{"id":1180,"date":"2022-03-20T22:47:30","date_gmt":"2022-03-20T22:47:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1180"},"modified":"2025-12-10T20:32:21","modified_gmt":"2025-12-10T20:32:21","slug":"3-1-polynomial-functions-and-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-1-polynomial-functions-and-graphs\/","title":{"raw":"3.1.2: Polynomial Functions and Their Graphs","rendered":"3.1.2: Polynomial Functions and Their Graphs"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-415\" class=\"standard post-415 chapter type-chapter status-publish hentry\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Define a polynomial function<\/li>\r\n \t<li>Identify the end behavior of the graph of a polynomial function based on the degree of the function and the sign of the leading coefficient<\/li>\r\n \t<li>Define zeros of a function<\/li>\r\n \t<li>Explain the relationship between zeros and [latex]x[\/latex]-intercepts on the graph of the function<\/li>\r\n \t<li>Explain the relationship between the degree of a polynomial function and the number of turning points on its graph<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"entry-content\">\r\n<h2>Polynomial Functions<\/h2>\r\nA polynomial function is any function that is a term or a sum (or difference) of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and variable with whole number exponents. In other words, a polynomial function consists of a polynomial.\r\n\r\nIn this chapter, we will focus on one-variable polynomial functions, like\u00a0[latex]f(x)=-2x+4[\/latex],\u00a0[latex]g(x)=2x^2-2x+4[\/latex], or\u00a0[latex]h(x)=x^5+2x^3-12x+3[\/latex]. Notice that these polynomial functions are written in <em><strong>descending order<\/strong><\/em> from the highest exponent to the lowest.\r\n<div class=\"textbox shaded\">\r\n<h3>polynomial functions in one variable<\/h3>\r\nA one-variable\u00a0<em><strong>polynomial function <\/strong><\/em>has\u00a0the form:\r\n<p id=\"eip-263\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\u2026+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\n<p class=\"equation unnumbered\" style=\"text-align: left;\" data-type=\"equation\">where [latex]a_0, a_1,..., a_n[\/latex] are real numbers, and [latex]n[\/latex] is a positive integer.<\/p>\r\n\r\n<\/div>\r\n<h2>Identifying\u00a0the Shape of the Graph of a Polynomial Function<\/h2>\r\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like.\u00a0The leading term has the highest exponent on the variable, meaning that the leading term will grow significantly faster than the other terms in the function as [latex]x[\/latex]<em>\u00a0<\/em>gets very large or very small. So, the behavior of the leading term will dominate the graph.<\/p>\r\nAs an example, we compare the outputs of a degree\u00a0[latex]2[\/latex] polynomial and a degree\u00a0[latex]5[\/latex] polynomial in table 1.\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"height: 15px; text-align: right;\">[latex]x[\/latex]<\/th>\r\n<th style=\"height: 15px; text-align: right;\">[latex]f(x)=2x^2-2x+4[\/latex]<\/th>\r\n<th style=\"height: 15px; text-align: right;\">[latex]g(x)=x^5+2x^3-12x+3[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]4[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: right;\">[latex]10[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]184[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]98,117[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: right;\">[latex]100[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]19,804[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]9,998,001,197[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: right;\">[latex]1000[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]1,998,004[\/latex]<\/td>\r\n<td style=\"height: 15px; text-align: right;\">[latex]9,999,980,000,000,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"3\">Table 1. Comparative growth of polynomial functions.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs the inputs for both functions get larger, the degree [latex]5[\/latex] polynomial outputs get much larger than the degree\u00a0[latex]2[\/latex] polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of the graphs of polynomial functions as the independent variable gets very large or very small. This is referred to as the <em><strong>end behavior<\/strong><\/em> of the graph.\r\n\r\nLet's take a look at the graphs of some polynomial functions and determine any patterns that exist in their graphs.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Odd degree<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Even degree<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1566\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1566 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-300x300.png\" alt=\"Graph of 1st degree polynomial Graph goes up on the right, down on the left\" width=\"300\" height=\"300\" \/> Figure 1. Degree 1 polynomial function.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1565\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1565 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-300x300.png\" alt=\"Degree 2 polynomial.  Graph goes up on the right, AND up on the left.\" width=\"300\" height=\"300\" \/> Figure 2. Degree 2 polynomial function.[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1564\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1564 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-300x300.png\" alt=\"Degree 3 polynomial function. Graph goes up on the right, down on the left.\" width=\"300\" height=\"300\" \/> Figure 3. Degree 3 polynomial function.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1563\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1563 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-300x300.png\" alt=\"Degree 4 polynomial function. Graph goes up on the right, AND up on the left.\" width=\"300\" height=\"300\" \/> Figure 4. Degree 4 polynomial function.[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1562\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1562 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-300x300.png\" alt=\"Degree 5 polynomial. Graph goes up on the right, down on the left.\" width=\"300\" height=\"300\" \/> Figure 5. Degree 5 polynomial function.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1561\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1561 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-300x300.png\" alt=\"Degree 6 polynomial. Graph goes up on the right, AND up on the left.