{"id":13838,"date":"2018-08-24T22:04:31","date_gmt":"2018-08-24T22:04:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13838"},"modified":"2025-02-05T05:19:15","modified_gmt":"2025-02-05T05:19:15","slug":"power-functions-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/power-functions-and-polynomial-functions\/","title":{"raw":"Power Functions and Polynomial Functions","rendered":"Power Functions and Polynomial Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\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 \t<li>Identify end behavior of polynomial functions.<\/li>\r\n \t<li>Identify intercepts of factored polynomial functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"CNX_Precalc_Figure_03_03_001.jpg\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010716\/CNX_Precalc_Figure_03_03_0012.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/> <b>Figure 1.<\/b> (credit: Jason Bay, Flickr)[\/caption]<\/figure>\r\n<p id=\"fs-id1165134540133\">Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.<\/p>\r\n\r\n<table id=\"Table_03_03_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<td>2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Bird Population<\/strong><\/td>\r\n<td>800<\/td>\r\n<td>897<\/td>\r\n<td>992<\/td>\r\n<td>1,083<\/td>\r\n<td>1,169<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137442798\">The population can be estimated using the function [latex]P\\left(t\\right)=-0.3{t}^{3}+97t+800[\/latex], where [latex]P\\left(t\\right)[\/latex] represents the bird population on the island <i>t<\/i>\u00a0years 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>\r\n\r\n<h2>Identify power functions<\/h2>\r\n<section id=\"fs-id1165137540446\">\r\n<p id=\"fs-id1165137570394\">In order to better understand the bird problem, we need to understand a specific 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. (A number that multiplies a variable raised to an exponent is known as a coefficient.)<\/p>\r\n<p id=\"fs-id1165135320417\">As an example, consider functions for area or volume. The function for the <strong>area of a circle<\/strong> with radius <em>r\u00a0<\/em>is<\/p>\r\n\r\n<div id=\"eip-544\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165135191346\">and the function for the <strong>volume of a sphere<\/strong> with radius <em>r<\/em>\u00a0is<\/p>\r\n\r\n<div id=\"eip-640\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/div>\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 <em>r<\/em>\u00a0raised to a power.<\/p>\r\n\r\n<div id=\"fs-id1165135356525\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Power Function<\/h3>\r\n<p id=\"fs-id1165137771947\">A <strong>power function<\/strong> is a function that can be represented in the form<\/p>\r\n\r\n<div id=\"eip-826\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\r\n<p id=\"eip-id1165135584093\">where <em>k<\/em>\u00a0and <em>p<\/em>\u00a0are real numbers, and <em>k<\/em>\u00a0is known as the <strong>coefficient<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137661479\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165137582131\"><strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong><\/p>\r\n<p id=\"fs-id1165137598469\"><em>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>\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_01\" class=\"example\">\r\n<div id=\"fs-id1165137745179\" class=\"exercise\">\r\n<div id=\"fs-id1165137742710\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Identifying Power Functions<\/h3>\r\n<p id=\"fs-id1165137824370\">Which of the following functions are power functions?<\/p>\r\n<p id=\"fs-id1165137422594\" style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(x\\right)=1 &amp;&amp; \\text{Constant function} \\\\ &amp;f\\left(x\\right)=x &amp;&amp; \\text{Identify function} \\\\ &amp;f\\left(x\\right)={x}^{2} &amp;&amp; \\text{Quadratic function} \\\\ &amp;f\\left(x\\right)={x}^{3} &amp;&amp; \\text{Cubic function} \\\\ &amp;f\\left(x\\right)=\\frac{1}{x} &amp;&amp; \\text{Reciprocal function} \\\\ &amp;f\\left(x\\right)=\\frac{1}{{x}^{2}} &amp;&amp; \\text{Reciprocal squared function} \\\\ &amp;f\\left(x\\right)=\\sqrt{x} &amp;&amp; \\text{Square root function} \\\\ &amp;f\\left(x\\right)=\\sqrt[3]{x} &amp;&amp; \\text{Cube root function} \\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"343941\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343941\"]\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\\left(x\\right)={x}^{0}[\/latex] and [latex]f\\left(x\\right)={x}^{1}[\/latex] respectively.<\/p>\r\n<p id=\"fs-id1165137411464\">The quadratic and cubic functions are power functions with whole number powers [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex].<\/p>\r\n<p id=\"fs-id1165137475956\">The <strong>reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].<\/p>\r\n<p id=\"fs-id1165135704907\">The square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137475225\">Which functions are power functions?<\/p>\r\n<p id=\"fs-id1165137824385\" style=\"text-align: center;\">[latex]\\begin{align}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3} \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4} \\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"475003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"475003\"]\r\n\r\n[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<h2>\u00a0Identify end behavior of power functions<\/h2>\r\n<section id=\"fs-id1165134269023\">\r\n<p id=\"fs-id1165135436540\">Figure 2\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex] and [latex]\\text{and}h\\left(x\\right)={x}^{6}[\/latex], which are all power functions with even, whole-number 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\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010716\/CNX_Precalc_Figure_03_03_0022.jpg\" alt=\"Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.\" width=\"487\" height=\"253\" \/> <b>Figure 2.<\/b> 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 \"<em>x<\/em> approaches infinity,\" which can be symbolically written as [latex]x\\to \\infty [\/latex], we are describing a behavior; we are saying that <em>x<\/em>\u00a0is increasing without bound.<\/p>\r\n<p id=\"fs-id1165137658268\">With the 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\\left(x\\right)[\/latex] values increase without bound. In symbolic form, we could write<\/p>\r\n\r\n<div id=\"eip-742\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{as }x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty\\\\ [\/latex]<\/div>\r\n<p id=\"fs-id1165137533222\">Figure 3\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},\\text{and}h\\left(x\\right)={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\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"312\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010716\/CNX_Precalc_Figure_03_03_0032.jpg\" alt=\"Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.\" width=\"312\" height=\"366\" \/> <b>Figure 3.<\/b> Odd-power function[\/caption]\r\n<p id=\"fs-id1165137730237\">These examples illustrate that functions of the form [latex]f\\left(x\\right)={x}^{n}[\/latex] reveal symmetry of one kind or another. First, in Figure 2\u00a0we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the <em>y<\/em>-axis. In Figure 3\u00a0we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.<\/p>\r\n<p id=\"fs-id1165137812578\">For these odd power functions, as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex]\u00a0decreases without bound. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex]\u00a0increases without bound. In symbolic form we write<\/p>\r\n\r\n<div id=\"eip-77\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&amp;\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\[1mm] &amp;\\text{as } x\\to \\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{ } \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values get very small ( [latex]x\\to -\\infty[\/latex] ) and get very large ( [latex]x\\to \\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\">The table\u00a0below shows the end behavior of power functions in the form [latex]f\\left(x\\right)=k{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.