{"id":1277,"date":"2019-03-07T15:23:15","date_gmt":"2019-03-07T15:23:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/power-functions-and-polynomial-functions\/"},"modified":"2025-03-31T20:16:51","modified_gmt":"2025-03-31T20:16:51","slug":"power-functions-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/power-functions-and-polynomial-functions\/","title":{"raw":"4.1 Power Functions and Polynomial Functions","rendered":"4.1 Power Functions and Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Identify power functions.<\/li>\r\n \t<li>Identify end behavior and local 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 and local behavior of a polynomial function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"CNX_Precalc_Figure_03_03_001.jpg\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152123\/CNX_Precalc_Figure_03_03_001.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/> <strong>Figure 1.\u00a0<\/strong>(credit: Jason Bay, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<div class=\"wp-caption-text\"><\/div>\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 in <a class=\"autogenerated-content\" href=\"#Table_03_03_01\">Table 1<\/a>.<\/p>\r\n\r\n<table id=\"Table_03_03_01\" summary=\"..\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>Year<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]2009[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]2010[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]2011[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]2012[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]2013[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>Bird Population<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]800[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]897[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]992[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]1,083[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">[latex]1,169[\/latex]<\/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 [latex]t[\/latex] years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.<\/p>\r\n\r\n<div id=\"fs-id1165137540446\" class=\"bc-section section\">\r\n<h3>Identifying Power Functions<\/h3>\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><span class=\"no-emphasis\">area of a circle<\/span><\/strong> with radius [latex]r[\/latex] is<\/p>\r\n\r\n<div id=\"eip-544\" class=\"unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135191346\">and the function for the <strong><span class=\"no-emphasis\">volume of a sphere<\/span><\/strong> with radius [latex]r[\/latex] is<\/p>\r\n\r\n<div id=\"eip-640\" class=\"unnumbered\" style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/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 [latex]r[\/latex] raised to a power.<\/p>\r\n\r\n<div id=\"fs-id1165135356525\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/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=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\r\n<p id=\"eip-id1165135584093\">where [latex]k[\/latex] and [latex]p[\/latex] are real numbers, and [latex]k[\/latex] is known as the <strong>coefficient<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137661479\" class=\"precalculus qa key-takeaways\">\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 an exponential function, not a power function.<\/em><\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_03_03_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137745179\">\r\n<div id=\"fs-id1165137742710\">\r\n<h3>Example 1:\u00a0 Identifying Power Functions<\/h3>\r\n<p id=\"fs-id1165137824370\">Which of the following functions are power functions?<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}f\\left(x\\right)=1\\hfill &amp; \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill &amp; \\text{Identify function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill &amp; \\text{Quadratic}\\text{\u200b}\\text{ function}\\hfill \\\\ f\\left(x\\right)={x}^{3}\\hfill &amp; \\text{Cubic function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{x} \\hfill &amp; \\text{Reciprocal function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{{x}^{2}}\\hfill &amp; \\text{Reciprocal squared function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\hfill &amp; \\text{Square root function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[3]{x}\\hfill &amp; \\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137422823\">[reveal-answer q=\"fs-id1165137422823\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137422823\"]\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><span class=\"no-emphasis\">reciprocal<\/span><\/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><span class=\"no-emphasis\">cube root<\/span><\/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 id=\"fs-id1165137660222\" class=\"precalculus tryit\">\r\n<h3>Try It #1<\/h3>\r\n<div id=\"ti_03_03_01\">\r\n<div id=\"fs-id1165137475224\">\r\n<p id=\"fs-id1165137475225\">Which functions are power functions?<\/p>\r\n<p id=\"fs-id1165137824385\" style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134312227\">[reveal-answer q=\"fs-id1165134312227\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134312227\"]\r\n<p id=\"fs-id1165134094624\">[latex]f\\left(x\\right)[\/latex] is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}.[\/latex] The other functions are not power functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134269023\" class=\"bc-section section\">\r\n<h3>Identifying End Behavior of Power Functions<\/h3>\r\n<p id=\"fs-id1165135436540\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">Figure 2<\/a> shows the graphs of [latex]f\\left(x\\right)={x}^{2},\\text{ }g\\left(x\\right)={x}^{4}[\/latex] and [latex]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<div id=\"Figure_03_03_002\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152126\/CNX_Precalc_Figure_03_03_002.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\" \/> <strong>Figure 2.<\/strong> Even-power functions[\/caption]\r\n\r\n<\/div>\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 symbolically write as [latex]x\\to \\infty ,[\/latex] we are describing a behavior; we are saying that [latex]x[\/latex] is increasing without bound.<\/p>\r\n<p id=\"fs-id1165137658268\">With the 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] increases or decreases without bound, 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=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \u00b1\\infty , f\\left(x\\right)\\to \\infty [\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137533222\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">Figure 3<\/a>\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},\\text{ }g\\left(x\\right)={x}^{5},\\text{ }\\text{and}\\text{ }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<div id=\"Figure_03_03_003\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"289\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152129\/CNX_Precalc_Figure_03_03_003.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=\"289\" height=\"366\" \/> <strong>Figure 3.<\/strong> Odd-power function[\/caption]\r\n\r\n<\/div>\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 <a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">Figure 2<\/a> we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n[\/latex] even, are symmetric about the [latex]y\\text{-}[\/latex]axis. In <a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">Figure 3<\/a> we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n[\/latex] odd, are symmetric about the origin.<\/p>\r\n<p id=\"fs-id1165137812578\">For these odd power functions, as [latex]x[\/latex] decreases without bound, [latex]f\\left(x\\right)[\/latex] decreases without bound. As [latex]x[\/latex] increases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. In symbolic form we write<\/p>\r\n\r\n<div id=\"eip-77\" class=\"unnumbered\" style=\"text-align: center;\">as [latex] x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty [\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values move far to the left of the origin ( [latex]x\\to -\\infty [\/latex] ) and move far to the right of the origin ( [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<a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">Figure 4<\/a> 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.\r\n<div id=\"Figure_03_03_004abcd\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152133\/CNX_Precalc_Figure_03_03_004abcd.jpg\" alt=\"Graph of an even-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to positive infinity. Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an even-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an odd-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity.\" width=\"731\" height=\"734\" \/> <strong>Figure 4.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135161436\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137415258\"><strong>Given a power function [latex]f\\left(x\\right)=k{x}^{n}[\/latex] where<\/strong> [latex]n[\/latex] <strong>is a non-negative integer, identify the end behavior.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137409522\" type=\"1\">\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 <a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">Figure 4<\/a>\u00a0to identify the end behavior.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_03_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137923491\">\r\n<div id=\"fs-id1165137599768\">\r\n<h3>Example 2:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165135169237\">[reveal-answer q=\"fs-id1165135169237\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135169237\"]\r\n<p id=\"fs-id1165137502379\">The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As [latex]x[\/latex] increases without bound, 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 [latex]x[\/latex] decreases without bound, 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 <a class=\"autogenerated-content\" href=\"#Figure_03_03_008\">Figure 5<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_03_008\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"348\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152136\/CNX_Precalc_Figure_03_03_008.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"348\" height=\"236\" \/> <strong>Figure 5.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137535914\">\r\n<div id=\"fs-id1165137811997\">\r\n<h3>Example 3:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165137722696\">[reveal-answer q=\"fs-id1165137722696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137722696\"]\r\n<p id=\"fs-id1165137409513\">The exponent of the power function is 9 (an odd number). Because the coefficient is [latex]\u20131[\/latex] (negative), the graph is the reflection about the [latex]x\\text{-}[\/latex]axis of the graph of [latex]f\\left(x\\right)={x}^{9}.