{"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":"2020-11-22T19:57:51","modified_gmt":"2020-11-22T19:57:51","slug":"power-functions-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/pdx-precalculus\/chapter\/power-functions-and-polynomial-functions\/","title":{"raw":"Walkthrough of Unit 6: Power Functions and Polynomial Functions","rendered":"Walkthrough of Unit 6: 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 \t<li style=\"font-weight: 400\">Recognize characteristics of graphs of polynomial functions.<\/li>\r\n \t<li style=\"font-weight: 400\">Identify zeros of polynomials and their multiplicities.<\/li>\r\n \t<li style=\"font-weight: 400\">Determine end behavior.<\/li>\r\n \t<li style=\"font-weight: 400\">Understand the relationship between degree and turning points.<\/li>\r\n \t<li style=\"font-weight: 400\">Graph polynomial functions.<\/li>\r\n \t<li>Write the formula for a polynomial function.<\/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\">Graphing Polynomials<\/span>\r\n\r\nThe revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b>\r\n<table id=\"Table_03_04_01\" summary=\"Two rows and nine columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/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>Revenues<\/strong><\/td>\r\n<td>52.4<\/td>\r\n<td>52.8<\/td>\r\n<td>51.2<\/td>\r\n<td>49.5<\/td>\r\n<td>48.6<\/td>\r\n<td>48.6<\/td>\r\n<td>48.7<\/td>\r\n<td>47.1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165134040487\">The revenue can be modeled by the polynomial function<\/p>\r\n\r\n<div id=\"eip-679\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[\/latex]<\/div>\r\n<p id=\"fs-id1165137659450\">where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\r\n\r\n<h2>Recognize characteristics of graphs of polynomial functions<\/h2>\r\n<p id=\"fs-id1165134352567\">Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<span id=\"fs-id1165135185916\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010727\/CNX_Precalc_Figure_03_04_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n<div id=\"Example_03_04_01\" class=\"example\">\r\n<div id=\"fs-id1165137643218\" class=\"exercise\">\r\n<div id=\"fs-id1165133360328\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Recognizing Polynomial Functions<\/h3>\r\nWhich of the graphs in Figure 2\u00a0represents a polynomial function?\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\/03010727\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"898519\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"898519\"]\r\n<p id=\"fs-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\r\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\r\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\r\n\r\n<\/div>\r\n<h2>\u00a0Use factoring to \ufb01nd zeros of polynomial functions<\/h2>\r\n<h3>Find zeros of polynomial functions<\/h3>\r\n<p id=\"fs-id1165134042185\">Recall that if <em>f<\/em>\u00a0is a polynomial function, the values of <em>x<\/em>\u00a0for which [latex]f\\left(x\\right)=0[\/latex] are called <strong>zeros<\/strong> of <em>f<\/em>. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros<strong>.<\/strong><\/p>\r\n<p id=\"fs-id1165134043725\">We can use this method to find <em>x<\/em>-intercepts because at the <em>x<\/em>-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:<\/p>\r\n\r\n<ol id=\"fs-id1165137733636\">\r\n \t<li>The polynomial can be factored using known methods: greatest common factor and trinomial factoring.<\/li>\r\n \t<li>The polynomial is given in factored form.<\/li>\r\n \t<li>Technology is used to determine the intercepts.<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165137640937\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137563367\">How To: Given a polynomial function <em>f<\/em>, find the <em>x<\/em>-intercepts by factoring.<\/h3>\r\n<ol id=\"fs-id1165134104993\">\r\n \t<li>Set [latex]f\\left(x\\right)=0[\/latex].<\/li>\r\n \t<li>If the polynomial function is not given in factored form:\r\n<ol id=\"fs-id1165137646354\">\r\n \t<li>Factor out any common monomial factors.<\/li>\r\n \t<li>Factor any factorable binomials or trinomials.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Set each factor equal to zero and solve to find the [latex]x\\text{-}[\/latex] intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_02\" class=\"example\">\r\n<div id=\"fs-id1165135191903\" class=\"exercise\">\r\n<div id=\"fs-id1165135179909\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\r\n<p id=\"fs-id1165137817691\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{6}-3{x}^{4}+2{x}^{2}[\/latex].<\/p>\r\n[reveal-answer q=\"546243\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"546243\"]\r\n<p id=\"fs-id1165137535791\">We can attempt to factor this polynomial to find solutions for [latex]f\\left(x\\right)=0[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align} &amp;{x}^{6}-3{x}^{4}+2{x}^{2}=0 &amp;&amp; \\\\ &amp;{x}^{2}\\left({x}^{4}-3{x}^{2}+2\\right)=0 &amp;&amp; \\text{Factor out the greatest common factor}. \\\\ &amp;{x}^{2}\\left({x}^{2}-1\\right)\\left({x}^{2}-2\\right)=0 &amp;&amp; \\text{Factor the trinomial}. \\\\ &amp;{x}^{2}\\left(x+1\\right)\\left(x-1\\right)\\left({x}^{2}-2\\right)=0 &amp;&amp; \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\r\nNow set each factor equal to zero and solve.\r\n<p style=\"text-align: center\">[latex]\\begin{align} &amp; {x}^{2}=0 &amp;&amp; x+1=0 &amp;&amp; x-1=0 &amp;&amp; {x}^{2}-2=0 \\\\ &amp;x=0 &amp;&amp; x=-1 &amp;&amp; x=1 &amp;&amp; x=\\pm \\sqrt{2} \\end{align}[\/latex]<\/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\/03010728\/CNX_Precalc_Figure_03_04_0032.jpg\" alt=\"Four graphs where the first graph is of an even-degree polynomial, the second graph is of an absolute function, the third graph is an odd-degree polynomial, and the fourth graph is a disjoint function.\" width=\"487\" height=\"224\" \/> <b>Figure 3<\/b>[\/caption]\r\n<p id=\"fs-id1165137932627\">This gives us five <em>x<\/em>-intercepts: [latex]\\left(0,0\\right),\\left(1,0\\right),\\left(-1,0\\right),\\left(\\sqrt{2},0\\right)[\/latex], and [latex]\\left(-\\sqrt{2},0\\right)[\/latex]. We can see that this is an even function.<\/p>\r\n[\/hidden-answer]<span id=\"fs-id1165134380378\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_04_03\" class=\"example\">\r\n<div id=\"fs-id1165137768835\" class=\"exercise\">\r\n<div id=\"fs-id1165137768837\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\r\n<p id=\"fs-id1165135254633\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}-x+5[\/latex].<\/p>\r\n[reveal-answer q=\"996911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"996911\"]\r\n<p id=\"fs-id1165137725387\">Find solutions for [latex]f\\left(x\\right)=0[\/latex]\u00a0by factoring.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align} &amp;{x}^{3}-5{x}^{2}-x+5=0 \\\\ &amp;{x}^{2}\\left(x - 5\\right)-1\\left(x - 5\\right)=0 &amp;&amp; \\text{Factor by grouping}. \\\\ &amp;\\left({x}^{2}-1\\right)\\left(x - 5\\right)=0 &amp;&amp; \\text{Factor out the common factor}. \\\\ &amp;\\left(x+1\\right)\\left(x - 1\\right)\\left(x - 5\\right)=0 &amp;&amp; \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\r\nNow we set each factor equal to 0.\r\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;x+1=0 &amp;&amp; x - 1=0 &amp;&amp; x - 5=0 \\\\ &amp;x=-1 &amp;&amp; x=1 &amp;&amp; x=5 \\end{align}[\/latex]<\/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\/03010728\/CNX_Precalc_Figure_03_04_0042.jpg\" alt=\"Graph of f(x)=x^6-3x^4+2x^2 with its five intercepts, (-sqrt(2), 0), (-1, 0), (0, 0), (1, 0), and (sqrt(2), 0).\" width=\"487\" height=\"402\" \/> <b>Figure 4<\/b>[\/caption]\r\n<p id=\"fs-id1165134541162\">There are three <em>x<\/em>-intercepts: [latex]\\left(-1,0\\right),\\left(1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex].<\/p>\r\n[\/hidden-answer]<span id=\"fs-id1165133344112\">\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_04_04\" class=\"example\">\r\n<div id=\"fs-id1165135154515\" class=\"exercise\">\r\n<div id=\"fs-id1165135154517\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Finding the <em>y<\/em>- and <em>x<\/em>-Intercepts of a Polynomial in Factored Form<\/h3>\r\n<p id=\"fs-id1165135528940\">Find the <i>y<\/i>-\u00a0and <em>x<\/em>-intercepts of [latex]g\\left(x\\right)={\\left(x - 2\\right)}^{2}\\left(2x+3\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"180029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"180029\"]\r\n<p id=\"fs-id1165135421555\">The <em>y<\/em>-intercept can be found by evaluating [latex]g\\left(0\\right)[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]g\\left(0\\right)={\\left(0 - 2\\right)}^{2}\\left(2\\left(0\\right)+3\\right)=12[\/latex]<\/p>\r\n<p id=\"eip-id1165134130215\">So the <em>y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165137870836\">The <em>x<\/em>-intercepts can be found by solving [latex]g\\left(x\\right)=0[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]{\\left(x - 2\\right)}^{2}\\left(2x+3\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;{\\left(x - 2\\right)}^{2}=0 &amp;&amp; 2x+3=0 \\\\ &amp;x=2 &amp;&amp;x=-\\frac{3}{2} \\end{align}[\/latex]<\/p>\r\n<p id=\"eip-id1165135518219\">So the <em>x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-\\frac{3}{2},0\\right)[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 5.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0052.jpg\" alt=\"Graph of f(x)=x^3-5x^2-x+5 with its three intercepts (-1, 0), (1, 0), and (5, 0).\" width=\"487\" height=\"670\" \/> <b>Figure 5<\/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_04_05\" class=\"example\">\r\n<div id=\"fs-id1165137415980\" class=\"exercise\">\r\n<div id=\"fs-id1165134381752\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Finding the <em>x<\/em>-Intercepts of a Polynomial Function Using a Graph<\/h3>\r\n<p id=\"fs-id1165137453950\">Find the <em>x<\/em>-intercepts of [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex].<\/p>\r\n[reveal-answer q=\"512408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512408\"]\r\n<p id=\"fs-id1165137895270\">This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.<\/p>\r\nLooking at the graph of this function, as shown in Figure 6, it appears that there are <em>x<\/em>-intercepts at [latex]x=-3,-2[\/latex], and 1.\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\/03010731\/CNX_Precalc_Figure_03_04_0062.jpg\" alt=\"Graph of g(x)=(x-2)^2(2x+3) with its two x-intercepts (2, 0) and (-3\/2, 0) and its y-intercept (0, 12).\" width=\"487\" height=\"440\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165131891784\">We can check whether these are correct by substituting these values for <em>x<\/em>\u00a0and verifying that the function is equal to 0.<\/p>\r\n<p id=\"fs-id1165135600839\">Since [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex], we have:<\/p>\r\n<p style=\"text-align: center\">[latex]h\\left(-3\\right)={\\left(-3\\right)}^{3}+4{\\left(-3\\right)}^{2}+\\left(-3\\right)-6=-27+36 - 3-6=0[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]h\\left(-2\\right)={\\left(-2\\right)}^{3}+4{\\left(-2\\right)}^{2}+\\left(-2\\right)-6=-8+16 - 2-6=0[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]h\\left(1\\right)={\\left(1\\right)}^{3}+4{\\left(1\\right)}^{2}+\\left(1\\right)-6=1+4+1 - 6=0[\/latex]<\/p>\r\n<p id=\"fs-id1165134129941\">Each <em>x<\/em>-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.<\/p>\r\n<p style=\"text-align: center\">[latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6=\\left(x+3\\right)\\left(x+2\\right)\\left(x - 1\\right)[\/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-id1165133238478\">Find the <em>y<\/em>-\u00a0and <em>x<\/em>-intercepts of the function [latex]f\\left(x\\right)={x}^{4}-19{x}^{2}+30x[\/latex].<\/p>\r\n[reveal-answer q=\"123401\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"123401\"]\r\n\r\ny-intercept [latex]\\left(0,0\\right)[\/latex]; x-intercepts [latex]\\left(0,0\\right),\\left(-5,0\\right),\\left(2,0\\right)[\/latex], and [latex]\\left(3,0\\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 2<\/h3>\r\n[ohm_question hide_question_numbers=1]66678[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Identify zeros and their multiplicities<\/h2>\r\n<p id=\"fs-id1165135581073\">Graphs behave differently at various <em>x<\/em>-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.<\/p>\r\n<p id=\"fs-id1165133092720\">Suppose, for example, we graph the function<\/p>\r\n\r\n<div id=\"eip-840\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/div>\r\nNotice in Figure 7\u00a0that the behavior of the function at each of the <em>x<\/em>-intercepts is different.\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\/03010731\/CNX_Precalc_Figure_03_04_0072.jpg\" alt=\"Graph of h(x)=x^3+4x^2+x-6.\" width=\"487\" height=\"329\" \/> <b>Figure 7.<\/b> Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.[\/caption]\r\n<p id=\"fs-id1165135407009\">The <em>x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution of equation [latex]x+3=0[\/latex]. The graph passes directly through the <em>x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line\u2014it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\r\n<p id=\"fs-id1165137897788\">The <em>x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution of the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\r\n\r\n<div id=\"eip-608\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1165137888924\">The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the <strong>multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\r\n<p id=\"fs-id1165133402140\">The <em>x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic\u2014with the same S-shape near the intercept as the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\r\nFor <strong>zeros<\/strong> with even multiplicities, the graphs <em>touch<\/em> or are tangent to the <em>x<\/em>-axis. For zeros with odd multiplicities, the graphs <em>cross<\/em> or intersect the <em>x<\/em>-axis. See Figure 8\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"874\"]<img src=\"https:\/\/cnx.org\/resources\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf\/CNX_Precalc_Figure_03_04_008_fixed.jpg\" alt=\"Three graphs, left to right, with zeros of multiplicity 1, 2, and 3.\" width=\"874\" height=\"324\" \/> <b>Figure 8<\/b>[\/caption]\r\n<p id=\"fs-id1165133078115\">For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\r\n<p id=\"fs-id1165133447988\">For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\r\n\r\n<div id=\"fs-id1165135620829\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Graphical Behavior of Polynomials at <em>x<\/em>-Intercepts<\/h3>\r\n<p id=\"fs-id1165134036762\">If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the <em>x<\/em>-intercept <em>h\u00a0<\/em>is determined by the power <em>p<\/em>. We say that [latex]x=h[\/latex] is a zero of <strong>multiplicity<\/strong> <em>p<\/em>.<\/p>\r\n<p id=\"fs-id1165137647546\">The graph of a polynomial function will touch the <em>x<\/em>-axis at zeros with even multiplicities. The graph will cross the <em>x<\/em>-axis at zeros with odd multiplicities.<\/p>\r\n<p id=\"fs-id1165135195405\">The sum of the multiplicities is the degree of the polynomial function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135195409\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135195416\">How To: Given a graph of a polynomial function of degree <i>n<\/i>, identify the zeros and their multiplicities.