{"id":265,"date":"2025-04-09T17:37:16","date_gmt":"2025-04-09T17:37:16","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-2-power-functions-and-polynomial-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-20T17:31:29","modified_gmt":"2025-08-20T17:31:29","slug":"5-2-power-functions-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-2-power-functions-and-polynomial-functions\/","title":{"raw":"5.2 Power Functions and Polynomial Functions","rendered":"5.2 Power Functions and Polynomial Functions"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_e017cb02-d494-4d95-8d09-966d0968a606\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Identify power functions.<\/li>\r\n \t<li>Identify end behavior of power functions.<\/li>\r\n \t<li>Identify polynomial functions.<\/li>\r\n \t<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_742\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-742\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1-300x225.jpeg\" alt=\"\" width=\"300\" height=\"225\" \/> Figure 1. (credit: Jason Bay, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134540133\">Suppose a certain species of bird thrives in the Morrison Nature Center in Aurora, CO. Its population over the last few years is shown in Table 1.<\/p>\r\n\r\n<div id=\"Table_03_03_01\" class=\"os-table\">\r\n<table class=\"grid\" style=\"height: 34px;\" data-id=\"Table_03_03_01\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"height: 17px; width: 220.733px;\" data-align=\"center\"><strong>Year<\/strong><\/td>\r\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex] 2009 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex] 2010 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex] 2011 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 83.9px;\" data-align=\"center\">[latex] 2012 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 84.3667px;\" data-align=\"center\">[latex] 2013 [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 17px;\">\r\n<td style=\"height: 17px; width: 220.733px;\" data-align=\"center\"><strong>Bird Population<\/strong><\/td>\r\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex] 800 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex] 897 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex] 992 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 83.9px;\" data-align=\"center\">[latex] 1,083 [\/latex]<\/td>\r\n<td style=\"height: 17px; width: 84.3667px;\" data-align=\"center\">[latex] 1,169 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nThe population can be estimated using the function [latex] P(t)=-0.3t^2+97t+800, [\/latex]where [latex] P(t) [\/latex] represents the bird population on the island [latex] t [\/latex] years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.\r\n\r\n<section id=\"fs-id1165137540446\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying Power Functions<\/h2>\r\n<p id=\"fs-id1165137570394\">Before we can understand the bird problem, it will be helpful to understand a different type of function. A <strong>power function <\/strong>is a function with a single term that is the product of a real number, a <strong>coefficient,<\/strong> and a variable raised to a fixed real number.<\/p>\r\n<p id=\"fs-id1165135320417\">As an example, consider functions for area or volume. The function for the area of a circle with radius [latex] r [\/latex] is<\/p>\r\n<p style=\"text-align: center;\">[latex] A(r)=\\pi r^2 [\/latex]<\/p>\r\n<p id=\"fs-id1165135191346\">and the function for the <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">volume of a sphere<\/span> with radius [latex] r [\/latex] is<\/p>\r\n<p style=\"text-align: center;\">[latex] V(r)=\\frac{4}{3}\\pi r^3 [\/latex]<\/p>\r\n<p id=\"fs-id1165137579058\">Both of these are examples of power functions because they consist of a coefficient, [latex] \\pi [\/latex] or [latex] \\frac{4}{3}\\pi [\/latex] multiplied by a variable [latex] r [\/latex] raised to a power.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Power Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>power function<\/strong> is a function that can be represented in the form\r\n<p style=\"text-align: center;\">[latex] f(x)=kx^p [\/latex]<\/p>\r\nwhere [latex] k [\/latex] and [latex] p [\/latex] are real numbers, and [latex] k [\/latex] is known as the <strong>coefficient<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Is [latex] f(x)=2^x [\/latex] a power function?<\/strong>\r\n\r\nA: <em data-effect=\"italics\">No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Identifying Power Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhich of the following functions are power functions?\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} f(x) &amp;=&amp; 1 &amp; \\text{Constant function} \\\\ f(x) &amp;=&amp; x &amp; \\text{Identity function} \\\\ f(x) &amp;=&amp; x^2 &amp; \\text{Quadratic function} \\\\ f(x) &amp;=&amp; x^3 &amp; \\text{Cubic function} \\\\ f(x) &amp;=&amp; \\frac{1}{x} &amp; \\text{Reciprocal function} \\\\ f(x) &amp;=&amp; \\frac{1}{x^2} &amp; \\text{Reciprocal squared function} \\\\ f(x) &amp;=&amp; \\sqrt{x} &amp; \\text{Square root function} \\\\ f(x) &amp;=&amp; \\sqrt[3]{x} &amp; \\text{Cube root function} \\end{array} [\/latex]<\/p>\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p id=\"fs-id1165137843987\">All of the listed functions are power functions.<\/p>\r\n<p id=\"fs-id1165135533093\">The constant and identity functions are power functions because they can be written as [latex] f(x)=x^0 [\/latex] and [latex] f(x)=x^1 [\/latex] respectively.<\/p>\r\nThe quadratic and cubic functions are power functions with whole number powers [latex]\u00a0 f(x)=x^2[\/latex] and [latex] f(x)=x^3. [\/latex]\r\n\r\nThe reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex] f(x)=x^{-1} [\/latex] and [latex] f(x)=x^{-2}. [\/latex]\r\n<p id=\"fs-id1165135704907\">The square and <span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">cube root<\/span> functions are power functions with fractional powers because they can be written as [latex] f(x)=x^{\\frac{1}{2}} [\/latex] or [latex] f(x)=x^{\\frac{1}{3}}. [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhich functions are power functions?\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(x) &amp;=&amp; 2x\\cdot 4x^3 \\\\ g(x) &amp;=&amp; -x^5+5x^3 \\\\ h(x) &amp;=&amp; \\frac{2x^5-1}{3x^2+4} \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134269023\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying End Behavior of Power Functions<\/h2>\r\n<p id=\"fs-id1165135436540\">Figure 2 shows the graphs of [latex] f(x)=x^2, g(x)=x^4, [\/latex] and [latex] h(x)=x^6, [\/latex] which are all power functions with even, positive integer powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_743\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-743\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2-300x156.jpeg\" alt=\"\" width=\"300\" height=\"156\" \/> Figure 2. Even-power functions[\/caption]\r\n<p id=\"fs-id1165137911555\">To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex] \\infty [\/latex] for positive infinity and [latex] -\\infty [\/latex] for negative infinity. When we say that \u201c[latex] x [\/latex] approaches infinity,\u201d which can be symbolically written as [latex] x\\rightarrow \\infty, [\/latex] we are describing a behavior; we are saying that [latex] x [\/latex] is increasing without bound.<\/p>\r\n<p id=\"fs-id1165137658268\">With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as [latex] x [\/latex] approaches positive or negative infinity, the [latex] f(x) [\/latex] values increase without bound. In symbolic form, we could write<\/p>\r\n<p style=\"text-align: center;\">[latex] \\text{as} \\ x\\rightarrow \\pm \\infty, f(x)\\rightarrow \\infty [\/latex]<\/p>\r\n<p id=\"fs-id1165137533222\">Figure 3 shows the graphs of [latex] f(x)=x^3, g(x)=x^5, [\/latex] and [latex] h(x)=x^7, [\/latex] which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_744\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-744\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3-300x225.jpeg\" alt=\"\" width=\"300\" height=\"225\" \/> Figure 3. Odd-power functions[\/caption]\r\n<p id=\"fs-id1165137730237\">These examples illustrate that functions of the form [latex] f(x)=x^n [\/latex] reveal symmetry of one kind or another. First, in Figure 2 we see that even functions of the form [latex] f(x)=x^n, n [\/latex] even, are symmetric about the <em>y<\/em>-axis. In Figure 3 we see that odd functions of the form [latex] f(x)-x^n, n [\/latex] odd, are symmetric about the origin.<\/p>\r\n<p id=\"fs-id1165137812578\">For these odd power functions, as [latex] x [\/latex] approaches negative infinity, [latex] f(x) [\/latex] decreases without bound. As [latex] x [\/latex] approaches positive infinity, [latex] f(x) [\/latex] increases without bound. In symbolic form we write<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}\\text{as } x \\rightarrow -\\infty, \\, f(x) \\rightarrow -\\infty \\, \\\\\\text{as } x \\rightarrow \\infty, \\, f(x) \\rightarrow \\infty\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values get very small [latex] (x\\rightarrow -\\infty) [\/latex] and get very large [latex] (x\\rightarrow \\infty) [\/latex] is referred to as the<strong> end behavior<\/strong> of the function. We can use words or symbols to describe end behavior.<\/p>\r\n<p id=\"fs-id1165137433212\"><a class=\"autogenerated-content\" href=\"5-2-power-functions-and-polynomial-functions#Figure_03_03_004abcd\">Figure 4<\/a> shows the end behavior of power functions in the form [latex] f(x)=kx^n [\/latex] where [latex] n [\/latex] is a non-negative integer depending on the power and the constant.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_745\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-745\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/> Figure 4[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a power function [latex] f(x)=kx^n [\/latex] where [latex] n [\/latex] is a positive integer, identify the end behavior.<\/strong>\r\n<ol>\r\n \t<li>Determine whether the power is even or odd.<\/li>\r\n \t<li>Determine whether the constant is positive or negative.<\/li>\r\n \t<li>Use Figure 4 to identify the end behavior.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Identifying the End Behavior of a Power Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDescribe the end behavior of the graph of [latex] f(x)=x^8. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As [latex] x [\/latex] approaches infinity, the output (value of [latex] f(x) [\/latex]) increases without bound. We write as [latex] x\\rightarrow \\infty, f(x)\\rightarrow \\infty. [\/latex] As [latex] x [\/latex] approaches negative infinity, the output increases without bound. In symbolic form, as [latex] x\\rightarrow -\\infty, f(x)\\rightarrow \\infty. [\/latex] We can graphically represent the function as shown in Figure 5.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_746\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-746\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5-300x203.jpeg\" alt=\"\" width=\"300\" height=\"203\" \/> Figure 5[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Identifying the End Behavior of a Power Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDescribe the end behavior of the graph of [latex] f(x)=-x^9. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The exponent of the power function is 9 (an odd number). Because the coefficient is [latex] -1 [\/latex] (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex] f(x)=x^9. [\/latex] Figure 6 shows that as [latex] x [\/latex] approaches infinity, the output decreases without bound. As [latex] x [\/latex] approaches negative infinity, the output increases without bound. In symbolic form, we would write\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}\\text{as } x \\rightarrow -\\infty, \\, f(x) \\rightarrow \\infty \\, \\\\\\text{as } x \\rightarrow \\infty, \\, f(x) \\rightarrow -\\infty \\end{array} [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_747\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-747\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6-288x300.png\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 6[\/caption]\r\n<h3>Analysis<\/h3>\r\nWe can check our work by using the table feature on a graphing utility.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex] f(x) [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] -10 [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex] 1,000,000,000 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] -5 [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex] 1,953,125 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] 0 [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex] 0 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] 5 [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex] -1,953,125 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">[latex] 10 [\/latex]<\/td>\r\n<td style=\"width: 50%; text-align: center;\">[latex] -1,000,000,000 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can see from Table 2 that, when we substitute very small values for [latex] x [\/latex] the output is very large, and when we substitute very large values for [latex] x [\/latex] the output is very small (meaning that it is a very large negative value).\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Describe in words and symbols the end behavior of [latex] f(x)=-5x^4. [\/latex]<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134069294\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165135689465\">An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex] r [\/latex] of the spill depends on the number of weeks [latex] w [\/latex] that have passed. This relationship is linear.<\/p>\r\n<p style=\"text-align: center;\">[latex] r(w)=24+8w [\/latex]<\/p>\r\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area [latex] A [\/latex] of a circle.