{"id":1524,"date":"2024-05-24T20:52:02","date_gmt":"2024-05-24T20:52:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1524"},"modified":"2025-08-13T16:31:13","modified_gmt":"2025-08-13T16:31:13","slug":"piecewise-functions-and-rates-of-change-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/piecewise-functions-and-rates-of-change-learn-it-4\/","title":{"raw":"Rates of Change and Behavior of Graphs: Learn It 4","rendered":"Rates of Change and Behavior of Graphs: Learn It 4"},"content":{"raw":"

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

Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h3>\r\nThere is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the absolute maximum <\/strong>and absolute minimum<\/strong>, respectively.\r\n\r\nTo locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a segment of a parabola[\/caption]\r\n\r\n
\r\n

absolute maxima and minima<\/h3>\r\n
    \r\n \t
  • The absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\r\n \t
  • The absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
    Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/section>
    For the function [latex]f[\/latex] shown below, find all absolute maxima and minima.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n[reveal-answer q=\"461473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"461473\"]Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

    Behaviors of Functions Cont.<\/h2>\n

    Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h3>\n

    There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the absolute maximum <\/strong>and absolute minimum<\/strong>, respectively.<\/p>\n

    To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function.<\/p>\n

    \"Graph
    Graph of a segment of a parabola<\/figcaption><\/figure>\n
    \n

    absolute maxima and minima<\/h3>\n
      \n
    • The absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\n
    • The absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\n<\/ul>\n<\/section>\n
      Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/section>\n
      For the function [latex]f[\/latex] shown below, find all absolute maxima and minima.<\/p>\n
      \"Graph
      Graph of a polynomial<\/figcaption><\/figure>\n