<\/h3>\r\n<ol id=\"fs-id1165135547216\">\r\n \t<li>If the graph crosses the <em>x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\r\n \t<li>If the graph touches the <em>x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\r\n \t<li>If the graph crosses the <em>x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\r\n \t<li>The sum of the multiplicities is <em>n<\/em>. This includes non-real zeros.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_06\" class=\"example\">\r\n<div id=\"fs-id1165137922408\" class=\"exercise\">\r\n<div id=\"fs-id1165135409401\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Identifying Zeros and Their Multiplicities<\/h3>\r\nUse the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.\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\/03010732\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"487\" height=\"628\" \/> <b>Figure 9<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"700901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"700901\"]\r\n<p id=\"fs-id1165135533055\">The polynomial function is of degree <em>n<\/em>. The sum of the multiplicities must be <em>n<\/em>.<\/p>\r\n<p id=\"fs-id1165135641694\">Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the <em>x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\r\n<p id=\"fs-id1165135369539\">The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\r\n<p id=\"fs-id1165135329820\">The last zero occurs at [latex]x=4[\/latex]. The graph crosses the<em> x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.<\/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\nUse the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010732\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/> <b>Figure 10<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"166598\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"166598\"]\r\n\r\nThe graph has a zero of \u20135 with multiplicity 1, a zero of \u20131 with multiplicity 2, and a zero of 3 with even multiplicity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Determine end behavior<\/h2>\r\n<p id=\"fs-id1165135514626\">As we have already learned, the behavior of a graph of a <strong>polynomial function<\/strong> of the form<\/p>\r\n\r\n<div id=\"eip-263\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\n<p id=\"eip-id1165134547362\">will either ultimately rise or fall as <em>x<\/em>\u00a0increases without bound and will either rise or fall as <em>x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\r\n<p id=\"fs-id1165132959259\">Recall that we call this behavior the <em>end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as <em>x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0<em>x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as <em>x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\r\n\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Even Degree<\/th>\r\n<th>Odd Degree<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\"><img class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\"><img class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\"><img class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\r\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\"><img class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Understand the relationship between degree and turning points<\/h2>\r\n<p id=\"fs-id1165135416524\">In addition to the end behavior, recall that we can analyze a polynomial function\u2019s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 11. The graph has three turning points.<span id=\"fs-id1165134155116\">\r\n<\/span><\/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\/03010733\/CNX_Precalc_Figure_03_04_0152.jpg\" alt=\"Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.\" width=\"487\" height=\"327\" \/> <b>Figure 11<\/b>[\/caption]\r\n<p id=\"fs-id1165137784439\">This function <em>f<\/em>\u00a0is a 4<sup>th<\/sup> degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.<\/p>\r\n\r\n<div id=\"fs-id1165135502799\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Interpreting Turning Points<\/h3>\r\n<p id=\"fs-id1165135469050\">A <strong>turning point<\/strong> is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).<\/p>\r\n<p id=\"fs-id1165135469055\">A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em> \u2013 1\u00a0turning points.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_03_04_07\" class=\"example\">\r\n<div id=\"fs-id1165134374690\" class=\"exercise\">\r\n<div id=\"fs-id1165134060420\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165134060425\">Find the maximum number of turning points of each polynomial function.<\/p>\r\n\r\n<ol id=\"fs-id1165134060428\">\r\n \t<li>[latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]<\/li>\r\n \t<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"157524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157524\"]\r\n<ol id=\"fs-id1165137784430\">\r\n \t<li>[latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\r\n<p id=\"fs-id1165135335895\">First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\r\n<p id=\"fs-id1165135453844\">Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\r\n<p id=\"fs-id1165135341233\">The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\r\n<\/li>\r\n \t<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\"><img class=\"aligncenter wp-image-15117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\" alt=\"Graphic of f(x) showing to multiply the first term of (x-1)^2 and 2x^2 to determine the leading term.\" width=\"487\" height=\"67\" \/><\/a>\r\n<p style=\"text-align: center\">[latex]a_{n}=-\\left(x^2\\right)\\left(2x^2\\right)=-2x^4[\/latex]<\/p>\r\n<p id=\"fs-id1165133104532\">First, identify the leading term of the polynomial function if the function were expanded.<span id=\"fs-id1165134130071\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135551181\">Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\r\n<p id=\"fs-id1165135551185\">The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>\u00a0Graph polynomial functions<\/h2>\r\n<p id=\"fs-id1165137843095\">We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.<\/p>\r\n\r\n<div id=\"fs-id1165137843101\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135449677\">How To: Given a polynomial function, sketch the graph.<\/h3>\r\n<ol id=\"fs-id1165135449683\">\r\n \t<li>Find the intercepts.<\/li>\r\n \t<li>Check for symmetry. If the function is an even function, its graph is symmetrical about the <em>y<\/em>-axis, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>f<\/em>(<em>x<\/em>).\r\nIf a function is an odd function, its graph is symmetrical about the origin, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>\u2013<\/em><em>f<\/em>(<em>x<\/em>).<\/li>\r\n \t<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the <em>x<\/em>-intercepts.<\/li>\r\n \t<li>Determine the end behavior by examining the leading term.<\/li>\r\n \t<li>Use the end behavior and the behavior at the intercepts to sketch a graph.<\/li>\r\n \t<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\r\n \t<li>Optionally, use technology to check the graph.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_08\" class=\"example\">\r\n<div id=\"fs-id1165135575951\" class=\"exercise\">\r\n<div id=\"fs-id1165135575953\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Sketching the Graph of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165135575958\">Sketch a graph of [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"892446\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"892446\"]\r\n<p id=\"fs-id1165135237929\">This graph has two <em>x-<\/em>intercepts. At <em>x\u00a0<\/em>= \u20133, the factor is squared, indicating a multiplicity of 2. The graph will bounce at this <em>x<\/em>-intercept. At <em>x\u00a0<\/em>= 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/p>\r\n<p id=\"fs-id1165135171021\">The <em>y<\/em>-intercept is found by evaluating <em>f<\/em>(0).