<\/p>\r\n<p style=\"text-align: center;\">[latex] A(r)=\\pi r^2 [\/latex]<\/p>\r\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} A(w) &amp;=&amp; A(r(w)) \\\\ &amp;=&amp; A(24+8w) \\\\ &amp;=&amp; \\pi(24+8w)^2 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\r\n<p style=\"text-align: center;\">[latex] A(w)=576\\pi +384\\pi w+64\\pi w ^2 [\/latex]<\/p>\r\n<p id=\"fs-id1165135205726\">This formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Polynomial Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nLet [latex] n [\/latex] be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form\r\n<p style=\"text-align: center;\">[latex] f(x)=a_nx^n+\\ldots+a_2x^2+a_1x+a_0 [\/latex]<\/p>\r\nThis is called the general form of a polynomial function. Each [latex] a_i [\/latex] is a coefficient and can be any real number, but [latex] a_n\\not=0. [\/latex] Each expression [latex] a_ix^i [\/latex] is a <strong>term of a polynomial function<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Identifying Polynomial Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhich of the following are polynomial functions?\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(x) &amp;=&amp; 2x^3\\cdot 3x+4 \\\\ g(x) &amp;=&amp; -x(x^2-4) \\\\ h(x) &amp;=&amp; 5\\sqrt{x+2} \\end{array} [\/latex]<\/p>\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The first two functions are examples of polynomial functions because they can be written in the form [latex] f(x)=a_nx^n+\\ldots+a_2x^2+a_1x+a_0, [\/latex] where the powers are non-negative integers and the coefficients are real numbers.\r\n<ul>\r\n \t<li>[latex] f(x) [\/latex] can be written as [latex] f(x)=6x^4+4. [\/latex]<\/li>\r\n \t<li>[latex] g(x) [\/latex] can be written as [latex] g(x)=-x^3+4x. [\/latex]<\/li>\r\n \t<li>[latex] h(x) [\/latex] cannot be written in this form and is therefore not a polynomial function.<\/li>\r\n<\/ul>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135508329\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h2>\r\n<p id=\"fs-id1165137831216\">Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The <strong><span id=\"term-00019\" data-type=\"term\">degree <\/span><\/strong>of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The- <strong><span id=\"term-00020\" data-type=\"term\">leading term <\/span><\/strong>is the term containing the highest power of the variable, or the term with the highest degree. The <strong><span id=\"term-00021\" data-type=\"term\">leading coefficient <\/span><\/strong>is the coefficient of the leading term.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Terminology of Polynomial Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">We often rearrange polynomials so that the powers are descending.<\/div>\r\n<div><\/div>\r\n<div class=\"textbox__content\"><img class=\"size-medium wp-image-748 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term-300x91.jpeg\" alt=\"\" width=\"300\" height=\"91\" \/><\/div>\r\n<div><\/div>\r\n<div class=\"textbox__content\">When a polynomial is written in this way, we say that it is in general form.<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a polynomial function, identify the degree and leading coefficient.<\/strong>\r\n<ol>\r\n \t<li>Find the highest power of [latex] x [\/latex] to determine the degree of the function.<\/li>\r\n \t<li>Identify the term containing the highest power of [latex] x [\/latex] to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIdentify the degree, leading term, and leading coefficient of the following polynomial functions.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(x) &amp;=&amp; 3+2x^2-4x^3 \\\\ g(t) &amp;=&amp; 5t^5-2t^3+7t \\\\ h(p) &amp;=&amp; 6p-p^3-2 \\end{array} [\/latex]<\/p>\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>For the function [latex] f(x), [\/latex] the highest power of [latex] x [\/latex] is [latex] 3, [\/latex] so the degree if 3. The leading term is the term containing that degree, [latex] -4x^3. [\/latex] The leading coefficient is the coefficient of that term, [latex] -4. [\/latex]\r\n\r\nFor the function [latex] g(t), [\/latex] the highest power of [latex] t [\/latex] is [latex] 5, [\/latex] so the degree is [latex] 5. [\/latex] The leading term is the term containing that degree, [latex] 5t^5. [\/latex] The leading coefficient is the coefficient of that term, [latex] 5. [\/latex]\r\n\r\nFor the function [latex] h(p) [\/latex] the highest power of [latex] p [\/latex] is [latex] 3, [\/latex] so the degree is [latex] 3. [\/latex] The leading term is the term containing that degree, [latex] -p^3. [\/latex] The leading coefficient is the coefficient of that term, [latex] -1. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Identify the degree, leading term, and leading coefficient of the polynomial [latex] f(x)=4x^2-x^6+2x-6. [\/latex]<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137702213\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Identifying End Behavior of Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as [latex] [\/latex] gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See Table 3.<\/p>\r\n\r\n<div id=\"Table_03_03_04\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_03_03_04\"><caption>Table 3<\/caption><colgroup> <col data-align=\"center\" \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th scope=\"col\" data-align=\"center\">Polynomial Function<\/th>\r\n<th scope=\"col\" data-align=\"center\">Leading Term<\/th>\r\n<th scope=\"col\" data-align=\"center\">Graph of Polynomial Function<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x)=5x^4+2x^3-x-4 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 5x^4 [\/latex]<\/td>\r\n<td data-align=\"center\"><span id=\"fs-id1165137768814\" data-type=\"media\" data-alt=\"Graph of f(x)=5x^4+2x^3-x-4.\"><img class=\"alignnone size-medium wp-image-749\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x)=-2x^6-x^5+3x^4+x^3 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -2x^6 [\/latex]<\/td>\r\n<td data-align=\"center\"><span id=\"fs-id1165137714206\" data-type=\"media\" data-alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\"><img class=\"alignnone size-medium wp-image-750\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x)=3x^5-4x^4+2x^2+1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 3x^5 [\/latex]<\/td>\r\n<td data-align=\"center\"><span id=\"fs-id1165137540879\" data-type=\"media\" data-alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\"><img class=\"alignnone size-medium wp-image-751\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x)=-6x^3+7x^2+3x+1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -6x^3 [\/latex]<\/td>\r\n<td data-align=\"center\"><span id=\"fs-id1165137600670\" data-type=\"media\" data-alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\"><img class=\"alignnone size-medium wp-image-752\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Identifying End Behavior and Degree of a Polynomial Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDescribe the end behavior and determine a possible degree of the polynomial function in Figure 7.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_753\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-753\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 7[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>As the input values [latex] x [\/latex] get very large, the output values [latex] f(x) [\/latex] increase without bound. As the input values [latex] x [\/latex] get very small, the output values [latex] f(x) [\/latex] decrease without bound. We can describe the end behavior symbolically by writing\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}\\text{as } x \\rightarrow -\\infty, \\, f(x) \\rightarrow -\\infty \\, \\\\\\text{as } x \\rightarrow \\infty, \\, f(x) \\rightarrow \\infty\\end{array} [\/latex]<\/p>\r\nIn words, we could say that as [latex] x [\/latex] values approach infinity, the function values approach infinity, and as [latex] x [\/latex] values approach negative infinity, the function values approach negative infinity.\r\n<p id=\"fs-id1165134113949\">We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDescribe the end behavior, and determine a possible degree of the polynomial function in Figure 8.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_754\" align=\"alignnone\" width=\"288\"]<img class=\"size-medium wp-image-754\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 8[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Identifying End Behavior and Degree of a Polynomial Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function [latex] f(x)=-3x^2(x-1)(x+4), [\/latex] express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Obtain the general form by expanding the given expression for [latex] f(x). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(x) &amp;=&amp; -3x^2(x-10(x+4) \\\\ &amp;=&amp; -3x^2(x^2+3x-4) \\\\ &amp;=&amp; -3x^4-9x^3+12x^2 \\end{array} [\/latex]<\/p>\r\nThe general form is [latex] f(x)=-3x^4-9x^3+12x^2. [\/latex] The leading term is [latex] -3x^4; [\/latex] therefore, the degree of the polynomial is [latex] 4. [\/latex] The degree is even [latex] (4) [\/latex] and the leading coefficient is negative [latex] (-3), [\/latex] so the end behavior is\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l} \\text{as } x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty \\\\ \\text{as } x\\rightarrow \\infty, f(x)\\rightarrow -\\infty \\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Given the function [latex] f(x)=0.2(x-2)(x+1)(x-5), [\/latex] express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137735781\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Identifying Local Behavior of Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. 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.<\/p>\r\n<p id=\"fs-id1165137417044\">We are also interested in the intercepts. As with all functions, the<em data-effect=\"italics\">y-<\/em>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 <em data-effect=\"italics\">y-<\/em>intercept [latex] (0, a_0). [\/latex] The <em data-effect=\"italics\">x-<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one <em data-effect=\"italics\">x-<\/em>intercept. See Figure 9<strong>.<\/strong><\/p>\r\n\r\n\r\n[caption id=\"attachment_755\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-755\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9-300x254.jpeg\" alt=\"\" width=\"300\" height=\"254\" \/> Figure 9[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Intercepts and Turning Points of Polynomial Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">A <strong><span id=\"term-00022\" data-type=\"term\">turning point <\/span><\/strong>of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The <em data-effect=\"italics\">y-<\/em>intercept is the point at which the function has an input value of zero. The <em data-effect=\"italics\">x<\/em>-intercepts are the points at which the output value is zero.<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a polynomial function, determine the intercepts.<\/strong>\r\n<ol>\r\n \t<li>Determine the <em data-effect=\"italics\">y-<\/em>intercept by setting [latex] x=0 [\/latex] and finding the corresponding output value.<\/li>\r\n \t<li>Determine the <em data-effect=\"italics\">x<\/em>-intercepts by solving for the input values that yield an output value of zero.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Determining the Intercepts of a Polynomial Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the polynomial function [latex] f(x)=(x-2)(x+1)(x-4), [\/latex] written in factored form for your convenience, determine the<em data-effect=\"italics\"> y<\/em>- and <em data-effect=\"italics\">x<\/em>-intercepts.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The<em data-effect=\"italics\">y-<\/em>intercept occurs when the input is zero so substitute 0 for [latex] x. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(0) &amp;=&amp; (0-2)(0+1)(-4) \\\\ &amp;=&amp; (-2)(1)(-4) \\\\ &amp;=&amp; 8 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165135689436\">The <em data-effect=\"italics\">y-<\/em>intercept is [latex] (0, 8). [\/latex]<\/p>\r\n<p id=\"fs-id1165137863224\">The <em data-effect=\"italics\">x<\/em>-intercepts occur when the output is zero.<\/p>\r\n<p style=\"text-align: center;\">[latex] 0=(x-2)(x+1)(x-4) [\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lllll} x-2 &amp;=&amp; 0 &amp; \\text{or} &amp; x+1 &amp;=&amp; 0 &amp; \\text{or} &amp; x-4 &amp;=&amp; 0 \\\\ x &amp;=&amp; 2 &amp; \\text{or} &amp; x &amp;=&amp; -1 &amp; \\text{or} &amp; x &amp;=&amp; 4 \\end{array} [\/latex]<\/p>\r\nThe <em data-effect=\"italics\">x<\/em>-intercepts are [latex] (2, 0), (-1, 0) [\/latex] and [latex] (4, 0). [\/latex]\r\n\r\nWe can see these intercepts on the graph of the function shown in Figure 10.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_756\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-756\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10-300x212.jpeg\" alt=\"\" width=\"300\" height=\"212\" \/> Figure 10[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Determining the Intercepts of a Polynomial Function with Factoring<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the polynomial function [latex] f(x)=x^4-4x^2-45, [\/latex] determine the <em data-effect=\"italics\">y<\/em>- and <em data-effect=\"italics\">x<\/em>-intercepts.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The <em data-effect=\"italics\">y-<\/em>intercept occurs when the input is zero.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(0) &amp;=&amp; (0)^4-4(0)^2-45 \\\\ &amp;=&amp; -45 \\end{array} [\/latex]<\/p>\r\nThe <em data-effect=\"italics\">y-<\/em>intercept is [latex] (0, -45). [\/latex]\r\n\r\nThe <em data-effect=\"italics\">x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(x) &amp;=&amp; x^4=4x^2-45 \\\\ &amp;=&amp; (x^2-9)(x^2+5) \\\\ &amp;=&amp; (x-3)(x+3)(x^2+5) \\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex] 0=(x-3)(x+3)(x^2+5) [\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lllll} x-3 &amp;=&amp; 0 &amp; \\text{or} &amp; x+3 &amp;=&amp; 0 &amp; \\text{or} &amp; x^2+5=0 \\\\ x &amp;=&amp; 3 &amp; \\text{or} &amp; x &amp;=&amp; -3 &amp; \\text{or} &amp; \\text{(no real solution)} \\end{array} [\/latex]<\/p>\r\nThe <em data-effect=\"italics\">x<\/em>-intercepts are [latex] (3, 0) [\/latex] and [latex] (-3, 0). [\/latex]\r\n\r\nWe can see these intercepts on the graph of the function shown in Figure 11. We can see that the function is even because [latex] f(x)=f(-x). [\/latex]\r\n\r\n[caption id=\"attachment_757\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-757\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11-300x256.jpeg\" alt=\"\" width=\"300\" height=\"256\" \/> Figure 11[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Given the polynomial function [latex] f(x)=2x^3-6x^2-20x, [\/latex] determine the <em data-effect=\"italics\">y<\/em>- and <em data-effect=\"italics\">x<\/em>-intercepts.