<span id=\"eip-id1165133101746\">\r\n<\/span><\/p>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/08\/15151124\/image0031.jpg\"><img class=\"alignnone size-full wp-image-15967\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/08\/15151124\/image0031.jpg\" alt=\"\" width=\"731\" height=\"734\" \/><\/a>\r\n<div id=\"fs-id1165135161436\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137415258\">How To: Given a power function [latex]f\\left(x\\right)=k{x}^{n}[\/latex] where <em>n<\/em>\u00a0is a non-negative integer, identify the end behavior.<\/h3>\r\n<ol id=\"fs-id1165137409522\">\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\u00a0to identify the end behavior.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_03_02\" class=\"example\">\r\n<div id=\"fs-id1165137923491\" class=\"exercise\">\r\n<div id=\"fs-id1165137599768\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Identifying the End Behavior of a Power Function<\/h3>\r\n<p id=\"fs-id1165137644554\">Describe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].<\/p>\r\n[reveal-answer q=\"122409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"122409\"]\r\n\r\nThe coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As <em>x<\/em>\u00a0approaches infinity, the output (value of [latex]f\\left(x\\right)[\/latex] ) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty [\/latex]. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty\\\\ [\/latex]. We can graphically represent the function as shown in Figure 5.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_0082.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/> <strong>Figure 4<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_03\" class=\"example\">\r\n<div id=\"fs-id1165137535914\" class=\"exercise\">\r\n<div id=\"fs-id1165137811997\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Identifying the End Behavior of a Power Function.<\/h3>\r\n<p id=\"fs-id1165137453217\">Describe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].<\/p>\r\n[reveal-answer q=\"929491\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"929491\"]\r\n\r\nThe exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/>\r\n<p style=\"text-align: center;\"><strong>Figure 5.\u00a0<\/strong>[latex]\\begin{cases}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{cases}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137548471\">We can check our work by using the table feature on a graphing utility.<\/p>\r\n\r\n<table id=\"Table_03_03_03\" summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th><em>x<\/em><\/th>\r\n<th><em>f<\/em>(<em>x<\/em>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>\u201310<\/td>\r\n<td>1,000,000,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u20135<\/td>\r\n<td>1,953,125<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>\u20131,953,125<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>\u20131,000,000,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137644426\">We can see from the table above\u00a0that, when we substitute very small values for <em>x<\/em>, the output is very large, and when we substitute very large values for <em>x<\/em>, the output is very small (meaning that it is a very large negative value).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137734868\">Describe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].<\/p>\r\n[reveal-answer q=\"472844\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"472844\"]\r\n\r\nAs <em>x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty\\\\ [\/latex] because of the negative coefficient.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<h2>\u00a0Identify 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 <em>r<\/em>\u00a0of the spill depends on the number of weeks <em>w<\/em>\u00a0that have passed. This relationship is linear.<\/p>\r\n\r\n<div id=\"eip-719\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\r\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area <em>A<\/em>\u00a0of a circle.<\/p>\r\n\r\n<div id=\"eip-731\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\r\n\r\n<div id=\"eip-645\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}A\\left(w\\right)&amp;=A\\left(r\\left(w\\right)\\right)\\\\ &amp;=A\\left(24+8w\\right)\\\\ &amp;=\\pi {\\left(24+8w\\right)}^{2}\\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\r\n\r\n<div id=\"eip-290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/div>\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 id=\"fs-id1165137715427\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165137823247\">Let <em>n<\/em>\u00a0be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\n<p id=\"eip-id1165137832690\">This is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_04\" class=\"example\">\r\n<div id=\"fs-id1165137817691\" class=\"exercise\">\r\n<div id=\"fs-id1165137817693\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Identifying Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4 \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right) \\\\ h\\left(x\\right)=5\\sqrt{x}+2 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"824812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"824812\"]\r\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.<\/p>\r\n\r\n<ul id=\"fs-id1165137864157\">\r\n \t<li>[latex]f\\left(x\\right)[\/latex]\r\ncan be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\r\n \t<li>[latex]g\\left(x\\right)[\/latex]\r\ncan be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\r\n \t<li>[latex]h\\left(x\\right)[\/latex]\r\ncannot be written in this form and is therefore not a polynomial function.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>\u00a0Identify the degree and leading coefficient of polynomial functions<\/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>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The <strong>leading term<\/strong> is the term containing the highest power of the variable, or the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\r\n\r\n<div id=\"fs-id1165135193124\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Terminology of Polynomial Functions<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_010n2.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"487\" height=\"147\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<span id=\"fs-id1165137406148\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165137482568\">When a polynomial is written in this way, we say that it is in general form.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134031372\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137803898\">How To: Given a polynomial function, identify the degree and leading coefficient.<\/h3>\r\n<ol id=\"fs-id1165135587816\">\r\n \t<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree function.<\/li>\r\n \t<li>Identify the term containing the highest power of <em>x\u00a0<\/em>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 id=\"Example_03_03_05\" class=\"example\">\r\n<div id=\"fs-id1165137401820\" class=\"exercise\">\r\n<div id=\"fs-id1165137862379\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165137435372\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{gathered} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\ g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\ h\\left(p\\right)=6p-{p}^{3}-2\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"951580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951580\"]\r\n<p id=\"fs-id1165137722510\">For the function [latex]f\\left(x\\right)[\/latex], the highest power of <em>x<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, \u20134.<\/p>\r\n<p id=\"fs-id1165135457771\">For the function [latex]g\\left(t\\right)[\/latex], the highest power of <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.<\/p>\r\n<p id=\"fs-id1165135503949\">For the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, \u20131.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137424484\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\r\n[reveal-answer q=\"148647\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"148647\"]\r\n\r\nThe degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is \u20131.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]34293[\/ohm_question]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165137702213\">\r\n<h2>Identifying End Behavior of Polynomial Functions<\/h2>\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 <em>x<\/em>\u00a0gets 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 term of highest degree.