\" width=\"300\" height=\"300\" \/> Figure 6. Degree 6 polynomial function[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAll of the graphs in figures 1 - 6, represent polynomial functions with positive <em><strong>leading coefficients<\/strong><\/em>. Notice that figures 1, 3, and 5 show graphs of functions with odd degrees, while figures 2, 4, and 6 show graphs of functions with even degrees. All of the odd degree polynomial functions have graphs that come from the bottom left and end up at the top right. Think John Travolta in <a href=\"https:\/\/www.amazon.com\/Saturday-Night-Fever-BD-Blu-ray\/dp\/B00AEBB82Y\">Saturday Night Fever<\/a>--Right arm up, left arm down! On the other hand, all of the even degree polynomial functions have graphs that start at the top left and end up at the top right. Think <a href=\"https:\/\/www.nfl.com\/photos\/ed-hochuli-through-the-years-0ap3000000919903#bbdef6d2-1755-40bb-b361-c0c9303a09b1\">football touchdown<\/a>!\r\n\r\nIf we look at graphs of the parent functions [latex]f(x)=x^n[\/latex], we will notice some major similarities.\r\n<h3>Odd Degree Polynomials<\/h3>\r\nFigure 7\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{1},\\;f\\left(x\\right)={x}^{3},\\;f\\left(x\\right)={x}^{5}[\/latex], and [latex]f(x)=x^{51}[\/latex], which all have odd degrees. When [latex]n=1[\/latex] we have the linear function [latex]f(x)=x[\/latex] whose graph is a line. The graphs when [latex]n=3, 5, ...[\/latex] all have similar end behavior that is right-up and left-down. This is because the value of the function goes to positive infinity as the value of [latex]x[\/latex] goes to positive infinity and the value of the function goes to negative infinity as the value of [latex]x[\/latex] goes to negative infinity.\r\n\r\nNotice that as the power increases, the graphs flatten somewhat near the origin ([latex]-1&lt;x&lt;1[\/latex]) and become steeper away from the origin ([latex]|\\,x\\,|&gt;1[\/latex]). This is because a proper fraction raised to any odd power has a value between \u20131 and 1, while an [latex]x[\/latex]-value greater than 1 in absolute value will get larger in absolute value as the exponent increases. Notice also that all of the graphs pass through the points (\u20131, 1), (0, 0), and (1, 1). This is because [latex](-1)^n=-1[\/latex] and [latex]1^n=1[\/latex] when [latex]n[\/latex] is odd, while [latex]0^n=0[\/latex] for all positive integer values of [latex]n[\/latex].\r\n\r\n[caption id=\"attachment_1568\" align=\"aligncenter\" width=\"283\"]<img class=\"wp-image-1568 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-283x300.png\" alt=\"graph of odd degree functions showing that All basic Odd degree functions  the right hand side goes to infinity, while the left hand side goes to negative infinity.\" width=\"283\" height=\"300\" \/> Figure 7. Odd degree polynomial functions[\/caption]\r\n<h3>Even Degree Polynomials<\/h3>\r\nFigure 8 shows the graphs of [latex]f\\left(x\\right)={x}^{2},\\;f\\left(x\\right)={x}^{4},\\;f\\left(x\\right)={x}^{6}[\/latex], and [latex]f(x)=x^{50}[\/latex], which all have even degrees. These graphs have similar end behavior that is right-up and left-up. This is because the value of the function goes to positive infinity as the value of [latex]x[\/latex] goes to either positive or negative infinity (e.g., a number to an even power, no matter the number is positive or negative, is always positive).\r\n\r\nNotice that as the power increases, the graphs flatten somewhat near the origin (between [latex]-1&lt;x&lt;1[\/latex]) and become steeper away from the origin ([latex]|\\,x\\,|&gt;1[\/latex]). This is because a proper fraction raised to any even power has a value less than one, while an [latex]x[\/latex]-value greater than 1 will get larger the higher the value of the exponent. Notice also that\u00a0<span style=\"font-size: 1em;\">the graph of\u00a0[latex]f(x)=x^n[\/latex] always passes through (0, 0), (1, 1) and (\u20131, 1) when\u00a0[latex]n[\/latex] is even.\u00a0<\/span>\r\n\r\n[caption id=\"attachment_1569\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1569 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-300x297.png\" alt=\"Several graphs showing that all basic even degree polynomial functions go up on the right AND up on the left.\" width=\"300\" height=\"297\" \/> Figure 8. Graph of even degree polynomial functions[\/caption]\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nIdentify whether each graph represents a polynomial function that has a degree that is even or odd.\r\n\r\n&nbsp;\r\n\r\na)\r\n<div class=\"wp-nocaption \"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201335\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"231\" height=\"245\" \/><\/div>\r\nb)\r\n<div class=\"wp-nocaption \"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201338\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"227\" height=\"241\" \/><\/div>\r\n<h4>Solution<\/h4>\r\na) Both arms of this polynomial point upward, therefore the degree must be even. \u00a0If we apply negative inputs to an even degree polynomial, you will get positive outputs back.\r\n\r\nb) As the inputs of this polynomial become more negative the outputs also become negative. The only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have an odd degree.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nIdentify whether each graph represents a polynomial function that has a degree that is even or odd.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Up on both sides<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Up on right, down on left<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1570\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1570 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/14231540\/even-example-300x300.png\" alt=\"Polynomial function\" width=\"300\" height=\"300\" \/> Graph 1[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1571\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1571 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/14231544\/odd-example-300x300.png\" alt=\"Graph\" width=\"300\" height=\"300\" \/> Graph 2[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm532\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm532\"]\r\n\r\nGraph 1: even degree\r\n\r\nGraph 2: odd degree\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>The Sign of the Leading Term<\/h3>\r\nWe saw in previous chapters that reflecting a function across the [latex]x[\/latex]-axis results in changing the sign of all the [latex]y[\/latex]-values on the graph to their opposites. If the parent function [latex]f(x)=x^n[\/latex] is reflected across the [latex]x[\/latex]-axis, the function [latex]f(x)=-x^n[\/latex] represents the new function. The reflected function now has a negative leading coefficient.