{"id":4983,"date":"2020-11-11T20:22:19","date_gmt":"2020-11-11T20:22:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebra\/?post_type=chapter&#038;p=4983"},"modified":"2024-02-02T19:21:49","modified_gmt":"2024-02-02T19:21:49","slug":"characteristics-of-power-and-polynomial-functions-custom-edit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/chapter\/characteristics-of-power-and-polynomial-functions-custom-edit\/","title":{"raw":"Characteristics of Power and Polynomial Functions","rendered":"Characteristics of Power and Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li class=\"li2\"><span class=\"s1\">Identify power functions.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Describe end behavior of power functions given its equation or graph.<\/span><\/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>Describe the end behavior of a polynomial function.<\/li>\r\n \t<li>Identify turning points of a polynomial function from its graph.<\/li>\r\n \t<li>Identify the number of turning points and intercepts of a polynomial function from its degree.<\/li>\r\n \t<li>Determine x and y-intercepts of a polynomial function given its equation in factored form.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this section we will examine functions that are\u00a0used to estimate and predict\u00a0things like changes in animal and bird populations or fluctuations in financial markets.\r\n\r\nWe will also continue to learn how to\u00a0analyze the behavior of functions by looking at their graphs. We will introduce and describe a new term called end behavior and show which parts of the function equation determine end behavior. We will also identify intercepts of polynomial functions.\r\n<h2>End Behavior of Power Functions<\/h2>\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194447\/CNX_Precalc_Figure_03_03_0012.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/> Three birds on a cliff with the sun rising in the background. Functions discussed in this module can be used to model populations of various animals, including birds. (credit: Jason Bay, Flickr)[\/caption]\r\n\r\nSuppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<td>2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Bird Population<\/strong><\/td>\r\n<td>800<\/td>\r\n<td>897<\/td>\r\n<td>992<\/td>\r\n<td>1,083<\/td>\r\n<td>1,169<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe population can be estimated using the function [latex]P\\left(t\\right)=-0.3{t}^{3}+97t+800[\/latex], where [latex]P\\left(t\\right)[\/latex] represents the bird population on the island <i>t<\/i>\u00a0years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island.\r\n<h3>Identifying Power Functions<\/h3>\r\nIn 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,\u00a0<strong>coefficient,<\/strong> and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a <strong>coefficient<\/strong>.\r\n\r\nAs an example, consider functions for area or volume. The function for the <strong>area of a circle<\/strong> with radius [latex]r[\/latex]<i> <\/i>is:\r\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\r\nand the function for the <strong>volume of a sphere<\/strong> with radius <em>r<\/em>\u00a0is:\r\n<p style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/p>\r\nBoth of these are examples of power functions because they consist of a coefficient, [latex]\\pi [\/latex] or [latex]\\frac{4}{3}\\pi [\/latex], multiplied by a variable <em>r<\/em>\u00a0raised to a power.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Power FunctionS<\/h3>\r\nA <strong>power function<\/strong> is a function that can be represented in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{n}[\/latex]<\/p>\r\nwhere <i>a<\/i>\u00a0and <i>n<\/i>\u00a0are real numbers and <em>a<\/em><i>\u00a0<\/i>is known as the <strong>coefficient<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong>\r\n\r\n<em>No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Power Functions<\/h3>\r\nWhich of the following functions are power functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=1\\hfill &amp; \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill &amp; \\text{Identity function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill &amp; \\text{Quadratic}\\text{ }\\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{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[reveal-answer q=\"82786\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"82786\"]\r\n\r\nAll of the listed functions are power functions.\r\n\r\nThe 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.\r\n\r\nThe 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].\r\n\r\nThe <strong>reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].\r\n\r\nThe square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWhich functions are power functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"105254\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"105254\"]\r\n\r\n[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Identifying End Behavior of Power Functions<\/h3>\r\nThe graph below shows the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], [latex]h\\left(x\\right)={x}^{6}[\/latex], [latex]k(x)=x^{8}[\/latex], and [latex]p(x)=x^{10}[\/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. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.\r\n\r\n<img class=\"alignnone size-medium wp-image-4764\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/07034858\/CNX_Precalc_Figure_03_03_002-300x156.jpg\" alt=\"power functions with even powers. \" width=\"300\" height=\"156\" \/>\r\n\r\nTo describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\\infty[\/latex] for positive infinity and [latex]-\\infty[\/latex] for negative infinity. When we say that \"<em>x<\/em> approaches infinity,\" which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that <em>x<\/em>\u00a0is increasing without bound.\r\n\r\nWith even-powered power functions, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as [latex]x[\/latex] approaches positive or negative infinity, the [latex]f\\left(x\\right)[\/latex] values increase without bound. In symbolic form, we could write\r\n<p style=\"text-align: center;\">[latex]\\text{as }x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/p>\r\nThe graph below shows [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},h\\left(x\\right)={x}^{7},k\\left(x\\right)={x}^{9},\\text{and }p\\left(x\\right)={x}^{11}[\/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function. As the power increases, the graphs flatten near the origin and become steeper away from the origin.\r\n\r\n<img class=\"alignnone size-medium wp-image-4985\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5479\/2020\/11\/13201740\/desmos-graph-odd-poly-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/>\r\n\r\nThese examples illustrate that functions of the form [latex]f\\left(x\\right)={x}^{n}[\/latex] reveal symmetry of one kind or another. First, in the even-powered power functions, we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the <em>y<\/em>-axis. In the odd-powered power functions, we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.\r\n\r\nFor these odd power functions, as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex]\u00a0decreases without bound. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex]\u00a0increases without bound. In symbolic form we write\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\r\nThe behavior of the graph of a function as the input values get very small ( [latex]x\\to -\\infty[\/latex] ) and get very large ( [latex]x\\to \\infty[\/latex] ) is referred to as the <strong>end behavior<\/strong> of the function. We can use words or symbols to describe end behavior.\r\n\r\nThe table\u00a0below shows the end behavior of power functions of the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"width: 135px;\"><\/th>\r\n<th style=\"text-align: center; width: 353px;\">Even Power<\/th>\r\n<th style=\"text-align: center; width: 336px;\">Odd Power<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 135px;\"><strong>Positive Constant<\/strong>\r\n\r\n<strong><i>a<\/i> &gt; 0<\/strong><\/td>\r\n<td style=\"width: 353px;\"><img class=\"alignnone size-full wp-image-4485\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231809\/Table1.png\" 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.