree is 3 so the graph has at most 2 turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], determine the local behavior.[reveal-answer q=\"104366\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"104366\"]The <em>x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the <em>y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.[\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<li>Identify end behavior of polynomial functions.<\/li>\n<li>Identify intercepts of factored polynomial functions.<\/li>\n<\/ul>\n<\/section>\n<h2>Power 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\">A power function is of the form [latex]f(x) = ax^n[\/latex], where [latex]a[\/latex] and [latex]n[\/latex] are real numbers.\n<ul>\n<li class=\"whitespace-normal break-words\">The &#8216;[latex]a[\/latex]&#8216; in the power function is called the coefficient.<\/li>\n<li class=\"whitespace-normal break-words\">The &#8216;[latex]n[\/latex]&#8216; in the power function can be any real number, not just positive integers.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Power functions include constant, identity, quadratic, cubic, reciprocal, and root functions.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Which functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q105254\">Show Solution<\/button><\/p>\n<div id=\"q105254\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.<\/p><\/div>\n<\/div>\n<\/section>\n<h2>Identifying End Behavior 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\">End behavior describes how a function behaves as [latex]x[\/latex] approaches positive or negative infinity.<\/li>\n<li class=\"whitespace-normal break-words\">Even-power functions ([latex]f(x) = ax^n[\/latex], [latex]n[\/latex] is even):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to \\infty[\/latex] if [latex]a > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]x \\to \\pm\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex] if [latex]a < 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Odd-power functions ([latex]f(x) = ax^n[\/latex], [latex]n[\/latex] is odd):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex] if [latex]a > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]x \\to \\infty[\/latex], [latex]f(x) \\to \\infty[\/latex] if [latex]a > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Behavior is reversed if [latex]a < 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Impact of exponent: Higher powers lead to flatter graphs near the origin and steeper graphs away from it.<\/li>\n<li class=\"whitespace-normal break-words\">Symmetry: Even-power functions are symmetric about the [latex]y[\/latex]-axis; odd-power functions are symmetric about the origin.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Describe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q631242\">Show Solution<\/button><\/p>\n<div id=\"q631242\" class=\"hidden-answer\" style=\"display: none\">The exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex] and as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194505\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/><figcaption class=\"wp-caption-text\">Graph of f(x)=-x^9<\/figcaption><\/figure>\n<h4>Analysis of the Solution<\/h4>\n<p>We can check our work by using the table feature on an online graphing calculator.<\/p>\n<ol>\n<li>Enter the function\u00a0[latex]f\\left(x\\right)=-{x}^{9}[\/latex] into an online graphing calculator<\/li>\n<li>Create a table with the following x values, and observe the sign of the outputs. [latex]-10,-5,0,5,10[\/latex]<\/li>\n<li>Now, enter the function\u00a0[latex]g\\left(x\\right)={x}^{9}[\/latex], and create a similar table. Compare the signs of the outputs for both functions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Describe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q582534\">Show Solution<\/button><\/p>\n<div id=\"q582534\" class=\"hidden-answer\" style=\"display: none\">As [latex]x[\/latex]\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.<\/div>\n<\/div>\n<\/section>\n<h2>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\">A polynomial function is of the form [latex]f(x) = a_nx^n + ... + a_2x^2 + a_1x + a_0[\/latex], where [latex]n[\/latex] is a non-negative integer and [latex]a_i[\/latex] are real number coefficients.<\/li>\n<li class=\"whitespace-normal break-words\">Polynomial functions can be created by combining simpler functions, including power functions.<\/li>\n<li class=\"whitespace-normal break-words\">Terms: Each [latex]a_ix^i[\/latex] is a term of the polynomial function.<\/li>\n<li class=\"whitespace-normal break-words\">Degree: The highest power of [latex]x[\/latex] in the polynomial is its degree.<\/li>\n<\/ul>\n<\/div>\n<h2>Degree and Leading Coefficient\u00a0of a Polynomial Function<\/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\">General Form: Polynomials are typically written in descending order of variable powers.<\/li>\n<li class=\"whitespace-normal break-words\">Degree: The highest power of the variable in the polynomial.<\/li>\n<li class=\"whitespace-normal break-words\">Leading Term: The term with the highest degree.<\/li>\n<li class=\"whitespace-normal break-words\">Leading Coefficient: The coefficient of the leading term.<\/li>\n<li class=\"whitespace-normal break-words\">Importance: These concepts help classify and analyze polynomial behavior.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q435637\">Show Solution<\/button><\/p>\n<div id=\"q435637\" class=\"hidden-answer\" style=\"display: none\">The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbgfbhbf-F_G_w82s0QA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbgfbhbf-F_G_w82s0QA\" 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=12843132&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bbgfbhbf-F_G_w82s0QA&amp;vembed=0&amp;video_id=F_G_w82s0QA&amp;video_target=tpm-plugin-bbgfbhbf-F_G_w82s0QA\" 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\/Degree%2C+Leading+Term%2C+and+Leading+Coefficient+of+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDegree, Leading Term, and Leading Coefficient of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Identifying End Behavior 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\">The end behavior of a polynomial is determined by its leading term.