<\/p>\r\n\r\n<table id=\"Table_03_03_04\" summary=\"..\"><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">Polynomial Function<\/th>\r\n<th style=\"text-align: center;\">Leading Term<\/th>\r\n<th style=\"text-align: center;\">Graph of Polynomial Function<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\r\n<td>[latex]5{x}^{4}[\/latex]<\/td>\r\n<td><span id=\"fs-id1165137768814\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\r\n<td>[latex]-2{x}^{6}[\/latex]<\/td>\r\n<td><span id=\"fs-id1165137714206\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\r\n<td>[latex]3{x}^{5}[\/latex]<\/td>\r\n<td><span id=\"fs-id1165137540879\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\r\n<td>[latex]-6{x}^{3}[\/latex]<\/td>\r\n<td><span id=\"fs-id1165137600670\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"Example_03_03_06\" class=\"example\">\r\n<div id=\"fs-id1165137452413\" class=\"exercise\">\r\n<div id=\"fs-id1165137452415\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\nDescribe the end behavior and determine a possible degree of the polynomial function in Figure 7.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0152.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"710491\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"710491\"]\r\n<p id=\"fs-id1165135251312\">As the input values <em>x<\/em>\u00a0get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values <em>x<\/em>\u00a0get very small, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ &amp;\\text{as } x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137454991\">In words, we could say that as <em>x<\/em>\u00a0values approach infinity, the function values approach infinity, and as <em>x<\/em>\u00a0values approach negative infinity, the function values approach negative infinity.<\/p>\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[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nDescribe the end behavior, and determine a possible degree of the polynomial function in Figure 9.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/> <b>Figure 9<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"361928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"361928\"]\r\n\r\nAs [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]. It has the shape of an even degree power function with a negative coefficient.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_07\" class=\"example\">\r\n<div id=\"fs-id1165137470361\" class=\"exercise\">\r\n<div id=\"fs-id1165137470363\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165132011287\">Given the function [latex]f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)[\/latex], express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.<\/p>\r\n[reveal-answer q=\"601270\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"601270\"]\r\n<p id=\"fs-id1165137401109\">Obtain the general form by expanding the given expression for [latex]f\\left(x\\right)[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(x\\right)&amp;=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ &amp;=-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ &amp;=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137634030\">The general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[\/latex].\u00a0The leading term is [latex]-3{x}^{4}[\/latex];\u00a0therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ &amp;\\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137416652\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\r\n[reveal-answer q=\"512714\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512714\"]\r\n\r\nThe general form is [latex]f(x)=0.2x^3-1.2x^2+0.6x-2[\/latex]\r\n\r\nThe leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137735781\">\r\n<h2>Identifying Local Behavior of Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \"middle\" 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\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0172.jpg\" alt=\"\" width=\"731\" height=\"629\" \/> <b>Figure 10<\/b>[\/caption]\r\n<p id=\"fs-id1165137417044\">We are also interested in the intercepts. As with all functions, the <em>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>y-<\/em>intercept, [latex]\\left(0,{a}_{0}\\right)[\/latex]. The <em>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>x-<\/em>intercept.\u00a0<span id=\"fs-id1165135511323\">\r\n<\/span><\/p>\r\n\r\n<div id=\"fs-id1165135378843\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Intercepts and Turning Points of Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165137638552\">A <strong>turning point<\/strong> of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The <em>y-<\/em>intercept is the point at which the function has an input value of zero. The <em>x<\/em>-intercepts are the points at which the output value is zero.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137766902\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137645233\">How To: Given a polynomial function, determine the intercepts.<\/h3>\r\n<ol id=\"fs-id1165137571388\">\r\n \t<li>Determine the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and finding the corresponding output value.<\/li>\r\n \t<li>Determine the <em>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 id=\"Example_03_03_08\" class=\"example\">\r\n<div id=\"fs-id1165137435581\" class=\"exercise\">\r\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Determining the Intercepts of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165137441767\">Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the <em>y<\/em>- and\u00a0<em>x<\/em>-intercepts.<\/p>\r\n[reveal-answer q=\"994834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"994834\"]\r\n<p id=\"fs-id1165135251468\">The <em>y-<\/em>intercept occurs when the input is zero so substitute 0 for <em>x<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(0\\right)&amp;=\\left(0 - 2\\right)\\left(0+1\\right)\\left(0 - 4\\right) \\\\ &amp;=\\left(-2\\right)\\left(1\\right)\\left(-4\\right) \\\\ &amp;=8 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135689436\">The <em>y-<\/em>intercept is (0, 8).<\/p>\r\n<p id=\"fs-id1165137863224\">The <em>x<\/em>-intercepts occur when the output is zero.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;x - 2=0 &amp;&amp; \\text{or} &amp;&amp; x+1=0 &amp;&amp; \\text{or} &amp;&amp; x - 4=0 \\\\ &amp;x=2 &amp;&amp; \\text{or} &amp;&amp; x=-1 &amp;&amp; \\text{or} &amp;&amp; x=4 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135316178\">The\u00a0<i>x<\/i>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\r\nWe can see these intercepts on the graph of the function shown in Figure 11.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0182.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"487\" height=\"630\" \/> <b>Figure 11<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_09\" class=\"example\">\r\n<div id=\"fs-id1165137834894\" class=\"exercise\">\r\n<div id=\"fs-id1165137834896\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Determining the Intercepts of a Polynomial Function with Factoring<\/h3>\r\n<p id=\"fs-id1165137628033\">Given the polynomial function [latex]f\\left(x\\right)={x}^{4}-4{x}^{2}-45[\/latex], determine the <em>y<\/em>- and\u00a0<em>x<\/em>-intercepts.<\/p>\r\n[reveal-answer q=\"133046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133046\"]\r\n<p id=\"fs-id1165137634475\">The <em>y-<\/em>intercept occurs when the input is zero.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(0\\right)&amp;={\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45 \\\\ &amp;=-45 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135653967\">The <em>y-<\/em>intercept is [latex]\\left(0,-45\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135152099\">The <em>x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(x\\right)&amp;={x}^{4}-4{x}^{2}-45 \\\\ &amp;=\\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right) \\\\ &amp;=\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\\\ \\text{ } \\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)=0[\/latex]<\/p>\r\n[latex]x^2+5[\/latex] can't be 0, so we only consider the first two factors.