[\/latex] <a class=\"autogenerated-content\" href=\"#Figure_03_03_009\">Figure 6<\/a> shows that as [latex]x[\/latex] increases without bound, the output decreases without bound. As [latex]x[\/latex] decreases without bound, the output increases without bound. In symbolic form, we would write<\/p>\r\n\r\n<div id=\"eip-id1165134384400\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/div>\r\n<div id=\"Figure_03_03_009\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"260\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152139\/CNX_Precalc_Figure_03_03_009.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"260\" height=\"356\" \/> <strong>Figure 6.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<h3>Analysis<\/h3>\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=\"..\"><caption>Table 2<\/caption>\r\n<thead>\r\n<tr>\r\n<th class=\"border\" style=\"text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"text-align: center;\">[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\">\u201310<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">1,000,000,000<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\">\u20135<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">1,953,125<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20131,953,125<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">\u20131,000,000,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137644426\">We can see from <a class=\"autogenerated-content\" href=\"#Table_03_03_03\">Table 2<\/a>\u00a0that, when we substitute more and more negative values for [latex]x,[\/latex] the output becomes very large, and when we substitute larger positive values for [latex]x,[\/latex] the output becomes more negative.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137626838\" class=\"precalculus tryit\">\r\n<h3>Try It #2<\/h3>\r\n<div id=\"ti_03_03_02\">\r\n<div id=\"fs-id1165137734867\">\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\r\n<\/div>\r\n<div id=\"fs-id1165137647550\">[reveal-answer q=\"fs-id1165137647550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137647550\"]\r\n<p id=\"fs-id1165137647551\">As [latex]x[\/latex] increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \u00b1\\infty , f\\left(x\\right)\\to -\\infty [\/latex] because of the negative coefficient.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134069294\" class=\"bc-section section\">\r\n<h3>Identifying Polynomial Functions<\/h3>\r\nAn oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex] of the spill depends on the number of weeks [latex]w[\/latex] that have passed. This relationship is linear.\r\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area [latex]A[\/latex] of a circle.<\/p>\r\n\r\n<div id=\"eip-731\" class=\"unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/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=\"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)\\\\ \\text{ }&amp;=\\pi {\\left(24+8w\\right)}^{2}\\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137835475\">Multiplying gives the formula<\/p>\r\n\r\n<div id=\"eip-290\" class=\"unnumbered\" style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}.[\/latex]<\/div>\r\n<div>[latex]\\\\[\/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 terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/p>\r\n\r\n<div id=\"fs-id1165137715427\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137823247\">Let [latex]n[\/latex] be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\r\n\r\n<div id=\"fs-id1165131937978\" style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"eip-id1165137832690\">This is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex] is a <strong>coefficient<\/strong> and can be any real number, but [latex]{a}_{n}[\/latex] cannot equal [latex]0[\/latex]. Each product [latex]{a}_{i}{x}^{i}[\/latex] is a <strong>term of a polynomial function<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137817691\">\r\n<div id=\"fs-id1165137817693\">\r\n<h3>Example 4:\u00a0 Identifying Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\r\n\r\n<div><\/div>\r\n<div id=\"eip-id1165134474011\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\end{array}\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt[\\leftroot{1}\\uproot{2} ]{x}+2\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134221783\">[reveal-answer q=\"fs-id1165134221783\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134221783\"]\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}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex] where 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] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4.[\/latex]<\/li>\r\n \t<li>[latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x.[\/latex]<\/li>\r\n \t<li>[latex]h\\left(x\\right)[\/latex] cannot 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<\/div>\r\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\r\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\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.\u00a0 For the constant function, the degree is zero, and for a non-constant linear function, the degree is one.<\/p>\r\n\r\n<div id=\"fs-id1165135193124\">\r\n<div class=\"textbox shaded\">\r\n<h3>Terminology of Polynomial Functions<\/h3>\r\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<\/p>\r\n<span id=\"fs-id1165137406148\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152142\/CNX_Precalc_Figure_03_03_010n.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.\" \/><\/span>\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>\r\n<div id=\"fs-id1165134031372\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137803898\"><strong>Given a polynomial function, identify the degree and leading coefficient.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135587816\" type=\"1\">\r\n \t<li>Find the highest power of [latex]x[\/latex] to determine the degree of the function.<\/li>\r\n \t<li>Identify the term containing the highest power of [latex]x[\/latex] to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_03_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137401820\">\r\n<div id=\"fs-id1165137862379\">\r\n<h3>Example 5:\u00a0 Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\r\nIdentify the degree, leading term, and leading coefficient of the following polynomial functions.\r\n<div id=\"eip-id1165134242117\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}\\begin{array}{l}\\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }f\\left(x\\right)=3+2{x}^{2}-4{x}^{3}\\end{array}\\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\ h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135527012\">\r\n\r\n[reveal-answer q=\"fs-id1165135527012\"]Show Solution[\/reveal-answer][hidden-answer a=\"fs-id1165135527012\"]\r\n\r\nFor the function [latex]f\\left(x\\right),[\/latex] the highest power of [latex]x[\/latex] is 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, [latex]-4.[\/latex]\r\n\r\nFor the function [latex]g\\left(t\\right),[\/latex] the highest power of [latex]t[\/latex] is [latex]5,[\/latex] so the degree is [latex]5.[\/latex] The leading term is the term containing that degree, [latex]5{t}^{5}.[\/latex] The leading coefficient is the coefficient of that term, [latex]5.[\/latex]\r\n\r\nFor the function [latex]h\\left(p\\right),[\/latex] the highest power of [latex]p[\/latex] is [latex]3,[\/latex] so the degree is [latex]3.[\/latex] The leading term is the term containing that degree, [latex]-{p}^{3};[\/latex] the leading coefficient is the coefficient of that term, [latex]-1.[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\r\n<h3>Try It #3<\/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\r\n<\/div>\r\n<div id=\"fs-id1165135701674\">[reveal-answer q=\"fs-id1165135701674\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135701674\"]\r\n<p id=\"fs-id1165135701675\">The degree is 6. The leading term is [latex]-{x}^{6}.[\/latex] The leading coefficient is [latex]-1.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135701674\">\r\n<div id=\"fs-id1165137702213\" class=\"bc-section section\">\r\n<h4>Identifying End Behavior of Polynomial Functions<\/h4>\r\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as [latex]x[\/latex] gets more and more positive or negative, 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. See <a class=\"autogenerated-content\" href=\"#Table_03_03_04\">Table 3<\/a>.<\/p>\r\n\r\n<table id=\"Table_03_03_04\" summary=\"..\"><caption>Table 3<\/caption><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th class=\"border\">Polynomial Function<\/th>\r\n<th class=\"border\">Leading Term<\/th>\r\n<th class=\"border\">Graph of Polynomial Function<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x-4[\/latex]<\/td>\r\n<td class=\"border\">[latex]5{x}^{4}[\/latex]<\/td>\r\n<td class=\"border\"><span id=\"fs-id1165137768814\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152145\/CNX_Precalc_Figure_03_03_011.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"259\" height=\"275\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\r\n<td class=\"border\">[latex]-2{x}^{6}[\/latex]<\/td>\r\n<td class=\"border\"><span id=\"fs-id1165137714206\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152148\/CNX_Precalc_Figure_03_03_012.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" width=\"259\" height=\"275\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\r\n<td class=\"border\">[latex]3{x}^{5}[\/latex]<\/td>\r\n<td class=\"border\"><span id=\"fs-id1165137540879\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152151\/CNX_Precalc_Figure_03_03_013.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"259\" height=\"275\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\r\n<td class=\"border\">[latex]-6{x}^{3}[\/latex]<\/td>\r\n<td class=\"border\"><span id=\"fs-id1165137600670\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152154\/CNX_Precalc_Figure_03_03_014.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" width=\"259\" height=\"275\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"Example_03_03_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137452413\">\r\n<div id=\"fs-id1165137452415\">\r\n<h3>Example 6:\u00a0 Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\n<p id=\"fs-id1165137831279\">Describe the end behavior and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_015\">Figure 7<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_03_015\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"302\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152157\/CNX_Precalc_Figure_03_03_015.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"302\" height=\"275\" \/> <strong>Figure 7.