\" width=\"356\" height=\"460\" \/><\/td>\r\n<td style=\"width: 336px;\"><img class=\"alignnone size-full wp-image-4487\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231957\/Table2.png\" alt=\"Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to positive infinity.\" width=\"359\" height=\"458\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 135px;\"><strong>Negative Constant<\/strong>\r\n\r\n<strong><i>a<\/i> &lt; 0<\/strong><\/td>\r\n<td style=\"width: 353px;\"><img class=\"alignnone size-full wp-image-4488\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232026\/Table3.png\" alt=\"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.\" width=\"375\" height=\"460\" \/><\/td>\r\n<td style=\"width: 336px;\"><img class=\"alignnone size-full wp-image-4489\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232106\/Table4.png\" alt=\"Graph of an odd-powered function with a negative 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.\" width=\"342\" height=\"464\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a power function [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex]\u00a0is a non-negative integer, identify the end behavior.<\/h3>\r\n<ol id=\"fs-id1165137409522\">\r\n \t<li>Determine whether the power is even or odd.<\/li>\r\n \t<li>Determine whether the constant is positive or negative.<\/li>\r\n \t<li>Use the above graphs to identify the end behavior.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\r\nDescribe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].\r\n\r\n[reveal-answer q=\"556064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556064\"]\r\n\r\nThe coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As <em>x\u00a0<\/em>(input)\u00a0approaches infinity, [latex]f\\left(x\\right)[\/latex] (output) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty [\/latex]. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex]. We can graphically represent the function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194503\/CNX_Precalc_Figure_03_03_0082.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\r\nDescribe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].\r\n\r\n[reveal-answer q=\"631242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"631242\"]\r\n\r\nThe exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex] and as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194505\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].\r\n\r\n[reveal-answer q=\"582534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"582534\"]\r\n\r\nAs <em>x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=69337&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15940&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>End Behavior of Polynomial Functions<\/h2>\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 <em>r<\/em>\u00a0of the spill depends on the number of weeks <em>w<\/em>\u00a0that have passed. This relationship is linear.\r\n<p style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/p>\r\nWe can combine this with the formula for the area <em>A<\/em>\u00a0of a circle.\r\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\r\nComposing these functions gives a formula for the area in terms of weeks.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ A\\left(w\\right)=A\\left(24+8w\\right)\\\\ A\\left(w\\right)=\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\r\nMultiplying gives the formula below.\r\n<p style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/p>\r\nThis formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Polynomial Functions<\/h3>\r\nLet <em>n<\/em>\u00a0be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\nThis is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Polynomial Functions<\/h3>\r\nWhich of the following are polynomial functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"906312\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"906312\"]\r\n\r\nThe first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.\r\n<ul>\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 class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=48358&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Defining the Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h3>\r\nBecause 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 based on the power on the variable. This is called writing a polynomial in general or standard 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 variable with the highest power, also called the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Terminology of Polynomial Functions<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194507\/CNX_Precalc_Figure_03_03_010n2.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"487\" height=\"147\" \/>\r\n\r\nWe often rearrange polynomials so that the powers on the variable are descending.\r\n\r\nWhen a polynomial is written in this way, we say that it is in general form.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial function, identify the degree and leading coefficient<\/h3>\r\n<ol>\r\n \t<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree of the function.<\/li>\r\n \t<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\r\n \t<li>The leading coefficient is the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: 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<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"632394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632394\"]\r\n\r\nFor the function [latex]f\\left(x\\right)[\/latex], the highest power of <em>x<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].\r\n\r\nFor the function [latex]g\\left(t\\right)[\/latex], the highest power of <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.\r\n\r\nFor the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, [latex]\u20131[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIdentify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].\r\n\r\n[reveal-answer q=\"435637\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"435637\"]\r\n\r\nThe degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.\r\n\r\nhttps:\/\/youtu.be\/F_G_w82s0QA\r\n<h3>Identifying End Behavior of Polynomial Functions<\/h3>\r\nKnowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.\r\n<table id=\"Table_03_03_04\" style=\"height: 1527px;\" summary=\"..\"><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"text-align: center; height: 15px; width: 354px;\">Polynomial Function<\/th>\r\n<th style=\"text-align: center; height: 15px; width: 106px;\">Leading Term<\/th>\r\n<th style=\"text-align: center; height: 15px; width: 364px;\">Graph of Polynomial Function<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 378px;\">\r\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]5{x}^{4}[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 364px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194510\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 378px;\">\r\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]-2{x}^{6}[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 364px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194512\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 378px;\">\r\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]3{x}^{5}[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 364px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194514\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 378px;\">\r\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]-6{x}^{3}[\/latex]<\/td>\r\n<td style=\"height: 378px; width: 364px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194516\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\nDescribe the end behavior and determine a possible degree of the polynomial function in the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194520\/CNX_Precalc_Figure_03_03_0152.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/>\r\n\r\n[reveal-answer q=\"626899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"626899\"]\r\n\r\nAs the input values <em>x<\/em>\u00a0get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values <em>x<\/em>\u00a0get very small, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\r\nIn words, we could say that as <em>x<\/em>\u00a0values approach infinity, the function values approach infinity, and as <em>x<\/em>\u00a0values approach negative infinity, the function values approach negative infinity.