{"id":4253,"date":"2024-09-30T12:55:49","date_gmt":"2024-09-30T12:55:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4253"},"modified":"2025-08-15T02:26:52","modified_gmt":"2025-08-15T02:26:52","slug":"introduction-to-power-and-polynomial-functions-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-power-and-polynomial-functions-learn-it-6\/","title":{"raw":"Introduction to Power and Polynomial Functions: Learn It 6","rendered":"Introduction to Power and Polynomial Functions: Learn It 6"},"content":{"raw":"

Identifying Local Behavior of Polynomial Functions<\/h2>\r\n[caption id=\"attachment_1911\" align=\"alignright\" width=\"400\"]\"\" Graph of a polynomial with key features labeled[\/caption]\r\n\r\nIn addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. <\/span>In particular, we are interested in locations where graph behavior changes. A turning point<\/strong> is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/span>\r\n\r\nWe are also interested in the intercepts<\/strong>. As with all functions, the [latex]y[\/latex]-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 [latex]y[\/latex]-intercept [latex]\\left(0,{a}_{0}\\right)[\/latex]. The [latex]x[\/latex]-<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one [latex]x[\/latex]-<\/em>intercept.\u00a0<\/span>\r\n\r\n
Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the [latex]y[\/latex]\u00a0and [latex]x[\/latex]-intercepts.[reveal-answer q=\"701514\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"701514\"]The y-<\/em>intercept occurs when the input is zero, so substitute [latex]0[\/latex] for [latex]x[\/latex].\r\n

[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 [latex]y[\/latex]-<\/em>intercept is [latex](0, 8)[\/latex].\r\n\r\nThe [latex]x[\/latex]-intercepts occur when the output [latex]f(x)[\/latex] is zero.\r\n

[latex]0=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex]<\/p>\r\n

[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>\r\nThe\u00a0[latex]x[\/latex]-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[caption id=\"\" align=\"aligncenter\" width=\"350\"]\"Graph Graph of a polynomial with intercepts labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

Determining the Number of Turning Points and Intercepts from the Degree of the Polynomial<\/h3>\r\nA continuous function<\/strong> has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A 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 [latex]x[\/latex]-intercepts and the number of turning points. A polynomial function of [latex]n[\/latex]th degree is the product of [latex]n[\/latex] factors, so it will have at most [latex]n[\/latex] roots or zeros, or [latex]x[\/latex]-intercepts. The graph of the polynomial function of degree [latex]n[\/latex] must have at most [latex]n-1[\/latex] turning 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\r\n
\r\n

intercepts and turning points of polynomial functions<\/h3>\r\n
    \r\n \t
  • A 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
  • The [latex]y[\/latex]-<\/em>intercept is the point where the function has an input value of zero.<\/li>\r\n \t
  • The [latex]x[\/latex]-intercepts are the points where the output value is zero.<\/li>\r\n \t
  • A polynomial of degree [latex]n[\/latex]\u00a0will have, at most, [latex]n[\/latex]\u00a0[latex]x[\/latex]-intercepts and [latex]n \u2013 1[\/latex] turning points.<\/li>\r\n<\/ul>\r\n<\/section>
    Why do we use the phrase \"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?[reveal-answer q=\"232068\"]more[\/reveal-answer]\r\n[hidden-answer a=\"232068\"]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 at most<\/em> [latex]n[\/latex] roots and [latex]n-1[\/latex] turning points.\r\n[latex]\\\\[\/latex]\r\nWe know this function has non-real roots since the discriminant of the quadratic formula is negative. This means that this [latex]2^{\\text{nd}}[\/latex] degree 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[latex]\\\\[\/latex]\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 [latex]x[\/latex]-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 [latex]x[\/latex]-axis more than twice.[\/hidden-answer]<\/section>
    Without graphing the function, determine the local behavior of the function by finding the maximum number of [latex]x[\/latex]-intercepts and turning points for [latex]f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}[\/latex].[reveal-answer q=\"96529\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96529\"]The polynomial has a degree of [latex]10[\/latex], so there are at most [latex]10[\/latex] [latex]x[\/latex]-intercepts and at most [latex]10 \u2013 1 = 9[\/latex] turning points.[\/hidden-answer]<\/section>\r\n

    The Whole Picture<\/h3>\r\nNow we can bring the two concepts of turning points and intercepts together to get a general picture of the behavior of polynomial functions. \u00a0These types of analyses on polynomials developed before the advent of mass computing as a way to quickly understand the general behavior of a polynomial function. We now have a quick way, with computers, to graph and calculate important characteristics of polynomials that once took a lot of algebra.\r\n\r\n
    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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n[reveal-answer q=\"200904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200904\"]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a polynomial with turning points and intercepts labeled[\/caption]\r\n\r\nThe end behavior of the graph tells us this is the graph of an even-degree polynomial. The graph has [latex]2[\/latex] [latex]x[\/latex]-intercepts, suggesting a degree of [latex]2[\/latex] or greater, and [latex]3[\/latex] turning points, suggesting a degree of [latex]4[\/latex] or greater. Based on this, it would be reasonable to conclude that the degree is even and at least [latex]4[\/latex].[\/hidden-answer]<\/section>Now you try to determine the least possible degree of a polynomial given its graph.\r\n\r\n
    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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n[reveal-answer q=\"492375\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"492375\"]The end behavior indicates an odd-degree polynomial function; there are [latex]3[\/latex] [latex]x[\/latex]-intercepts and [latex]2[\/latex] turning points, so the degree is odd and at least [latex]3[\/latex]. Because of the end behavior, we know that the leading coefficient must be negative.[\/hidden-answer]<\/section>
    [ohm2_question hide_question_numbers=1]24606[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]24607[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]24608[\/ohm2_question]<\/section><\/section>","rendered":"

    Identifying Local Behavior of Polynomial Functions<\/h2>\n
    \"\"
    Graph of a polynomial with key features labeled<\/figcaption><\/figure>\n

    In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. <\/span>In particular, we are interested in locations where graph behavior changes. A turning point<\/strong> is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/span><\/p>\n

    We are also interested in the intercepts<\/strong>. As with all functions, the [latex]y[\/latex]-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 [latex]y[\/latex]-intercept [latex]\\left(0,{a}_{0}\\right)[\/latex]. The [latex]x[\/latex]–<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one [latex]x[\/latex]–<\/em>intercept.\u00a0<\/span><\/p>\n

    Given the polynomial function [latex]f\\left(x\\right)=\\left(x - 2\\right)\\left(x+1\\right)\\left(x - 4\\right)[\/latex], written in factored form for your convenience, determine the [latex]y[\/latex]\u00a0and [latex]x[\/latex]-intercepts.<\/p>\n