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135251309\">[reveal-answer q=\"fs-id1165135251309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135251309\"]\r\n<p id=\"fs-id1165135251312\">As the input values [latex]x[\/latex] get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values [latex]x[\/latex] get more and more negative, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\r\n\r\n<div id=\"eip-id1165137778911\" class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty. [\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137454991\">In words, we read this notation, \"as [latex]x[\/latex] values increase without bound, the function values increase without bound and as [latex]x[\/latex] values decrease without bound, the function values decrease without bound.\"<\/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 id=\"fs-id1165137470875\" class=\"precalculus tryit\">\r\n<h3>Try It #4<\/h3>\r\n<div id=\"ti_03_03_04\">\r\n<div id=\"fs-id1165137732301\">\r\n<p id=\"fs-id1165135460938\">Describe the end behavior, and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_016\">Figure<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_03_016\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"304\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152200\/CNX_Precalc_Figure_03_03_016n.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"304\" height=\"275\" \/> <strong>Figure 8.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134047710\">[reveal-answer q=\"fs-id1165134047710\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134047710\"]\r\n<p id=\"fs-id1165134047711\">As the input increases without bound, the output decreases without bound.\u00a0 As the input decreases without bound, the output decreases without bound. As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ;[\/latex] as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty .[\/latex] It has the shape of an even degree power function with a negative coefficient.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_07\" class=\"textbox examples\">\r\n<div>\r\n<div id=\"fs-id1165137470363\">\r\n<h3>Example 7:\u00a0 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\r\n<\/div>\r\n<div id=\"fs-id1165137401107\">[reveal-answer q=\"fs-id1165137401107\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137401107\"]\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\r\n<div id=\"eip-id1165132051075\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(x\\right)&amp;=-3{x}^{2}\\left(x-1\\right)\\left(x+4\\right)\\\\ \\text{ }&amp;=-3{x}^{2}\\left({x}^{2}+3x-4\\right)\\\\ \\text{ }&amp;=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\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] The leading term is [latex]-3{x}^{4};[\/latex] therefore, the degree of the polynomial is 4. The degree is even [latex]\\left(4\\right)[\/latex] and the leading coefficient is negative [latex]\\left(-3\\right),[\/latex] so the end behavior is<\/p>\r\n[latex]\\\\[\/latex]\r\n<div id=\"eip-id1165133007607\" class=\"unnumbered\" style=\"text-align: center;\">\u00a0as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]<\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty. [\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"precalculus tryit\">\r\n<h3>Try It #5<\/h3>\r\n<div id=\"ti_03_03_05\">\r\n<div id=\"fs-id1165137722131\">\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\r\n<\/div>\r\n<div id=\"fs-id1165135409431\">[reveal-answer q=\"fs-id1165135409431\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135409431\"]\r\n<p id=\"fs-id1165137749856\">The function in general form is [latex]f\\left(x\\right)=0.2x^3-1.2x^2+0.6x+2.[\/latex] The leading term is [latex]0.2{x}^{3},[\/latex] so it is a degree 3 polynomial. As [latex]x[\/latex] increases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound; as [latex]x[\/latex] decreases without bound, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137735781\" class=\"bc-section section\">\r\n<h4>Identifying Local Behavior of Polynomial Functions<\/h4>\r\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. In particular, we are interested in locations where graph behavior changes. A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing (<strong>local maximum<\/strong>) or decreasing to increasing (<strong>local minimum<\/strong>).<\/p>\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. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_017\">Figure 9<\/a><strong>.<\/strong><\/p>\r\n\r\n<div id=\"Figure_03_03_017\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"294\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152203\/CNX_Precalc_Figure_03_03_017.jpg\" alt=\"A cubic function with the interceps and turning points labeled\" width=\"294\" height=\"462\" \/> <strong>Figure 9.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\nA <strong>turning point<\/strong> of a graph is a point at which the graph changes direction from increasing to decreasing (<strong>local maximum<\/strong>) or decreasing to increasing (<strong>local minimum<\/strong>). The <strong><em>y-<\/em>intercept<\/strong> is the point at which the function has an input value of zero. The <strong><em>x<\/em>-intercepts<\/strong> are the points at which the output value is zero.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137766902\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137645233\"><strong>Given a polynomial function, determine the intercepts.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137571388\" type=\"1\">\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 [latex]x\\text{-}[\/latex]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=\"textbox examples\">\r\n<div id=\"fs-id1165137435581\">\r\n<div id=\"fs-id1165137803210\">\r\n<h3>Example 8:\u00a0 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 [latex]y\\text{-}[\/latex]and [latex]x\\text{-}[\/latex]intercepts.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135251466\">[reveal-answer q=\"fs-id1165135251466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135251466\"]\r\n<p id=\"fs-id1165135251468\">The <em>y-<\/em>intercept occurs when the input is zero so substitute 0 for [latex]x.[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165133032876\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(0\\right)&amp;=\\left(0-2\\right)\\left(0+1\\right)\\left(0-4\\right)\\hfill \\\\ \\text{ }&amp;=\\left(-2\\right)\\left(1\\right)\\left(-4\\right)\\hfill \\\\ \\text{ }&amp;=8\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\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. To solve [latex]0=\\left(x-2\\right)\\left(x+1\\right)\\left(x-4\\right)[\/latex] set each factor equal to zero and simplify.<\/p>\r\n\r\n<div id=\"eip-id1165134380311\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*} x-2&amp;=0&amp;\\text{ }\\text{ }\\text{ } \\text{ or }\\text{ }\\text{ }\\text{ }&amp; x+1=0&amp;\\text{ }\\text{ }\\text{ }\\text{ or }\\text{ }\\text{ }\\text{ } &amp;x-4=0\\\\ \\text{ }x&amp;=2&amp;\\text{ }&amp;x=-1&amp; \\text{ }&amp;x=4 \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135316178\">The [latex]x\\text{-}[\/latex]intercepts are [latex]\\left(2,0\\right),\\left(\u20131,0\\right),[\/latex] and [latex]\\left(4,0\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165134380385\">We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_018\">Figure 10<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_03_018\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"298\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152206\/CNX_Precalc_Figure_03_03_018.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"298\" height=\"386\" \/> <strong>Figure 10.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_09\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137834894\">\r\n<div id=\"fs-id1165137834896\">\r\n<h3>Example 9:\u00a0 Determining the Intercepts of a Polynomial Function with Factoring<\/h3>\r\nGiven the polynomial function [latex]f\\left(x\\right)={x}^{4}-4{x}^{2}-45,[\/latex] determine the [latex]y\\text{-}[\/latex]and [latex]x\\text{-}[\/latex]intercepts.\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137634473\">[reveal-answer q=\"fs-id1165137634473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137634473\"]The <em>y-<\/em>intercept occurs when the input is zero.\r\n<div id=\"eip-id1165132943488\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*} \\\\ f\\left(0\\right)&amp;={\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45 \\\\ \\text{ }&amp;=-45\\hfill \\end{align*}[\/latex]<\/div>\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\r\n<div id=\"eip-id1165135376171\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(x\\right)&amp;={x}^{4}-4{x}^{2}-45\\hfill \\\\\\text{ }&amp;=\\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right)\\hfill \\\\ \\text{ }&amp;=\\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div id=\"eip-id1165135684199\" class=\"unnumbered\" style=\"text-align: center;\">[latex]0=\\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div id=\"eip-id1165134383791\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}x-3=0\\hfill &amp; \\text{or}\\hfill &amp; x+3=0\\hfill &amp; \\text{or}\\hfill &amp; {x}^{2}+5=0\\hfill \\\\ \\text{ }x=3\\hfill &amp; \\text{or}\\hfill &amp; \\text{ }x=-3\\hfill &amp; \\text{or}\\hfill &amp; \\text{(no real solution)}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135436471\">The <em>x<\/em>-intercepts are [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(\u20133,0\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165137444727\">We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_019\">Figure 11<\/a>. We can see that the function is even because [latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_03_03_019\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"310\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152209\/CNX_Precalc_Figure_03_03_019.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=\"310\" height=\"271\" \/> <strong>Figure 11.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137749604\" class=\"precalculus tryit\">\r\n<h3>Try It #6<\/h3>\r\n<div id=\"ti_03_03_06\">\r\n<div id=\"fs-id1165137405243\">\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 [latex]y\\text{-}[\/latex]and [latex]x\\text{-}[\/latex]intercepts.