\r\n\r\nWe 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.\r\n\r\nhttps:\/\/youtu.be\/y78Dpr9LLN0\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe the end behavior of the polynomial function in the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/>\r\n[reveal-answer q=\"304329\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304329\"]\r\n\r\nAs [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty [\/latex]. It has the shape of an even degree power function with a negative coefficient.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\r\nGiven 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.\r\n\r\n[reveal-answer q=\"76137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"76137\"]\r\n\r\nObtain the general form by expanding the given expression [latex]f\\left(x\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ f\\left(x\\right)=-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{array}[\/latex]<\/p>\r\nThe general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[\/latex].\u00a0The leading term is [latex]-3{x}^{4}[\/latex];\u00a0therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven 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.\r\n\r\n[reveal-answer q=\"657153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"657153\"]\r\n\r\nThe leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Local Behavior of Polynomial Functions<\/h2>\r\n<h3>Identifying Local Behavior of Polynomial Functions<\/h3>\r\nIn addition to the end behavior of polynomial functions, we are also interested in what happens in the \"middle\" of the function. In particular, we are interested in locations where graph behavior changes. A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing or decreasing to increasing.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194524\/CNX_Precalc_Figure_03_03_0172.jpg\" width=\"731\" height=\"629\" \/>\r\n\r\nWe 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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Intercepts and Turning Points of Polynomial Functions<\/h3>\r\n<ul>\r\n \t<li>A <strong>turning point<\/strong> of a graph is a point where the graph changes from increasing to decreasing or decreasing to increasing.<\/li>\r\n \t<li>The <em>y-<\/em>intercept is the point where the function has an input value of zero.<\/li>\r\n \t<li>The <em>x<\/em>-intercepts are the points where the output value is zero.<\/li>\r\n \t<li>A polynomial of degree <em>n<\/em>\u00a0will have, at most, <em>n<\/em>\u00a0<em>x<\/em>-intercepts and <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h3>\r\nA <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.\r\n\r\nThe degree of a polynomial function helps us to determine the number of <em>x<\/em>-intercepts and the number of turning points. A polynomial function of\u00a0<em>n<\/em>th degree is the product of <em>n<\/em>\u00a0factors, so it will have at most <em>n<\/em>\u00a0roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree <em>n<\/em>\u00a0must have at most <em>n<\/em> \u2013 1\u00a0turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining the Number of Intercepts and Turning Points of a Polynomial<\/h3>\r\nWithout graphing the function, determine the local behavior of the function by finding the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].\r\n\r\n[reveal-answer q=\"96529\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96529\"]\r\n\r\nThe polynomial has a degree of 10, so there are at most <i>10<\/i>\u00a0<em>x<\/em>-intercepts and at most [latex]10 \u2013 1 = 9[\/latex] turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given its degree.\r\n\r\n[embed]https:\/\/youtu.be\/9WW0EetLD4Q[\/embed]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWithout graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]\r\n\r\n[reveal-answer q=\"304362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304362\"]\r\n\r\nThere are at most 12 <em>x<\/em>-intercepts and at most 11 turning points.[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=123739&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial function, determine the intercepts<\/h3>\r\n<ol>\r\n \t<li>Determine the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and finding the corresponding output value.<\/li>\r\n \t<li>Determine the <em>x<\/em>-intercepts by setting the function equal to zero and solving for the input values.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h3>Using the Principle of Zero Products to Find the Roots of a Polynomial in Factored Form<\/h3>\r\nThe Principle of Zero Products states that if the product of n\u00a0numbers is 0, then at least one of the factors is 0. If [latex]ab=0[\/latex], then either [latex]a=0[\/latex] or [latex]b=0[\/latex], or both a and b are 0. We will use this idea to find the zeros of a polynomial that is either in factored form or can be written in factored form. For example, the polynomial\r\n<p style=\"text-align: center;\">[latex]P(x)=(x-4)^2(x+1)(x-7)[\/latex]<\/p>\r\nis in factored form. In the following examples, we will show the process of factoring a polynomial and calculating its x and y-intercepts.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining the Intercepts of a Polynomial Function<\/h3>\r\nGiven 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 <i>y<\/i>\u00a0and <i>x<\/i>-intercepts.\r\n\r\n[reveal-answer q=\"701514\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"701514\"]\r\n\r\nThe <em>y-<\/em>intercept occurs when the input is zero, so substitute 0 for <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(0\\right)=\\left(0 - 2\\right)\\left(0+1\\right)\\left(0 - 4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=\\left(-2\\right)\\left(1\\right)\\left(-4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=8\\hfill \\end{array}[\/latex]<\/p>\r\nThe <em>y-<\/em>intercept is (0, 8).\r\n\r\nThe <em>x<\/em>-intercepts occur when the output [latex]f(x)[\/latex] is zero.\r\n<p style=\"text-align: center;\">[latex]0=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllllllll}x - 2=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x+1=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x - 4=0\\hfill \\\\ \\text{}x=2\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{ }x=-1\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x=4 \\end{array}[\/latex]<\/p>\r\nThe\u00a0<i>x<\/i>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].\r\n\r\nWe can see these intercepts on the graph of the function shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194527\/CNX_Precalc_Figure_03_03_0182.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"487\" height=\"630\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining the Intercepts of a Polynomial Function BY Factoring<\/h3>\r\nGiven the polynomial function [latex]f\\left(x\\right)={x}^{4}-4{x}^{2}-45[\/latex], determine the <em>y<\/em>\u00a0and\u00a0<em>x<\/em>-intercepts.\r\n\r\n[reveal-answer q=\"492513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"492513\"]\r\n\r\nThe <em>y-<\/em>intercept occurs when the input is zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\\\ f\\left(0\\right)={\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45\\hfill \\hfill \\\\ \\text{}f\\left(0\\right)=-45\\hfill \\end{array}[\/latex]<\/p>\r\nThe <em>y-<\/em>intercept is [latex]\\left(0,-45\\right)[\/latex].\r\n\r\nThe <em>x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(x\\right)={x}^{4}-4{x}^{2}-45\\hfill \\\\ f\\left(x\\right)=\\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right)\\hfill \\\\ f\\left(x\\right)=\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\hfill \\end{array}[\/latex]<\/p>\r\nThen set the polynomial function equal to 0.\r\n<p style=\"text-align: center;\">[latex]0=\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllll}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]<\/p>\r\nThe <em>x<\/em>-intercepts are [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex].\r\n\r\nWe can see these intercepts on the graph of the function shown\u00a0below. We can see that the function has y-axis symmetry or is even because [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194529\/CNX_Precalc_Figure_03_03_0192.jpg\" alt=\"Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).\" width=\"487\" height=\"426\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the polynomial function [latex]f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x[\/latex], determine the <em>y\u00a0<\/em>and<em> x-<\/em>intercepts.\r\n\r\n[reveal-answer q=\"743200\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"743200\"]\r\n\r\n<em>y<\/em>-intercept [latex]\\left(0,0\\right)[\/latex]; <em>x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(-2,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>The Whole Picture<\/h3>\r\nNow we can bring the two concepts of turning points and intercepts together to get a general picture of the behavior of polynomial functions. \u00a0These types of analyses on polynomials developed before the advent of mass computing as a way to quickly understand the general behavior of a polynomial function. We now have a quick way, with computers, to graph and calculate important characteristics of polynomials that once took a lot of algebra.\r\n\r\nIn the first example, we will determine the least degree of a polynomial based on the number of turning points and intercepts.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Drawing Conclusions about a Polynomial Function from Its Graph<\/h3>\r\nGiven the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194531\/CNX_Precalc_Figure_03_03_0202.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/>\r\n\r\n[reveal-answer q=\"200904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200904\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194532\/CNX_Precalc_Figure_03_03_0212.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/>\r\n\r\nThe end behavior of the graph tells us this is the graph of an even-degree polynomial.\r\n\r\nThe graph has 2 <em>x<\/em>-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try to determine the least possible degree of a polynomial given its graph.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194534\/CNX_Precalc_Figure_03_03_0224.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/>\r\n[reveal-answer q=\"492375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"492375\"]\r\n\r\nThe end behavior indicates an odd-degree polynomial function; there are 3 <em>x<\/em>-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the leading coefficient must be negative.[\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15937&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nNow we will show that you can also determine the least possible degree and number of turning points\u00a0of a polynomial function given in factored form.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Drawing Conclusions about a Polynomial Function from ITS Factors<\/h3>\r\nGiven the function [latex]f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex],\u00a0determine the local behavior.\r\n\r\n[reveal-answer q=\"978752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"978752\"]\r\n\r\nThe <em>y<\/em>-intercept is found by evaluating [latex]f\\left(0\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(0\\right)=-4\\left(0\\right)\\left(0+3\\right)\\left(0 - 4\\right)\\hfill \\hfill \\\\ \\text{}f\\left(0\\right)=0\\hfill \\end{array}[\/latex]<\/p>\r\nThe <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].\r\n\r\nThe <em>x<\/em>-intercepts are found by setting the function equal to 0.\r\n<p style=\"text-align: center;\">[latex]0=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllllllllll}-4x=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x+3=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x - 4=0\\hfill \\\\ x=0\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{}x=-3\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; \\text{}x=4\\end{array}[\/latex]<\/p>\r\nThe <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(-3,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].\r\n\r\nThe degree is 3 so the graph has at most 2 turning points.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow it is your turn to determine the local behavior of a polynomial given in factored form.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], determine the local behavior.\r\n\r\n[reveal-answer q=\"104366\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"104366\"]\r\n\r\nThe <em>x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the <em>y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.[\/hidden-answer]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165137724050\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165134063974\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>general form of a polynomial function<\/td>\r\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137731646\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135438864\">\r\n \t<li>A power function is a variable base raised to a number power.<\/li>\r\n \t<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\r\n \t<li>The end behavior depends on whether the power is even or odd.<\/li>\r\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with non-negative integer powers.<\/li>\r\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\r\n \t<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\r\n \t<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em>x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\r\n<\/ul>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137668266\" class=\"definition\">\r\n \t<dt><strong>coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135194918\" class=\"definition\">\r\n \t<dt><strong>continuous function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832108\" class=\"definition\">\r\n \t<dt><strong>degree<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137832115\" class=\"definition\">\r\n \t<dt><strong>end behavior<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>leading coefficient<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>leading term<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943528\" class=\"definition\">\r\n \t<dt><strong>polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134297646\" class=\"definition\">\r\n \t<dt><strong>power function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex]\u00a0where <em>a\u00a0<\/em>is a constant, the base is a variable, and the exponent is\u00a0<i>n<\/i>,\u00a0is a smooth curve represented by a graph with no sharp corners<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137644987\" class=\"definition\">\r\n \t<dt><strong>term of a polynomial function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133085661\" class=\"definition\">\r\n \t<dt><strong>turning point<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165133085665\">the location where the graph of a function changes direction<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li class=\"li2\"><span class=\"s1\">Identify power functions.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Describe end behavior of power functions given its equation or graph.<\/span><\/li>\n<li>Identify polynomial functions.<\/li>\n<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<li>Describe the end behavior of a polynomial function.<\/li>\n<li>Identify turning points of a polynomial function from its graph.<\/li>\n<li>Identify the number of turning points and intercepts of a polynomial function from its degree.<\/li>\n<li>Determine x and y-intercepts of a polynomial function given its equation in factored form.<\/li>\n<\/ul>\n<\/div>\n<p>In this section we will examine functions that are\u00a0used to estimate and predict\u00a0things like changes in animal and bird populations or fluctuations in financial markets.<\/p>\n<p>We will also continue to learn how to\u00a0analyze the behavior of functions by looking at their graphs. We will introduce and describe a new term called end behavior and show which parts of the function equation determine end behavior. We will also identify intercepts of polynomial functions.<\/p>\n<h2>End Behavior of Power Functions<\/h2>\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\/896\/2016\/11\/02194447\/CNX_Precalc_Figure_03_03_0012.