\r\n\r\nConsider the two functions [latex]f(x)=x^2[\/latex] and [latex]f(x)=x^3[\/latex], one even and one odd. If we change the sign of the leading term, the functions become [latex]f(x)=-x^2[\/latex] and [latex]f(x)=-x^3[\/latex]. Now let's look at the graphs:\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Negative leading coefficient<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Negative leading coefficient<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_1573\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1573 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-300x300.png\" alt=\"Making the leading coefficient negative changes end behavior of an odd degree function to down on right, up on left.\" width=\"300\" height=\"300\" \/> FIgure 9. [latex]f(x)=x^3[\/latex] and [latex]f(x)=-x^3[\/latex][\/caption]<\/td>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_1574\" align=\"aligncenter\" width=\"298\"]<img class=\"wp-image-1574 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-298x300.png\" alt=\"Making the leading coefficient negative changes end behavior of an even degree function to down on right, down on left.\" width=\"298\" height=\"300\" \/> Figure 10. [latex]f(x)=x^2[\/latex] and [latex]f(x)-x^2[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFigures 9 and 10 show that a negative leading term changes the end behavior of the graphs, equivalent to reflection across the [latex]x[\/latex]-axis.\r\n<div class=\"textbox examples\">\r\n<h3>example 2<\/h3>\r\nIdentify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Right side down, left side down<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">Right side down, left side up<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"214\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201336\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.  Goes down on right and goes down on left.\" width=\"214\" height=\"227\" \/> Graph 1.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"217\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201339\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.  Goes down on right and goes up on left.\" width=\"217\" height=\"230\" \/> Graph 2.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<h4>Solution<\/h4>\r\nGraph 1: Both arms of the graph are heading in the same direction, so the degree of the function is even. SInce the arms are both pointing downwards, the leading coefficient is negative.\r\n\r\nGraph 2: The arms of the graph are going in opposite directions, so the degree of the function is odd. Since the graph is moving from top left to bottom right, the leading coefficient is negative.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-1310\" class=\"standard post-1310 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nIdentify whether the leading coefficient is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Graph 1<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Graph 2<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Graph 3<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><img class=\"wp-image-1585\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15003354\/Fx-x%5E45x-300x300.png\" alt=\"F(x)=-x^4+5x. Down on right, down on left\" width=\"222\" height=\"222\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"wp-image-1583\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15003350\/gx-x%5E54x%5E2-300x300.png\" alt=\"g(x)=-x^5+4x^2.  Down on right, up on left.\" width=\"228\" height=\"228\" \/><\/td>\r\n<td style=\"width: 33.3333%;\"><img class=\"wp-image-1586\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15003357\/fxx%5E43x%5E3-300x300.png\" alt=\"Graph goes up on right and up on left.\" width=\"226\" height=\"226\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm314\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm314\"]\r\n\r\nGraph 1: Even degree; negative leading coefficient\r\n\r\nGraph 2: Odd degree; negative leading coefficient\r\n\r\nGraph 3: Even degree: positive leading coefficient\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Zeros<\/h2>\r\nA zero of a function [latex]f(x)[\/latex] is any value of the independent variable [latex]x[\/latex] that causes the function value to equal zero. i.e., [latex]f(x) = 0[\/latex]. On the coordinate plane, zeros are the intersection points between the graph of a function [latex]y=f(x)[\/latex] and the [latex]x[\/latex]-axis. In other words, they are the [latex]x[\/latex]-intercepts. Since\u00a0[latex]f(x) = 0[\/latex],\u00a0each intersection point has a [latex]y[\/latex]-coordinate of 0.\r\n\r\n[caption id=\"attachment_4245\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-4245 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-300x300.png\" alt=\"Graph with zeros at -3, -2, and 1 as described above.\" width=\"300\" height=\"300\" \/> Figure 11. Graph of [latex]h(x)=x^3+4x^2+x-6[\/latex][\/caption]\r\n<p id=\"fs-id1165135407009\">Figure 11 shows the graph of [latex] h(x)=x^3+4x^2+x-6[\/latex]. The graph of the function has [latex]x[\/latex]-intercepts at the points (\u20133, 0), (\u20132, 0), and (1, 0). Therefore, the zeros of the function are [latex] x = \u20133, \u20132[\/latex] and [latex]1[\/latex].<\/p>\r\nNotice that when we identify an [latex]x[\/latex]-intercept we write it as a point: e.g., (\u20132, 0). However, when we identify a zero, we write is as a solution of the equation [latex]f(x)=0[\/latex]: e.g., [latex]x=\u20132[\/latex].\r\n<div class=\"textbox examples\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-1310\" class=\"standard post-1310 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<h3 data-type=\"title\">Example 3<\/h3>\r\nUse the graph of the polynomial function of degree 6 to identify the zeros of the function.\r\n<div class=\"wp-caption aligncenter\" style=\"width: 277px;\">\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201430\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"277\" height=\"358\" data-media-type=\"image\/jpg\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h4>Solution<\/h4>\r\nThe [latex]x[\/latex]-intercepts are (\u20133, 0), (\u20131, 0), and (4, 0). Therefore, the zeros are [latex]x=[\/latex] \u20133, \u20131, and 4.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nUse the graph of the polynomial function of degree 6 to identify the zeros of the function.\r\n\r\n<img class=\"aligncenter wp-image-1593 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-300x300.png\" alt=\"f\\left(x\\right)=x\\left(x+2\\right)\\left(x-3\\right)\\left(x-4\\right). With x intercepts (-2,0), (0,0), (3,0), and (4,0).\" width=\"300\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"hjm188\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm188\"]\r\n\r\n[latex]x=\u20132, \\;0, \\;3, \\;4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3 data-type=\"title\"><span style=\"color: #077fab; font-size: 1.15em;\">The Number of Turning Points<\/span><\/h3>\r\n<p data-type=\"title\">When the [latex]y[\/latex]-values of a graph increase as [latex]x[\/latex] increases, we say the graph is increasing over the given [latex]x[\/latex]-values. Likewise, when the [latex]y[\/latex]-values of a graph decrease as [latex]x[\/latex] increases, we say the graph is decreasing over the given [latex]x[\/latex]-values. The point where a graph changes from increasing to decreasing or vice versa is called a turning point. This is illustrated in figure 12 where the graph of the function has 3 turning points.