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137762370\">[reveal-answer q=\"fs-id1165137762370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137762370\"]\r\n<p id=\"fs-id1165137762371\"><em>y<\/em>-intercept [latex]\\left(0,0\\right);[\/latex] <em>x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(\u20132,0\\right),[\/latex] and [latex]\\left(5,0\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134080932\" class=\"bc-section section\">\r\n<h4>Comparing Smooth and Continuous Graphs<\/h4>\r\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of [latex]x\\text{-}[\/latex]intercepts and the number of turning points. A polynomial function of [latex]n\\text{th}[\/latex] degree is the product of at most [latex]n[\/latex] factors, so it will have at most [latex]n[\/latex] roots or zeros, or [latex]x\\text{-}[\/latex]intercepts. The graph of the polynomial function of degree [latex]n[\/latex] must have at most [latex]n\u20131[\/latex] turning points. This means the graph has at most one fewer turning points than the degree of the polynomial.<\/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\">\r\n<div class=\"textbox shaded\">\r\n<h3>Intercepts and Turning Points of Polynomials<\/h3>\r\nA polynomial of degree [latex]n[\/latex] will have, at most, [latex]n[\/latex] <em>x<\/em>-intercepts and [latex]n-1[\/latex] turning points.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135237034\">\r\n<div id=\"fs-id1165135237036\">\r\n<h3>Example 10:\u00a0 Determining the Maximum Possible Number of Intercepts and Turning Points of a Polynomial<\/h3>\r\n<p id=\"fs-id1165134152759\">Without graphing the function, determine the maximum number of possible [latex]x\\text{-}[\/latex]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\r\n<\/div>\r\n<div id=\"fs-id1165135414339\">[reveal-answer q=\"fs-id1165135414339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135414339\"]\r\n<p id=\"fs-id1165135414341\">The polynomial has a degree of [latex]10,[\/latex] so there are at most [latex]10[\/latex] [latex]x[\/latex]-intercepts and at most [latex]\\mathrm{10}-1=9[\/latex] turning points.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137628834\" class=\"precalculus tryit\">\r\n<h3>Try It #7<\/h3>\r\n<div id=\"ti_03_03_07\">\r\n<div id=\"fs-id1165135188273\">\r\n\r\nWithout graphing the function, determine the maximum number of possible [latex]x\\text{-}[\/latex]intercepts and turning points for [latex]f\\left(x\\right)=108-13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}.[\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137660801\">[reveal-answer q=\"fs-id1165137660801\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137660801\"]\r\n<p id=\"fs-id1165137660802\">There are at most 12 [latex]x\\text{-}[\/latex]intercepts and at most 11 turning points.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_11\" class=\"textbox examples\">\r\n<div>\r\n<div id=\"fs-id1165137435066\">\r\n<h3>Example 11:\u00a0 Drawing Conclusions about a Polynomial Function from the Graph<\/h3>\r\n<p id=\"fs-id1165137843783\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_020\">Figure 12<\/a>\u00a0based on its intercepts and turning points?<\/p>\r\n\r\n<div id=\"Figure_03_03_020\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"330\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152213\/CNX_Precalc_Figure_03_03_020.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"330\" height=\"249\" \/> <strong>Figure 12.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137737264\">[reveal-answer q=\"fs-id1165137737264\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137737264\"]\r\n<p id=\"fs-id1165131926327\">The end behavior of the graph tells us this is the graph of an even-degree polynomial. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_021\">Figure 13<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_03_021\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"332\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152216\/CNX_Precalc_Figure_03_03_021.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"332\" height=\"251\" \/> <strong>Figure 13.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135670389\">The graph has 2 [latex]x\\text{-}[\/latex]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 id=\"fs-id1165137871106\" class=\"precalculus tryit\">\r\n<h3>Try It #8<\/h3>\r\n<div id=\"ti_03_03_08\">\r\n<div id=\"fs-id1165137834183\">\r\n<p id=\"fs-id1165137454180\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_022\">Figure 14<\/a>\u00a0based on its intercepts and turning points?<\/p>\r\n\r\n<div id=\"Figure_03_03_022\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"326\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152219\/CNX_Precalc_Figure_03_03_022.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"326\" height=\"296\" \/> <strong>Figure 14.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137666790\">[reveal-answer q=\"fs-id1165137666790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137666790\"]The end behavior indicates an odd-degree polynomial function; there are 3 [latex]x\\text{-}[\/latex]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.[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_03_12\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135184013\">\r\n<div id=\"fs-id1165137725458\">\r\n<h3>Example 12:\u00a0 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] determine the x and y intercepts and the maximum number of turning points possible.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135457721\">[reveal-answer q=\"fs-id1165135457721\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135457721\"]\r\n<p id=\"fs-id1165135457723\">The [latex]y\\text{-}[\/latex]intercept is found by evaluating [latex]f\\left(0\\right).[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165134587897\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(0\\right)&amp;=-4\\left(0\\right)\\left(0+3\\right)\\left(0-4\\right)\\hfill \\\\ \\text{ }&amp;=0\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135245749\">The [latex]y\\text{-}[\/latex]intercept is [latex]\\left(0,0\\right).[\/latex]<\/p>\r\n<p id=\"fs-id1165135203755\">The [latex]x\\text{-}[\/latex]intercepts are found by determining the zeros of the function.<\/p>\r\n<p style=\"text-align: center;\">[latex]0=-4x\\left(x+3\\right)\\left(x-4\\right)[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165135401630\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l} \\begin{array}{lllllllll}x=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x+3=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x-4=0\\hfill \\\\ x=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{ }x=-3\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{ }\\text{ }x=4\\hfill \\end{array}\\end{array}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135431016\">The [latex]x\\text{-}[\/latex]intercepts are [latex]\\left(0,0\\right),\\left(\u20133,0\\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 id=\"fs-id1165137661075\" class=\"precalculus tryit\">\r\n<h3>Try It #9<\/h3>\r\n<div id=\"ti_03_03_09\">\r\n<div id=\"fs-id1165137575430\">\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 x and y intercepts and the maximum number of turning points possible.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137833005\">[reveal-answer q=\"fs-id1165137833005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137833005\"]\r\n<p id=\"fs-id1165137833006\">The [latex]x\\text{-}[\/latex]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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137653058\" class=\"precalculus media\">\r\n<div class=\"textbox shaded\">\r\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\r\n<div id=\"fs-id1165134080932\" class=\"bc-section section\">\r\n<div id=\"fs-id1165137653058\" class=\"precalculus media\">\r\n<h3>Media:<\/h3>\r\n<p id=\"fs-id1165135456729\">Access these online resources for additional instruction and practice with power and polynomial functions.<\/p>\r\n\r\n<ul id=\"fs-id1165137410802\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/keyinfopoly\">Find Key Information about a Given Polynomial Function<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/endbehavior\">End Behavior of a Polynomial Function<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/turningpoints\">Turning Points and [latex]x\\text{-}[\/latex]intercepts of Polynomial Functions<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/leastposdegree\">Least Possible Degree of a Polynomial Functio<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">Key Equations<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"key-equations\">\r\n<table id=\"eip-id1165134063974\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"width: 200.234px;\">general form of a polynomial function<\/td>\r\n<td class=\"border\" style=\"width: 504.766px;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137731646\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165135438864\">\r\n \t<li>A power function is a coefficient multiplied by a variable base raised to a number power.<\/li>\r\n \t<li>The behavior of a graph as the input decreases without bound and increases without bound is called the end behavior.<\/li>\r\n \t<li>The end behavior depends on whether the power is even or odd.<\/li>\r\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\r\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\r\n \t<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\r\n \t<li>A polynomial of degree [latex]n[\/latex] will have at most [latex]n[\/latex] <em>x-<\/em>intercepts and at most [latex]n-1[\/latex] turning points.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137668266\">\r\n \t<dt>coefficient<\/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\">\r\n \t<dt>continuous function<\/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\">\r\n \t<dt>degree<\/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\">\r\n \t<dt>end behavior<\/dt>\r\n \t<dd>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\">\r\n \t<dt>leading coefficient<\/dt>\r\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\">\r\n \t<dt>leading term<\/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\">\r\n \t<dt>polynomial function<\/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 terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\">\r\n \t<dt>power function<\/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] where [latex]k[\/latex] is a constant, the base is a variable, and the exponent, [latex]p,[\/latex] is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137833929\">\r\n \t<dt>smooth curve<\/dt>\r\n \t<dd>a graph with no sharp corners<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\">\r\n \t<dt>term of a polynomial function<\/dt>\r\n \t<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex] of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{1}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133085661\">\r\n \t<dt>turning point<\/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=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify power functions.<\/li>\n<li>Identify end behavior and local 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 and local behavior of a polynomial function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"CNX_Precalc_Figure_03_03_001.jpg\" class=\"wp-caption aligncenter\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152123\/CNX_Precalc_Figure_03_03_001.