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\">Three birds on a cliff with the sun rising in the background. Functions discussed in this module can be used to model populations of various animals, including birds. (credit: Jason Bay, Flickr)<\/p>\n<\/div>\n<p>Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr>\n<tr>\n<td><strong>Bird Population<\/strong><\/td>\n<td>800<\/td>\n<td>897<\/td>\n<td>992<\/td>\n<td>1,083<\/td>\n<td>1,169<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The population can be estimated using the function [latex]P\\left(t\\right)=-0.3{t}^{3}+97t+800[\/latex], where [latex]P\\left(t\\right)[\/latex] represents the bird population on the island <i>t<\/i>\u00a0years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island.<\/p>\n<h3>Identifying Power Functions<\/h3>\n<p>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,\u00a0<strong>coefficient,<\/strong> and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a <strong>coefficient<\/strong>.<\/p>\n<p>As an example, consider functions for area or volume. The function for the <strong>area of a circle<\/strong> with radius [latex]r[\/latex]<i> <\/i>is:<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\n<p>and the function for the <strong>volume of a sphere<\/strong> with radius <em>r<\/em>\u00a0is:<\/p>\n<p style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/p>\n<p>Both of these are examples of power functions because they consist of a coefficient, [latex]\\pi[\/latex] or [latex]\\frac{4}{3}\\pi[\/latex], multiplied by a variable <em>r<\/em>\u00a0raised to a power.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Power FunctionS<\/h3>\n<p>A <strong>power function<\/strong> is a function that can be represented in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{n}[\/latex]<\/p>\n<p>where <i>a<\/i>\u00a0and <i>n<\/i>\u00a0are real numbers and <em>a<\/em><i>\u00a0<\/i>is known as the <strong>coefficient<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong><\/p>\n<p><em>No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Power Functions<\/h3>\n<p>Which of the following functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=1\\hfill & \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill & \\text{Identity function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill & \\text{Quadratic}\\text{ }\\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{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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q82786\">Show Solution<\/span><\/p>\n<div id=\"q82786\" class=\"hidden-answer\" style=\"display: none\">\n<p>All of the listed functions are power functions.<\/p>\n<p>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>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>The <strong>reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].<\/p>\n<p>The square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Which functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q105254\">Show Solution<\/span><\/p>\n<div id=\"q105254\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.<\/p><\/div>\n<\/div>\n<\/div>\n<h3>Identifying End Behavior of Power Functions<\/h3>\n<p>The graph below shows the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], [latex]h\\left(x\\right)={x}^{6}[\/latex], [latex]k(x)=x^{8}[\/latex], and [latex]p(x)=x^{10}[\/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. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4764\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/07034858\/CNX_Precalc_Figure_03_03_002-300x156.jpg\" alt=\"power functions with even powers.\" width=\"300\" height=\"156\" \/><\/p>\n<p>To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol [latex]\\infty[\/latex] for positive infinity and [latex]-\\infty[\/latex] for negative infinity. When we say that &#8220;<em>x<\/em> approaches infinity,&#8221; which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that <em>x<\/em>\u00a0is increasing without bound.<\/p>\n<p>With even-powered power functions, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as [latex]x[\/latex] approaches positive or negative infinity, the [latex]f\\left(x\\right)[\/latex] values increase without bound. In symbolic form, we could write<\/p>\n<p style=\"text-align: center;\">[latex]\\text{as }x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p>The graph below shows [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},h\\left(x\\right)={x}^{7},k\\left(x\\right)={x}^{9},\\text{and }p\\left(x\\right)={x}^{11}[\/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function. As the power increases, the graphs flatten near the origin and become steeper away from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4985\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5479\/2020\/11\/13201740\/desmos-graph-odd-poly-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p>\n<p>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 the even-powered power functions, we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the <em>y<\/em>-axis. In the odd-powered power functions, we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.<\/p>\n<p>For these odd power functions, as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex]\u00a0decreases without bound. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex]\u00a0increases without bound. In symbolic form we write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\n<p>The behavior of the graph of a function as the input values get very small ( [latex]x\\to -\\infty[\/latex] ) and get very large ( [latex]x\\to \\infty[\/latex] ) is referred to as the <strong>end behavior<\/strong> of the function. We can use words or symbols to describe end behavior.<\/p>\n<p>The table\u00a0below shows the end behavior of power functions of the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"width: 135px;\"><\/th>\n<th style=\"text-align: center; width: 353px;\">Even Power<\/th>\n<th style=\"text-align: center; width: 336px;\">Odd Power<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 135px;\"><strong>Positive Constant<\/strong><\/p>\n<p><strong><i>a<\/i> &gt; 0<\/strong><\/td>\n<td style=\"width: 353px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4485\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231809\/Table1.png\" 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.\" width=\"356\" height=\"460\" \/><\/td>\n<td style=\"width: 336px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4487\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231957\/Table2.png\" alt=\"Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to positive infinity.\" width=\"359\" height=\"458\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 135px;\"><strong>Negative Constant<\/strong><\/p>\n<p><strong><i>a<\/i> &lt; 0<\/strong><\/td>\n<td style=\"width: 353px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4488\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232026\/Table3.png\" alt=\"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.\" width=\"375\" height=\"460\" \/><\/td>\n<td style=\"width: 336px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4489\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232106\/Table4.png\" alt=\"Graph of an odd-powered function with a negative 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.\" width=\"342\" height=\"464\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>How To: Given a power function [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex]\u00a0is a non-negative integer, identify the end behavior.<\/h3>\n<ol id=\"fs-id1165137409522\">\n<li>Determine whether the power is even or odd.<\/li>\n<li>Determine whether the constant is positive or negative.<\/li>\n<li>Use the above graphs to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\n<p>Describe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q556064\">Show Solution<\/span><\/p>\n<div id=\"q556064\" class=\"hidden-answer\" style=\"display: none\">\n<p>The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As <em>x\u00a0<\/em>(input)\u00a0approaches infinity, [latex]f\\left(x\\right)[\/latex] (output) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty[\/latex]. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex]. We can graphically represent the function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194503\/CNX_Precalc_Figure_03_03_0082.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\n<p>Describe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q631242\">Show Solution<\/span><\/p>\n<div id=\"q631242\" class=\"hidden-answer\" style=\"display: none\">\n<p>The exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex] and as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194505\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q582534\">Show Solution<\/span><\/p>\n<div id=\"q582534\" class=\"hidden-answer\" style=\"display: none\">\n<p>As <em>x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=69337&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15940&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>End Behavior of Polynomial Functions<\/h2>\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 <em>r<\/em>\u00a0of the spill depends on the number of weeks <em>w<\/em>\u00a0that have passed. This relationship is linear.<\/p>\n<p style=\"text-align: center;\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/p>\n<p>We can combine this with the formula for the area <em>A<\/em>\u00a0of a circle.<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\n<p>Composing these functions gives a formula for the area in terms of weeks.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A\\left(w\\right)=A\\left(r\\left(w\\right)\\right)\\\\ A\\left(w\\right)=A\\left(24+8w\\right)\\\\ A\\left(w\\right)=\\pi {\\left(24+8w\\right)}^{2}\\end{array}[\/latex]<\/p>\n<p>Multiplying gives the formula below.<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/p>\n<p>This formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Polynomial Functions<\/h3>\n<p>Let <em>n<\/em>\u00a0be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p>This is called the general form of a polynomial function. Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Polynomial Functions<\/h3>\n<p>Which of the following are polynomial functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q906312\">Show Solution<\/span><\/p>\n<div id=\"q906312\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first two functions are examples of polynomial functions because they can be written in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex],\u00a0where the powers are non-negative integers and the coefficients are real numbers.<\/p>\n<ul>\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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=48358&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Defining the Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h3>\n<p>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 based on the power on the variable. This is called writing a polynomial in general or standard 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 variable with the highest power, also called the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Terminology of Polynomial Functions<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194507\/CNX_Precalc_Figure_03_03_010n2.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" width=\"487\" height=\"147\" \/><\/p>\n<p>We often rearrange polynomials so that the powers on the variable are descending.<\/p>\n<p>When a polynomial is written in this way, we say that it is in general form.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial function, identify the degree and leading coefficient<\/h3>\n<ol>\n<li>Find the highest power of <em>x\u00a0<\/em>to determine the degree of the function.<\/li>\n<li>Identify the term containing the highest power of <em>x\u00a0<\/em>to find the leading term.<\/li>\n<li>The leading coefficient is the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: 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<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q632394\">Show Solution<\/span><\/p>\n<div id=\"q632394\" class=\"hidden-answer\" style=\"display: none\">\n<p>For the function [latex]f\\left(x\\right)[\/latex], the highest power of <em>x<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].<\/p>\n<p>For the function [latex]g\\left(t\\right)[\/latex], the highest power of <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.<\/p>\n<p>For the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, [latex]\u20131[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q435637\">Show Solution<\/span><\/p>\n<div id=\"q435637\" class=\"hidden-answer\" style=\"display: none\">\n<p>The degree is 6. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Identifying End Behavior of Polynomial Functions<\/h3>\n<p>Knowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree.<\/p>\n<table id=\"Table_03_03_04\" style=\"height: 1527px;\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"text-align: center; height: 15px; width: 354px;\">Polynomial Function<\/th>\n<th style=\"text-align: center; height: 15px; width: 106px;\">Leading Term<\/th>\n<th style=\"text-align: center; height: 15px; width: 364px;\">Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 378px;\">\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x - 4[\/latex]<\/td>\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]5{x}^{4}[\/latex]<\/td>\n<td style=\"height: 378px; width: 364px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194510\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/td>\n<\/tr>\n<tr style=\"height: 378px;\">\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]-2{x}^{6}[\/latex]<\/td>\n<td style=\"height: 378px; width: 364px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194512\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/td>\n<\/tr>\n<tr style=\"height: 378px;\">\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]3{x}^{5}[\/latex]<\/td>\n<td style=\"height: 378px; width: 364px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194514\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/td>\n<\/tr>\n<tr style=\"height: 378px;\">\n<td style=\"height: 378px; width: 354px; text-align: center;\">[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td style=\"height: 378px; width: 106px; text-align: center;\">[latex]-6{x}^{3}[\/latex]<\/td>\n<td style=\"height: 378px; width: 364px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194516\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p>Describe the end behavior and determine a possible degree of the polynomial function in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194520\/CNX_Precalc_Figure_03_03_0152.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q626899\">Show Solution<\/span><\/p>\n<div id=\"q626899\" class=\"hidden-answer\" style=\"display: none\">\n<p>As the input values <em>x<\/em>\u00a0get very large, the output values [latex]f\\left(x\\right)[\/latex] increase without bound. As the input values <em>x<\/em>\u00a0get very small, the output values [latex]f\\left(x\\right)[\/latex] decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\n<p>In words, we could say that as <em>x<\/em>\u00a0values approach infinity, the function values approach infinity, and as <em>x<\/em>\u00a0values approach negative infinity, the function values approach negative infinity.<\/p>\n<p>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<p>In the following video, we show more examples that summarize the end behavior of polynomial functions and which components of the function contribute to it.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Summary of End Behavior or Long Run Behavior of Polynomial Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/y78Dpr9LLN0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe the end behavior of the polynomial function in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194522\/CNX_Precalc_Figure_03_03_016n2.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304329\">Show Solution<\/span><\/p>\n<div id=\"q304329\" class=\"hidden-answer\" style=\"display: none\">\n<p>As [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to -\\infty , f\\left(x\\right)\\to -\\infty[\/latex]. It has the shape of an even degree power function with a negative coefficient.