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"wp-caption aligncenter\" style=\"width: 497px;\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"299\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201439\/CNX_Precalc_Figure_03_04_0152.jpg\" alt=\"Graph of an odd-degree polynomial with a negative leading coefficient. It has three turning points as described above.\" width=\"299\" height=\"201\" data-media-type=\"image\/jpg\" \/> Figure 12. A function with turning points[\/caption]\r\n\r\n<\/div>\r\nThe number of turning points the graph of a polynomial function has is dependent upon the degree of the polynomial.\r\n<div class=\"textbox shaded\">\r\n\r\nNumber of turning points\r\n<p style=\"text-align: center;\">\u00a0A polynomial function [latex]f(x)[\/latex] of degree [latex]n[\/latex], has at most [latex]n-1[\/latex] turning points.<\/p>\r\n\r\n<\/div>\r\nThis does not mean that, for example\u00a0the polynomial function [latex]f(x)=x^8[\/latex] has 7 turning points. Rather, the function has <em>at most<\/em> 7 turning points. In reality, the graph of the function\u00a0[latex]f(x)=x^8[\/latex] has only one turning point (figure 13).\r\n\r\n[caption id=\"attachment_1596\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1596 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-300x300.png\" alt=\"Graph of f(x)=x^8 only has one turning point.\" width=\"300\" height=\"300\" \/> Figure 13. Graph of [latex]f(x)=x^8[\/latex][\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\n<ol>\r\n \t<li>\u00a0Determine the maximum number of turning points of the graph of the function [latex]f\\left(x\\right)=x^{5}-3x^{4}+x^{3}+2x^{2}+4[\/latex].<\/li>\r\n \t<li>\u00a0Use Desmos to graph the function.<\/li>\r\n \t<li>\u00a0Show the intervals of [latex]x[\/latex] on the graph when the function is increasing and when it is decreasing.<\/li>\r\n \t<li>\u00a0How many turning points are on the graph?<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n1.[latex]f\\left(x\\right)=x^{5}-3x^{4}+x^{3}+2x^{2}+4[\/latex] has degree 5, so the graph has at most 4 turning points.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">How many turning points?<\/th>\r\n<th style=\"width: 50%; text-align: center;\">Increasing or decreasing?<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2.\u00a0<img class=\"aligncenter size-medium wp-image-1598\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15013001\/rightx%5E5-3x%5E4x%5E32x%5E24-300x300.png\" alt=\"Graph of function\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 50%;\">3.<img class=\"aligncenter wp-image-1599 size-medium\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-300x300.png\" alt=\"Graph of function with increasing and decreasing parts labeled.\" width=\"300\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n4. There are 4 turning points.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nThe graph shows the function [latex]f\\left(x\\right)=x^{4}+2x^{3}+4[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-1600 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-300x300.png\" alt=\"A fourth degree polynomial decreasing on negative infinity to negative .5, and increasing on negative .5 to infinity.\" width=\"300\" height=\"300\" \/>\r\n<ol>\r\n \t<li>\u00a0What is the maximum number of turning points of the graph of [latex]f\\left(x\\right)=x^{4}+2x^{3}+4[\/latex]?<\/li>\r\n \t<li>\u00a0Show the intervals of [latex]x[\/latex] on the graph when the function is increasing and when it is decreasing.<\/li>\r\n \t<li>\u00a0How many turning points are on the graph?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm272\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm272\"]\r\n<ol>\r\n \t<li>\u00a03<\/li>\r\n \t<li><\/li>\r\n \t<li><img class=\"aligncenter size-medium wp-image-1601\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15014955\/Deg-4-markup-300x300.png\" alt=\"Degree 4 polynomial\" width=\"300\" height=\"300\" \/><\/li>\r\n \t<li>3. 1<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice in the last Try It, there is a flat area on the graph at [latex]x=0[\/latex]. At this point, the graph is neither increasing nor decreasing. It goes from increasing to taking a breath to increasing again. The same thing can happen when the graph is decreasing then stalls for a moment then decreases again. Such a point is called a <em><strong>point of inflection<\/strong><\/em>.\r\n\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-415\" class=\"standard post-415 chapter type-chapter status-publish hentry\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Define a polynomial function<\/li>\n<li>Identify the end behavior of the graph of a polynomial function based on the degree of the function and the sign of the leading coefficient<\/li>\n<li>Define zeros of a function<\/li>\n<li>Explain the relationship between zeros and [latex]x[\/latex]-intercepts on the graph of the function<\/li>\n<li>Explain the relationship between the degree of a polynomial function and the number of turning points on its graph<\/li>\n<\/ul>\n<\/div>\n<div class=\"entry-content\">\n<h2>Polynomial Functions<\/h2>\n<p>A polynomial function is any function that is a term or a sum (or difference) of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and variable with whole number exponents. In other words, a polynomial function consists of a polynomial.<\/p>\n<p>In this chapter, we will focus on one-variable polynomial functions, like\u00a0[latex]f(x)=-2x+4[\/latex],\u00a0[latex]g(x)=2x^2-2x+4[\/latex], or\u00a0[latex]h(x)=x^5+2x^3-12x+3[\/latex]. Notice that these polynomial functions are written in <em><strong>descending order<\/strong><\/em> from the highest exponent to the lowest.<\/p>\n<div class=\"textbox shaded\">\n<h3>polynomial functions in one variable<\/h3>\n<p>A one-variable\u00a0<em><strong>polynomial function <\/strong><\/em>has\u00a0the form:<\/p>\n<p id=\"eip-263\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\u2026+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p class=\"equation unnumbered\" style=\"text-align: left;\" data-type=\"equation\">where [latex]a_0, a_1,..., a_n[\/latex] are real numbers, and [latex]n[\/latex] is a positive integer.<\/p>\n<\/div>\n<h2>Identifying\u00a0the Shape of the Graph of a Polynomial Function<\/h2>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like.\u00a0The leading term has the highest exponent on the variable, meaning that the leading term will grow significantly faster than the other terms in the function as [latex]x[\/latex]<em>\u00a0<\/em>gets very large or very small. So, the behavior of the leading term will dominate the graph.