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\"><strong>Figure 1.\u00a0<\/strong>(credit: Jason Bay, Flickr)<\/p>\n<\/div>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\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 in <a class=\"autogenerated-content\" href=\"#Table_03_03_01\">Table 1<\/a>.<\/p>\n<table id=\"Table_03_03_01\" summary=\"..\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td class=\"border\"><strong>Year<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]2009[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]2010[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]2011[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]2012[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]2013[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>Bird Population<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]800[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]897[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]992[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]1,083[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center;\">[latex]1,169[\/latex]<\/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 [latex]t[\/latex] years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.<\/p>\n<div id=\"fs-id1165137540446\" class=\"bc-section section\">\n<h3>Identifying Power Functions<\/h3>\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><span class=\"no-emphasis\">area of a circle<\/span><\/strong> with radius [latex]r[\/latex] is<\/p>\n<div id=\"eip-544\" class=\"unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135191346\">and the function for the <strong><span class=\"no-emphasis\">volume of a sphere<\/span><\/strong> with radius [latex]r[\/latex] is<\/p>\n<div id=\"eip-640\" class=\"unnumbered\" style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/div>\n<div>[latex]\\\\[\/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 [latex]r[\/latex] raised to a power.<\/p>\n<div id=\"fs-id1165135356525\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/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=\"unnumbered\" style=\"text-align: center;\">[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\n<p id=\"eip-id1165135584093\">where [latex]k[\/latex] and [latex]p[\/latex] are real numbers, and [latex]k[\/latex] is known as the <strong>coefficient<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661479\" class=\"precalculus qa key-takeaways\">\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 an exponential function, not a power function.<\/em><\/p>\n<\/div>\n<div id=\"Example_03_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137745179\">\n<div id=\"fs-id1165137742710\">\n<h3>Example 1:\u00a0 Identifying Power Functions<\/h3>\n<p id=\"fs-id1165137824370\">Which of the following functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}f\\left(x\\right)=1\\hfill & \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill & \\text{Identify function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill & \\text{Quadratic}\\text{\u200b}\\text{ function}\\hfill \\\\ f\\left(x\\right)={x}^{3}\\hfill & \\text{Cubic function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{x} \\hfill & \\text{Reciprocal function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{{x}^{2}}\\hfill & \\text{Reciprocal squared function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\hfill & \\text{Square root function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[3]{x}\\hfill & \\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137422823\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137422823\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137422823\" 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><span class=\"no-emphasis\">reciprocal<\/span><\/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><span class=\"no-emphasis\">cube root<\/span><\/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 id=\"fs-id1165137660222\" class=\"precalculus tryit\">\n<h3>Try It #1<\/h3>\n<div id=\"ti_03_03_01\">\n<div id=\"fs-id1165137475224\">\n<p id=\"fs-id1165137475225\">Which functions are power functions?<\/p>\n<p id=\"fs-id1165137824385\" style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134312227\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134312227\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134312227\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134094624\">[latex]f\\left(x\\right)[\/latex] is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}.[\/latex] The other functions are not power functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134269023\" class=\"bc-section section\">\n<h3>Identifying End Behavior of Power Functions<\/h3>\n<p id=\"fs-id1165135436540\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">Figure 2<\/a> shows the graphs of [latex]f\\left(x\\right)={x}^{2},\\text{ }g\\left(x\\right)={x}^{4}[\/latex] and [latex]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 id=\"Figure_03_03_002\" 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\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152126\/CNX_Precalc_Figure_03_03_002.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\"><strong>Figure 2.<\/strong> Even-power functions<\/p>\n<\/div>\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 symbolically write as [latex]x\\to \\infty ,[\/latex] we are describing a behavior; we are saying that [latex]x[\/latex] is increasing without bound.<\/p>\n<p id=\"fs-id1165137658268\">With the 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] increases or decreases without bound, the [latex]f\\left(x\\right)[\/latex] values increase without bound. In symbolic form, we could write<\/p>\n<div id=\"eip-742\" class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \u00b1\\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137533222\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">Figure 3<\/a>\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},\\text{ }g\\left(x\\right)={x}^{5},\\text{ }\\text{and}\\text{ }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 id=\"Figure_03_03_003\" class=\"small\">\n<div style=\"width: 299px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152129\/CNX_Precalc_Figure_03_03_003.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=\"289\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong> Odd-power function<\/p>\n<\/div>\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 <a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">Figure 2<\/a> we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n[\/latex] even, are symmetric about the [latex]y\\text{-}[\/latex]axis. In <a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">Figure 3<\/a> we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n[\/latex] odd, are symmetric about the origin.<\/p>\n<p id=\"fs-id1165137812578\">For these odd power functions, as [latex]x[\/latex] decreases without bound, [latex]f\\left(x\\right)[\/latex] decreases without bound. As [latex]x[\/latex] increases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. In symbolic form we write<\/p>\n<div id=\"eip-77\" class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values move far to the left of the origin ( [latex]x\\to -\\infty[\/latex] ) and move far to the right of the origin ( [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><a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">Figure 4<\/a> 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.<\/p>\n<div id=\"Figure_03_03_004abcd\" class=\"wp-caption aligncenter\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152133\/CNX_Precalc_Figure_03_03_004abcd.jpg\" alt=\"Graph of an even-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to positive infinity. Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an even-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an odd-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity.\" width=\"731\" height=\"734\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135161436\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137415258\"><strong>Given a power function [latex]f\\left(x\\right)=k{x}^{n}[\/latex] where<\/strong> [latex]n[\/latex] <strong>is a non-negative integer, identify the end behavior.<\/strong><\/p>\n<ol id=\"fs-id1165137409522\" type=\"1\">\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 <a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">Figure 4<\/a>\u00a0to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137923491\">\n<div id=\"fs-id1165137599768\">\n<h3>Example 2:\u00a0 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>\n<div id=\"fs-id1165135169237\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135169237\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135169237\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137502379\">The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As [latex]x[\/latex] increases without bound, 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 [latex]x[\/latex] decreases without bound, 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 <a class=\"autogenerated-content\" href=\"#Figure_03_03_008\">Figure 5<\/a>.<\/p>\n<div id=\"Figure_03_03_008\" class=\"wp-caption aligncenter\">\n<div style=\"width: 358px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152136\/CNX_Precalc_Figure_03_03_008.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"348\" height=\"236\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137535914\">\n<div id=\"fs-id1165137811997\">\n<h3>Example 3:\u00a0 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>\n<div id=\"fs-id1165137722696\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137722696\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137722696\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137409513\">The exponent of the power function is 9 (an odd number). Because the coefficient is [latex]\u20131[\/latex] (negative), the graph is the reflection about the [latex]x\\text{-}[\/latex]axis of the graph of [latex]f\\left(x\\right)={x}^{9}.[\/latex] <a class=\"autogenerated-content\" href=\"#Figure_03_03_009\">Figure 6<\/a> shows that as [latex]x[\/latex] increases without bound, the output decreases without bound. As [latex]x[\/latex] decreases without bound, the output increases without bound. In symbolic form, we would write<\/p>\n<div id=\"eip-id1165134384400\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{as } x\\to -\\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{as } x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/div>\n<div id=\"Figure_03_03_009\" class=\"wp-caption aligncenter\">\n<div style=\"width: 270px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152139\/CNX_Precalc_Figure_03_03_009.