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q76137\">Show Solution<\/span><\/p>\n<div id=\"q76137\" class=\"hidden-answer\" style=\"display: none\">\n<p>Obtain the general form by expanding the given expression [latex]f\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} f\\left(x\\right)=-3{x}^{2}\\left(x - 1\\right)\\left(x+4\\right)\\\\ f\\left(x\\right)=-3{x}^{2}\\left({x}^{2}+3x - 4\\right)\\\\ f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}\\end{array}[\/latex]<\/p>\n<p>The general form is [latex]f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}[\/latex].\u00a0The leading term is [latex]-3{x}^{4}[\/latex];\u00a0therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q657153\">Show Solution<\/span><\/p>\n<div id=\"q657153\" class=\"hidden-answer\" style=\"display: none\">\n<p>The leading term is [latex]0.2{x}^{3}[\/latex], so it is a degree 3 polynomial. As x approaches positive infinity, [latex]f\\left(x\\right)[\/latex] increases without bound; as x approaches negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound.<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Local Behavior of Polynomial Functions<\/h2>\n<h3>Identifying Local Behavior of Polynomial Functions<\/h3>\n<p>In addition to the end behavior of polynomial functions, we are also interested in what happens in the &#8220;middle&#8221; of the function. In particular, we are interested in locations where graph behavior changes. A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194524\/CNX_Precalc_Figure_03_03_0172.jpg\" width=\"731\" height=\"629\" alt=\"image\" \/><\/p>\n<p>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.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Intercepts and Turning Points of Polynomial Functions<\/h3>\n<ul>\n<li>A <strong>turning point<\/strong> of a graph is a point where the graph changes from increasing to decreasing or decreasing to increasing.<\/li>\n<li>The <em>y-<\/em>intercept is the point where the function has an input value of zero.<\/li>\n<li>The <em>x<\/em>-intercepts are the points where the output value is zero.<\/li>\n<li>A polynomial of degree <em>n<\/em>\u00a0will have, at most, <em>n<\/em>\u00a0<em>x<\/em>-intercepts and <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\n<\/ul>\n<\/div>\n<h3>Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h3>\n<p>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<p>The degree of a polynomial function helps us to determine the number of <em>x<\/em>-intercepts and the number of turning points. A polynomial function of\u00a0<em>n<\/em>th degree is the product of <em>n<\/em>\u00a0factors, so it will have at most <em>n<\/em>\u00a0roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree <em>n<\/em>\u00a0must have at most <em>n<\/em> \u2013 1\u00a0turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Determining the Number of Intercepts and Turning Points of a Polynomial<\/h3>\n<p>Without graphing the function, determine the local behavior of the function by finding the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96529\">Show Solution<\/span><\/p>\n<div id=\"q96529\" class=\"hidden-answer\" style=\"display: none\">\n<p>The polynomial has a degree of 10, so there are at most <i>10<\/i>\u00a0<em>x<\/em>-intercepts and at most [latex]10 \u2013 1 = 9[\/latex] turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video gives a 5 minute lesson on how to determine the number of intercepts and turning points of a polynomial function given its degree.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Turning Points and X Intercepts of a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9WW0EetLD4Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for [latex]f\\left(x\\right)=108 - 13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q304362\">Show Solution<\/span><\/p>\n<div id=\"q304362\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are at most 12 <em>x<\/em>-intercepts and at most 11 turning points.<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=123739&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial function, determine the intercepts<\/h3>\n<ol>\n<li>Determine the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and finding the corresponding output value.<\/li>\n<li>Determine the <em>x<\/em>-intercepts by setting the function equal to zero and solving for the input values.<\/li>\n<\/ol>\n<\/div>\n<h3>Using the Principle of Zero Products to Find the Roots of a Polynomial in Factored Form<\/h3>\n<p>The Principle of Zero Products states that if the product of n\u00a0numbers is 0, then at least one of the factors is 0. If [latex]ab=0[\/latex], then either [latex]a=0[\/latex] or [latex]b=0[\/latex], or both a and b are 0. We will use this idea to find the zeros of a polynomial that is either in factored form or can be written in factored form. For example, the polynomial<\/p>\n<p style=\"text-align: center;\">[latex]P(x)=(x-4)^2(x+1)(x-7)[\/latex]<\/p>\n<p>is in factored form. In the following examples, we will show the process of factoring a polynomial and calculating its x and y-intercepts.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Determining the Intercepts of a Polynomial Function<\/h3>\n<p>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 <i>y<\/i>\u00a0and <i>x<\/i>-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q701514\">Show Solution<\/span><\/p>\n<div id=\"q701514\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em>y-<\/em>intercept occurs when the input is zero, so substitute 0 for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(0\\right)=\\left(0 - 2\\right)\\left(0+1\\right)\\left(0 - 4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=\\left(-2\\right)\\left(1\\right)\\left(-4\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=8\\hfill \\end{array}[\/latex]<\/p>\n<p>The <em>y-<\/em>intercept is (0, 8).<\/p>\n<p>The <em>x<\/em>-intercepts occur when the output [latex]f(x)[\/latex] is zero.<\/p>\n<p style=\"text-align: center;\">[latex]0=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{llllllllllll}x - 2=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x+1=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x - 4=0\\hfill \\\\ \\text{}x=2\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{ }x=-1\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x=4 \\end{array}[\/latex]<\/p>\n<p>The\u00a0<i>x<\/i>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\n<p>We can see these intercepts on the graph of the function shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194527\/CNX_Precalc_Figure_03_03_0182.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"487\" height=\"630\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining the Intercepts of a Polynomial Function BY Factoring<\/h3>\n<p>Given the polynomial function [latex]f\\left(x\\right)={x}^{4}-4{x}^{2}-45[\/latex], determine the <em>y<\/em>\u00a0and\u00a0<em>x<\/em>-intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q492513\">Show Solution<\/span><\/p>\n<div id=\"q492513\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em>y-<\/em>intercept occurs when the input is zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} \\\\ f\\left(0\\right)={\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45\\hfill \\hfill \\\\ \\text{}f\\left(0\\right)=-45\\hfill \\end{array}[\/latex]<\/p>\n<p>The <em>y-<\/em>intercept is [latex]\\left(0,-45\\right)[\/latex].<\/p>\n<p>The <em>x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}f\\left(x\\right)={x}^{4}-4{x}^{2}-45\\hfill \\\\ f\\left(x\\right)=\\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right)\\hfill \\\\ f\\left(x\\right)=\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\hfill \\end{array}[\/latex]<\/p>\n<p>Then set the polynomial function equal to 0.<\/p>\n<p style=\"text-align: center;\">[latex]0=\\left(x - 3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllll}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]<\/p>\n<p>The <em>x<\/em>-intercepts are [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex].<\/p>\n<p>We can see these intercepts on the graph of the function shown\u00a0below. We can see that the function has y-axis symmetry or is even because [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194529\/CNX_Precalc_Figure_03_03_0192.jpg\" alt=\"Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).