<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165134080932\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Comparing Smooth and Continuous Graphs<\/h3>\r\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of <em data-effect=\"italics\">x<\/em>-intercepts and the number of turning points. A polynomial function of [latex] n\\text{th} [\/latex] degree is the product of [latex] n [\/latex] factors, so it will have at most [latex] n [\/latex] roots or zeros, or <em data-effect=\"italics\">x<\/em>-intercepts. The graph of the polynomial function of degree [latex] n [\/latex] must have at most [latex] n-1 [\/latex] turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\r\n<p id=\"fs-id1165137657937\">A <strong><span id=\"term-00023\" data-type=\"term\">continuous function <\/span><\/strong>has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A <strong><span id=\"term-00024\" data-type=\"term\">smooth curve <\/span><\/strong>is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Intercepts and Turning Points of Polynomials<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">A polynomial of degree [latex] n [\/latex] will have, at most, [latex] n [\/latex] <em data-effect=\"italics\">x<\/em>-intercepts and [latex] n-1 [\/latex] turning points.<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Determining the Number of Intercepts and Turning Points of a Polynomial<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWithout graphing the function, determine the local behavior of the function by finding the maximum number of <em data-effect=\"italics\">x<\/em>-intercepts and turning points for [latex] f(x)=-3x^{10}+4x^7-x^4+2x^3. [\/latex]\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The polynomial has a degree of [latex] 10, [\/latex] so there are at most 10 <em data-effect=\"italics\">x<\/em>-intercepts and at most 9 turning points.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Without graphing the function, determine the maximum number of <em data-effect=\"italics\">x<\/em>-intercepts and turning points for [latex] f(x)=108-13x^9-8x^4+14x^{12}+2x^3. [\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Drawing Conclusions about a Polynomial Function from the Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat can we conclude about the polynomial represented by the graph shown in Figure 12 based on its intercepts and turning points?\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_758\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-758\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12-300x259.jpeg\" alt=\"\" width=\"300\" height=\"259\" \/> Figure 12[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure 13.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_759\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-759\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13-300x290.jpeg\" alt=\"\" width=\"300\" height=\"290\" \/> Figure 13[\/caption]\r\n\r\nThe graph has 2 <em data-effect=\"italics\">x<\/em>-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat can we conclude about the polynomial represented by the graph shown in Figure 14 based on its intercepts and turning points?\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_760\" align=\"aligncenter\" width=\"288\"]<img class=\"size-medium wp-image-760\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/> Figure 14[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 12: Drawing Conclusions about a Polynomial Function from the Factors<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function [latex] f(x)=-4x(x+3)(x-4), [\/latex] determine the local behavior.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The <em data-effect=\"italics\">y<\/em>-intercept is found by evaluating [latex] f(0). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(0) &amp;=&amp; -4(0)(0+3)(0-4) \\\\ &amp;=&amp; 0 \\end{array} [\/latex]<\/p>\r\nThe <em data-effect=\"italics\">y<\/em>-intercept is [latex] (0, 0). [\/latex]\r\n\r\nThe <em data-effect=\"italics\">x<\/em>-intercepts are found by determining the zeros of the function.\r\n<p style=\"text-align: center;\">[latex] 0=-4x(x+3)(x-4) [\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lllll} x &amp;=&amp; 0 &amp; \\text{or} &amp; x+3 &amp;=&amp; 0 &amp; \\text{or} &amp; x-4 &amp;=&amp; 0 \\\\ x &amp;=&amp; 0 &amp; \\text{or} &amp; \\quad x &amp;=&amp; -3 &amp; \\text{or} &amp; \\quad x &amp;=&amp; 4 \\end{array} [\/latex]<\/p>\r\nThe <em data-effect=\"italics\">x<\/em>-intercepts are [latex] (0, 0), (-3, 0), [\/latex] and [latex] (4, 0). [\/latex]\r\n\r\nThe degree is 3 so the graph has at most 2 turning points.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">Given the function [latex] f(x)=0.2(x-2)(x+1)(x-5), [\/latex] determine the local behavior.<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with power and polynomial functions.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=_C6uzPpJuiE\">Find Key Information about a Given Polynomial Function<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=y78Dpr9LLN0\">End Behavior of a Polynomial Function<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=9WW0EetLD4Q\">Turning Points and x-intercepts of Polynomial Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=LU1nLawYyH0\">Least Possible Degree of a Polynomial Function<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">5.2 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165137553381\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137761271\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137651029\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137651030\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137651029-solution\">1<\/a><span class=\"os-divider\">. <\/span>Explain the difference between the coefficient of a power function and its degree.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137784642\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137784643\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137794232\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137794233\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137794232-solution\">3<\/a><span class=\"os-divider\">. <\/span>In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137409144\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137409146\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137628462\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137628463\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137628462-solution\">5<\/a><span class=\"os-divider\">. <\/span>What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As [latex] x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty [\/latex] and as [latex] x\\rightarrow \\infty, f(x)\\rightarrow -\\infty. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135444041\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165137807114\">For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\r\n\r\n<div id=\"fs-id1165137807118\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137807119\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137585053\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137585054\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137585053-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=(x^2)^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137771538\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137771539\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x-x^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135339598\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135339599\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135339598-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{x^2}{x^2-1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137723156\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137723157\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2x(x+2)(x-1)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134149794\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134149795\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134149794-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3^{x+1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137619583\">For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\r\n\r\n<div id=\"fs-id1165137464314\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137464315\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] -3x^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165134380913\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134380914\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134380913-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] 7-2x^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137424478\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137425870\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] -2x^2-3x^5+x-6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137742433\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137742434\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137742433-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] x(4-x^2)(2x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132943536\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132943537\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] x^2(2x-3)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137455116\">For the following exercises, determine the end behavior of the functions.<\/p>\r\n\r\n<div id=\"fs-id1165137455119\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133078101\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137455119-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137423948\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137423949\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137678592\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137678593\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137678592-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-x^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135557775\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135457738\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-x^9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137526614\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137526615\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137526614-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-2x^4-3x^2+x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135696746\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135696747\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3x^2+x-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137627936\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137627937\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137627936-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2(2x^3-x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137761673\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137761674\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=(2-x)^7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135650636\">For the following exercises, find the intercepts of the functions.<\/p>\r\n\r\n<div id=\"fs-id1165135650639\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135650640\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135650639-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] f(t)=2(t-1)(t+1)(t-3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137634317\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137634318\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] g(n)=-2(3n-1)(2n+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134113935\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137645540\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134113935-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4-16 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137783610\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137783611\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3+27 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137823331\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137734444\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137823331-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x(x^2-2x-8) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134223322\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134223324\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=(x+3)(4x^2-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135416569\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165134486753\">For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\r\n\r\n<div id=\"fs-id1165134486757\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134486758\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134486757-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137942457\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-761\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137823453\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137823454\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135530394\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-762\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135436619\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135436620\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135436619-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135543346\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-763\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137781545\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137781546\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137433119\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-764\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34-300x201.jpeg\" alt=\"\" width=\"300\" height=\"201\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135457734\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137833940\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135457734-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137833945\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-765\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137410264\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137410266\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134323766\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-766\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134129751\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134129752\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134129751-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137405932\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-767\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135191392\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135191393\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137653349\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\">\r\n<img class=\"alignnone size-medium wp-image-768\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134262467\">For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\r\n\r\n<div id=\"fs-id1165135332833\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135332834\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135332833-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135678606\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-full wp-image-769\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.39.