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x - 3=0 &amp;&amp; \\text{or} &amp;&amp; x+3=0 \\\\ x=3 &amp;&amp; \\text{or} &amp;&amp; x=-3 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135436471\">The <em>x<\/em>-intercepts are [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex].<\/p>\r\nWe can see these intercepts on the graph of the function shown in Figure 12. We can see that the function is even because [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0192.jpg\" alt=\"Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).\" width=\"487\" height=\"426\" \/> <b>Figure 12<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137405244\">Given the polynomial function [latex]f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x[\/latex], determine the <em>y<\/em>- and<em> x<\/em>-intercepts.<\/p>\r\n[reveal-answer q=\"512961\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512961\"]\r\n\r\n<em>y<\/em>-intercept [latex]\\left(0,0\\right)[\/latex]; <em>x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(-2,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]99335[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Comparing Smooth and Continuous Graphs<\/span>\r\n\r\n<\/section><section id=\"fs-id1165134080932\">\r\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of <em>x<\/em>-intercepts and the number of turning points. A polynomial function of\u00a0<em>n<\/em>th degree is the product of <em>n<\/em>\u00a0factors, so it will have at most <em>n<\/em>\u00a0roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree <em>n<\/em>\u00a0must have at most <em>n<\/em> \u2013 1\u00a0turning 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>continuous function<\/strong> has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A <strong>smooth curve<\/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 id=\"fs-id1165137847104\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Intercepts and Turning Points of Polynomials<\/h3>\r\n<p id=\"fs-id1165137405499\">A polynomial of degree <em>n<\/em>\u00a0will have, at most, <em>n<\/em>\u00a0<em>x<\/em>-intercepts and <em>n<\/em> \u2013 1\u00a0turning points.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_10\" class=\"example\">\r\n<div id=\"fs-id1165135237034\" class=\"exercise\">\r\n<div id=\"fs-id1165135237036\" class=\"problem textbox shaded\">\r\n<h3>Example 10: Determining the Number of Intercepts and Turning Points of a Polynomial<\/h3>\r\n<p id=\"fs-id1165134152759\">Without graphing the function, determine the local behavior of the function by finding the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].<\/p>\r\n[reveal-answer q=\"308403\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"308403\"]\r\n\r\nThe polynomial has a degree of 10, so there are at most <em>10<\/em>\u00a0[latex]x[\/latex]-intercepts and at most <i>9<\/i>\u00a0turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135188274\">Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]<\/p>\r\n[reveal-answer q=\"515707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"515707\"]\r\n\r\nThere are at most 12 <em>x<\/em>-intercepts and at most 11 turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_11\" class=\"example\">\r\n<div id=\"fs-id1165137435064\" class=\"exercise\">\r\n<div id=\"fs-id1165137435066\" class=\"problem textbox shaded\">\r\n<h3>Example 11: Drawing Conclusions about a Polynomial Function from the Graph<\/h3>\r\nWhat can we conclude about the polynomial represented by the graph shown in the graph in Figure 13\u00a0based on its intercepts and turning points?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0202.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/> <b>Figure 13<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"236792\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236792\"]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010720\/CNX_Precalc_Figure_03_03_0212.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/> <b>Figure 14<\/b>[\/caption]\r\n<p id=\"fs-id1165131926327\">The end behavior of the graph tells us this is the graph of an even-degree polynomial.\u00a0<span id=\"fs-id1165137883772\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135670389\">The graph has 2 <em>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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWhat can we conclude about the polynomial represented by Figure 15\u00a0based on its intercepts and turning points?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010720\/CNX_Precalc_Figure_03_03_0224.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/> <b>Figure 15<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"587065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587065\"]\r\n\r\nThe end behavior indicates an odd-degree polynomial function; there are 3 <em>x<\/em>-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_12\" class=\"example\">\r\n<div id=\"fs-id1165135184013\" class=\"exercise\">\r\n<div id=\"fs-id1165137725458\" class=\"problem textbox shaded\">\r\n<h3>Example 12: Drawing Conclusions about a Polynomial Function from the Factors<\/h3>\r\n<p id=\"fs-id1165135435639\">Given the function [latex]f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex],\u00a0determine the local behavior.<\/p>\r\n[reveal-answer q=\"141768\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"141768\"]\r\n<p id=\"fs-id1165135457723\">The <em>y<\/em>-intercept is found by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(0\\right)=-4\\left(0\\right)\\left(0+3\\right)\\left(0 - 4\\right)=0 [\/latex]<\/p>\r\n<p id=\"fs-id1165135245749\">The <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135203755\">The <em>x<\/em>-intercepts are found by determining the zeros of the function.<\/p>\r\n<p style=\"text-align: center;\">[latex]-4x\\left(x+3\\right)\\left(x - 4\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x=0 &amp;&amp; \\text{or} &amp;&amp; x+3=0 &amp;&amp; \\text{or} &amp;&amp; x - 4=0 \\\\ x=0 &amp;&amp; \\text{or} &amp;&amp; x=-3 &amp;&amp; \\text{or} &amp;&amp; x=4\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135431016\">The <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(-3,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165137472984\">The degree is 3 so the graph has at most 2 turning points.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137575431\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], determine the local behavior.<\/p>\r\n[reveal-answer q=\"617003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617003\"]\r\n\r\nThe <em>x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the <em>y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question hide_question_numbers=1]66677[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span>\r\n\r\n<section id=\"fs-id1165137724050\" class=\"key-equations\">\r\n<table id=\"eip-id1165134063974\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>general form of a polynomial function<\/td>\r\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137731646\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135438864\">\r\n \t<li>A power function is a variable base raised to a number power.<\/li>\r\n \t<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\r\n \t<li>The end behavior depends on whether the power is even or odd.<\/li>\r\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\r\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\r\n \t<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\r\n \t<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em>x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137668266\" class=\"definition\">\r\n \t<dt><strong>coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135194918\" class=\"definition\">\r\n \t<dt><strong>continuous function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832108\" class=\"definition\">\r\n \t<dt><strong>degree<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832115\" class=\"definition\">\r\n \t<dt><strong>end behavior<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>leading coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>leading term<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a function that 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.