{"id":935,"date":"2015-11-12T18:37:58","date_gmt":"2015-11-12T18:37:58","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=935"},"modified":"2017-03-31T20:36:14","modified_gmt":"2017-03-31T20:36:14","slug":"graph-functions-using-reflections-about-the-x-axis-and-the-y-axis","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/graph-functions-using-reflections-about-the-x-axis-and-the-y-axis\/","title":{"raw":"Graph functions using reflections about the x-axis and the y-axis","rendered":"Graph functions using reflections about the x-axis and the y-axis"},"content":{"raw":"

Another transformation that can be applied to a function is a reflection over the x<\/em>- or y<\/em>-axis. A vertical reflection<\/strong> reflects a graph vertically across the x<\/em>-axis, while a horizontal reflection<\/strong> reflects a graph horizontally across the y<\/em>-axis. The reflections are shown in Figure 9.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 9.<\/b> Vertical and horizontal reflections of a function.[\/caption]\r\n

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y<\/em>-axis.<\/p>\r\n\r\n

\r\n

A General Note: Reflections<\/h3>\r\n

Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] is a vertical reflection<\/span> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about (or over, or through) the x<\/em>-axis.<\/p>\r\n

Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] is a horizontal reflection<\/span> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about the y<\/em>-axis.<\/p>\r\n\r\n<\/div>\r\n

\r\n

How To: Given a function, reflect the graph both vertically and horizontally.\r\n<\/strong><\/h3>\r\n
  1. Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the x<\/em>-axis.<\/li>\r\n\t
  2. Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the y<\/em>-axis.<\/li>\r\n<\/ol><\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 7: Reflecting a Graph Horizontally and Vertically<\/h3>\r\n

    Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex] (a) vertically and (b) horizontally.<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solution<\/h3>\r\na. Reflecting the graph vertically means that each output value will be reflected over the horizontal t-<\/em>axis as shown in Figure 10.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"GraphFigure 10.\u00a0<\/b>Vertical reflection of the square root function[\/caption]\r\n

    Because each output value is the opposite of the original output value, we can write<\/p>\r\n\r\n

    [latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt{t}[\/latex]<\/div>\r\n

    Notice that this is an outside change, or vertical shift, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.<\/p>\r\nb.\r\n\r\nReflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 11.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"GraphFigure 11.<\/b> Horizontal reflection of the square root function[\/caption]\r\n

    Because each input value is the opposite of the original input value, we can write<\/p>\r\n\r\n

    [latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/div>\r\n

    Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.<\/p>\r\n

    Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\\left[0,\\infty \\right)[\/latex] and range [latex]\\left[0,\\infty \\right)[\/latex], the vertical reflection gives the [latex]V\\left(t\\right)[\/latex] function the range [latex]\\left(-\\infty ,0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,0\\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Try It 2<\/h3>\r\n

    Reflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

    \r\n
    \r\n
    \r\n

    Example 8: Reflecting a Tabular Function Horizontally and Vertically<\/h3>\r\n

    A function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.<\/p>\r\n\r\n

    1. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\r\n\t
    2. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\r\n<\/ol><\/colgroup>
      [latex]x[\/latex]<\/strong><\/td>\r\n2<\/td>\r\n4<\/td>\r\n6<\/td>\r\n8<\/td>\r\n<\/tr>
      [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n1<\/td>\r\n3<\/td>\r\n7<\/td>\r\n11<\/td>\r\n<\/tr><\/tbody><\/table><\/div>\r\n
      \r\n

      Solution<\/h3>\r\n
      1. \r\n

        For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the x<\/em>-values stay the same and each output value will be the opposite of the original output value.<\/p>\r\n\r\n<\/colgroup>
        [latex]x[\/latex]<\/strong><\/td>\r\n2<\/td>\r\n4<\/td>\r\n6<\/td>\r\n8<\/td>\r\n<\/tr>
        [latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n\u20131<\/td>\r\n\u20133<\/td>\r\n\u20137<\/td>\r\n\u201311<\/td>\r\n<\/tr><\/tbody><\/table><\/li>\r\n\t
      2. \r\n

        For [latex]h\\left(x\\right)[\/latex], the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\\left(x\\right)[\/latex] values stay the same as the [latex]f\\left(x\\right)[\/latex] values.<\/p>\r\n\r\n<\/colgroup>
        [latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22124<\/td>\r\n\u22126<\/td>\r\n\u22128<\/td>\r\n<\/tr>
        [latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n1<\/td>\r\n3<\/td>\r\n7<\/td>\r\n11<\/td>\r\n<\/tr><\/tbody><\/table><\/li>\r\n<\/ol><\/div>\r\n<\/div>\r\n<\/div>\r\n
        \r\n

        Try It 3<\/h3>\r\n<\/colgroup>
        [latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n0<\/td>\r\n2<\/td>\r\n4<\/td>\r\n<\/tr>
        [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n5<\/td>\r\n10<\/td>\r\n15<\/td>\r\n20<\/td>\r\n<\/tr><\/tbody><\/table>