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137444870\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137444871\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135264631\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an equation.\">\r\n<img class=\"alignnone size-full wp-image-770\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.40.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137704867\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137704868\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137704867-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134089461\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\">\r\n<img class=\"alignnone size-full wp-image-771\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.41.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135191925\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135191926\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135591035\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-full wp-image-772\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.42.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134313927\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137476631\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134313927-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135319603\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-full wp-image-773\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.43.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137436429\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137436430\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137772222\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an equation.\">\r\n<img class=\"alignnone size-full wp-image-774\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.44.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133310456\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135191886\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133310456-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137642864\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\">\r\n<img class=\"alignnone size-full wp-image-775\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.45.jpeg\" alt=\"\" width=\"287\" height=\"297\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134130094\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165137828131\">For the following exercises, make a table to confirm the end behavior of the function.<\/p>\r\n\r\n<div id=\"fs-id1165137828134\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137416865\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-x^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137734861\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137734862\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137734861-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4-5x^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133045350\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133045351\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2(1-x)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135456684\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135456685\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135456684-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=(x-1)(x-2)(3-x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134122886\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134122887\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{x^5}{10}-x^4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137755588\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165133238486\">For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\r\n\r\n<div id=\"fs-id1165135317475\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135317476\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135317475-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3(x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135388458\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135388459\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x(x-3)(x+3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137920714\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137920716\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137920714-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x(14-2x)(10-2x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133448049\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133448050\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x(14-2x)(10-2x)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134417902\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134417903\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134417902-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-16x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137837875\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137837876\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^4-81 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135160696\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135160697\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135160696-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-x^3+x^2+2x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135484565\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135484566\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-x^3-2x^2-15x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133092744\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133092745\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133092744-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-2x^2-15x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134378662\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134378663\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^3-0.01x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137811752\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165133111641\">For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\r\n\r\n<div id=\"fs-id1165133448009\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133448010\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133448009-solution\">61<\/a><span class=\"os-divider\">. <\/span>The <em>y<\/em>-intercept is [latex] (0, 04). [\/latex] The <em>x<\/em>-intercepts are [latex] (-2, 0), (2, 0). [\/latex] Degree is 2.\r\n<div class=\"os-problem-container\">\r\n<p id=\"eip-id1165134540124\">End behavior: as [latex] x\\rightarrow -\\infty, f(x)\\rightarrow \\infty; [\/latex] as [latex] x\\rightarrow \\infty, f(x)\\rightarrow \\infty. [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137894256\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137894257\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>The <em>y<\/em>-intercept is [latex] (0, 9). [\/latex] The <em>x<\/em>-intercepts are [latex] (-3, 0), (3, 0). [\/latex] Degree is 2.\r\n<div class=\"os-problem-container\">\r\n<p id=\"eip-id1165134566570\">End behavior: as [latex] x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty; [\/latex] as [latex] x\\rightarrow \\infty, f(x)\\rightarrow -\\infty. [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137897907\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137897908\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137897907-solution\">63<\/a><span class=\"os-divider\">. <\/span>The<em> y<\/em>-intercept is [latex] (0, 0). [\/latex] The <em>x<\/em>-intercepts are [latex] (0, 0), (2, 0). [\/latex] Degree is 3.\r\n<div class=\"os-problem-container\">\r\n<p id=\"eip-id1165137749966\">End behavior: as [latex] x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty; [\/latex] as [latex] x\\rightarrow \\infty, f(x)\\rightarrow \\infty. [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137921541\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137921542\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>The<em> y<\/em>-intercept is [latex] (0, 1). [\/latex] The <em>x<\/em>-intercept is [latex] (1, 0). [\/latex] Degree is 3.\r\n<div class=\"os-problem-container\">\r\n<p id=\"eip-id1165137895228\">End behavior: as [latex] x\\rightarrow -\\infty, f(x)\\rightarrow \\infty; [\/latex] as [latex] x\\rightarrow \\infty, f(x)\\rightarrow -\\infty. [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135536216\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135536217\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135536216-solution\">65<\/a><span class=\"os-divider\">. <\/span>The y-intercept is [latex] (0, 1). [\/latex] There is no x-intercept. Degree is 4.\r\n<div class=\"os-problem-container\">\r\n<p id=\"eip-id1165135434911\">End behavior: as [latex] x\\rightarrow -\\infty, f(x)\\rightarrow \\infty; [\/latex] as [latex] x\\rightarrow \\infty, f(x)\\rightarrow \\infty. [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134085635\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1165134085641\">For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\r\n\r\n<div id=\"fs-id1165135581116\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135581117\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of [latex] [\/latex] the number of days elapsed.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137892519\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137892520\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137892519-solution\">67<\/a><span class=\"os-divider\">. <\/span>A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of [latex] m, [\/latex] the number of minutes elapsed.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137848042\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137848043\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by [latex] x [\/latex] inches and the width increased by twice that amount, express the area of the rectangle as a function of [latex] x. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137891257\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137891258\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137891257-solution\">69<\/a><span class=\"os-divider\">. <\/span>An open box is to be constructed by cutting out square corners of [latex] x [\/latex]-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of [latex] x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135190825\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135190826\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width [latex] (x). [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_e017cb02-d494-4d95-8d09-966d0968a606\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify power functions.<\/li>\n<li>Identify end behavior of power functions.<\/li>\n<li>Identify polynomial functions.<\/li>\n<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure id=\"attachment_742\" aria-describedby=\"caption-attachment-742\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-742\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1-300x225.jpeg\" alt=\"\" width=\"300\" height=\"225\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1-300x225.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1-65x49.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1-225x169.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1-350x263.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-1.jpeg 488w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-742\" class=\"wp-caption-text\">Figure 1. (credit: Jason Bay, Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134540133\">Suppose a certain species of bird thrives in the Morrison Nature Center in Aurora, CO. Its population over the last few years is shown in Table 1.<\/p>\n<div id=\"Table_03_03_01\" class=\"os-table\">\n<table class=\"grid\" style=\"height: 34px;\" data-id=\"Table_03_03_01\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr style=\"height: 17px;\">\n<td style=\"height: 17px; width: 220.733px;\" data-align=\"center\"><strong>Year<\/strong><\/td>\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex]2009[\/latex]<\/td>\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex]2010[\/latex]<\/td>\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex]2011[\/latex]<\/td>\n<td style=\"height: 17px; width: 83.9px;\" data-align=\"center\">[latex]2012[\/latex]<\/td>\n<td style=\"height: 17px; width: 84.3667px;\" data-align=\"center\">[latex]2013[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 17px;\">\n<td style=\"height: 17px; width: 220.733px;\" data-align=\"center\"><strong>Bird Population<\/strong><\/td>\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex]800[\/latex]<\/td>\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex]897[\/latex]<\/td>\n<td style=\"height: 17px; width: 73.5667px;\" data-align=\"center\">[latex]992[\/latex]<\/td>\n<td style=\"height: 17px; width: 83.9px;\" data-align=\"center\">[latex]1,083[\/latex]<\/td>\n<td style=\"height: 17px; width: 84.3667px;\" data-align=\"center\">[latex]1,169[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>The population can be estimated using the function [latex]P(t)=-0.3t^2+97t+800,[\/latex]where [latex]P(t)[\/latex] represents the bird population on the island [latex]t[\/latex] years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.