{"id":9438,"date":"2023-10-20T19:21:49","date_gmt":"2023-10-20T19:21:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=9438"},"modified":"2023-12-13T17:31:03","modified_gmt":"2023-12-13T17:31:03","slug":"quadratic-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/quadratic-functions-learn-it-2\/","title":{"raw":"Quadratic Functions: Learn It 2","rendered":"Quadratic Functions: Learn It 2"},"content":{"raw":"

Finding the Domain and Range of a Quadratic Function<\/h2>\r\n

Any number can be the input value of a quadratic function. Therefore the domain<\/strong> of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range<\/strong> will consist of all [latex]y[\/latex]-values greater than or equal to the [latex]y[\/latex]-coordinate of the vertex or less than or equal to the [latex]y[\/latex]-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\r\n

\r\n

domain and range of a quadratic function<\/h3>\r\n

The domain of any quadratic function<\/strong> is all real numbers.<\/p>\r\n

 <\/p>\r\n\r\nDetermining the range of a quadratic formula<\/strong> is different depending on which form the quadratic function is in:\r\n\r\n

 <\/p>\r\n

General Form<\/strong><\/p>\r\n

    \r\n\t
  • The range of a quadratic function written in general form with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left[f\\left(-\\frac{b}{2a}\\right),\\infty \\right)[\/latex]<\/li>\r\n\t
  • The range of a quadratic function written in general form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left(-\\infty ,f\\left(-\\frac{b}{2a}\\right)\\right][\/latex].<\/li>\r\n<\/ul>\r\n

    Standard Form<\/strong><\/p>\r\n

      \r\n\t
    • The range of a quadratic function written in standard form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge k[\/latex] or [latex][k,\\infty)[\/latex].<\/li>\r\n\t
    • The range of a quadratic function written in standard form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le k[\/latex] or or [latex](-\\infty,k][\/latex].<\/li>\r\n<\/ul>\r\n<\/section>\r\n
      How to: Determine the Domain and Range from the Vertex<\/strong>\r\n
        \r\n\t
      1. The domain of any quadratic function is all real numbers.<\/li>\r\n\t
      2. Determine whether [latex]a[\/latex] is positive or negative.
        \r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If [latex]a[\/latex] is positive, the parabola has a minimum.
        \r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If [latex]a[\/latex] is negative, the parabola has a maximum.<\/li>\r\n\t
      3. Determine the maximum or minimum value of the parabola, [latex]k[\/latex].
        \r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex].
        \r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n
        \r\n

        Find the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].<\/p>\r\n

        [reveal-answer q=\"40392\"]Show Solution[\/reveal-answer]<\/p>\r\n

        [hidden-answer a=\"40392\"]<\/p>\r\n

        As with any quadratic function, the domain is all real numbers or [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/p>\r\n

        Because [latex]a[\/latex] is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the [latex]x[\/latex]-value of the vertex.<\/p>\r\n

        [latex]h=-\\dfrac{b}{2a}=-\\dfrac{9}{2\\left(-5\\right)}=\\dfrac{9}{10}[\/latex]<\/p>\r\n

        The maximum value is given by [latex]f\\left(h\\right)[\/latex].<\/p>\r\n

        [latex]f\\left(\\dfrac{9}{10}\\right)=5{\\left(\\dfrac{9}{10}\\right)}^{2}+9\\left(\\dfrac{9}{10}\\right)-1=\\dfrac{61}{20}[\/latex]<\/p>\r\n

        The range is [latex]f\\left(x\\right)\\le \\dfrac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\dfrac{61}{20}\\right][\/latex].<\/p>\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        \r\n

        [ohm2_question hide_question_numbers=1]13807[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"

        Finding the Domain and Range of a Quadratic Function<\/h2>\n

        Any number can be the input value of a quadratic function. Therefore the domain<\/strong> of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range<\/strong> will consist of all [latex]y[\/latex]-values greater than or equal to the [latex]y[\/latex]-coordinate of the vertex or less than or equal to the [latex]y[\/latex]-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\n

        \n

        domain and range of a quadratic function<\/h3>\n

        The domain of any quadratic function<\/strong> is all real numbers.<\/p>\n

         <\/p>\n

        Determining the range of a quadratic formula<\/strong> is different depending on which form the quadratic function is in:<\/p>\n

         <\/p>\n

        General Form<\/strong><\/p>\n

          \n
        • The range of a quadratic function written in general form with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left[f\\left(-\\frac{b}{2a}\\right),\\infty \\right)[\/latex]<\/li>\n
        • The range of a quadratic function written in general form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left(-\\infty ,f\\left(-\\frac{b}{2a}\\right)\\right][\/latex].<\/li>\n<\/ul>\n

          Standard Form<\/strong><\/p>\n

            \n
          • The range of a quadratic function written in standard form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] with a positive [latex]a[\/latex] value is [latex]f\\left(x\\right)\\ge k[\/latex] or [latex][k,\\infty)[\/latex].<\/li>\n
          • The range of a quadratic function written in standard form with a negative [latex]a[\/latex] value is [latex]f\\left(x\\right)\\le k[\/latex] or or [latex](-\\infty,k][\/latex].<\/li>\n<\/ul>\n<\/section>\n
            How to: Determine the Domain and Range from the Vertex<\/strong><\/p>\n
              \n
            1. The domain of any quadratic function is all real numbers.<\/li>\n
            2. Determine whether [latex]a[\/latex] is positive or negative.
              \n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If [latex]a[\/latex] is positive, the parabola has a minimum.
              \n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If [latex]a[\/latex] is negative, the parabola has a maximum.<\/li>\n
            3. Determine the maximum or minimum value of the parabola, [latex]k[\/latex].
              \n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex].
              \n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0If the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\n<\/ol>\n<\/section>\n
              \n

              Find the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].<\/p>\n