<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align} f\\left(0\\right)&amp;=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right) \\\\ &amp;=-2\\cdot 9\\cdot \\left(-5\\right) \\\\ &amp;=90 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165134374772\">The <em>y<\/em>-intercept is (0, 90).<\/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\/03010733\/CNX_Precalc_Figure_03_04_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"487\" height=\"362\" \/> <b>Figure 13<\/b>[\/caption]\r\n<p id=\"fs-id1165134381522\">Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex],\r\nso the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity.<span id=\"fs-id1165135646080\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165134374738\">To sketch this, we consider that:<\/p>\r\n\r\n<ul id=\"fs-id1165134374741\">\r\n \t<li>As [latex]x\\to -\\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\infty [\/latex], so we know the graph starts in the second quadrant and is decreasing toward the <em>x<\/em>-axis.<\/li>\r\n \t<li>Since [latex]f\\left(-x\\right)=-2{\\left(-x+3\\right)}^{2}\\left(-x - 5\\right)[\/latex]\r\nis not equal to <em>f<\/em>(<em>x<\/em>), the graph does not display symmetry.<\/li>\r\n \t<li>At (-3,0), the graph bounces off of the <em>x<\/em>-axis, so the function must start increasing.\r\n<p id=\"fs-id1165135536183\" style=\"text-align: left\">At (0, 90), the graph crosses the <em>y<\/em>-axis at the <em>y<\/em>-intercept.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<figure id=\"Figure_03_04_018\" class=\"small\">\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\/03010733\/CNX_Precalc_Figure_03_04_0182.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/> <b>Figure 14<\/b>[\/caption]<\/figure>\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\/03010734\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/> <b>Figure 15<\/b>[\/caption]\r\n<p id=\"fs-id1165135241000\">Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).\u00a0<span id=\"fs-id1165135241013\">\r\n<\/span><\/p>\r\n<p id=\"fs-id1165135613608\">As [latex]x\\to \\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\mathrm{-\\infty }[\/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.<\/p>\r\n<p id=\"fs-id1165135574296\">Using technology, we can create the graph for the polynomial function, shown in Figure 16, and verify that the resulting graph looks like our sketch in Figure 15.<\/p>\r\n\r\n<figure id=\"Figure_03_04_020\" class=\"small\"><figcaption>The complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/figcaption>\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\/03010734\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/> <b>Figure 16<\/b>[\/caption]<\/figure>\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-id1165133065140\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].<\/p>\r\n[reveal-answer q=\"408253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"408253\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010734\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165135369116\">\r\n<h2>Writing Formulas for Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165135369122\">Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a <strong>polynomial function<\/strong> written in factored form will have an <em>x<\/em>-intercept where each factor is equal to zero, we can form a function that will pass through a set of <em>x<\/em>-intercepts by introducing a corresponding set of factors.<\/p>\r\n\r\n<div id=\"fs-id1165133320785\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Factored Form of Polynomials<\/h3>\r\n<p id=\"fs-id1165133320793\">If a polynomial of lowest degree <em>p<\/em>\u00a0has horizontal intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form: [latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]\u00a0where the powers [latex]{p}_{i}[\/latex]\u00a0on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor <em>a<\/em>\u00a0can be determined given a value of the function other than the <em>x<\/em>-intercept.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135580289\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135580296\">How To: Given a graph of a polynomial function, write a formula for the function.<\/h3>\r\n<ol id=\"fs-id1165133309878\">\r\n \t<li>Identify the <em>x<\/em>-intercepts of the graph to find the factors of the polynomial.<\/li>\r\n \t<li>Examine the behavior of the graph at the <em>x<\/em>-intercepts to determine the multiplicity of each factor.<\/li>\r\n \t<li>Find the polynomial of least degree containing all the factors found in the previous step.<\/li>\r\n \t<li>Use any other point on the graph (the <em>y<\/em>-intercept may be easiest) to determine the stretch factor.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_04_10\" class=\"example\">\r\n<div id=\"fs-id1165134043949\" class=\"exercise\">\r\n<div id=\"fs-id1165134043951\" class=\"problem textbox shaded\">\r\n<h3>Example 13: Writing a Formula for a Polynomial Function from the Graph<\/h3>\r\nWrite a formula for the polynomial function shown in Figure 19.\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\/03010735\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/> <b>Figure 19<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"574656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"574656\"]\r\n<p id=\"fs-id1165135621955\">his graph has three <em>x<\/em>-intercepts: <em>x\u00a0<\/em>= \u20133, 2, and 5. The <em>y<\/em>-intercept is located at (0, 2). At <em>x\u00a0<\/em>= \u20133 and <em>x\u00a0<\/em>= 5,\u00a0the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At <em>x\u00a0<\/em>= 2, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us<\/p>\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\r\n<p id=\"fs-id1165135575901\">To determine the stretch factor, we utilize another point on the graph. We will use the <em>y<\/em>-intercept (0, \u20132), to solve for <em>a<\/em>.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}f\\left(0\\right)&amp;=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&amp;=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&amp;=-60a \\\\ a&amp;=\\frac{1}{30} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165133437286\">The graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/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\nGiven the graph in Figure 20, write a formula for the function shown.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010735\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/> <b>Figure 20<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"412515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"412515\"]\r\n\r\n[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135440065\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]15942[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section>\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165137846272\">\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 pf a power function 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 \t<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\r\n \t<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\r\n \t<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the <em>x<\/em>-axis.<\/li>\r\n \t<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercepts.<\/li>\r\n \t<li>The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.<\/li>\r\n \t<li>The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.<\/li>\r\n \t<li>The end behavior of a polynomial function depends on the leading term.