<\/p>\n<section id=\"fs-id1165137540446\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying Power Functions<\/h2>\n<p id=\"fs-id1165137570394\">Before we can understand the bird problem, it will be helpful to understand a different type of function. A <strong>power function <\/strong>is a function with a single term that is the product of a real number, a <strong>coefficient,<\/strong> and a variable raised to a fixed real number.<\/p>\n<p id=\"fs-id1165135320417\">As an example, consider functions for area or volume. The function for the area of a circle with radius [latex]r[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]A(r)=\\pi r^2[\/latex]<\/p>\n<p id=\"fs-id1165135191346\">and the function for the <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">volume of a sphere<\/span> with radius [latex]r[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]V(r)=\\frac{4}{3}\\pi r^3[\/latex]<\/p>\n<p id=\"fs-id1165137579058\">Both of these are examples of power functions because they consist of a coefficient, [latex]\\pi[\/latex] or [latex]\\frac{4}{3}\\pi[\/latex] multiplied by a variable [latex]r[\/latex] raised to a power.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Power Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>power function<\/strong> is a function that can be represented in the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=kx^p[\/latex]<\/p>\n<p>where [latex]k[\/latex] and [latex]p[\/latex] are real numbers, and [latex]k[\/latex] is known as the <strong>coefficient<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Is [latex]f(x)=2^x[\/latex] a power function?<\/strong><\/p>\n<p>A: <em data-effect=\"italics\">No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Identifying Power Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Which of the following functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} f(x) &=& 1 & \\text{Constant function} \\\\ f(x) &=& x & \\text{Identity function} \\\\ f(x) &=& x^2 & \\text{Quadratic function} \\\\ f(x) &=& x^3 & \\text{Cubic function} \\\\ f(x) &=& \\frac{1}{x} & \\text{Reciprocal function} \\\\ f(x) &=& \\frac{1}{x^2} & \\text{Reciprocal squared function} \\\\ f(x) &=& \\sqrt{x} & \\text{Square root function} \\\\ f(x) &=& \\sqrt[3]{x} & \\text{Cube root function} \\end{array}[\/latex]<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p id=\"fs-id1165137843987\">All of the listed functions are power functions.<\/p>\n<p id=\"fs-id1165135533093\">The constant and identity functions are power functions because they can be written as [latex]f(x)=x^0[\/latex] and [latex]f(x)=x^1[\/latex] respectively.<\/p>\n<p>The quadratic and cubic functions are power functions with whole number powers [latex]\u00a0 f(x)=x^2[\/latex] and [latex]f(x)=x^3.[\/latex]<\/p>\n<p>The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f(x)=x^{-1}[\/latex] and [latex]f(x)=x^{-2}.[\/latex]<\/p>\n<p id=\"fs-id1165135704907\">The square and <span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">cube root<\/span> functions are power functions with fractional powers because they can be written as [latex]f(x)=x^{\\frac{1}{2}}[\/latex] or [latex]f(x)=x^{\\frac{1}{3}}.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Which functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(x) &=& 2x\\cdot 4x^3 \\\\ g(x) &=& -x^5+5x^3 \\\\ h(x) &=& \\frac{2x^5-1}{3x^2+4} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134269023\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying End Behavior of Power Functions<\/h2>\n<p id=\"fs-id1165135436540\">Figure 2 shows the graphs of [latex]f(x)=x^2, g(x)=x^4,[\/latex] and [latex]h(x)=x^6,[\/latex] which are all power functions with even, positive integer powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_743\" aria-describedby=\"caption-attachment-743\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-743\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2-300x156.jpeg\" alt=\"\" width=\"300\" height=\"156\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2-300x156.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2-65x34.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2-225x117.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2-350x182.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-2.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-743\" class=\"wp-caption-text\">Figure 2. Even-power functions<\/figcaption><\/figure>\n<p id=\"fs-id1165137911555\">To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\\infty[\/latex] for positive infinity and [latex]-\\infty[\/latex] for negative infinity. When we say that \u201c[latex]x[\/latex] approaches infinity,\u201d which can be symbolically written as [latex]x\\rightarrow \\infty,[\/latex] we are describing a behavior; we are saying that [latex]x[\/latex] is increasing without bound.<\/p>\n<p id=\"fs-id1165137658268\">With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as [latex]x[\/latex] approaches positive or negative infinity, the [latex]f(x)[\/latex] values increase without bound. In symbolic form, we could write<\/p>\n<p style=\"text-align: center;\">[latex]\\text{as} \\ x\\rightarrow \\pm \\infty, f(x)\\rightarrow \\infty[\/latex]<\/p>\n<p id=\"fs-id1165137533222\">Figure 3 shows the graphs of [latex]f(x)=x^3, g(x)=x^5,[\/latex] and [latex]h(x)=x^7,[\/latex] which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_744\" aria-describedby=\"caption-attachment-744\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-744\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3-300x225.jpeg\" alt=\"\" width=\"300\" height=\"225\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3-300x225.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3-65x49.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3-225x169.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3-350x263.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-3.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-744\" class=\"wp-caption-text\">Figure 3. Odd-power functions<\/figcaption><\/figure>\n<p id=\"fs-id1165137730237\">These examples illustrate that functions of the form [latex]f(x)=x^n[\/latex] reveal symmetry of one kind or another. First, in Figure 2 we see that even functions of the form [latex]f(x)=x^n, n[\/latex] even, are symmetric about the <em>y<\/em>-axis. In Figure 3 we see that odd functions of the form [latex]f(x)-x^n, n[\/latex] odd, are symmetric about the origin.<\/p>\n<p id=\"fs-id1165137812578\">For these odd power functions, as [latex]x[\/latex] approaches negative infinity, [latex]f(x)[\/latex] decreases without bound. As [latex]x[\/latex] approaches positive infinity, [latex]f(x)[\/latex] increases without bound. In symbolic form we write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{as } x \\rightarrow -\\infty, \\, f(x) \\rightarrow -\\infty \\, \\\\\\text{as } x \\rightarrow \\infty, \\, f(x) \\rightarrow \\infty\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values get very small [latex](x\\rightarrow -\\infty)[\/latex] and get very large [latex](x\\rightarrow \\infty)[\/latex] is referred to as the<strong> end behavior<\/strong> of the function. We can use words or symbols to describe end behavior.<\/p>\n<p id=\"fs-id1165137433212\"><a class=\"autogenerated-content\" href=\"5-2-power-functions-and-polynomial-functions#Figure_03_03_004abcd\">Figure 4<\/a> shows the end behavior of power functions in the form [latex]f(x)=kx^n[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_745\" aria-describedby=\"caption-attachment-745\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-745\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4-350x351.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-4.jpeg 731w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-745\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a power function [latex]f(x)=kx^n[\/latex] where [latex]n[\/latex] is a positive integer, identify the end behavior.<\/strong><\/p>\n<ol>\n<li>Determine whether the power is even or odd.<\/li>\n<li>Determine whether the constant is positive or negative.<\/li>\n<li>Use Figure 4 to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Identifying the End Behavior of a Power Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Describe the end behavior of the graph of [latex]f(x)=x^8.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As [latex]x[\/latex] approaches infinity, the output (value of [latex]f(x)[\/latex]) increases without bound. We write as [latex]x\\rightarrow \\infty, f(x)\\rightarrow \\infty.[\/latex] As [latex]x[\/latex] approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\\rightarrow -\\infty, f(x)\\rightarrow \\infty.[\/latex] We can graphically represent the function as shown in Figure 5.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_746\" aria-describedby=\"caption-attachment-746\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-746\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5-300x203.jpeg\" alt=\"\" width=\"300\" height=\"203\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5-300x203.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5-225x152.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5-350x237.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-5.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-746\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Identifying the End Behavior of a Power Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Describe the end behavior of the graph of [latex]f(x)=-x^9.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The exponent of the power function is 9 (an odd number). Because the coefficient is [latex]-1[\/latex] (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f(x)=x^9.[\/latex] Figure 6 shows that as [latex]x[\/latex] approaches infinity, the output decreases without bound. As [latex]x[\/latex] approaches negative infinity, the output increases without bound. In symbolic form, we would write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{as } x \\rightarrow -\\infty, \\, f(x) \\rightarrow \\infty \\, \\\\\\text{as } x \\rightarrow \\infty, \\, f(x) \\rightarrow -\\infty \\end{array}[\/latex]<\/p>\n<figure id=\"attachment_747\" aria-describedby=\"caption-attachment-747\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-747\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6-288x300.png\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6-288x300.png 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6-65x68.png 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6-225x234.png 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6-350x365.png 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-6.png 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-747\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>We can check our work by using the table feature on a graphing utility.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]f(x)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]-10[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]1,000,000,000[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]1,953,125[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]-1,953,125[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 50%; text-align: center;\">[latex]-1,000,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can see from Table 2 that, when we substitute very small values for [latex]x[\/latex] the output is very large, and when we substitute very large values for [latex]x[\/latex] the output is very small (meaning that it is a very large negative value).<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">Describe in words and symbols the end behavior of [latex]f(x)=-5x^4.[\/latex]<\/div>\n<\/div>\n<section id=\"fs-id1165134069294\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying Polynomial Functions<\/h2>\n<p id=\"fs-id1165135689465\">An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex] of the spill depends on the number of weeks [latex]w[\/latex] that have passed. This relationship is linear.<\/p>\n<p style=\"text-align: center;\">[latex]r(w)=24+8w[\/latex]<\/p>\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area [latex]A[\/latex] of a circle.<\/p>\n<p style=\"text-align: center;\">[latex]A(r)=\\pi r^2[\/latex]<\/p>\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} A(w) &=& A(r(w)) \\\\ &=& A(24+8w) \\\\ &=& \\pi(24+8w)^2 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\n<p style=\"text-align: center;\">[latex]A(w)=576\\pi +384\\pi w+64\\pi w ^2[\/latex]<\/p>\n<p id=\"fs-id1165135205726\">This formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Polynomial Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Let [latex]n[\/latex] be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a_nx^n+\\ldots+a_2x^2+a_1x+a_0[\/latex]<\/p>\n<p>This is called the general form of a polynomial function. Each [latex]a_i[\/latex] is a coefficient and can be any real number, but [latex]a_n\\not=0.[\/latex] Each expression [latex]a_ix^i[\/latex] is a <strong>term of a polynomial function<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Identifying Polynomial Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Which of the following are polynomial functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(x) &=& 2x^3\\cdot 3x+4 \\\\ g(x) &=& -x(x^2-4) \\\\ h(x) &=& 5\\sqrt{x+2} \\end{array}[\/latex]<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The first two functions are examples of polynomial functions because they can be written in the form [latex]f(x)=a_nx^n+\\ldots+a_2x^2+a_1x+a_0,[\/latex] where the powers are non-negative integers and the coefficients are real numbers.<\/p>\n<ul>\n<li>[latex]f(x)[\/latex] can be written as [latex]f(x)=6x^4+4.[\/latex]<\/li>\n<li>[latex]g(x)[\/latex] can be written as [latex]g(x)=-x^3+4x.[\/latex]<\/li>\n<li>[latex]h(x)[\/latex] cannot be written in this form and is therefore not a polynomial function.<\/li>\n<\/ul>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135508329\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h2>\n<p id=\"fs-id1165137831216\">Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The <strong><span id=\"term-00019\" data-type=\"term\">degree <\/span><\/strong>of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The- <strong><span id=\"term-00020\" data-type=\"term\">leading term <\/span><\/strong>is the term containing the highest power of the variable, or the term with the highest degree. The <strong><span id=\"term-00021\" data-type=\"term\">leading coefficient <\/span><\/strong>is the coefficient of the leading term.