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"260\" height=\"356\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/p>\n<\/div>\n<\/div>\n<h3>Analysis<\/h3>\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<caption>Table 2<\/caption>\n<thead>\n<tr>\n<th class=\"border\" style=\"text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"text-align: center;\">[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\" style=\"text-align: center;\">\u201310<\/td>\n<td class=\"border\" style=\"text-align: center;\">1,000,000,000<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\">\u20135<\/td>\n<td class=\"border\" style=\"text-align: center;\">1,953,125<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20131,953,125<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">\u20131,000,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137644426\">We can see from <a class=\"autogenerated-content\" href=\"#Table_03_03_03\">Table 2<\/a>\u00a0that, when we substitute more and more negative values for [latex]x,[\/latex] the output becomes very large, and when we substitute larger positive values for [latex]x,[\/latex] the output becomes more negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137626838\" class=\"precalculus tryit\">\n<h3>Try It #2<\/h3>\n<div id=\"ti_03_03_02\">\n<div id=\"fs-id1165137734867\">\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>\n<div id=\"fs-id1165137647550\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137647550\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137647550\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137647551\">As [latex]x[\/latex] increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \u00b1\\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134069294\" class=\"bc-section section\">\n<h3>Identifying Polynomial Functions<\/h3>\n<p>An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[\/latex] of the spill depends on the number of weeks [latex]w[\/latex] that have passed. This relationship is linear.<\/p>\n<div class=\"unnumbered\" style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area [latex]A[\/latex] of a circle.<\/p>\n<div id=\"eip-731\" class=\"unnumbered\" style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<div>[latex]\\\\[\/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=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}A\\left(w\\right)&=A\\left(r\\left(w\\right)\\right)\\\\ &=A\\left(24+8w\\right)\\\\ \\text{ }&=\\pi {\\left(24+8w\\right)}^{2}\\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137835475\">Multiplying gives the formula<\/p>\n<div id=\"eip-290\" class=\"unnumbered\" style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}.[\/latex]<\/div>\n<div>[latex]\\\\[\/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 terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/p>\n<div id=\"fs-id1165137715427\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137823247\">Let [latex]n[\/latex] be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<div id=\"fs-id1165131937978\" style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"eip-id1165137832690\">This is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex] is a <strong>coefficient<\/strong> and can be any real number, but [latex]{a}_{n}[\/latex] cannot equal [latex]0[\/latex]. Each product [latex]{a}_{i}{x}^{i}[\/latex] is a <strong>term of a polynomial function<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137817691\">\n<div id=\"fs-id1165137817693\">\n<h3>Example 4:\u00a0 Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n<div><\/div>\n<div id=\"eip-id1165134474011\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\end{array}\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt[\\leftroot{1}\\uproot{2} ]{x}+2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134221783\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134221783\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134221783\" 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}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex] where 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] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4.[\/latex]<\/li>\n<li>[latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x.[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)[\/latex] cannot 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<\/div>\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\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.\u00a0 For the constant function, the degree is zero, and for a non-constant linear function, the degree is one.<\/p>\n<div id=\"fs-id1165135193124\">\n<div class=\"textbox shaded\">\n<h3>Terminology of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<\/p>\n<p><span id=\"fs-id1165137406148\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152142\/CNX_Precalc_Figure_03_03_010n.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.\" \/><\/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>\n<div id=\"fs-id1165134031372\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137803898\"><strong>Given a polynomial function, identify the degree and leading coefficient.<\/strong><\/p>\n<ol id=\"fs-id1165135587816\" type=\"1\">\n<li>Find the highest power of [latex]x[\/latex] to determine the degree of the function.<\/li>\n<li>Identify the term containing the highest power of [latex]x[\/latex] to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137401820\">\n<div id=\"fs-id1165137862379\">\n<h3>Example 5:\u00a0 Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p>Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<div id=\"eip-id1165134242117\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}\\begin{array}{l}\\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }f\\left(x\\right)=3+2{x}^{2}-4{x}^{3}\\end{array}\\\\ \\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\ h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135527012\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135527012\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135527012\" class=\"hidden-answer\" style=\"display: none\">\n<p>For the function [latex]f\\left(x\\right),[\/latex] the highest power of [latex]x[\/latex] is 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, [latex]-4.[\/latex]<\/p>\n<p>For the function [latex]g\\left(t\\right),[\/latex] the highest power of [latex]t[\/latex] is [latex]5,[\/latex] so the degree is [latex]5.[\/latex] The leading term is the term containing that degree, [latex]5{t}^{5}.[\/latex] The leading coefficient is the coefficient of that term, [latex]5.[\/latex]<\/p>\n<p>For the function [latex]h\\left(p\\right),[\/latex] the highest power of [latex]p[\/latex] is [latex]3,[\/latex] so the degree is [latex]3.[\/latex] The leading term is the term containing that degree, [latex]-{p}^{3};[\/latex] the leading coefficient is the coefficient of that term, [latex]-1.[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\n<h3>Try It #3<\/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>\n<div id=\"fs-id1165135701674\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135701674\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135701674\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135701675\">The degree is 6. The leading term is [latex]-{x}^{6}.[\/latex] The leading coefficient is [latex]-1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135701674\">\n<div id=\"fs-id1165137702213\" class=\"bc-section section\">\n<h4>Identifying End Behavior of Polynomial Functions<\/h4>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as [latex]x[\/latex] gets more and more positive or negative, 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. See <a class=\"autogenerated-content\" href=\"#Table_03_03_04\">Table 3<\/a>.<\/p>\n<table id=\"Table_03_03_04\" summary=\"..\">\n<caption>Table 3<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th class=\"border\">Polynomial Function<\/th>\n<th class=\"border\">Leading Term<\/th>\n<th class=\"border\">Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x-4[\/latex]<\/td>\n<td class=\"border\">[latex]5{x}^{4}[\/latex]<\/td>\n<td class=\"border\"><span id=\"fs-id1165137768814\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152145\/CNX_Precalc_Figure_03_03_011.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"259\" height=\"275\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td class=\"border\">[latex]-2{x}^{6}[\/latex]<\/td>\n<td class=\"border\"><span id=\"fs-id1165137714206\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152148\/CNX_Precalc_Figure_03_03_012.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" width=\"259\" height=\"275\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td class=\"border\">[latex]3{x}^{5}[\/latex]<\/td>\n<td class=\"border\"><span id=\"fs-id1165137540879\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152151\/CNX_Precalc_Figure_03_03_013.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"259\" height=\"275\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td class=\"border\">[latex]-6{x}^{3}[\/latex]<\/td>\n<td class=\"border\"><span id=\"fs-id1165137600670\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152154\/CNX_Precalc_Figure_03_03_014.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" width=\"259\" height=\"275\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Example_03_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137452413\">\n<div id=\"fs-id1165137452415\">\n<h3>Example 6:\u00a0 Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137831279\">Describe the end behavior and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_015\">Figure 7<\/a>.<\/p>\n<div id=\"Figure_03_03_015\" class=\"wp-caption aligncenter\">\n<div style=\"width: 312px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152157\/CNX_Precalc_Figure_03_03_015.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"302\" height=\"275\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251309\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135251309\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135251309\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135251312\">As the input values [latex]x[\/latex] get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values [latex]x[\/latex] get more and more negative, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n<div id=\"eip-id1165137778911\" class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty.[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137454991\">In words, we read this notation, &#8220;as [latex]x[\/latex] values increase without bound, the function values increase without bound and as [latex]x[\/latex] values decrease without bound, the function values decrease without bound.