<\/li>\r\n \t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\r\n \t<li>A polynomial function of degree <em>n<\/em>\u00a0has at most\u00a0<em>n <\/em>\u2013\u00a01 turning points.<\/li>\r\n \t<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>\u00a0n <\/em>\u2013\u00a01 turning points.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165134112772\" class=\"definition\">\r\n \t<dt><strong>multiplicity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134112776\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity <em>p<\/em>.<\/dd>\r\n<\/dl>\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>","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<li style=\"font-weight: 400\">Recognize characteristics of graphs of polynomial functions.<\/li>\n<li style=\"font-weight: 400\">Identify zeros of polynomials and their multiplicities.<\/li>\n<li style=\"font-weight: 400\">Determine end behavior.<\/li>\n<li style=\"font-weight: 400\">Understand the relationship between degree and turning points.<\/li>\n<li style=\"font-weight: 400\">Graph polynomial functions.<\/li>\n<li>Write the formula for a polynomial function.<\/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\">Graphing Polynomials<\/span><\/p>\n<p>The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b><\/p>\n<table id=\"Table_03_04_01\" summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/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>Revenues<\/strong><\/td>\n<td>52.4<\/td>\n<td>52.8<\/td>\n<td>51.2<\/td>\n<td>49.5<\/td>\n<td>48.6<\/td>\n<td>48.6<\/td>\n<td>48.7<\/td>\n<td>47.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165134040487\">The revenue can be modeled by the polynomial function<\/p>\n<div id=\"eip-679\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[\/latex]<\/div>\n<p id=\"fs-id1165137659450\">where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\n<h2>Recognize characteristics of graphs of polynomial functions<\/h2>\n<p id=\"fs-id1165134352567\">Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<span id=\"fs-id1165135185916\"><br \/>\n<\/span><\/p>\n<div style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010727\/CNX_Precalc_Figure_03_04_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<div id=\"Example_03_04_01\" class=\"example\">\n<div id=\"fs-id1165137643218\" class=\"exercise\">\n<div id=\"fs-id1165133360328\" class=\"problem textbox shaded\">\n<h3>Example 1: Recognizing Polynomial Functions<\/h3>\n<p>Which of the graphs in Figure 2\u00a0represents a polynomial function?<\/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\/03010727\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q898519\">Show Solution<\/span><\/p>\n<div id=\"q898519\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\n<\/div>\n<h2>\u00a0Use factoring to \ufb01nd zeros of polynomial functions<\/h2>\n<h3>Find zeros of polynomial functions<\/h3>\n<p id=\"fs-id1165134042185\">Recall that if <em>f<\/em>\u00a0is a polynomial function, the values of <em>x<\/em>\u00a0for which [latex]f\\left(x\\right)=0[\/latex] are called <strong>zeros<\/strong> of <em>f<\/em>. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros<strong>.<\/strong><\/p>\n<p id=\"fs-id1165134043725\">We can use this method to find <em>x<\/em>-intercepts because at the <em>x<\/em>-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:<\/p>\n<ol id=\"fs-id1165137733636\">\n<li>The polynomial can be factored using known methods: greatest common factor and trinomial factoring.<\/li>\n<li>The polynomial is given in factored form.<\/li>\n<li>Technology is used to determine the intercepts.<\/li>\n<\/ol>\n<div id=\"fs-id1165137640937\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137563367\">How To: Given a polynomial function <em>f<\/em>, find the <em>x<\/em>-intercepts by factoring.<\/h3>\n<ol id=\"fs-id1165134104993\">\n<li>Set [latex]f\\left(x\\right)=0[\/latex].<\/li>\n<li>If the polynomial function is not given in factored form:\n<ol id=\"fs-id1165137646354\">\n<li>Factor out any common monomial factors.<\/li>\n<li>Factor any factorable binomials or trinomials.<\/li>\n<\/ol>\n<\/li>\n<li>Set each factor equal to zero and solve to find the [latex]x\\text{-}[\/latex] intercepts.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_02\" class=\"example\">\n<div id=\"fs-id1165135191903\" class=\"exercise\">\n<div id=\"fs-id1165135179909\" class=\"problem textbox shaded\">\n<h3>Example 2: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\n<p id=\"fs-id1165137817691\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{6}-3{x}^{4}+2{x}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q546243\">Show Solution<\/span><\/p>\n<div id=\"q546243\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137535791\">We can attempt to factor this polynomial to find solutions for [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} &{x}^{6}-3{x}^{4}+2{x}^{2}=0 && \\\\ &{x}^{2}\\left({x}^{4}-3{x}^{2}+2\\right)=0 && \\text{Factor out the greatest common factor}. \\\\ &{x}^{2}\\left({x}^{2}-1\\right)\\left({x}^{2}-2\\right)=0 && \\text{Factor the trinomial}. \\\\ &{x}^{2}\\left(x+1\\right)\\left(x-1\\right)\\left({x}^{2}-2\\right)=0 && \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\n<p>Now set each factor equal to zero and solve.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} & {x}^{2}=0 && x+1=0 && x-1=0 && {x}^{2}-2=0 \\\\ &x=0 && x=-1 && x=1 && x=\\pm \\sqrt{2} \\end{align}[\/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\/03010728\/CNX_Precalc_Figure_03_04_0032.jpg\" alt=\"Four graphs where the first graph is of an even-degree polynomial, the second graph is of an absolute function, the third graph is an odd-degree polynomial, and the fourth graph is a disjoint function.\" width=\"487\" height=\"224\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137932627\">This gives us five <em>x<\/em>-intercepts: [latex]\\left(0,0\\right),\\left(1,0\\right),\\left(-1,0\\right),\\left(\\sqrt{2},0\\right)[\/latex], and [latex]\\left(-\\sqrt{2},0\\right)[\/latex]. We can see that this is an even function.<\/p>\n<\/div>\n<\/div>\n<p><span id=\"fs-id1165134380378\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_03\" class=\"example\">\n<div id=\"fs-id1165137768835\" class=\"exercise\">\n<div id=\"fs-id1165137768837\" class=\"problem textbox shaded\">\n<h3>Example 3: Finding the <em>x<\/em>-Intercepts of a Polynomial Function by Factoring<\/h3>\n<p id=\"fs-id1165135254633\">Find the <em>x<\/em>-intercepts of [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}-x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q996911\">Show Solution<\/span><\/p>\n<div id=\"q996911\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137725387\">Find solutions for [latex]f\\left(x\\right)=0[\/latex]\u00a0by factoring.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} &{x}^{3}-5{x}^{2}-x+5=0 \\\\ &{x}^{2}\\left(x - 5\\right)-1\\left(x - 5\\right)=0 && \\text{Factor by grouping}. \\\\ &\\left({x}^{2}-1\\right)\\left(x - 5\\right)=0 && \\text{Factor out the common factor}. \\\\ &\\left(x+1\\right)\\left(x - 1\\right)\\left(x - 5\\right)=0 && \\text{Factor the difference of squares}. \\end{align}[\/latex]<\/p>\n<p>Now we set each factor equal to 0.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&x+1=0 && x - 1=0 && x - 5=0 \\\\ &x=-1 && x=1 && x=5 \\end{align}[\/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\/03010728\/CNX_Precalc_Figure_03_04_0042.jpg\" alt=\"Graph of f(x)=x^6-3x^4+2x^2 with its five intercepts, (-sqrt(2), 0), (-1, 0), (0, 0), (1, 0), and (sqrt(2), 0).