        Using the function [latex]f\\left(x\\right)[\/latex] given in the table above, create a table for the functions below.<\/p>\r\n

        a. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n

        b. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

        Another transformation that can be applied to a function is a reflection over the x<\/em>– or y<\/em>-axis. A vertical reflection<\/strong> reflects a graph vertically across the x<\/em>-axis, while a horizontal reflection<\/strong> reflects a graph horizontally across the y<\/em>-axis. The reflections are shown in Figure 9.<\/p>\n

        \"Graph<\/p>\n

        Figure 9.<\/b> Vertical and horizontal reflections of a function.<\/p>\n<\/div>\n

        Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x<\/em>-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y<\/em>-axis.<\/p>\n

        \n

        A General Note: Reflections<\/h3>\n

        Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex] is a vertical reflection<\/span> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about (or over, or through) the x<\/em>-axis.<\/p>\n

        Given a function [latex]f\\left(x\\right)[\/latex], a new function [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex] is a horizontal reflection<\/span> of the function [latex]f\\left(x\\right)[\/latex], sometimes called a reflection about the y<\/em>-axis.<\/p>\n<\/div>\n

        \n

        How To: Given a function, reflect the graph both vertically and horizontally.
        \n<\/strong><\/h3>\n
          \n
        1. Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the x<\/em>-axis.<\/li>\n
        2. Multiply all inputs by \u20131 for a horizontal reflection. The new graph is a reflection of the original graph about the y<\/em>-axis.<\/li>\n<\/ol>\n<\/div>\n
          \n
          \n
          \n

          Example 7: Reflecting a Graph Horizontally and Vertically<\/h3>\n

          Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex] (a) vertically and (b) horizontally.<\/p>\n<\/div>\n

          \n

          Solution<\/h3>\n

          a. Reflecting the graph vertically means that each output value will be reflected over the horizontal t-<\/em>axis as shown in Figure 10.<\/p>\n

          \"Graph<\/p>\n

          Figure 10.\u00a0<\/b>Vertical reflection of the square root function<\/p>\n<\/div>\n

          Because each output value is the opposite of the original output value, we can write<\/p>\n

          [latex]V\\left(t\\right)=-s\\left(t\\right)\\text{ or }V\\left(t\\right)=-\\sqrt{t}[\/latex]<\/div>\n

          Notice that this is an outside change, or vertical shift, that affects the output [latex]s\\left(t\\right)[\/latex] values, so the negative sign belongs outside of the function.<\/p>\n

          b.<\/p>\n

          Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 11.<\/p>\n

          \"Graph<\/p>\n

          Figure 11.<\/b> Horizontal reflection of the square root function<\/p>\n<\/div>\n

          Because each input value is the opposite of the original input value, we can write<\/p>\n

          [latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/div>\n

          Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.<\/p>\n

          Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [latex]\\left[0,\\infty \\right)[\/latex] and range [latex]\\left[0,\\infty \\right)[\/latex], the vertical reflection gives the [latex]V\\left(t\\right)[\/latex] function the range [latex]\\left(-\\infty ,0\\right][\/latex] and the horizontal reflection gives the [latex]H\\left(t\\right)[\/latex] function the domain [latex]\\left(-\\infty ,0\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          \n

          Try It 2<\/h3>\n

          Reflect the graph of [latex]f\\left(x\\right)=|x - 1|[\/latex] (a) vertically and (b) horizontally.<\/p>\n

          Solution<\/a><\/p>\n<\/div>\n

          \n
          \n
          \n

          Example 8: Reflecting a Tabular Function Horizontally and Vertically<\/h3>\n

          A function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.<\/p>\n

            \n
          1. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/li>\n
          2. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/li>\n<\/ol>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
            [latex]x[\/latex]<\/strong><\/td>\n2<\/td>\n4<\/td>\n6<\/td>\n8<\/td>\n<\/tr>\n
            [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n1<\/td>\n3<\/td>\n7<\/td>\n11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
            \n

            Solution<\/h3>\n
              \n
            1. \n

              For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the x<\/em>-values stay the same and each output value will be the opposite of the original output value.<\/p>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
              [latex]x[\/latex]<\/strong><\/td>\n2<\/td>\n4<\/td>\n6<\/td>\n8<\/td>\n<\/tr>\n
              [latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\n\u20131<\/td>\n\u20133<\/td>\n\u20137<\/td>\n\u201311<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
            2. \n

              For [latex]h\\left(x\\right)[\/latex], the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the [latex]h\\left(x\\right)[\/latex] values stay the same as the [latex]f\\left(x\\right)[\/latex] values.<\/p>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
              [latex]x[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22124<\/td>\n\u22126<\/td>\n\u22128<\/td>\n<\/tr>\n
              [latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\n1<\/td>\n3<\/td>\n7<\/td>\n11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
              \n

              Try It 3<\/h3>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
              [latex]x[\/latex]<\/strong><\/td>\n\u22122<\/td>\n0<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n
              [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\n5<\/td>\n10<\/td>\n15<\/td>\n20<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Using the function [latex]f\\left(x\\right)[\/latex] given in the table above, create a table for the functions below.<\/p>\n

              a. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n

              b. [latex]h\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n

              Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

              \n\t\t\t

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