\" width=\"487\" height=\"426\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the polynomial function [latex]f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x[\/latex], determine the <em>y\u00a0<\/em>and<em> x-<\/em>intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q743200\">Show Solution<\/span><\/p>\n<div id=\"q743200\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>y<\/em>-intercept [latex]\\left(0,0\\right)[\/latex]; <em>x<\/em>-intercepts [latex]\\left(0,0\\right),\\left(-2,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<h3>The Whole Picture<\/h3>\n<p>Now we can bring the two concepts of turning points and intercepts together to get a general picture of the behavior of polynomial functions. \u00a0These types of analyses on polynomials developed before the advent of mass computing as a way to quickly understand the general behavior of a polynomial function. We now have a quick way, with computers, to graph and calculate important characteristics of polynomials that once took a lot of algebra.<\/p>\n<p>In the first example, we will determine the least degree of a polynomial based on the number of turning points and intercepts.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Drawing Conclusions about a Polynomial Function from Its Graph<\/h3>\n<p>Given the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194531\/CNX_Precalc_Figure_03_03_0202.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200904\">Show Solution<\/span><\/p>\n<div id=\"q200904\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194532\/CNX_Precalc_Figure_03_03_0212.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/><\/p>\n<p>The end behavior of the graph tells us this is the graph of an even-degree polynomial.<\/p>\n<p>The graph has 2 <em>x<\/em>-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try to determine the least possible degree of a polynomial given its graph.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph of the polynomial function below, determine the least possible degree of the polynomial and whether it is even or odd. Use end behavior, the number of intercepts, and the number of turning points to help you.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194534\/CNX_Precalc_Figure_03_03_0224.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q492375\">Show Solution<\/span><\/p>\n<div id=\"q492375\" class=\"hidden-answer\" style=\"display: none\">\n<p>The end behavior indicates an odd-degree polynomial function; there are 3 <em>x<\/em>-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the leading coefficient must be negative.<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15937&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>Now we will show that you can also determine the least possible degree and number of turning points\u00a0of a polynomial function given in factored form.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Drawing Conclusions about a Polynomial Function from ITS Factors<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex],\u00a0determine the local behavior.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q978752\">Show Solution<\/span><\/p>\n<div id=\"q978752\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em>y<\/em>-intercept is found by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(0\\right)=-4\\left(0\\right)\\left(0+3\\right)\\left(0 - 4\\right)\\hfill \\hfill \\\\ \\text{}f\\left(0\\right)=0\\hfill \\end{array}[\/latex]<\/p>\n<p>The <em>y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex].<\/p>\n<p>The <em>x<\/em>-intercepts are found by setting the function equal to 0.<\/p>\n<p style=\"text-align: center;\">[latex]0=-4x\\left(x+3\\right)\\left(x - 4\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lllllllllllllll}-4x=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x+3=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x - 4=0\\hfill \\\\ x=0\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{}x=-3\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & \\text{}x=4\\end{array}[\/latex]<\/p>\n<p>The <em>x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(-3,0\\right)[\/latex], and [latex]\\left(4,0\\right)[\/latex].<\/p>\n<p>The degree is 3 so the graph has at most 2 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now it is your turn to determine the local behavior of a polynomial given in factored form.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the function [latex]f\\left(x\\right)=0.2\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 5\\right)[\/latex], determine the local behavior.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q104366\">Show Solution<\/span><\/p>\n<div id=\"q104366\" class=\"hidden-answer\" style=\"display: none\">\n<p>The <em>x<\/em>-intercepts are [latex]\\left(2,0\\right),\\left(-1,0\\right)[\/latex], and [latex]\\left(5,0\\right)[\/latex], the <em>y-<\/em>intercept is [latex]\\left(0,\\text{2}\\right)[\/latex], and the graph has at most 2 turning points.<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137724050\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td>general form of a polynomial function<\/td>\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137731646\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165135438864\">\n<li>A power function is a variable base raised to a number power.<\/li>\n<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\n<li>The end behavior depends on whether the power is even or odd.<\/li>\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with non-negative integer powers.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.<\/li>\n<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.<\/li>\n<li>A polynomial of degree <em>n<\/em>\u00a0will have at most <em>n<\/em>\u00a0<em>x-<\/em>intercepts and at most <em>n<\/em> \u2013 1\u00a0turning points.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137668266\" class=\"definition\">\n<dt><strong>coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\" class=\"definition\">\n<dt><strong>continuous function<\/strong><\/dt>\n<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\" class=\"definition\">\n<dt><strong>degree<\/strong><\/dt>\n<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\" class=\"definition\">\n<dt><strong>end behavior<\/strong><\/dt>\n<dd id=\"fs-id1165131990654\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>leading coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>leading term<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero\u00a0terms, each of which is a product of a number, called the\u00a0coefficient\u00a0of the term, and a variable raised to a non-negative integer power.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>power function<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">a function that can be represented in the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex]\u00a0where <em>a\u00a0<\/em>is a constant, the base is a variable, and the exponent is\u00a0<i>n<\/i>,\u00a0is a smooth curve represented by a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>term of a polynomial function<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">any [latex]{a}_{i}{x}^{i}[\/latex]\u00a0of a polynomial function in the form [latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>turning point<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">the location where the graph of a function changes direction<\/dd>\n<\/dl>\n<\/div>\n<\/section>\n","protected":false},"author":167848,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4983","chapter","type-chapter","status-publish","hentry"],"part":764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4983","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4983\/revisions"}],"predecessor-version":[{"id":5275,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4983\/revisions\/5275"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/parts\/764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4983\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/media?parent=4983"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapter-type?post=4983"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/contributor?post=4983"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/license?post=4983"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}