<\/p>\n<p>As an example, we compare the outputs of a degree\u00a0[latex]2[\/latex] polynomial and a degree\u00a0[latex]5[\/latex] polynomial in table 1.<\/p>\n<table>\n<tbody>\n<tr style=\"height: 15px;\">\n<th style=\"height: 15px; text-align: right;\">[latex]x[\/latex]<\/th>\n<th style=\"height: 15px; text-align: right;\">[latex]f(x)=2x^2-2x+4[\/latex]<\/th>\n<th style=\"height: 15px; text-align: right;\">[latex]g(x)=x^5+2x^3-12x+3[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]4[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: right;\">[latex]10[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]184[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]98,117[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: right;\">[latex]100[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]19,804[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]9,998,001,197[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: right;\">[latex]1000[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]1,998,004[\/latex]<\/td>\n<td style=\"height: 15px; text-align: right;\">[latex]9,999,980,000,000,000[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\">Table 1. Comparative growth of polynomial functions.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As the inputs for both functions get larger, the degree [latex]5[\/latex] polynomial outputs get much larger than the degree\u00a0[latex]2[\/latex] polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of the graphs of polynomial functions as the independent variable gets very large or very small. This is referred to as the <em><strong>end behavior<\/strong><\/em> of the graph.<\/p>\n<p>Let&#8217;s take a look at the graphs of some polynomial functions and determine any patterns that exist in their graphs.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Odd degree<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Even degree<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1566\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1566\" class=\"wp-image-1566 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-300x300.png\" alt=\"Graph of 1st degree polynomial Graph goes up on the right, down on the left\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/fx1st-degree-poly.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1566\" class=\"wp-caption-text\">Figure 1. Degree 1 polynomial function.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1565\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1565\" class=\"wp-image-1565 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-300x300.png\" alt=\"Degree 2 polynomial.  Graph goes up on the right, AND up on the left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2nd-degree-polynomial.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1565\" class=\"wp-caption-text\">Figure 2. Degree 2 polynomial function.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1564\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1564\" class=\"wp-image-1564 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-300x300.png\" alt=\"Degree 3 polynomial function. Graph goes up on the right, down on the left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-3-poly.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1564\" class=\"wp-caption-text\">Figure 3. Degree 3 polynomial function.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1563\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1563\" class=\"wp-image-1563 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-300x300.png\" alt=\"Degree 4 polynomial function. Graph goes up on the right, AND up on the left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-4-poly.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1563\" class=\"wp-caption-text\">Figure 4. Degree 4 polynomial function.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1562\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1562\" class=\"wp-image-1562 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-300x300.png\" alt=\"Degree 5 polynomial. Graph goes up on the right, down on the left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/Degree-5-poly.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1562\" class=\"wp-caption-text\">Figure 5. Degree 5 polynomial function.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1561\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1561\" class=\"wp-image-1561 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-300x300.png\" alt=\"Degree 6 polynomial. Graph goes up on the right, AND up on the left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/degree-6-poly.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1561\" class=\"wp-caption-text\">Figure 6. Degree 6 polynomial function<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>All of the graphs in figures 1 &#8211; 6, represent polynomial functions with positive <em><strong>leading coefficients<\/strong><\/em>. Notice that figures 1, 3, and 5 show graphs of functions with odd degrees, while figures 2, 4, and 6 show graphs of functions with even degrees. All of the odd degree polynomial functions have graphs that come from the bottom left and end up at the top right. Think John Travolta in <a href=\"https:\/\/www.amazon.com\/Saturday-Night-Fever-BD-Blu-ray\/dp\/B00AEBB82Y\">Saturday Night Fever<\/a>&#8211;Right arm up, left arm down! On the other hand, all of the even degree polynomial functions have graphs that start at the top left and end up at the top right. Think <a href=\"https:\/\/www.nfl.com\/photos\/ed-hochuli-through-the-years-0ap3000000919903#bbdef6d2-1755-40bb-b361-c0c9303a09b1\">football touchdown<\/a>!<\/p>\n<p>If we look at graphs of the parent functions [latex]f(x)=x^n[\/latex], we will notice some major similarities.<\/p>\n<h3>Odd Degree Polynomials<\/h3>\n<p>Figure 7\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{1},\\;f\\left(x\\right)={x}^{3},\\;f\\left(x\\right)={x}^{5}[\/latex], and [latex]f(x)=x^{51}[\/latex], which all have odd degrees. When [latex]n=1[\/latex] we have the linear function [latex]f(x)=x[\/latex] whose graph is a line. The graphs when [latex]n=3, 5, ...[\/latex] all have similar end behavior that is right-up and left-down. This is because the value of the function goes to positive infinity as the value of [latex]x[\/latex] goes to positive infinity and the value of the function goes to negative infinity as the value of [latex]x[\/latex] goes to negative infinity.