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Terminology of Polynomial Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">We often rearrange polynomials so that the powers are descending.<\/div>\n<div><\/div>\n<div class=\"textbox__content\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-748 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term-300x91.jpeg\" alt=\"\" width=\"300\" height=\"91\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term-300x91.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term-65x20.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term-225x68.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term-350x106.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-term.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/div>\n<div><\/div>\n<div class=\"textbox__content\">When a polynomial is written in this way, we say that it is in general form.<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a polynomial function, identify the degree and leading coefficient.<\/strong><\/p>\n<ol>\n<li>Find the highest power of [latex]x[\/latex] to determine the degree of the function.<\/li>\n<li>Identify the term containing the highest power of [latex]x[\/latex] to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(x) &=& 3+2x^2-4x^3 \\\\ g(t) &=& 5t^5-2t^3+7t \\\\ h(p) &=& 6p-p^3-2 \\end{array}[\/latex]<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>For the function [latex]f(x),[\/latex] the highest power of [latex]x[\/latex] is [latex]3,[\/latex] so the degree if 3. The leading term is the term containing that degree, [latex]-4x^3.[\/latex] The leading coefficient is the coefficient of that term, [latex]-4.[\/latex]<\/p>\n<p>For the function [latex]g(t),[\/latex] the highest power of [latex]t[\/latex] is [latex]5,[\/latex] so the degree is [latex]5.[\/latex] The leading term is the term containing that degree, [latex]5t^5.[\/latex] The leading coefficient is the coefficient of that term, [latex]5.[\/latex]<\/p>\n<p>For the function [latex]h(p)[\/latex] the highest power of [latex]p[\/latex] is [latex]3,[\/latex] so the degree is [latex]3.[\/latex] The leading term is the term containing that degree, [latex]-p^3.[\/latex] The leading coefficient is the coefficient of that term, [latex]-1.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f(x)=4x^2-x^6+2x-6.[\/latex]<\/div>\n<\/div>\n<section id=\"fs-id1165137702213\" data-depth=\"2\">\n<h3 data-type=\"title\">Identifying End Behavior of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as [latex][\/latex] gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See Table 3.<\/p>\n<div id=\"Table_03_03_04\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_03_03_04\">\n<caption>Table 3<\/caption>\n<colgroup>\n<col data-align=\"center\" \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th scope=\"col\" data-align=\"center\">Polynomial Function<\/th>\n<th scope=\"col\" data-align=\"center\">Leading Term<\/th>\n<th scope=\"col\" data-align=\"center\">Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]f(x)=5x^4+2x^3-x-4[\/latex]<\/td>\n<td data-align=\"center\">[latex]5x^4[\/latex]<\/td>\n<td data-align=\"center\"><span id=\"fs-id1165137768814\" data-type=\"media\" data-alt=\"Graph of f(x)=5x^4+2x^3-x-4.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-749\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.1.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)=-2x^6-x^5+3x^4+x^3[\/latex]<\/td>\n<td data-align=\"center\">[latex]-2x^6[\/latex]<\/td>\n<td data-align=\"center\"><span id=\"fs-id1165137714206\" data-type=\"media\" data-alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-750\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.2.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)=3x^5-4x^4+2x^2+1[\/latex]<\/td>\n<td data-align=\"center\">[latex]3x^5[\/latex]<\/td>\n<td data-align=\"center\"><span id=\"fs-id1165137540879\" data-type=\"media\" data-alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-751\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.3-table-3.3.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]f(x)=-6x^3+7x^2+3x+1[\/latex]<\/td>\n<td data-align=\"center\">[latex]-6x^3[\/latex]<\/td>\n<td data-align=\"center\"><span id=\"fs-id1165137600670\" data-type=\"media\" data-alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-752\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-table-3.4.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Identifying End Behavior and Degree of a Polynomial Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_753\" aria-describedby=\"caption-attachment-753\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-753\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-7.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-753\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>As the input values [latex]x[\/latex] get very large, the output values [latex]f(x)[\/latex] increase without bound. As the input values [latex]x[\/latex] get very small, the output values [latex]f(x)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{as } x \\rightarrow -\\infty, \\, f(x) \\rightarrow -\\infty \\, \\\\\\text{as } x \\rightarrow \\infty, \\, f(x) \\rightarrow \\infty\\end{array}[\/latex]<\/p>\n<p>In words, we could say that as [latex]x[\/latex] values approach infinity, the function values approach infinity, and as [latex]x[\/latex] values approach negative infinity, the function values approach negative infinity.<\/p>\n<p id=\"fs-id1165134113949\">We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Describe the end behavior, and determine a possible degree of the polynomial function in Figure 8.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_754\" aria-describedby=\"caption-attachment-754\" style=\"width: 288px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-754\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-8.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-754\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Identifying End Behavior and Degree of a Polynomial Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function [latex]f(x)=-3x^2(x-1)(x+4),[\/latex] express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Obtain the general form by expanding the given expression for [latex]f(x).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(x) &=& -3x^2(x-10(x+4) \\\\ &=& -3x^2(x^2+3x-4) \\\\ &=& -3x^4-9x^3+12x^2 \\end{array}[\/latex]<\/p>\n<p>The general form is [latex]f(x)=-3x^4-9x^3+12x^2.[\/latex] The leading term is [latex]-3x^4;[\/latex] therefore, the degree of the polynomial is [latex]4.[\/latex] The degree is even [latex](4)[\/latex] and the leading coefficient is negative [latex](-3),[\/latex] so the end behavior is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\text{as } x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty \\\\ \\text{as } x\\rightarrow \\infty, f(x)\\rightarrow -\\infty \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">Given the function [latex]f(x)=0.2(x-2)(x+1)(x-5),[\/latex] express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137735781\" data-depth=\"2\">\n<h3 data-type=\"title\">Identifying Local Behavior of Polynomial Functions<\/h3>\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. 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.<\/p>\n<p id=\"fs-id1165137417044\">We are also interested in the intercepts. As with all functions, the<em data-effect=\"italics\">y-<\/em>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 <em data-effect=\"italics\">y-<\/em>intercept [latex](0, a_0).[\/latex] The <em data-effect=\"italics\">x-<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one <em data-effect=\"italics\">x-<\/em>intercept. See Figure 9<strong>.<\/strong><\/p>\n<figure id=\"attachment_755\" aria-describedby=\"caption-attachment-755\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-755\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9-300x254.jpeg\" alt=\"\" width=\"300\" height=\"254\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9-300x254.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9-65x55.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9-225x190.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9-350x296.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-9.jpeg 540w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-755\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Intercepts and Turning Points of Polynomial Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">A <strong><span id=\"term-00022\" data-type=\"term\">turning point <\/span><\/strong>of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The <em data-effect=\"italics\">y-<\/em>intercept is the point at which the function has an input value of zero. The <em data-effect=\"italics\">x<\/em>-intercepts are the points at which the output value is zero.<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a polynomial function, determine the intercepts.<\/strong><\/p>\n<ol>\n<li>Determine the <em data-effect=\"italics\">y-<\/em>intercept by setting [latex]x=0[\/latex] and finding the corresponding output value.<\/li>\n<li>Determine the <em data-effect=\"italics\">x<\/em>-intercepts by solving for the input values that yield an output value of zero.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Determining the Intercepts of a Polynomial Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the polynomial function [latex]f(x)=(x-2)(x+1)(x-4),[\/latex] written in factored form for your convenience, determine the<em data-effect=\"italics\"> y<\/em>&#8211; and <em data-effect=\"italics\">x<\/em>-intercepts.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The<em data-effect=\"italics\">y-<\/em>intercept occurs when the input is zero so substitute 0 for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(0) &=& (0-2)(0+1)(-4) \\\\ &=& (-2)(1)(-4) \\\\ &=& 8 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165135689436\">The <em data-effect=\"italics\">y-<\/em>intercept is [latex](0, 8).[\/latex]<\/p>\n<p id=\"fs-id1165137863224\">The <em data-effect=\"italics\">x<\/em>-intercepts occur when the output is zero.<\/p>\n<p style=\"text-align: center;\">[latex]0=(x-2)(x+1)(x-4)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll} x-2 &=& 0 & \\text{or} & x+1 &=& 0 & \\text{or} & x-4 &=& 0 \\\\ x &=& 2 & \\text{or} & x &=& -1 & \\text{or} & x &=& 4 \\end{array}[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">x<\/em>-intercepts are [latex](2, 0), (-1, 0)[\/latex] and [latex](4, 0).[\/latex]<\/p>\n<p>We can see these intercepts on the graph of the function shown in Figure 10.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_756\" aria-describedby=\"caption-attachment-756\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-756\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10-300x212.jpeg\" alt=\"\" width=\"300\" height=\"212\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10-300x212.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10-65x46.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10-225x159.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10-350x248.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-10.jpeg 561w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-756\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Determining the Intercepts of a Polynomial Function with Factoring<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the polynomial function [latex]f(x)=x^4-4x^2-45,[\/latex] determine the <em data-effect=\"italics\">y<\/em>&#8211; and <em data-effect=\"italics\">x<\/em>-intercepts.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The <em data-effect=\"italics\">y-<\/em>intercept occurs when the input is zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(0) &=& (0)^4-4(0)^2-45 \\\\ &=& -45 \\end{array}[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">y-<\/em>intercept is [latex](0, -45).[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(x) &=& x^4=4x^2-45 \\\\ &=& (x^2-9)(x^2+5) \\\\ &=& (x-3)(x+3)(x^2+5) \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]0=(x-3)(x+3)(x^2+5)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll} x-3 &=& 0 & \\text{or} & x+3 &=& 0 & \\text{or} & x^2+5=0 \\\\ x &=& 3 & \\text{or} & x &=& -3 & \\text{or} & \\text{(no real solution)} \\end{array}[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">x<\/em>-intercepts are [latex](3, 0)[\/latex] and [latex](-3, 0).[\/latex]<\/p>\n<p>We can see these intercepts on the graph of the function shown in Figure 11. We can see that the function is even because [latex]f(x)=f(-x).[\/latex]<\/p>\n<figure id=\"attachment_757\" aria-describedby=\"caption-attachment-757\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-757\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11-300x256.jpeg\" alt=\"\" width=\"300\" height=\"256\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11-300x256.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11-65x56.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11-225x192.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11-350x299.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-11.jpeg 438w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-757\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">Given the polynomial function [latex]f(x)=2x^3-6x^2-20x,[\/latex] determine the <em data-effect=\"italics\">y<\/em>&#8211; and <em data-effect=\"italics\">x<\/em>-intercepts.<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134080932\" data-depth=\"2\">\n<h3 data-type=\"title\">Comparing Smooth and Continuous Graphs<\/h3>\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of <em data-effect=\"italics\">x<\/em>-intercepts and the number of turning points. A polynomial function of [latex]n\\text{th}[\/latex] degree is the product of [latex]n[\/latex] factors, so it will have at most [latex]n[\/latex] roots or zeros, or <em data-effect=\"italics\">x<\/em>-intercepts. The graph of the polynomial function of degree [latex]n[\/latex] must have at most [latex]n-1[\/latex] turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\n<p id=\"fs-id1165137657937\">A <strong><span id=\"term-00023\" data-type=\"term\">continuous function <\/span><\/strong>has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A <strong><span id=\"term-00024\" data-type=\"term\">smooth curve <\/span><\/strong>is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Intercepts and Turning Points of Polynomials<\/p>\n<\/header>\n<div class=\"textbox__content\">A polynomial of degree [latex]n[\/latex] will have, at most, [latex]n[\/latex] <em data-effect=\"italics\">x<\/em>-intercepts and [latex]n-1[\/latex] turning points.