{"id":211,"date":"2023-06-21T13:22:44","date_gmt":"2023-06-21T13:22:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/local-behavior-of-polynomial-functions\/"},"modified":"2023-07-04T03:57:37","modified_gmt":"2023-07-04T03:57:37","slug":"local-behavior-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/local-behavior-of-polynomial-functions\/","title":{"raw":"\u25aa   Local Behavior of Polynomial Functions","rendered":"\u25aa   Local Behavior of Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\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\n<h2>Identifying Local Behavior of Polynomial Functions<\/h2>\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<h2>Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h2>\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 examples\">\r\n<h3>tip for success<\/h3>\r\nWhy do we use the phrase \"<em>at most<\/em> [latex]n[\/latex]\" when describing the number of real roots (x-intercepts) of the graph of an [latex]n^{\\text{th}}[\/latex] degree polynomial? Can it have fewer?\r\n\r\n[reveal-answer q=\"232068\"]more[\/reveal-answer]\r\n[hidden-answer a=\"232068\"]\r\n\r\nEx. Consider the graph of the polynomial function [latex]f(x)=x^2-x+1[\/latex]. The function is a [latex]2^{\\text{nd}}[\/latex] degree polynomial, so it must have\u00a0<em>at most<\/em> [latex]n[\/latex] roots and [latex]n-1[\/latex] turning points.\r\n\r\nWe know this function has non-real roots since the discriminant of the quadratic formula is negative. This means that this\u00a0[latex]2^{\\text{nd}}[\/latex] polynomial has no real roots (apply the quadratic formula to prove this to yourself if needed). That is, it has no x-intercepts. But it does have two distinct complex roots.\r\n\r\nCan you picture the graph of a quadratic function with one distinct real root? Two? But you can also see that there will never be more than two x-intercepts. Since a parabola (the graph of a [latex]2^{\\text{nd}}[\/latex] degree polynomial) has only one turning point, it can't cross the x-axis more than twice.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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\r\n[ohm_question]123739[\/ohm_question]\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<h2>Using the Principle of Zero Products to Find the Roots of a Polynomial in Factored Form<\/h2>\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 examples\">\r\n<h3>recall special factoring techniques<\/h3>\r\nIn the example above, you needed to factor a [latex]4^{\\text{th}}[\/latex] polynomial in order to use the zero-product principle. Some, but not all, polynomials of degree higher than [latex]2[\/latex] are factorable using special techniques you learned earlier in the course. What technique was used to factor [latex]x^4-4x^2-45[\/latex]?\r\n\r\n[reveal-answer q=\"854513\"]more[\/reveal-answer]\r\n[hidden-answer a=\"854513\"]\r\n\r\nFirst, we can recognize that the smallest power in this trinomial is exactly half the larger power. This allows us to make a substitution. We can let [latex]u=x^2[\/latex]. Then [latex]u^2 = x^4[\/latex].\r\n\r\nWe now have the factorable quadratic equation\r\n\r\n[latex]\\begin{align} 0 &amp;= u^2-4u-45 \\\\ 0 &amp;= (u-9)(u+5)\\end{align}[\/latex].\r\n\r\nBack substituting [latex]u=x^2[\/latex] into the function yields the factored form of the equation to\u00a0which we can apply the zero-product principle.\r\n\r\n[latex]\\begin{align}0 &amp;= (x^2 - 9)(x^2+5) \\\\ 0 &amp;= (x+3)(x-3)(x^2+5)\\end{align}[\/latex].\r\n\r\nSo, [latex]x=-3, 3, \\text{ and } \\pm \\sqrt{5}i[\/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\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<h2>The Whole Picture<\/h2>\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<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\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\r\n[ohm_question]15937[\/ohm_question]\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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\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<h2>Identifying Local Behavior of Polynomial Functions<\/h2>\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<h2>Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h2>\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 examples\">\n<h3>tip for success<\/h3>\n<p>Why do we use the phrase &#8220;<em>at most<\/em> [latex]n[\/latex]&#8221; when describing the number of real roots (x-intercepts) of the graph of an [latex]n^{\\text{th}}[\/latex] degree polynomial? Can it have fewer?