\" width=\"487\" height=\"402\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134541162\">There are three <em>x<\/em>-intercepts: [latex]\\left(-1,0\\right),\\left(1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><span id=\"fs-id1165133344112\"><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_04\" class=\"example\">\n<div id=\"fs-id1165135154515\" class=\"exercise\">\n<div id=\"fs-id1165135154517\" class=\"problem textbox shaded\">\n<h3>Example 4: Finding the <em>y<\/em>&#8211; and <em>x<\/em>-Intercepts of a Polynomial in Factored Form<\/h3>\n<p id=\"fs-id1165135528940\">Find the <i>y<\/i>&#8211;\u00a0and <em>x<\/em>-intercepts of [latex]g\\left(x\\right)={\\left(x - 2\\right)}^{2}\\left(2x+3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q180029\">Show Solution<\/span><\/p>\n<div id=\"q180029\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135421555\">The <em>y<\/em>-intercept can be found by evaluating [latex]g\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]g\\left(0\\right)={\\left(0 - 2\\right)}^{2}\\left(2\\left(0\\right)+3\\right)=12[\/latex]<\/p>\n<p id=\"eip-id1165134130215\">So the <em>y<\/em>-intercept is [latex]\\left(0,12\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137870836\">The <em>x<\/em>-intercepts can be found by solving [latex]g\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x - 2\\right)}^{2}\\left(2x+3\\right)=0[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}&{\\left(x - 2\\right)}^{2}=0 && 2x+3=0 \\\\ &x=2 &&x=-\\frac{3}{2} \\end{align}[\/latex]<\/p>\n<p id=\"eip-id1165135518219\">So the <em>x<\/em>-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-\\frac{3}{2},0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown in Figure 5.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010731\/CNX_Precalc_Figure_03_04_0052.jpg\" alt=\"Graph of f(x)=x^3-5x^2-x+5 with its three intercepts (-1, 0), (1, 0), and (5, 0).\" width=\"487\" height=\"670\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_05\" class=\"example\">\n<div id=\"fs-id1165137415980\" class=\"exercise\">\n<div id=\"fs-id1165134381752\" class=\"problem textbox shaded\">\n<h3>Example 5: Finding the <em>x<\/em>-Intercepts of a Polynomial Function Using a Graph<\/h3>\n<p id=\"fs-id1165137453950\">Find the <em>x<\/em>-intercepts of [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512408\">Show Solution<\/span><\/p>\n<div id=\"q512408\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137895270\">This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.<\/p>\n<p>Looking at the graph of this function, as shown in Figure 6, it appears that there are <em>x<\/em>-intercepts at [latex]x=-3,-2[\/latex], and 1.<\/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\/03010731\/CNX_Precalc_Figure_03_04_0062.jpg\" alt=\"Graph of g(x)=(x-2)^2(2x+3) with its two x-intercepts (2, 0) and (-3\/2, 0) and its y-intercept (0, 12).\" width=\"487\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165131891784\">We can check whether these are correct by substituting these values for <em>x<\/em>\u00a0and verifying that the function is equal to 0.<\/p>\n<p id=\"fs-id1165135600839\">Since [latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6[\/latex], we have:<\/p>\n<p style=\"text-align: center\">[latex]h\\left(-3\\right)={\\left(-3\\right)}^{3}+4{\\left(-3\\right)}^{2}+\\left(-3\\right)-6=-27+36 - 3-6=0[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]h\\left(-2\\right)={\\left(-2\\right)}^{3}+4{\\left(-2\\right)}^{2}+\\left(-2\\right)-6=-8+16 - 2-6=0[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]h\\left(1\\right)={\\left(1\\right)}^{3}+4{\\left(1\\right)}^{2}+\\left(1\\right)-6=1+4+1 - 6=0[\/latex]<\/p>\n<p id=\"fs-id1165134129941\">Each <em>x<\/em>-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.<\/p>\n<p style=\"text-align: center\">[latex]h\\left(x\\right)={x}^{3}+4{x}^{2}+x - 6=\\left(x+3\\right)\\left(x+2\\right)\\left(x - 1\\right)[\/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-id1165133238478\">Find the <em>y<\/em>&#8211;\u00a0and <em>x<\/em>-intercepts of the function [latex]f\\left(x\\right)={x}^{4}-19{x}^{2}+30x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q123401\">Show Solution<\/span><\/p>\n<div id=\"q123401\" class=\"hidden-answer\" style=\"display: none\">\n<p>y-intercept [latex]\\left(0,0\\right)[\/latex]; x-intercepts [latex]\\left(0,0\\right),\\left(-5,0\\right),\\left(2,0\\right)[\/latex], and [latex]\\left(3,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm66678\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=66678&theme=oea&iframe_resize_id=ohm66678\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Identify zeros and their multiplicities<\/h2>\n<p id=\"fs-id1165135581073\">Graphs behave differently at various <em>x<\/em>-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.<\/p>\n<p id=\"fs-id1165133092720\">Suppose, for example, we graph the function<\/p>\n<div id=\"eip-840\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/div>\n<p>Notice in Figure 7\u00a0that the behavior of the function at each of the <em>x<\/em>-intercepts is different.<\/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\/03010731\/CNX_Precalc_Figure_03_04_0072.jpg\" alt=\"Graph of h(x)=x^3+4x^2+x-6.\" width=\"487\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7.<\/b> Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.<\/p>\n<\/div>\n<p id=\"fs-id1165135407009\">The <em>x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution of equation [latex]x+3=0[\/latex]. The graph passes directly through the <em>x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line\u2014it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\n<p id=\"fs-id1165137897788\">The <em>x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution of the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\n<div id=\"eip-608\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137888924\">The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the <strong>multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\n<p id=\"fs-id1165133402140\">The <em>x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic\u2014with the same S-shape near the intercept as the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\n<p>For <strong>zeros<\/strong> with even multiplicities, the graphs <em>touch<\/em> or are tangent to the <em>x<\/em>-axis. For zeros with odd multiplicities, the graphs <em>cross<\/em> or intersect the <em>x<\/em>-axis. See Figure 8\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.<\/p>\n<div style=\"width: 884px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/404d5117e8c2b2cc187c001d0fcf267e8d3c7bbf\/CNX_Precalc_Figure_03_04_008_fixed.jpg\" alt=\"Three graphs, left to right, with zeros of multiplicity 1, 2, and 3.\" width=\"874\" height=\"324\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165133078115\">For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\n<p id=\"fs-id1165133447988\">For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\n<div id=\"fs-id1165135620829\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Graphical Behavior of Polynomials at <em>x<\/em>-Intercepts<\/h3>\n<p id=\"fs-id1165134036762\">If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the <em>x<\/em>-intercept <em>h\u00a0<\/em>is determined by the power <em>p<\/em>. We say that [latex]x=h[\/latex] is a zero of <strong>multiplicity<\/strong> <em>p<\/em>.<\/p>\n<p id=\"fs-id1165137647546\">The graph of a polynomial function will touch the <em>x<\/em>-axis at zeros with even multiplicities. The graph will cross the <em>x<\/em>-axis at zeros with odd multiplicities.<\/p>\n<p id=\"fs-id1165135195405\">The sum of the multiplicities is the degree of the polynomial function.<\/p>\n<\/div>\n<div id=\"fs-id1165135195409\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135195416\">How To: Given a graph of a polynomial function of degree <i>n<\/i>, identify the zeros and their multiplicities.