&#8221;<\/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 id=\"fs-id1165137470875\" class=\"precalculus tryit\">\n<h3>Try It #4<\/h3>\n<div id=\"ti_03_03_04\">\n<div id=\"fs-id1165137732301\">\n<p id=\"fs-id1165135460938\">Describe the end behavior, and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_016\">Figure<\/a>.<\/p>\n<div id=\"Figure_03_03_016\" class=\"wp-caption aligncenter\">\n<div style=\"width: 314px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152200\/CNX_Precalc_Figure_03_03_016n.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"304\" height=\"275\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134047710\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134047710\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134047710\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134047711\">As the input increases without bound, the output decreases without bound.\u00a0 As the input decreases without bound, the output decreases without bound. As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty ;[\/latex] as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty .[\/latex] It has the shape of an even degree power function with a negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_07\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137470363\">\n<h3>Example 7:\u00a0 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>\n<div id=\"fs-id1165137401107\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137401107\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137401107\" 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<div id=\"eip-id1165132051075\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(x\\right)&=-3{x}^{2}\\left(x-1\\right)\\left(x+4\\right)\\\\ \\text{ }&=-3{x}^{2}\\left({x}^{2}+3x-4\\right)\\\\ \\text{ }&=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137634030\">The general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}.[\/latex] The leading term is [latex]-3{x}^{4};[\/latex] therefore, the degree of the polynomial is 4. The degree is even [latex]\\left(4\\right)[\/latex] and the leading coefficient is negative [latex]\\left(-3\\right),[\/latex] so the end behavior is<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<div id=\"eip-id1165133007607\" class=\"unnumbered\" style=\"text-align: center;\">\u00a0as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]<\/div>\n<div class=\"unnumbered\" style=\"text-align: center;\">as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"precalculus tryit\">\n<h3>Try It #5<\/h3>\n<div id=\"ti_03_03_05\">\n<div id=\"fs-id1165137722131\">\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>\n<div id=\"fs-id1165135409431\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135409431\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135409431\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137749856\">The function in general form is [latex]f\\left(x\\right)=0.2x^3-1.2x^2+0.6x+2.[\/latex] The leading term is [latex]0.2{x}^{3},[\/latex] so it is a degree 3 polynomial. As [latex]x[\/latex] increases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound; as [latex]x[\/latex] decreases without bound, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735781\" class=\"bc-section section\">\n<h4>Identifying Local Behavior of Polynomial Functions<\/h4>\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. In particular, we are interested in locations where graph behavior changes. A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing (<strong>local maximum<\/strong>) or decreasing to increasing (<strong>local minimum<\/strong>).<\/p>\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. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_017\">Figure 9<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_03_03_017\" class=\"wp-caption aligncenter\">\n<div style=\"width: 304px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152203\/CNX_Precalc_Figure_03_03_017.jpg\" alt=\"A cubic function with the interceps and turning points labeled\" width=\"294\" height=\"462\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p>A <strong>turning point<\/strong> of a graph is a point at which the graph changes direction from increasing to decreasing (<strong>local maximum<\/strong>) or decreasing to increasing (<strong>local minimum<\/strong>). The <strong><em>y-<\/em>intercept<\/strong> is the point at which the function has an input value of zero. The <strong><em>x<\/em>-intercepts<\/strong> are the points at which the output value is zero.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137766902\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137645233\"><strong>Given a polynomial function, determine the intercepts.<\/strong><\/p>\n<ol id=\"fs-id1165137571388\" type=\"1\">\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 [latex]x\\text{-}[\/latex]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=\"textbox examples\">\n<div id=\"fs-id1165137435581\">\n<div id=\"fs-id1165137803210\">\n<h3>Example 8:\u00a0 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 [latex]y\\text{-}[\/latex]and [latex]x\\text{-}[\/latex]intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165135251466\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135251466\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135251466\" 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 [latex]x.[\/latex]<\/p>\n<div id=\"eip-id1165133032876\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(0\\right)&=\\left(0-2\\right)\\left(0+1\\right)\\left(0-4\\right)\\hfill \\\\ \\text{ }&=\\left(-2\\right)\\left(1\\right)\\left(-4\\right)\\hfill \\\\ \\text{ }&=8\\hfill \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\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. To solve [latex]0=\\left(x-2\\right)\\left(x+1\\right)\\left(x-4\\right)[\/latex] set each factor equal to zero and simplify.<\/p>\n<div id=\"eip-id1165134380311\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*} x-2&=0&\\text{ }\\text{ }\\text{ } \\text{ or }\\text{ }\\text{ }\\text{ }& x+1=0&\\text{ }\\text{ }\\text{ }\\text{ or }\\text{ }\\text{ }\\text{ } &x-4=0\\\\ \\text{ }x&=2&\\text{ }&x=-1& \\text{ }&x=4 \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135316178\">The [latex]x\\text{-}[\/latex]intercepts are [latex]\\left(2,0\\right),\\left(\u20131,0\\right),[\/latex] and [latex]\\left(4,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165134380385\">We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_018\">Figure 10<\/a>.<\/p>\n<div id=\"Figure_03_03_018\" class=\"wp-caption aligncenter\">\n<div style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152206\/CNX_Precalc_Figure_03_03_018.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"298\" height=\"386\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_09\" class=\"textbox examples\">\n<div id=\"fs-id1165137834894\">\n<div id=\"fs-id1165137834896\">\n<h3>Example 9:\u00a0 Determining the Intercepts of a Polynomial Function with Factoring<\/h3>\n<p>Given the polynomial function [latex]f\\left(x\\right)={x}^{4}-4{x}^{2}-45,[\/latex] determine the [latex]y\\text{-}[\/latex]and [latex]x\\text{-}[\/latex]intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165137634473\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137634473\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137634473\" class=\"hidden-answer\" style=\"display: none\">The <em>y-<\/em>intercept occurs when the input is zero.<\/p>\n<div id=\"eip-id1165132943488\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*} \\\\ f\\left(0\\right)&={\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45 \\\\ \\text{ }&=-45\\hfill \\end{align*}[\/latex]<\/div>\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<div id=\"eip-id1165135376171\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(x\\right)&={x}^{4}-4{x}^{2}-45\\hfill \\\\\\text{ }&=\\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right)\\hfill \\\\ \\text{ }&=\\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\hfill \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div id=\"eip-id1165135684199\" class=\"unnumbered\" style=\"text-align: center;\">[latex]0=\\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div id=\"eip-id1165134383791\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}x-3=0\\hfill & \\text{or}\\hfill & x+3=0\\hfill & \\text{or}\\hfill & {x}^{2}+5=0\\hfill \\\\ \\text{ }x=3\\hfill & \\text{or}\\hfill & \\text{ }x=-3\\hfill & \\text{or}\\hfill & \\text{(no real solution)}\\hfill \\end{array}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135436471\">The <em>x<\/em>-intercepts are [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(\u20133,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137444727\">We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_019\">Figure 11<\/a>. We can see that the function is even because [latex]f\\left(x\\right)=f\\left(-x\\right).[\/latex]<\/p>\n<div id=\"Figure_03_03_019\" class=\"wp-caption aligncenter\">\n<div style=\"width: 320px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152209\/CNX_Precalc_Figure_03_03_019.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=\"310\" height=\"271\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749604\" class=\"precalculus tryit\">\n<h3>Try It #6<\/h3>\n<div id=\"ti_03_03_06\">\n<div id=\"fs-id1165137405243\">\n<p id=\"fs-id1165137405244\">Given the polynomial function [latex]f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x,[\/latex] determine the [latex]y\\text{-}[\/latex]and [latex]x\\text{-}[\/latex]intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165137762370\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137762370\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137762370\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137762371\"><em>y<\/em>-intercept [latex]\\left(0,0\\right);[\/latex] <em>x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(\u20132,0\\right),[\/latex] and [latex]\\left(5,0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134080932\" class=\"bc-section section\">\n<h4>Comparing Smooth and Continuous Graphs<\/h4>\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of [latex]x\\text{-}[\/latex]intercepts and the number of turning points. A polynomial function of [latex]n\\text{th}[\/latex] degree is the product of at most [latex]n[\/latex] factors, so it will have at most [latex]n[\/latex] roots or zeros, or [latex]x\\text{-}[\/latex]intercepts. The graph of the polynomial function of degree [latex]n[\/latex] must have at most [latex]n\u20131[\/latex] turning points. This means the graph has at most one fewer turning points than the degree of the polynomial.