<\/p>\n<p>Notice that as the power increases, the graphs flatten somewhat near the origin ([latex]-1<x<1[\/latex]) and become steeper away from the origin ([latex]|\\,x\\,|>1[\/latex]). This is because a proper fraction raised to any odd power has a value between \u20131 and 1, while an [latex]x[\/latex]-value greater than 1 in absolute value will get larger in absolute value as the exponent increases. Notice also that all of the graphs pass through the points (\u20131, 1), (0, 0), and (1, 1). This is because [latex](-1)^n=-1[\/latex] and [latex]1^n=1[\/latex] when [latex]n[\/latex] is odd, while [latex]0^n=0[\/latex] for all positive integer values of [latex]n[\/latex].<\/p>\n<div id=\"attachment_1568\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1568\" class=\"wp-image-1568 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-283x300.png\" alt=\"graph of odd degree functions showing that All basic Odd degree functions  the right hand side goes to infinity, while the left hand side goes to negative infinity.\" width=\"283\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-283x300.png 283w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-768x813.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-967x1024.png 967w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-65x69.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-225x238.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions-350x371.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/odd-functions.png 980w\" sizes=\"auto, (max-width: 283px) 100vw, 283px\" \/><\/p>\n<p id=\"caption-attachment-1568\" class=\"wp-caption-text\">Figure 7. Odd degree polynomial functions<\/p>\n<\/div>\n<h3>Even Degree Polynomials<\/h3>\n<p>Figure 8 shows the graphs of [latex]f\\left(x\\right)={x}^{2},\\;f\\left(x\\right)={x}^{4},\\;f\\left(x\\right)={x}^{6}[\/latex], and [latex]f(x)=x^{50}[\/latex], which all have even degrees. These graphs have similar end behavior that is right-up and left-up. This is because the value of the function goes to positive infinity as the value of [latex]x[\/latex] goes to either positive or negative infinity (e.g., a number to an even power, no matter the number is positive or negative, is always positive).<\/p>\n<p>Notice that as the power increases, the graphs flatten somewhat near the origin (between [latex]-1<x<1[\/latex]) and become steeper away from the origin ([latex]|\\,x\\,|>1[\/latex]). This is because a proper fraction raised to any even power has a value less than one, while an [latex]x[\/latex]-value greater than 1 will get larger the higher the value of the exponent. Notice also that\u00a0<span style=\"font-size: 1em;\">the graph of\u00a0[latex]f(x)=x^n[\/latex] always passes through (0, 0), (1, 1) and (\u20131, 1) when\u00a0[latex]n[\/latex] is even.\u00a0<\/span><\/p>\n<div id=\"attachment_1569\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1569\" class=\"wp-image-1569 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-300x297.png\" alt=\"Several graphs showing that all basic even degree polynomial functions go up on the right AND up on the left.\" width=\"300\" height=\"297\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-300x297.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-768x760.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-65x64.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-225x223.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions-350x346.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/even-functions.png 978w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1569\" class=\"wp-caption-text\">Figure 8. Graph of even degree polynomial functions<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Identify whether each graph represents a polynomial function that has a degree that is even or odd.<\/p>\n<p>&nbsp;<\/p>\n<p>a)<\/p>\n<div class=\"wp-nocaption\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201335\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"231\" height=\"245\" \/><\/div>\n<p>b)<\/p>\n<div class=\"wp-nocaption\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201338\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"227\" height=\"241\" \/><\/div>\n<h4>Solution<\/h4>\n<p>a) Both arms of this polynomial point upward, therefore the degree must be even. \u00a0If we apply negative inputs to an even degree polynomial, you will get positive outputs back.<\/p>\n<p>b) As the inputs of this polynomial become more negative the outputs also become negative. The only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have an odd degree.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Identify whether each graph represents a polynomial function that has a degree that is even or odd.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Up on both sides<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Up on right, down on left<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1570\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1570\" class=\"wp-image-1570 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/14231540\/even-example-300x300.png\" alt=\"Polynomial function\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1570\" class=\"wp-caption-text\">Graph 1<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1571\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1571\" class=\"wp-image-1571 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/14231544\/odd-example-300x300.png\" alt=\"Graph\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1571\" class=\"wp-caption-text\">Graph 2<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm532\">Show Answer<\/span><\/p>\n<div id=\"qhjm532\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph 1: even degree<\/p>\n<p>Graph 2: odd degree<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>The Sign of the Leading Term<\/h3>\n<p>We saw in previous chapters that reflecting a function across the [latex]x[\/latex]-axis results in changing the sign of all the [latex]y[\/latex]-values on the graph to their opposites. If the parent function [latex]f(x)=x^n[\/latex] is reflected across the [latex]x[\/latex]-axis, the function [latex]f(x)=-x^n[\/latex] represents the new function. The reflected function now has a negative leading coefficient.<\/p>\n<p>Consider the two functions [latex]f(x)=x^2[\/latex] and [latex]f(x)=x^3[\/latex], one even and one odd. If we change the sign of the leading term, the functions become [latex]f(x)=-x^2[\/latex] and [latex]f(x)=-x^3[\/latex]. Now let&#8217;s look at the graphs:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Negative leading coefficient<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Negative leading coefficient<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1573\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1573\" class=\"wp-image-1573 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-300x300.png\" alt=\"Making the leading coefficient negative changes end behavior of an odd degree function to down on right, up on left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^3-and-x^3.png 976w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1573\" class=\"wp-caption-text\">FIgure 9. [latex]f(x)=x^3[\/latex] and [latex]f(x)=-x^3[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1574\" style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1574\" class=\"wp-image-1574 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-298x300.png\" alt=\"Making the leading coefficient negative changes end behavior of an even degree function to down on right, down on left.