<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Determining the Number of Intercepts and Turning Points of a Polynomial<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Without graphing the function, determine the local behavior of the function by finding the maximum number of <em data-effect=\"italics\">x<\/em>-intercepts and turning points for [latex]f(x)=-3x^{10}+4x^7-x^4+2x^3.[\/latex]<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The polynomial has a degree of [latex]10,[\/latex] so there are at most 10 <em data-effect=\"italics\">x<\/em>-intercepts and at most 9 turning points.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">Without graphing the function, determine the maximum number of <em data-effect=\"italics\">x<\/em>-intercepts and turning points for [latex]f(x)=108-13x^9-8x^4+14x^{12}+2x^3.[\/latex]<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Drawing Conclusions about a Polynomial Function from the Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What can we conclude about the polynomial represented by the graph shown in Figure 12 based on its intercepts and turning points?<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_758\" aria-describedby=\"caption-attachment-758\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-758\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12-300x259.jpeg\" alt=\"\" width=\"300\" height=\"259\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12-300x259.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12-65x56.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12-225x194.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12-350x302.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-12.jpeg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-758\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure 13.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_759\" aria-describedby=\"caption-attachment-759\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-759\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13-300x290.jpeg\" alt=\"\" width=\"300\" height=\"290\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13-300x290.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13-65x63.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13-225x217.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13-350x338.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-13.jpeg 378w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-759\" class=\"wp-caption-text\">Figure 13<\/figcaption><\/figure>\n<p>The graph has 2 <em data-effect=\"italics\">x<\/em>-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What can we conclude about the polynomial represented by the graph shown in Figure 14 based on its intercepts and turning points?<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_760\" aria-describedby=\"caption-attachment-760\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-760\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2-fig-14.jpeg 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-760\" class=\"wp-caption-text\">Figure 14<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 12: Drawing Conclusions about a Polynomial Function from the Factors<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function [latex]f(x)=-4x(x+3)(x-4),[\/latex] determine the local behavior.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The <em data-effect=\"italics\">y<\/em>-intercept is found by evaluating [latex]f(0).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(0) &=& -4(0)(0+3)(0-4) \\\\ &=& 0 \\end{array}[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">y<\/em>-intercept is [latex](0, 0).[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">x<\/em>-intercepts are found by determining the zeros of the function.<\/p>\n<p style=\"text-align: center;\">[latex]0=-4x(x+3)(x-4)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllll} x &=& 0 & \\text{or} & x+3 &=& 0 & \\text{or} & x-4 &=& 0 \\\\ x &=& 0 & \\text{or} & \\quad x &=& -3 & \\text{or} & \\quad x &=& 4 \\end{array}[\/latex]<\/p>\n<p>The <em data-effect=\"italics\">x<\/em>-intercepts are [latex](0, 0), (-3, 0),[\/latex] and [latex](4, 0).[\/latex]<\/p>\n<p>The degree is 3 so the graph has at most 2 turning points.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #9<\/p>\n<\/header>\n<div class=\"textbox__content\">Given the function [latex]f(x)=0.2(x-2)(x+1)(x-5),[\/latex] determine the local behavior.<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with power and polynomial functions.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=_C6uzPpJuiE\">Find Key Information about a Given Polynomial Function<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=y78Dpr9LLN0\">End Behavior of a Polynomial Function<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=9WW0EetLD4Q\">Turning Points and x-intercepts of Polynomial Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=LU1nLawYyH0\">Least Possible Degree of a Polynomial Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">5.2 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165137553381\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137761271\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137651029\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137651030\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137651029-solution\">1<\/a><span class=\"os-divider\">. <\/span>Explain the difference between the coefficient of a power function and its degree.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137784642\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137784643\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137794232\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137794233\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137794232-solution\">3<\/a><span class=\"os-divider\">. <\/span>In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137409144\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137409146\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137628462\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137628463\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137628462-solution\">5<\/a><span class=\"os-divider\">. <\/span>What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As [latex]x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty[\/latex] and as [latex]x\\rightarrow \\infty, f(x)\\rightarrow -\\infty.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135444041\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165137807114\">For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\n<div id=\"fs-id1165137807118\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137807119\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137585053\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137585054\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137585053-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=(x^2)^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137771538\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137771539\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x-x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135339598\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135339599\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135339598-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{x^2}{x^2-1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137723156\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137723157\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2x(x+2)(x-1)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134149794\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134149795\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134149794-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3^{x+1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137619583\">For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\n<div id=\"fs-id1165137464314\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137464315\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]-3x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134380913\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134380914\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134380913-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]7-2x^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137424478\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137425870\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]-2x^2-3x^5+x-6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137742433\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137742434\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137742433-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]x(4-x^2)(2x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132943536\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132943537\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]x^2(2x-3)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137455116\">For the following exercises, determine the end behavior of the functions.<\/p>\n<div id=\"fs-id1165137455119\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133078101\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137455119-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137423948\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137423949\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137678592\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137678593\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137678592-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135557775\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135457738\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-x^9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137526614\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137526615\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137526614-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-2x^4-3x^2+x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135696746\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135696747\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3x^2+x-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137627936\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137627937\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137627936-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2(2x^3-x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137761673\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137761674\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=(2-x)^7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135650636\">For the following exercises, find the intercepts of the functions.<\/p>\n<div id=\"fs-id1165135650639\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135650640\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135650639-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]f(t)=2(t-1)(t+1)(t-3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137634317\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137634318\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]g(n)=-2(3n-1)(2n+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134113935\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137645540\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134113935-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4-16[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137783610\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137783611\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3+27[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137823331\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137734444\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137823331-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x(x^2-2x-8)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134223322\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134223324\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=(x+3)(4x^2-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135416569\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165134486753\">For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n<div id=\"fs-id1165134486757\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134486758\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134486757-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137942457\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-761\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.31.jpeg 359w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137823453\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137823454\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135530394\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-762\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.32.jpeg 359w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135436619\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135436620\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135436619-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135543346\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-763\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.33.jpeg 359w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137781545\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137781546\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137433119\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-764\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34-300x201.jpeg\" alt=\"\" width=\"300\" height=\"201\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34-300x201.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34-65x44.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34-225x151.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34-350x235.