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232068\">more<\/span><\/p>\n<div id=\"q232068\" class=\"hidden-answer\" style=\"display: none\">\n<p>Ex. Consider the graph of the polynomial function [latex]f(x)=x^2-x+1[\/latex]. The function is a [latex]2^{\\text{nd}}[\/latex] degree polynomial, so it must have\u00a0<em>at most<\/em> [latex]n[\/latex] roots and [latex]n-1[\/latex] turning points.<\/p>\n<p>We know this function has non-real roots since the discriminant of the quadratic formula is negative. This means that this\u00a0[latex]2^{\\text{nd}}[\/latex] polynomial has no real roots (apply the quadratic formula to prove this to yourself if needed). That is, it has no x-intercepts. But it does have two distinct complex roots.<\/p>\n<p>Can you picture the graph of a quadratic function with one distinct real root? Two? But you can also see that there will never be more than two x-intercepts. Since a parabola (the graph of a [latex]2^{\\text{nd}}[\/latex] degree polynomial) has only one turning point, it can&#8217;t cross the x-axis more than twice.<\/p>\n<\/div>\n<\/div>\n<\/div>\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=\"ohm123739\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=123739&theme=oea&iframe_resize_id=ohm123739&show_question_numbers\" width=\"100%\" height=\"150\"><\/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<h2>Using the Principle of Zero Products to Find the Roots of a Polynomial in Factored Form<\/h2>\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 examples\">\n<h3>recall special factoring techniques<\/h3>\n<p>In the example above, you needed to factor a [latex]4^{\\text{th}}[\/latex] polynomial in order to use the zero-product principle. Some, but not all, polynomials of degree higher than [latex]2[\/latex] are factorable using special techniques you learned earlier in the course. What technique was used to factor [latex]x^4-4x^2-45[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q854513\">more<\/span><\/p>\n<div id=\"q854513\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we can recognize that the smallest power in this trinomial is exactly half the larger power. This allows us to make a substitution. We can let [latex]u=x^2[\/latex]. Then [latex]u^2 = x^4[\/latex].<\/p>\n<p>We now have the factorable quadratic equation<\/p>\n<p>[latex]\\begin{align} 0 &= u^2-4u-45 \\\\ 0 &= (u-9)(u+5)\\end{align}[\/latex].<\/p>\n<p>Back substituting [latex]u=x^2[\/latex] into the function yields the factored form of the equation to\u00a0which we can apply the zero-product principle.<\/p>\n<p>[latex]\\begin{align}0 &= (x^2 - 9)(x^2+5) \\\\ 0 &= (x+3)(x-3)(x^2+5)\\end{align}[\/latex].<\/p>\n<p>So, [latex]x=-3, 3, \\text{ and } \\pm \\sqrt{5}i[\/latex].<\/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<h2>The Whole Picture<\/h2>\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<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=\"ohm15937\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15937&theme=oea&iframe_resize_id=ohm15937&show_question_numbers\" width=\"100%\" height=\"150\"><\/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\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-211\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 123739. <strong>Provided by<\/strong>: Lumen Learning. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 15937. <strong>Authored by<\/strong>: Sousa, James. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: iMathAS\/ WAMAP\/ MyOpenMath Community License (GPL + CC-BY)<\/li><li>Turning Points and X-Intercepts of a Polynomial Function. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9WW0EetLD4Q\">https:\/\/youtu.be\/9WW0EetLD4Q<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 15937\",\"author\":\"Sousa, James\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"iMathAS\/ WAMAP\/ MyOpenMath Community License (GPL + CC-BY)\"},{\"type\":\"cc\",\"description\":\"Turning Points and X-Intercepts of a Polynomial Function\",\"author\":\"Sousa, James 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