<\/h3>\n<ol id=\"fs-id1165135547216\">\n<li>If the graph crosses the <em>x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\n<li>If the graph touches the <em>x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\n<li>If the graph crosses the <em>x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\n<li>The sum of the multiplicities is <em>n<\/em>. This includes non-real zeros.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_06\" class=\"example\">\n<div id=\"fs-id1165137922408\" class=\"exercise\">\n<div id=\"fs-id1165135409401\" class=\"problem textbox shaded\">\n<h3>Example 6: Identifying Zeros and Their Multiplicities<\/h3>\n<p>Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.<\/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\/03010732\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"487\" height=\"628\" \/><\/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=\"q700901\">Show Solution<\/span><\/p>\n<div id=\"q700901\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135533055\">The polynomial function is of degree <em>n<\/em>. The sum of the multiplicities must be <em>n<\/em>.<\/p>\n<p id=\"fs-id1165135641694\">Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the <em>x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\n<p id=\"fs-id1165135369539\">The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\n<p id=\"fs-id1165135329820\">The last zero occurs at [latex]x=4[\/latex]. The graph crosses the<em> x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.<\/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>Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"small\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010732\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q166598\">Show Solution<\/span><\/p>\n<div id=\"q166598\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has a zero of \u20135 with multiplicity 1, a zero of \u20131 with multiplicity 2, and a zero of 3 with even multiplicity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Determine end behavior<\/h2>\n<p id=\"fs-id1165135514626\">As we have already learned, the behavior of a graph of a <strong>polynomial function<\/strong> of the form<\/p>\n<div id=\"eip-263\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"eip-id1165134547362\">will either ultimately rise or fall as <em>x<\/em>\u00a0increases without bound and will either rise or fall as <em>x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\n<p id=\"fs-id1165132959259\">Recall that we call this behavior the <em>end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as <em>x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0<em>x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as <em>x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\n<table>\n<thead>\n<tr>\n<th>Even Degree<\/th>\n<th>Odd Degree<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\n<td><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012927\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Understand the relationship between degree and turning points<\/h2>\n<p id=\"fs-id1165135416524\">In addition to the end behavior, recall that we can analyze a polynomial function\u2019s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 11. The graph has three turning points.<span id=\"fs-id1165134155116\"><br \/>\n<\/span><\/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\/03010733\/CNX_Precalc_Figure_03_04_0152.jpg\" alt=\"Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.\" width=\"487\" height=\"327\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137784439\">This function <em>f<\/em>\u00a0is a 4<sup>th<\/sup> degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.<\/p>\n<div id=\"fs-id1165135502799\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Interpreting Turning Points<\/h3>\n<p id=\"fs-id1165135469050\">A <strong>turning point<\/strong> is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).<\/p>\n<p id=\"fs-id1165135469055\">A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em> \u2013 1\u00a0turning points.<\/p>\n<\/div>\n<div id=\"Example_03_04_07\" class=\"example\">\n<div id=\"fs-id1165134374690\" class=\"exercise\">\n<div id=\"fs-id1165134060420\" class=\"problem textbox shaded\">\n<h3>Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165134060425\">Find the maximum number of turning points of each polynomial function.<\/p>\n<ol id=\"fs-id1165134060428\">\n<li>[latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]<\/li>\n<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157524\">Show Solution<\/span><\/p>\n<div id=\"q157524\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137784430\">\n<li>[latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\n<p id=\"fs-id1165135335895\">First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\n<p id=\"fs-id1165135453844\">Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\n<p id=\"fs-id1165135341233\">The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\n<\/li>\n<li>[latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15117 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2019\/04\/01021335\/CNX_Precalc_Figure_03_04_0162.jpg\" alt=\"Graphic of f(x) showing to multiply the first term of (x-1)^2 and 2x^2 to determine the leading term.\" width=\"487\" height=\"67\" \/><\/a><\/p>\n<p style=\"text-align: center\">[latex]a_{n}=-\\left(x^2\\right)\\left(2x^2\\right)=-2x^4[\/latex]<\/p>\n<p id=\"fs-id1165133104532\">First, identify the leading term of the polynomial function if the function were expanded.<span id=\"fs-id1165134130071\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135551181\">Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\n<p id=\"fs-id1165135551185\">The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Graph polynomial functions<\/h2>\n<p id=\"fs-id1165137843095\">We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.<\/p>\n<div id=\"fs-id1165137843101\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135449677\">How To: Given a polynomial function, sketch the graph.<\/h3>\n<ol id=\"fs-id1165135449683\">\n<li>Find the intercepts.<\/li>\n<li>Check for symmetry. If the function is an even function, its graph is symmetrical about the <em>y<\/em>-axis, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>f<\/em>(<em>x<\/em>).<br \/>\nIf a function is an odd function, its graph is symmetrical about the origin, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>\u2013<\/em><em>f<\/em>(<em>x<\/em>).<\/li>\n<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the <em>x<\/em>-intercepts.<\/li>\n<li>Determine the end behavior by examining the leading term.<\/li>\n<li>Use the end behavior and the behavior at the intercepts to sketch a graph.<\/li>\n<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\n<li>Optionally, use technology to check the graph.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_08\" class=\"example\">\n<div id=\"fs-id1165135575951\" class=\"exercise\">\n<div id=\"fs-id1165135575953\" class=\"problem textbox shaded\">\n<h3>Example 8: Sketching the Graph of a Polynomial Function<\/h3>\n<p id=\"fs-id1165135575958\">Sketch a graph of [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q892446\">Show Solution<\/span><\/p>\n<div id=\"q892446\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135237929\">This graph has two <em>x-<\/em>intercepts. At <em>x\u00a0<\/em>= \u20133, the factor is squared, indicating a multiplicity of 2. The graph will bounce at this <em>x<\/em>-intercept. At <em>x\u00a0<\/em>= 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/p>\n<p id=\"fs-id1165135171021\">The <em>y<\/em>-intercept is found by evaluating <em>f<\/em>(0).