\" width=\"298\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-298x300.png 298w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-768x773.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-225x226.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2-350x352.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^2-and-x^2.png 966w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<p id=\"caption-attachment-1574\" class=\"wp-caption-text\">Figure 10. [latex]f(x)=x^2[\/latex] and [latex]f(x)-x^2[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Figures 9 and 10 show that a negative leading term changes the end behavior of the graphs, equivalent to reflection across the [latex]x[\/latex]-axis.<\/p>\n<div class=\"textbox examples\">\n<h3>example 2<\/h3>\n<p>Identify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Right side down, left side down<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">Right side down, left side up<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div style=\"width: 224px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201336\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.  Goes down on right and goes down on left.\" width=\"214\" height=\"227\" \/><\/p>\n<p class=\"wp-caption-text\">Graph 1.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div style=\"width: 227px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201339\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.  Goes down on right and goes up on left.\" width=\"217\" height=\"230\" \/><\/p>\n<p class=\"wp-caption-text\">Graph 2.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<h4>Solution<\/h4>\n<p>Graph 1: Both arms of the graph are heading in the same direction, so the degree of the function is even. SInce the arms are both pointing downwards, the leading coefficient is negative.<\/p>\n<p>Graph 2: The arms of the graph are going in opposite directions, so the degree of the function is odd. Since the graph is moving from top left to bottom right, the leading coefficient is negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-1310\" class=\"standard post-1310 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Identify whether the leading coefficient is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Graph 1<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Graph 2<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Graph 3<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1585\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15003354\/Fx-x%5E45x-300x300.png\" alt=\"F(x)=-x^4+5x. Down on right, down on left\" width=\"222\" height=\"222\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1583\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15003350\/gx-x%5E54x%5E2-300x300.png\" alt=\"g(x)=-x^5+4x^2.  Down on right, up on left.\" width=\"228\" height=\"228\" \/><\/td>\n<td style=\"width: 33.3333%;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1586\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15003357\/fxx%5E43x%5E3-300x300.png\" alt=\"Graph goes up on right and up on left.\" width=\"226\" height=\"226\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm314\">Show Answer<\/span><\/p>\n<div id=\"qhjm314\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph 1: Even degree; negative leading coefficient<\/p>\n<p>Graph 2: Odd degree; negative leading coefficient<\/p>\n<p>Graph 3: Even degree: positive leading coefficient<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Zeros<\/h2>\n<p>A zero of a function [latex]f(x)[\/latex] is any value of the independent variable [latex]x[\/latex] that causes the function value to equal zero. i.e., [latex]f(x) = 0[\/latex]. On the coordinate plane, zeros are the intersection points between the graph of a function [latex]y=f(x)[\/latex] and the [latex]x[\/latex]-axis. In other words, they are the [latex]x[\/latex]-intercepts. Since\u00a0[latex]f(x) = 0[\/latex],\u00a0each intersection point has a [latex]y[\/latex]-coordinate of 0.<\/p>\n<div id=\"attachment_4245\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4245\" class=\"wp-image-4245 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-300x300.png\" alt=\"Graph with zeros at -3, -2, and 1 as described above.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/desmos-graph-2022-09-22T130315.804.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-4245\" class=\"wp-caption-text\">Figure 11. Graph of [latex]h(x)=x^3+4x^2+x-6[\/latex]<\/p>\n<\/div>\n<p id=\"fs-id1165135407009\">Figure 11 shows the graph of [latex]h(x)=x^3+4x^2+x-6[\/latex]. The graph of the function has [latex]x[\/latex]-intercepts at the points (\u20133, 0), (\u20132, 0), and (1, 0). Therefore, the zeros of the function are [latex]x = \u20133, \u20132[\/latex] and [latex]1[\/latex].<\/p>\n<p>Notice that when we identify an [latex]x[\/latex]-intercept we write it as a point: e.g., (\u20132, 0). However, when we identify a zero, we write is as a solution of the equation [latex]f(x)=0[\/latex]: e.g., [latex]x=\u20132[\/latex].<\/p>\n<div class=\"textbox examples\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-1310\" class=\"standard post-1310 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<h3 data-type=\"title\">Example 3<\/h3>\n<p>Use the graph of the polynomial function of degree 6 to identify the zeros of the function.<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 277px;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201430\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"277\" height=\"358\" data-media-type=\"image\/jpg\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h4>Solution<\/h4>\n<p>The [latex]x[\/latex]-intercepts are (\u20133, 0), (\u20131, 0), and (4, 0). Therefore, the zeros are [latex]x=[\/latex] \u20133, \u20131, and 4.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Use the graph of the polynomial function of degree 6 to identify the zeros of the function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1593 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-300x300.png\" alt=\"f\\left(x\\right)=x\\left(x+2\\right)\\left(x-3\\right)\\left(x-4\\right). With x intercepts (-2,0), (0,0), (3,0), and (4,0).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/right.