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong>power function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=k{x}^{p}[\/latex]\u00a0where <em>k\u00a0<\/em>is a constant, the base is a variable, and the exponent, <em>p<\/em>,\u00a0is a constant\u00a0smooth curve\u00a0a graph with no sharp corners<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>term of a polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133085661\" class=\"definition\">\r\n \t<dt><strong>turning point<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\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<li>Identify end behavior of polynomial functions.<\/li>\n<li>Identify intercepts of factored polynomial functions.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"CNX_Precalc_Figure_03_03_001.jpg\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010716\/CNX_Precalc_Figure_03_03_0012.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> (credit: Jason Bay, Flickr)<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165134540133\">Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.<\/p>\n<table id=\"Table_03_03_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr>\n<tr>\n<td><strong>Bird Population<\/strong><\/td>\n<td>800<\/td>\n<td>897<\/td>\n<td>992<\/td>\n<td>1,083<\/td>\n<td>1,169<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137442798\">The population can be estimated using the function [latex]P\\left(t\\right)=-0.3{t}^{3}+97t+800[\/latex], where [latex]P\\left(t\\right)[\/latex] represents the bird population on the island <i>t<\/i>\u00a0years 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<h2>Identify power functions<\/h2>\n<section id=\"fs-id1165137540446\">\n<p id=\"fs-id1165137570394\">In order to better understand the bird problem, we need to understand a specific 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. (A number that multiplies a variable raised to an exponent is known as a coefficient.)<\/p>\n<p id=\"fs-id1165135320417\">As an example, consider functions for area or volume. The function for the <strong>area of a circle<\/strong> with radius <em>r\u00a0<\/em>is<\/p>\n<div id=\"eip-544\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165135191346\">and the function for the <strong>volume of a sphere<\/strong> with radius <em>r<\/em>\u00a0is<\/p>\n<div id=\"eip-640\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/div>\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 <em>r<\/em>\u00a0raised to a power.<\/p>\n<div id=\"fs-id1165135356525\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Power Function<\/h3>\n<p id=\"fs-id1165137771947\">A <strong>power function<\/strong> is a function that can be represented in the form<\/p>\n<div id=\"eip-826\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\n<p id=\"eip-id1165135584093\">where <em>k<\/em>\u00a0and <em>p<\/em>\u00a0are real numbers, and <em>k<\/em>\u00a0is known as the <strong>coefficient<\/strong>.<\/p>\n<\/div>\n<div id=\"fs-id1165137661479\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165137582131\"><strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong><\/p>\n<p id=\"fs-id1165137598469\"><em>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 id=\"Example_03_03_01\" class=\"example\">\n<div id=\"fs-id1165137745179\" class=\"exercise\">\n<div id=\"fs-id1165137742710\" class=\"problem textbox shaded\">\n<h3>Example 1: Identifying Power Functions<\/h3>\n<p id=\"fs-id1165137824370\">Which of the following functions are power functions?<\/p>\n<p id=\"fs-id1165137422594\" style=\"text-align: center;\">[latex]\\begin{align}&f\\left(x\\right)=1 && \\text{Constant function} \\\\ &f\\left(x\\right)=x && \\text{Identify function} \\\\ &f\\left(x\\right)={x}^{2} && \\text{Quadratic function} \\\\ &f\\left(x\\right)={x}^{3} && \\text{Cubic function} \\\\ &f\\left(x\\right)=\\frac{1}{x} && \\text{Reciprocal function} \\\\ &f\\left(x\\right)=\\frac{1}{{x}^{2}} && \\text{Reciprocal squared function} \\\\ &f\\left(x\\right)=\\sqrt{x} && \\text{Square root function} \\\\ &f\\left(x\\right)=\\sqrt[3]{x} && \\text{Cube root function} \\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q343941\">Show Solution<\/span><\/p>\n<div id=\"q343941\" class=\"hidden-answer\" style=\"display: none\">\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\\left(x\\right)={x}^{0}[\/latex] and [latex]f\\left(x\\right)={x}^{1}[\/latex] respectively.<\/p>\n<p id=\"fs-id1165137411464\">The quadratic and cubic functions are power functions with whole number powers [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex].<\/p>\n<p id=\"fs-id1165137475956\">The <strong>reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].<\/p>\n<p id=\"fs-id1165135704907\">The square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137475225\">Which functions are power functions?<\/p>\n<p id=\"fs-id1165137824385\" style=\"text-align: center;\">[latex]\\begin{align}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3} \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4} \\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q475003\">Show Solution<\/span><\/p>\n<div id=\"q475003\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>\u00a0Identify end behavior of power functions<\/h2>\n<section id=\"fs-id1165134269023\">\n<p id=\"fs-id1165135436540\">Figure 2\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex] and [latex]\\text{and}h\\left(x\\right)={x}^{6}[\/latex], which are all power functions with even, whole-number 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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010716\/CNX_Precalc_Figure_03_03_0022.jpg\" alt=\"Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.\" width=\"487\" height=\"253\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> Even-power functions<\/p>\n<\/div>\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 &#8220;<em>x<\/em> approaches infinity,&#8221; which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that <em>x<\/em>\u00a0is increasing without bound.<\/p>\n<p id=\"fs-id1165137658268\">With the 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\\left(x\\right)[\/latex] values increase without bound. In symbolic form, we could write<\/p>\n<div id=\"eip-742\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{as }x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137533222\">Figure 3\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},\\text{and}h\\left(x\\right)={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<div style=\"width: 322px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010716\/CNX_Precalc_Figure_03_03_0032.jpg\" alt=\"Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.\" width=\"312\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3.<\/b> Odd-power function<\/p>\n<\/div>\n<p id=\"fs-id1165137730237\">These examples illustrate that functions of the form [latex]f\\left(x\\right)={x}^{n}[\/latex] reveal symmetry of one kind or another. First, in Figure 2\u00a0we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the <em>y<\/em>-axis. In Figure 3\u00a0we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.<\/p>\n<p id=\"fs-id1165137812578\">For these odd power functions, as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex]\u00a0decreases without bound. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex]\u00a0increases without bound. In symbolic form we write<\/p>\n<div id=\"eip-77\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}&\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\[1mm] &\\text{as } x\\to \\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values get very small ( [latex]x\\to -\\infty[\/latex] ) and get very large ( [latex]x\\to \\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\">The table\u00a0below shows the end behavior of power functions in the form [latex]f\\left(x\\right)=k{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.<span id=\"eip-id1165133101746\"><br \/>\n<\/span><\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/08\/15151124\/image0031.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-15967\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/08\/15151124\/image0031.jpg\" alt=\"\" width=\"731\" height=\"734\" \/><\/a><\/p>\n<div id=\"fs-id1165135161436\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137415258\">How To: Given a power function [latex]f\\left(x\\right)=k{x}^{n}[\/latex] where <em>n<\/em>\u00a0is a non-negative integer, identify the end behavior.