<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} f\\left(0\\right)&=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right) \\\\ &=-2\\cdot 9\\cdot \\left(-5\\right) \\\\ &=90 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165134374772\">The <em>y<\/em>-intercept is (0, 90).<\/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\/03010733\/CNX_Precalc_Figure_03_04_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"487\" height=\"362\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134381522\">Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex],<br \/>\nso the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity.<span id=\"fs-id1165135646080\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165134374738\">To sketch this, we consider that:<\/p>\n<ul id=\"fs-id1165134374741\">\n<li>As [latex]x\\to -\\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\infty[\/latex], so we know the graph starts in the second quadrant and is decreasing toward the <em>x<\/em>-axis.<\/li>\n<li>Since [latex]f\\left(-x\\right)=-2{\\left(-x+3\\right)}^{2}\\left(-x - 5\\right)[\/latex]<br \/>\nis not equal to <em>f<\/em>(<em>x<\/em>), the graph does not display symmetry.<\/li>\n<li>At (-3,0), the graph bounces off of the <em>x<\/em>-axis, so the function must start increasing.\n<p id=\"fs-id1165135536183\" style=\"text-align: left\">At (0, 90), the graph crosses the <em>y<\/em>-axis at the <em>y<\/em>-intercept.<\/p>\n<\/li>\n<\/ul>\n<figure id=\"Figure_03_04_018\" class=\"small\">\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\/03010733\/CNX_Precalc_Figure_03_04_0182.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<\/figure>\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\/03010734\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165135241000\">Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).\u00a0<span id=\"fs-id1165135241013\"><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165135613608\">As [latex]x\\to \\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\mathrm{-\\infty }[\/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.<\/p>\n<p id=\"fs-id1165135574296\">Using technology, we can create the graph for the polynomial function, shown in Figure 16, and verify that the resulting graph looks like our sketch in Figure 15.<\/p>\n<figure id=\"Figure_03_04_020\" class=\"small\"><figcaption>The complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/figcaption><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\/03010734\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<\/figure>\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-id1165133065140\">Sketch a graph of [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q408253\">Show Solution<\/span><\/p>\n<div id=\"q408253\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010734\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135369116\">\n<h2>Writing Formulas for Polynomial Functions<\/h2>\n<p id=\"fs-id1165135369122\">Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a <strong>polynomial function<\/strong> written in factored form will have an <em>x<\/em>-intercept where each factor is equal to zero, we can form a function that will pass through a set of <em>x<\/em>-intercepts by introducing a corresponding set of factors.<\/p>\n<div id=\"fs-id1165133320785\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Factored Form of Polynomials<\/h3>\n<p id=\"fs-id1165133320793\">If a polynomial of lowest degree <em>p<\/em>\u00a0has horizontal intercepts at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form: [latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]\u00a0where the powers [latex]{p}_{i}[\/latex]\u00a0on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor <em>a<\/em>\u00a0can be determined given a value of the function other than the <em>x<\/em>-intercept.<\/p>\n<\/div>\n<div id=\"fs-id1165135580289\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135580296\">How To: Given a graph of a polynomial function, write a formula for the function.<\/h3>\n<ol id=\"fs-id1165133309878\">\n<li>Identify the <em>x<\/em>-intercepts of the graph to find the factors of the polynomial.<\/li>\n<li>Examine the behavior of the graph at the <em>x<\/em>-intercepts to determine the multiplicity of each factor.<\/li>\n<li>Find the polynomial of least degree containing all the factors found in the previous step.<\/li>\n<li>Use any other point on the graph (the <em>y<\/em>-intercept may be easiest) to determine the stretch factor.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_04_10\" class=\"example\">\n<div id=\"fs-id1165134043949\" class=\"exercise\">\n<div id=\"fs-id1165134043951\" class=\"problem textbox shaded\">\n<h3>Example 13: Writing a Formula for a Polynomial Function from the Graph<\/h3>\n<p>Write a formula for the polynomial function shown in Figure 19.<\/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\/03010735\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 19<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q574656\">Show Solution<\/span><\/p>\n<div id=\"q574656\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135621955\">his graph has three <em>x<\/em>-intercepts: <em>x\u00a0<\/em>= \u20133, 2, and 5. The <em>y<\/em>-intercept is located at (0, 2). At <em>x\u00a0<\/em>= \u20133 and <em>x\u00a0<\/em>= 5,\u00a0the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At <em>x\u00a0<\/em>= 2, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\n<p id=\"fs-id1165135575901\">To determine the stretch factor, we utilize another point on the graph. We will use the <em>y<\/em>-intercept (0, \u20132), to solve for <em>a<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}f\\left(0\\right)&=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right) \\\\ -2&=-60a \\\\ a&=\\frac{1}{30} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165133437286\">The graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/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>Given the graph in Figure 20, write a formula for the function shown.<\/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\/03010735\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q412515\">Show Solution<\/span><\/p>\n<div id=\"q412515\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135440065\">\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm15942\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15942&theme=oea&iframe_resize_id=ohm15942\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137846272\">\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 pf a power function 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<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\n<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\n<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the <em>x<\/em>-axis.<\/li>\n<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercepts.<\/li>\n<li>The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.<\/li>\n<li>The end behavior of a polynomial function depends on the leading term.<\/li>\n<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>A polynomial function of degree <em>n<\/em>\u00a0has at most\u00a0<em>n <\/em>\u2013\u00a01 turning points.<\/li>\n<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>\u00a0n <\/em>\u2013\u00a01 turning points.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165134112772\" class=\"definition\">\n<dt><strong>multiplicity<\/strong><\/dt>\n<dd id=\"fs-id1165134112776\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity <em>p<\/em>.<\/dd>\n<\/dl>\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\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 <\/section>","protected":false},"author":359553,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax 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