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm188\">Show Answer<\/span><\/p>\n<div id=\"qhjm188\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=\u20132, \\;0, \\;3, \\;4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3 data-type=\"title\"><span style=\"color: #077fab; font-size: 1.15em;\">The Number of Turning Points<\/span><\/h3>\n<p data-type=\"title\">When the [latex]y[\/latex]-values of a graph increase as [latex]x[\/latex] increases, we say the graph is increasing over the given [latex]x[\/latex]-values. Likewise, when the [latex]y[\/latex]-values of a graph decrease as [latex]x[\/latex] increases, we say the graph is decreasing over the given [latex]x[\/latex]-values. The point where a graph changes from increasing to decreasing or vice versa is called a turning point. This is illustrated in figure 12 where the graph of the function has 3 turning points.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"wp-caption aligncenter\" style=\"width: 497px;\">\n<div style=\"width: 309px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201439\/CNX_Precalc_Figure_03_04_0152.jpg\" alt=\"Graph of an odd-degree polynomial with a negative leading coefficient. It has three turning points as described above.\" width=\"299\" height=\"201\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 12. A function with turning points<\/p>\n<\/div>\n<\/div>\n<p>The number of turning points the graph of a polynomial function has is dependent upon the degree of the polynomial.<\/p>\n<div class=\"textbox shaded\">\n<p>Number of turning points<\/p>\n<p style=\"text-align: center;\">\u00a0A polynomial function [latex]f(x)[\/latex] of degree [latex]n[\/latex], has at most [latex]n-1[\/latex] turning points.<\/p>\n<\/div>\n<p>This does not mean that, for example\u00a0the polynomial function [latex]f(x)=x^8[\/latex] has 7 turning points. Rather, the function has <em>at most<\/em> 7 turning points. In reality, the graph of the function\u00a0[latex]f(x)=x^8[\/latex] has only one turning point (figure 13).<\/p>\n<div id=\"attachment_1596\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1596\" class=\"wp-image-1596 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-300x300.png\" alt=\"Graph of f(x)=x^8 only has one turning point.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/x^8.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1596\" class=\"wp-caption-text\">Figure 13. Graph of [latex]f(x)=x^8[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<ol>\n<li>\u00a0Determine the maximum number of turning points of the graph of the function [latex]f\\left(x\\right)=x^{5}-3x^{4}+x^{3}+2x^{2}+4[\/latex].<\/li>\n<li>\u00a0Use Desmos to graph the function.<\/li>\n<li>\u00a0Show the intervals of [latex]x[\/latex] on the graph when the function is increasing and when it is decreasing.<\/li>\n<li>\u00a0How many turning points are on the graph?<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>1.[latex]f\\left(x\\right)=x^{5}-3x^{4}+x^{3}+2x^{2}+4[\/latex] has degree 5, so the graph has at most 4 turning points.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">How many turning points?<\/th>\n<th style=\"width: 50%; text-align: center;\">Increasing or decreasing?<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1598\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15013001\/rightx%5E5-3x%5E4x%5E32x%5E24-300x300.png\" alt=\"Graph of function\" width=\"300\" height=\"300\" \/><\/td>\n<td style=\"width: 50%;\">3.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1599 size-medium\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-300x300.png\" alt=\"Graph of function with increasing and decreasing parts labeled.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-768x770.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec-350x351.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/inc-and-dec.png 972w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>4. There are 4 turning points.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>The graph shows the function [latex]f\\left(x\\right)=x^{4}+2x^{3}+4[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1600 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-300x300.png\" alt=\"A fourth degree polynomial decreasing on negative infinity to negative .5, and increasing on negative .5 to infinity.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/rightx^42x^34.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<ol>\n<li>\u00a0What is the maximum number of turning points of the graph of [latex]f\\left(x\\right)=x^{4}+2x^{3}+4[\/latex]?<\/li>\n<li>\u00a0Show the intervals of [latex]x[\/latex] on the graph when the function is increasing and when it is decreasing.<\/li>\n<li>\u00a0How many turning points are on the graph?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm272\">Show Answer<\/span><\/p>\n<div id=\"qhjm272\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\u00a03<\/li>\n<li><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1601\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/15014955\/Deg-4-markup-300x300.png\" alt=\"Degree 4 polynomial\" width=\"300\" height=\"300\" \/><\/li>\n<li>3. 1<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice in the last Try It, there is a flat area on the graph at [latex]x=0[\/latex]. At this point, the graph is neither increasing nor decreasing. It goes from increasing to taking a breath to increasing again. The same thing can happen when the graph is decreasing then stalls for a moment then decreases again. Such a point is called a <em><strong>point of inflection<\/strong><\/em>.<\/p>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1180\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.1.2: Polynomial Functions and Their Graphs. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using Desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/Desmos.com\">http:\/\/Desmos.com<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"3.1.2: Polynomial Functions and Their Graphs\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\"},{\"type\":\"original\",\"description\":\"All graphs created using Desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"Desmos.com\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1180","chapter","type-chapter","status-publish","hentry"],"part":1176,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1180","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":32,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1180\/revisions"}],"predecessor-version":[{"id":4735,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1180\/revisions\/4735"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/1176"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1180\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=1180"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1180"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=1180"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=1180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}