<\/h3>\n<ol id=\"fs-id1165137409522\">\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\u00a0to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_02\" class=\"example\">\n<div id=\"fs-id1165137923491\" class=\"exercise\">\n<div id=\"fs-id1165137599768\" class=\"problem textbox shaded\">\n<h3>Example 2: Identifying the End Behavior of a Power Function<\/h3>\n<p id=\"fs-id1165137644554\">Describe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q122409\">Show Solution<\/span><\/p>\n<div id=\"q122409\" class=\"hidden-answer\" style=\"display: none\">\n<p>The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As <em>x<\/em>\u00a0approaches infinity, the output (value of [latex]f\\left(x\\right)[\/latex] ) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty[\/latex]. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty\\\\[\/latex]. We can graphically represent the function as shown in Figure 5.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_0082.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_03\" class=\"example\">\n<div id=\"fs-id1165137535914\" class=\"exercise\">\n<div id=\"fs-id1165137811997\" class=\"problem textbox shaded\">\n<h3>Example 3: Identifying the End Behavior of a Power Function.<\/h3>\n<p id=\"fs-id1165137453217\">Describe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q929491\">Show Solution<\/span><\/p>\n<div id=\"q929491\" class=\"hidden-answer\" style=\"display: none\">\n<p>The exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Figure 5.\u00a0<\/strong>[latex]\\begin{cases}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{cases}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137548471\">We can check our work by using the table feature on a graphing utility.<\/p>\n<table id=\"Table_03_03_03\" summary=\"..\">\n<thead>\n<tr>\n<th><em>x<\/em><\/th>\n<th><em>f<\/em>(<em>x<\/em>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u201310<\/td>\n<td>1,000,000,000<\/td>\n<\/tr>\n<tr>\n<td>\u20135<\/td>\n<td>1,953,125<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>\u20131,953,125<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>\u20131,000,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137644426\">We can see from the table above\u00a0that, when we substitute very small values for <em>x<\/em>, the output is very large, and when we substitute very large values for <em>x<\/em>, the output is very small (meaning that it is a very large negative value).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137734868\">Describe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q472844\">Show Solution<\/span><\/p>\n<div id=\"q472844\" class=\"hidden-answer\" style=\"display: none\">\n<p>As <em>x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty\\\\[\/latex] because of the negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>\u00a0Identify 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 <em>r<\/em>\u00a0of the spill depends on the number of weeks <em>w<\/em>\u00a0that have passed. This relationship is linear.<\/p>\n<div id=\"eip-719\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area <em>A<\/em>\u00a0of a circle.<\/p>\n<div id=\"eip-731\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\n<div id=\"eip-645\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}A\\left(w\\right)&=A\\left(r\\left(w\\right)\\right)\\\\ &=A\\left(24+8w\\right)\\\\ &=\\pi {\\left(24+8w\\right)}^{2}\\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\n<div id=\"eip-290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/div>\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 id=\"fs-id1165137715427\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Polynomial Functions<\/h3>\n<p id=\"fs-id1165137823247\">Let <em>n<\/em>\u00a0be 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\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p id=\"eip-id1165137832690\">This is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.<\/p>\n<\/div>\n<div id=\"Example_03_03_04\" class=\"example\">\n<div id=\"fs-id1165137817691\" class=\"exercise\">\n<div id=\"fs-id1165137817693\" class=\"problem textbox shaded\">\n<h3>Example 4: Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4 \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right) \\\\ h\\left(x\\right)=5\\sqrt{x}+2 \\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q824812\">Show Solution<\/span><\/p>\n<div id=\"q824812\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.<\/p>\n<ul id=\"fs-id1165137864157\">\n<li>[latex]f\\left(x\\right)[\/latex]<br \/>\ncan be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\n<li>[latex]g\\left(x\\right)[\/latex]<br \/>\ncan be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\n<li>[latex]h\\left(x\\right)[\/latex]<br \/>\ncannot be written in this form and is therefore not a polynomial function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Identify the degree and leading coefficient of polynomial functions<\/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>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The <strong>leading term<\/strong> is the term containing the highest power of the variable, or the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<div id=\"fs-id1165135193124\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Terminology of Polynomial Functions<\/h3>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_010n2.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"487\" height=\"147\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<span id=\"fs-id1165137406148\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137482568\">When a polynomial is written in this way, we say that it is in general form.<\/p>\n<\/div>\n<div id=\"fs-id1165134031372\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137803898\">How To: Given a polynomial function, identify the degree and leading coefficient.<\/h3>\n<ol id=\"fs-id1165135587816\">\n<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree function.<\/li>\n<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_05\" class=\"example\">\n<div id=\"fs-id1165137401820\" class=\"exercise\">\n<div id=\"fs-id1165137862379\" class=\"problem textbox shaded\">\n<h3>Example 5: Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137435372\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\ g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\ h\\left(p\\right)=6p-{p}^{3}-2\\end{gathered}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951580\">Show Solution<\/span><\/p>\n<div id=\"q951580\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137722510\">For the function [latex]f\\left(x\\right)[\/latex], the highest power of <em>x<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, \u20134.<\/p>\n<p id=\"fs-id1165135457771\">For the function [latex]g\\left(t\\right)[\/latex], the highest power of <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.<\/p>\n<p id=\"fs-id1165135503949\">For the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, \u20131.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137424484\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q148647\">Show Solution<\/span><\/p>\n<div id=\"q148647\" class=\"hidden-answer\" style=\"display: none\">\n<p>The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is \u20131.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm34293\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34293&theme=oea&iframe_resize_id=ohm34293\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<section id=\"fs-id1165137702213\">\n<h2>Identifying End Behavior of Polynomial Functions<\/h2>\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 <em>x<\/em>\u00a0gets 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 term of highest degree.