<\/li>\n<li class=\"whitespace-normal break-words\">The degree (even or odd) and sign of the leading coefficient determine the specific end behavior pattern.<\/li>\n<li class=\"whitespace-normal break-words\">End behavior is often described using limit notation, such as [latex]\\text{as } x \\to \\infty, f(x) \\to \\infty[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">To identify end behavior, polynomials should be in expanded (general) form.<\/li>\n<li class=\"whitespace-normal break-words\">The graph of a polynomial function reflects its end behavior.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Describe the end behavior of the polynomial function in the graph below.<\/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\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/><figcaption class=\"wp-caption-text\">Graph of a polynomial<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304329\">Show Solution<\/button><\/p>\n<div id=\"q304329\" class=\"hidden-answer\" style=\"display: none\">As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you&#8217;ll see more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbbeeccc-y78Dpr9LLN0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/y78Dpr9LLN0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbbeeccc-y78Dpr9LLN0\" 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=12845030&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fbbeeccc-y78Dpr9LLN0&amp;vembed=0&amp;video_id=y78Dpr9LLN0&amp;video_target=tpm-plugin-fbbeeccc-y78Dpr9LLN0\" 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\/Summary+of+End+Behavior+or+Long+Run+Behavior+of+Polynomial+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSummary of End Behavior or Long Run Behavior of Polynomial Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q657153\">Show Solution<\/button><\/p>\n<div id=\"q657153\" class=\"hidden-answer\" style=\"display: none\">The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/div>\n<\/div>\n<\/section>\n<h2>Identifying Local Behavior 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\">Turning Points: Locations where the function changes from increasing to decreasing or vice versa.<\/li>\n<li class=\"whitespace-normal break-words\">Intercepts:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-intercept: Point where the graph crosses the [latex]y[\/latex]-axis [latex](0, a_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts: Points where the graph crosses the [latex]x[\/latex]-axis (roots of the polynomial)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Continuity and Smoothness: Polynomial functions are both continuous and smooth.<\/li>\n<li class=\"whitespace-normal break-words\">Degree-Behavior Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Maximum number of [latex]x[\/latex]-intercepts = degree of polynomial<\/li>\n<li class=\"whitespace-normal break-words\">Maximum number of turning points = degree of polynomial &#8211; 1<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">End Behavior: Determined by the degree (odd or even) and the sign of the leading coefficient.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304362\">Show Solution<\/button><\/p>\n<div id=\"q304362\" class=\"hidden-answer\" style=\"display: none\">There are at most 12 <em>x<\/em>-intercepts and at most 11 turning points.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given its degree.<br \/>\n<script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bagddhhg-9WW0EetLD4Q\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/9WW0EetLD4Q?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bagddhhg-9WW0EetLD4Q\" 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=12846610&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bagddhhg-9WW0EetLD4Q&amp;vembed=0&amp;video_id=9WW0EetLD4Q&amp;video_target=tpm-plugin-bagddhhg-9WW0EetLD4Q\" 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\/Turning+Points+and+X+Intercepts+of+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTurning Points and X Intercepts of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex],\u00a0determine the local behavior.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q978752\">Show Solution<\/button><\/p>\n<div id=\"q978752\" class=\"hidden-answer\" style=\"display: none\">The <em>y<\/em>-intercept is found by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(0\\right)=-4\\left(0\\right)\\left(0+3\\right)\\left(0 - 4\\right)\\hfill \\hfill \\\\ \\text{}f\\left(0\\right)=0\\hfill \\end{array}[\/latex]<\/p>\n<p>The <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].<\/p>\n<p>The <em>x<\/em>-intercepts are found by setting the function equal to 0.<\/p>\n<p style=\"text-align: center;\">[latex]0=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllllllllll}-4x=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x+3=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x - 4=0\\hfill \\\\ x=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{}x=-3\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{}x=4\\end{array}[\/latex]<\/p>\n<p>The <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(-3,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\n<p>The degree is 3 so the graph has at most 2 turning points.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], determine the local behavior.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q104366\">Show Solution<\/button><\/p>\n<div id=\"q104366\" class=\"hidden-answer\" style=\"display: none\">The <em>x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the <em>y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/F_G_w82s0QA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Summary of End Behavior or Long Run Behavior of Polynomial Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/y78Dpr9LLN0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Turning Points and X Intercepts of a Polynomial 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