{"id":1512,"date":"2024-05-24T20:04:15","date_gmt":"2024-05-24T20:04:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1512"},"modified":"2025-08-13T16:24:49","modified_gmt":"2025-08-13T16:24:49","slug":"piecewise-functions-and-rates-of-change-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/piecewise-functions-and-rates-of-change-learn-it-2\/","title":{"raw":"Rates of Change and Behavior of Graphs: Learn It 2","rendered":"Rates of Change and Behavior of Graphs: Learn It 2"},"content":{"raw":"

Behaviors of Functions<\/h2>\r\n

Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h3>\r\nAs part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.\r\n\r\nWe say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below\u00a0shows examples of increasing and decreasing intervals on a function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].[\/caption]While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a local minimum<\/strong>. The plural form is \"local minima.\" Together, local maxima and minima are called local extrema<\/strong>, or local extreme values, of the function. (The singular form is \"extremum.\") Often, the term local<\/em> is replaced by the term relative<\/em>. In this text, we will use the term local<\/em>.\r\n\r\n
A function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/section>
\r\n

local minima and local maxima<\/h3>\r\n
    \r\n \t
  • A function [latex]f[\/latex] is an increasing function<\/strong> on an open interval if [latex]f\\left(b\\right) > f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b > a[\/latex].<\/li>\r\n \t
  • A function [latex]f[\/latex] is a decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right) < f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b > a[\/latex].<\/li>\r\n<\/ul>\r\n
      \r\n \t
    • A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a < b < c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex].<\/li>\r\n \t
    • Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a < b < c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].<\/li>\r\n<\/ul>\r\n \r\n\r\n<\/section>
      For the function below, the local maximum is [latex]16[\/latex], and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n<\/section>To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below\u00a0illustrates these ideas for a local maximum.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Definition of a local maximum.[\/caption]\r\n\r\n
      Given the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n[reveal-answer q=\"927495\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"927495\"]We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.In interval notation, we would say the function appears to be increasing on the interval [latex](1,3)[\/latex] and the interval [latex]\\left(4,\\infty \\right)[\/latex].\r\n[latex]\\\\[\/latex]\r\nAnalysis of the Solution\r\n<\/strong>[latex]\\\\[\/latex]Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).[\/hidden-answer]<\/section>
      [ohm_question hide_question_numbers=1]292207[\/ohm_question]<\/section>
      The behavior of the function values of the graph of a function is read over the x-axis, from left to right. That is,\r\n
        \r\n \t
      • a function is said to be increasing if its function values increase as x increases;<\/li>\r\n \t
      • a function is said to be decreasing if its function values decrease as x increases.<\/li>\r\n<\/ul>\r\n<\/section>
        Graph the function [latex]f\\left(x\\right)=\\dfrac{2}{x}+\\dfrac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.[reveal-answer q=\"818075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"818075\"]Using technology, we find that the graph of the function looks like that below. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a reciprocal function[\/caption]\r\n\r\nAnalysis of the Solution<\/strong>[latex]\\\\[\/latex]Most graphing calculators and graphing utilities can estimate the location of maxima and minima. The graph below\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Graph Graph of the reciprocal function on a graphing calculator[\/caption]\r\n\r\nBased on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex] and [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact locations of the extrema are at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)[\/hidden-answer]\r\n\r\n<\/section>
        Recall that points on the graph of a function are ordered pairs in the form of\r\n

        [latex]\\left(\\text{input, output}\\right) \\quad = \\quad \\left(x, f(x)\\right)[\/latex].<\/p>\r\nIf a function's graph has a local minimum or maximum at some point [latex]\\left(x, f(x)\\right)[\/latex], we say\r\n

        \"the extrema\u00a0occurs at <\/em>[latex]x[\/latex], and that the minimum or maximum\u00a0is\u00a0<\/em>[latex]f(x)[\/latex].\"<\/p>\r\n\r\n<\/section>

        Graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.[reveal-answer q=\"466198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"466198\"]The local maximum appears to occur at [latex]\\left(-1,28\\right)[\/latex], and the local minimum occurs at [latex]\\left(5,-80\\right)[\/latex]. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].\r\n\r\n[caption id=\"attachment_6728\" align=\"aligncenter\" width=\"490\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n\u00a0<\/span>[\/hidden-answer]<\/section>
        [ohm_question hide_question_numbers=1]293825[\/ohm_question]<\/section>","rendered":"

        Behaviors of Functions<\/h2>\n

        Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h3>\n

        As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.<\/p>\n

        We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below\u00a0shows examples of increasing and decreasing intervals on a function.<\/p>\n

        \"Graph
        The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].<\/figcaption><\/figure>\n

        While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a local minimum<\/strong>. The plural form is “local minima.” Together, local maxima and minima are called local extrema<\/strong>, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local<\/em> is replaced by the term relative<\/em>. In this text, we will use the term local<\/em>.<\/p>\n

        A function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/section>\n
        \n

        local minima and local maxima<\/h3>\n
          \n
        • A function [latex]f[\/latex] is an increasing function<\/strong> on an open interval if [latex]f\\left(b\\right) > f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b > a[\/latex].<\/li>\n
        • A function [latex]f[\/latex] is a decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right) < f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b > a[\/latex].<\/li>\n<\/ul>\n
            \n
          • A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a < b < c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex].<\/li>\n
          • Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a < b < c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].<\/li>\n<\/ul>\n

             <\/p>\n<\/section>\n

            For the function below, the local maximum is [latex]16[\/latex], and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].<\/p>\n
            \"Graph
            Graph of a polynomial<\/figcaption><\/figure>\n<\/section>\n

            To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below\u00a0illustrates these ideas for a local maximum.<\/p>\n

            \"Graph
            Definition of a local maximum.<\/figcaption><\/figure>\n
            Given the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.<\/p>\n
            \"Graph
            Graph of a polynomial<\/figcaption><\/figure>\n