<\/p>\n<table id=\"Table_03_03_04\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Polynomial Function<\/th>\n<th style=\"text-align: center;\">Leading Term<\/th>\n<th style=\"text-align: center;\">Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\n<td>[latex]5{x}^{4}[\/latex]<\/td>\n<td><span id=\"fs-id1165137768814\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010717\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{6}[\/latex]<\/td>\n<td><span id=\"fs-id1165137714206\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td>[latex]3{x}^{5}[\/latex]<\/td>\n<td><span id=\"fs-id1165137540879\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td>[latex]-6{x}^{3}[\/latex]<\/td>\n<td><span id=\"fs-id1165137600670\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Example_03_03_06\" class=\"example\">\n<div id=\"fs-id1165137452413\" class=\"exercise\">\n<div id=\"fs-id1165137452415\" class=\"problem textbox shaded\">\n<h3>Example 6: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p>Describe the end behavior and determine a possible degree of the polynomial function in Figure 7.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010718\/CNX_Precalc_Figure_03_03_0152.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q710491\">Show Solution<\/span><\/p>\n<div id=\"q710491\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135251312\">As the input values <em>x<\/em>\u00a0get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values <em>x<\/em>\u00a0get very small, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ &\\text{as } x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137454991\">In words, we could say that as <em>x<\/em>\u00a0values approach infinity, the function values approach infinity, and as <em>x<\/em>\u00a0values 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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Describe the end behavior, and determine a possible degree of the polynomial function in Figure 9.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q361928\">Show Solution<\/span><\/p>\n<div id=\"q361928\" class=\"hidden-answer\" style=\"display: none\">\n<p>As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_07\" class=\"example\">\n<div id=\"fs-id1165137470361\" class=\"exercise\">\n<div id=\"fs-id1165137470363\" class=\"problem textbox shaded\">\n<h3>Example 7: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165132011287\">Given the function [latex]f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)[\/latex], express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q601270\">Show Solution<\/span><\/p>\n<div id=\"q601270\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137401109\">Obtain the general form by expanding the given expression for [latex]f\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(x\\right)&=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ &=-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ &=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137634030\">The general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[\/latex].\u00a0The leading term is [latex]-3{x}^{4}[\/latex];\u00a0therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\text{as } x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ &\\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137416652\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512714\">Show Solution<\/span><\/p>\n<div id=\"q512714\" class=\"hidden-answer\" style=\"display: none\">\n<p>The general form is [latex]f(x)=0.2x^3-1.2x^2+0.6x-2[\/latex]<\/p>\n<p>The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137735781\">\n<h2>Identifying Local Behavior of Polynomial Functions<\/h2>\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the &#8220;middle&#8221; 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<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0172.jpg\" alt=\"\" width=\"731\" height=\"629\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137417044\">We are also interested in the intercepts. As with all functions, the <em>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>y-<\/em>intercept, [latex]\\left(0,{a}_{0}\\right)[\/latex]. The <em>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>x-<\/em>intercept.\u00a0<span id=\"fs-id1165135511323\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1165135378843\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Intercepts and Turning Points of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137638552\">A <strong>turning point<\/strong> of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The <em>y-<\/em>intercept is the point at which the function has an input value of zero. The <em>x<\/em>-intercepts are the points at which the output value is zero.<\/p>\n<\/div>\n<div id=\"fs-id1165137766902\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137645233\">How To: Given a polynomial function, determine the intercepts.<\/h3>\n<ol id=\"fs-id1165137571388\">\n<li>Determine the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and finding the corresponding output value.<\/li>\n<li>Determine the <em>x<\/em>-intercepts by solving for the input values that yield an output value of zero.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_08\" class=\"example\">\n<div id=\"fs-id1165137435581\" class=\"exercise\">\n<div id=\"fs-id1165137803210\" class=\"problem textbox shaded\">\n<h3>Example 8: Determining the Intercepts of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137441767\">Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the <em>y<\/em>&#8211; and\u00a0<em>x<\/em>-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q994834\">Show Solution<\/span><\/p>\n<div id=\"q994834\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135251468\">The <em>y-<\/em>intercept occurs when the input is zero so substitute 0 for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(0\\right)&=\\left(0 - 2\\right)\\left(0+1\\right)\\left(0 - 4\\right) \\\\ &=\\left(-2\\right)\\left(1\\right)\\left(-4\\right) \\\\ &=8 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135689436\">The <em>y-<\/em>intercept is (0, 8).<\/p>\n<p id=\"fs-id1165137863224\">The <em>x<\/em>-intercepts occur when the output is zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &x - 2=0 && \\text{or} && x+1=0 && \\text{or} && x - 4=0 \\\\ &x=2 && \\text{or} && x=-1 && \\text{or} && x=4 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135316178\">The\u00a0<i>x<\/i>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\n<p>We can see these intercepts on the graph of the function shown in Figure 11.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0182.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"487\" height=\"630\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_09\" class=\"example\">\n<div id=\"fs-id1165137834894\" class=\"exercise\">\n<div id=\"fs-id1165137834896\" class=\"problem textbox shaded\">\n<h3>Example 9: Determining the Intercepts of a Polynomial Function with Factoring<\/h3>\n<p id=\"fs-id1165137628033\">Given the polynomial function [latex]f\\left(x\\right)={x}^{4}-4{x}^{2}-45[\/latex], determine the <em>y<\/em>&#8211; and\u00a0<em>x<\/em>-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133046\">Show Solution<\/span><\/p>\n<div id=\"q133046\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137634475\">The <em>y-<\/em>intercept occurs when the input is zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(0\\right)&={\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45 \\\\ &=-45 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135653967\">The <em>y-<\/em>intercept is [latex]\\left(0,-45\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135152099\">The <em>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{align}f\\left(x\\right)&={x}^{4}-4{x}^{2}-45 \\\\ &=\\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right) \\\\ &=\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\\\ \\text{ } \\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)=0[\/latex]<\/p>\n<p>[latex]x^2+5[\/latex] can&#8217;t be 0, so we only consider the first two factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x - 3=0 && \\text{or} && x+3=0 \\\\ x=3 && \\text{or} && x=-3 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135436471\">The <em>x<\/em>-intercepts are [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex].<\/p>\n<p>We can see these intercepts on the graph of the function shown in Figure 12. We can see that the function is even because [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0192.jpg\" alt=\"Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).