<\/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\">\n<div class=\"textbox shaded\">\n<h3>Intercepts and Turning Points of Polynomials<\/h3>\n<p>A polynomial of degree [latex]n[\/latex] will have, at most, [latex]n[\/latex] <em>x<\/em>-intercepts and [latex]n-1[\/latex] turning points.<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165135237034\">\n<div id=\"fs-id1165135237036\">\n<h3>Example 10:\u00a0 Determining the Maximum Possible Number of Intercepts and Turning Points of a Polynomial<\/h3>\n<p id=\"fs-id1165134152759\">Without graphing the function, determine the maximum number of possible [latex]x\\text{-}[\/latex]intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135414339\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135414339\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135414339\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135414341\">The polynomial has a degree of [latex]10,[\/latex] so there are at most [latex]10[\/latex] [latex]x[\/latex]-intercepts and at most [latex]\\mathrm{10}-1=9[\/latex] turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628834\" class=\"precalculus tryit\">\n<h3>Try It #7<\/h3>\n<div id=\"ti_03_03_07\">\n<div id=\"fs-id1165135188273\">\n<p>Without graphing the function, determine the maximum number of possible [latex]x\\text{-}[\/latex]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>\n<div id=\"fs-id1165137660801\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137660801\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137660801\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137660802\">There are at most 12 [latex]x\\text{-}[\/latex]intercepts and at most 11 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_11\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137435066\">\n<h3>Example 11:\u00a0 Drawing Conclusions about a Polynomial Function from the Graph<\/h3>\n<p id=\"fs-id1165137843783\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_020\">Figure 12<\/a>\u00a0based on its intercepts and turning points?<\/p>\n<div id=\"Figure_03_03_020\" class=\"wp-caption aligncenter\">\n<div style=\"width: 340px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152213\/CNX_Precalc_Figure_03_03_020.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"330\" height=\"249\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737264\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137737264\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137737264\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165131926327\">The end behavior of the graph tells us this is the graph of an even-degree polynomial. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_021\">Figure 13<\/a>.<\/p>\n<div id=\"Figure_03_03_021\" class=\"wp-caption aligncenter\">\n<div style=\"width: 342px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152216\/CNX_Precalc_Figure_03_03_021.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"332\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135670389\">The graph has 2 [latex]x\\text{-}[\/latex]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 id=\"fs-id1165137871106\" class=\"precalculus tryit\">\n<h3>Try It #8<\/h3>\n<div id=\"ti_03_03_08\">\n<div id=\"fs-id1165137834183\">\n<p id=\"fs-id1165137454180\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_022\">Figure 14<\/a>\u00a0based on its intercepts and turning points?<\/p>\n<div id=\"Figure_03_03_022\" class=\"wp-caption aligncenter\">\n<div style=\"width: 336px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07152219\/CNX_Precalc_Figure_03_03_022.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"326\" height=\"296\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137666790\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137666790\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137666790\" class=\"hidden-answer\" style=\"display: none\">The end behavior indicates an odd-degree polynomial function; there are 3 [latex]x\\text{-}[\/latex]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.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135184013\">\n<div id=\"fs-id1165137725458\">\n<h3>Example 12:\u00a0 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] determine the x and y intercepts and the maximum number of turning points possible.<\/p>\n<\/div>\n<div id=\"fs-id1165135457721\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135457721\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135457721\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135457723\">The [latex]y\\text{-}[\/latex]intercept is found by evaluating [latex]f\\left(0\\right).[\/latex]<\/p>\n<div id=\"eip-id1165134587897\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(0\\right)&=-4\\left(0\\right)\\left(0+3\\right)\\left(0-4\\right)\\hfill \\\\ \\text{ }&=0\\hfill \\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135245749\">The [latex]y\\text{-}[\/latex]intercept is [latex]\\left(0,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135203755\">The [latex]x\\text{-}[\/latex]intercepts are found by determining the zeros of the function.<\/p>\n<p style=\"text-align: center;\">[latex]0=-4x\\left(x+3\\right)\\left(x-4\\right)[\/latex]<\/p>\n<div id=\"eip-id1165135401630\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{l} \\begin{array}{lllllllll}x=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x+3=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x-4=0\\hfill \\\\ x=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{ }x=-3\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{ }\\text{ }x=4\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135431016\">The [latex]x\\text{-}[\/latex]intercepts are [latex]\\left(0,0\\right),\\left(\u20133,0\\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 id=\"fs-id1165137661075\" class=\"precalculus tryit\">\n<h3>Try It #9<\/h3>\n<div id=\"ti_03_03_09\">\n<div id=\"fs-id1165137575430\">\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 x and y intercepts and the maximum number of turning points possible.<\/p>\n<\/div>\n<div id=\"fs-id1165137833005\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137833005\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137833005\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137833006\">The [latex]x\\text{-}[\/latex]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<\/div>\n<\/div>\n<div id=\"fs-id1165137653058\" class=\"precalculus media\">\n<div class=\"textbox shaded\">\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\n<div id=\"fs-id1165134080932\" class=\"bc-section section\">\n<div id=\"fs-id1165137653058\" class=\"precalculus media\">\n<h3>Media:<\/h3>\n<p id=\"fs-id1165135456729\">Access these online resources for additional instruction and practice with power and polynomial functions.<\/p>\n<ul id=\"fs-id1165137410802\">\n<li><a href=\"http:\/\/openstax.org\/l\/keyinfopoly\">Find Key Information about a Given Polynomial Function<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/endbehavior\">End Behavior of a Polynomial Function<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/turningpoints\">Turning Points and [latex]x\\text{-}[\/latex]intercepts of Polynomial Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/leastposdegree\">Least Possible Degree of a Polynomial Functio<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">Key Equations<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"key-equations\">\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\" style=\"width: 200.234px;\">general form of a polynomial function<\/td>\n<td class=\"border\" style=\"width: 504.766px;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137731646\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135438864\">\n<li>A power function is a coefficient multiplied by a variable base raised to a number power.<\/li>\n<li>The behavior of a graph as the input decreases without bound and increases without bound is called the end behavior.<\/li>\n<li>The end behavior depends on whether the power is even or odd.<\/li>\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\n<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\n<li>A polynomial of degree [latex]n[\/latex] will have at most [latex]n[\/latex] <em>x-<\/em>intercepts and at most [latex]n-1[\/latex] turning points.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137668266\">\n<dt>coefficient<\/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\">\n<dt>continuous function<\/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\">\n<dt>degree<\/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\">\n<dt>end behavior<\/dt>\n<dd>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\">\n<dt>leading coefficient<\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\">\n<dt>leading term<\/dt>\n<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\">\n<dt>polynomial function<\/dt>\n<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\">\n<dt>power function<\/dt>\n<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=k{x}^{p}[\/latex] where [latex]k[\/latex] is a constant, the base is a variable, and the exponent, [latex]p,[\/latex] is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137833929\">\n<dt>smooth curve<\/dt>\n<dd>a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\">\n<dt>term of a polynomial function<\/dt>\n<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex] of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{1}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\">\n<dt>turning point<\/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-1277\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Power Functions and Polynomial Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:wMXABFlZ@14\/Power-Functions-and-Polynomial-Functions\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:wMXABFlZ@14\/Power-Functions-and-Polynomial-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <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":311,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Power Functions and Polynomial 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1\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1277","chapter","type-chapter","status-publish","hentry"],"part":1198,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1277","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":22,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1277\/revisions"}],"predecessor-version":[{"id":3283,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1277\/revisions\/3283"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/1198"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1277\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=1277"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1277"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=1277"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=1277"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}