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.34.jpeg 377w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135457734\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137833940\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135457734-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137833945\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-765\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.35.jpeg 359w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137410264\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137410266\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134323766\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-766\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36-288x300.jpeg\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36-288x300.jpeg 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36-225x234.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36-350x365.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.36.jpeg 359w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134129751\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134129752\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134129751-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137405932\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-767\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37-290x300.jpeg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37-225x233.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37-350x363.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.37.jpeg 359w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135191392\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135191393\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137653349\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-768\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38-290x300.jpeg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38-225x233.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38-350x363.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.38.jpeg 359w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134262467\">For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\n<div id=\"fs-id1165135332833\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135332834\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135332833-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135678606\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-769\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.39.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.39.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.39-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.39-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137444870\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137444871\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135264631\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an equation.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-770\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.40.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.40.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.40-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.40-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137704867\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137704868\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137704867-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134089461\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an even-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-771\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.41.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.41.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.41-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.41-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135191925\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135191926\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135591035\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-772\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.42.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.42.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.42-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.42-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134313927\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137476631\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134313927-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165135319603\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-773\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.43.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.43.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.43-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.43-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137436429\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137436430\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137772222\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an equation.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-774\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.44.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.44.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.44-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.44-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133310456\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135191886\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133310456-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137642864\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an odd-degree polynomial.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-775\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.45.jpeg\" alt=\"\" width=\"287\" height=\"297\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.45.jpeg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.45-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.2.45-225x233.jpeg 225w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134130094\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165137828131\">For the following exercises, make a table to confirm the end behavior of the function.<\/p>\n<div id=\"fs-id1165137828134\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137416865\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-x^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137734861\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137734862\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137734861-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4-5x^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133045350\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133045351\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2(1-x)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135456684\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135456685\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135456684-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=(x-1)(x-2)(3-x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134122886\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134122887\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{x^5}{10}-x^4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137755588\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165133238486\">For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\n<div id=\"fs-id1165135317475\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135317476\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135317475-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3(x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135388458\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135388459\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x(x-3)(x+3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137920714\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137920716\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137920714-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x(14-2x)(10-2x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133448049\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133448050\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x(14-2x)(10-2x)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134417902\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134417903\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134417902-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-16x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137837875\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137837876\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^4-81[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135160696\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135160697\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135160696-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-x^3+x^2+2x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135484565\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135484566\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-x^3-2x^2-15x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133092744\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133092745\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133092744-solution\">59<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-2x^2-15x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134378662\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134378663\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^3-0.01x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137811752\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165133111641\">For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\n<div id=\"fs-id1165133448009\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133448010\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133448009-solution\">61<\/a><span class=\"os-divider\">. <\/span>The <em>y<\/em>-intercept is [latex](0, 04).[\/latex] The <em>x<\/em>-intercepts are [latex](-2, 0), (2, 0).[\/latex] Degree is 2.<\/p>\n<div class=\"os-problem-container\">\n<p id=\"eip-id1165134540124\">End behavior: as [latex]x\\rightarrow -\\infty, f(x)\\rightarrow \\infty;[\/latex] as [latex]x\\rightarrow \\infty, f(x)\\rightarrow \\infty.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137894256\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137894257\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>The <em>y<\/em>-intercept is [latex](0, 9).[\/latex] The <em>x<\/em>-intercepts are [latex](-3, 0), (3, 0).[\/latex] Degree is 2.<\/p>\n<div class=\"os-problem-container\">\n<p id=\"eip-id1165134566570\">End behavior: as [latex]x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty;[\/latex] as [latex]x\\rightarrow \\infty, f(x)\\rightarrow -\\infty.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137897907\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137897908\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137897907-solution\">63<\/a><span class=\"os-divider\">. <\/span>The<em> y<\/em>-intercept is [latex](0, 0).[\/latex] The <em>x<\/em>-intercepts are [latex](0, 0), (2, 0).[\/latex] Degree is 3.<\/p>\n<div class=\"os-problem-container\">\n<p id=\"eip-id1165137749966\">End behavior: as [latex]x\\rightarrow -\\infty, f(x)\\rightarrow -\\infty;[\/latex] as [latex]x\\rightarrow \\infty, f(x)\\rightarrow \\infty.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137921541\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137921542\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>The<em> y<\/em>-intercept is [latex](0, 1).[\/latex] The <em>x<\/em>-intercept is [latex](1, 0).[\/latex] Degree is 3.<\/p>\n<div class=\"os-problem-container\">\n<p id=\"eip-id1165137895228\">End behavior: as [latex]x\\rightarrow -\\infty, f(x)\\rightarrow \\infty;[\/latex] as [latex]x\\rightarrow \\infty, f(x)\\rightarrow -\\infty.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135536216\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135536217\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135536216-solution\">65<\/a><span class=\"os-divider\">. <\/span>The y-intercept is [latex](0, 1).[\/latex] There is no x-intercept. Degree is 4.<\/p>\n<div class=\"os-problem-container\">\n<p id=\"eip-id1165135434911\">End behavior: as [latex]x\\rightarrow -\\infty, f(x)\\rightarrow \\infty;[\/latex] as [latex]x\\rightarrow \\infty, f(x)\\rightarrow \\infty.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134085635\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1165134085641\">For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\n<div id=\"fs-id1165135581116\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135581117\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of [latex][\/latex] the number of days elapsed.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137892519\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137892520\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137892519-solution\">67<\/a><span class=\"os-divider\">. <\/span>A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of [latex]m,[\/latex] the number of minutes elapsed.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137848042\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137848043\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by [latex]x[\/latex] inches and the width increased by twice that amount, express the area of the rectangle as a function of [latex]x.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137891257\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137891258\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137891257-solution\">69<\/a><span class=\"os-divider\">. <\/span>An open box is to be constructed by cutting out square corners of [latex]x[\/latex]-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of [latex]x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135190825\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135190826\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width [latex](x).[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-265","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/265","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":22,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/265\/revisions"}],"predecessor-version":[{"id":1529,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/265\/revisions\/1529"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/158"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/265\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=265"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=265"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=265"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}