\" width=\"487\" height=\"426\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137405244\">Given the polynomial function [latex]f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x[\/latex], determine the <em>y<\/em>&#8211; and<em> x<\/em>-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512961\">Show Solution<\/span><\/p>\n<div id=\"q512961\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>y<\/em>-intercept [latex]\\left(0,0\\right)[\/latex]; <em>x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(-2,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm99335\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99335&theme=oea&iframe_resize_id=ohm99335\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Comparing Smooth and Continuous Graphs<\/span><\/p>\n<\/section>\n<section id=\"fs-id1165134080932\">\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of <em>x<\/em>-intercepts and the number of turning points. A polynomial function of\u00a0<em>n<\/em>th degree is the product of <em>n<\/em>\u00a0factors, so it will have at most <em>n<\/em>\u00a0roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree <em>n<\/em>\u00a0must have at most <em>n<\/em> \u2013 1\u00a0turning 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>continuous function<\/strong> has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A <strong>smooth curve<\/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 id=\"fs-id1165137847104\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Intercepts and Turning Points of Polynomials<\/h3>\n<p id=\"fs-id1165137405499\">A polynomial of degree <em>n<\/em>\u00a0will have, at most, <em>n<\/em>\u00a0<em>x<\/em>-intercepts and <em>n<\/em> \u2013 1\u00a0turning points.<\/p>\n<\/div>\n<div id=\"Example_03_03_10\" class=\"example\">\n<div id=\"fs-id1165135237034\" class=\"exercise\">\n<div id=\"fs-id1165135237036\" class=\"problem textbox shaded\">\n<h3>Example 10: Determining the Number of Intercepts and Turning Points of a Polynomial<\/h3>\n<p id=\"fs-id1165134152759\">Without graphing the function, determine the local behavior of the function by finding the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q308403\">Show Solution<\/span><\/p>\n<div id=\"q308403\" class=\"hidden-answer\" style=\"display: none\">\n<p>The polynomial has a degree of 10, so there are at most <em>10<\/em>\u00a0[latex]x[\/latex]-intercepts and at most <i>9<\/i>\u00a0turning points.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135188274\">Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q515707\">Show Solution<\/span><\/p>\n<div id=\"q515707\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are at most 12 <em>x<\/em>-intercepts and at most 11 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_11\" class=\"example\">\n<div id=\"fs-id1165137435064\" class=\"exercise\">\n<div id=\"fs-id1165137435066\" class=\"problem textbox shaded\">\n<h3>Example 11: Drawing Conclusions about a Polynomial Function from the Graph<\/h3>\n<p>What can we conclude about the polynomial represented by the graph shown in the graph in Figure 13\u00a0based on its intercepts and turning points?<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010719\/CNX_Precalc_Figure_03_03_0202.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236792\">Show Solution<\/span><\/p>\n<div id=\"q236792\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010720\/CNX_Precalc_Figure_03_03_0212.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165131926327\">The end behavior of the graph tells us this is the graph of an even-degree polynomial.\u00a0<span id=\"fs-id1165137883772\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135670389\">The graph has 2 <em>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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>What can we conclude about the polynomial represented by Figure 15\u00a0based on its intercepts and turning points?<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010720\/CNX_Precalc_Figure_03_03_0224.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587065\">Show Solution<\/span><\/p>\n<div id=\"q587065\" class=\"hidden-answer\" style=\"display: none\">\n<p>The end behavior indicates an odd-degree polynomial function; there are 3 <em>x<\/em>-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_12\" class=\"example\">\n<div id=\"fs-id1165135184013\" class=\"exercise\">\n<div id=\"fs-id1165137725458\" class=\"problem textbox shaded\">\n<h3>Example 12: Drawing Conclusions about a Polynomial Function from the Factors<\/h3>\n<p id=\"fs-id1165135435639\">Given the function [latex]f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex],\u00a0determine the local behavior.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q141768\">Show Solution<\/span><\/p>\n<div id=\"q141768\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135457723\">The <em>y<\/em>-intercept is found by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(0\\right)=-4\\left(0\\right)\\left(0+3\\right)\\left(0 - 4\\right)=0[\/latex]<\/p>\n<p id=\"fs-id1165135245749\">The <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135203755\">The <em>x<\/em>-intercepts are found by determining the zeros of the function.<\/p>\n<p style=\"text-align: center;\">[latex]-4x\\left(x+3\\right)\\left(x - 4\\right)=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x=0 && \\text{or} && x+3=0 && \\text{or} && x - 4=0 \\\\ x=0 && \\text{or} && x=-3 && \\text{or} && x=4\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135431016\">The <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(-3,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137472984\">The degree is 3 so the graph has at most 2 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137575431\">Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], determine the local behavior.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617003\">Show Solution<\/span><\/p>\n<div id=\"q617003\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em>x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the <em>y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm66677\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=66677&theme=oea&iframe_resize_id=ohm66677\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Key Equations<\/span><\/p>\n<section id=\"fs-id1165137724050\" class=\"key-equations\">\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td>general form of a polynomial function<\/td>\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137731646\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135438864\">\n<li>A power function is a variable base raised to a number power.<\/li>\n<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\n<li>The end behavior depends on whether the power is even or odd.<\/li>\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\n<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\n<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em>x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137668266\" class=\"definition\">\n<dt><strong>coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\" class=\"definition\">\n<dt><strong>continuous function<\/strong><\/dt>\n<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\" class=\"definition\">\n<dt><strong>degree<\/strong><\/dt>\n<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\" class=\"definition\">\n<dt><strong>end behavior<\/strong><\/dt>\n<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>leading coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>leading term<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a function that 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.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>power function<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=k{x}^{p}[\/latex]\u00a0where <em>k\u00a0<\/em>is a constant, the base is a variable, and the exponent, <em>p<\/em>,\u00a0is a constant\u00a0smooth curve\u00a0a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>term of a polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>turning point<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\n<\/dl>\n<\/div>\n<\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-13838\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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