{"id":1600,"date":"2024-05-30T00:42:59","date_gmt":"2024-05-30T00:42:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1600"},"modified":"2025-08-13T23:01:52","modified_gmt":"2025-08-13T23:01:52","slug":"transformations-of-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/transformations-of-functions-learn-it-3\/","title":{"raw":"Transformations of Functions: Learn It 3","rendered":"Transformations of Functions: Learn It 3"},"content":{"raw":"

Graphing Functions Using Reflections about the Axes<\/h2>\r\nAnother transformation that can be applied to a function is a reflection over the [latex]x[\/latex]- or [latex]y[\/latex]-axis. A vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis, while a horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the vertical and horizontal reflection of a function[\/caption]\r\n\r\n
Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the [latex]x[\/latex]-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the [latex]y[\/latex]-axis.<\/section>
\r\n

reflections<\/h3>\r\nA vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis. <\/span>This transformation changes the sign of the output values of [latex]f(x)[\/latex].\r\n
    \r\n \t
  • If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]x[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\r\n<\/ul>\r\n

    [latex]g(x) = -f(x)[\/latex]<\/p>\r\n \r\n\r\nA horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis. <\/span>This transformation changes the sign of the input values of [latex]f(x)[\/latex].\r\n

      \r\n \t
    • If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]y[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\r\n<\/ul>\r\n

      [latex]g(x) = f(-x)[\/latex]<\/p>\r\n\r\n<\/section>

      How To: Given a function, reflect the graph both vertically and horizontally.<\/strong>\r\n
        \r\n \t
      1. Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the [latex]x[\/latex]-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 [latex]y[\/latex]-axis.<\/li>\r\n<\/ol>\r\n<\/section>
        Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex]\r\n
          \r\n \t
        1. vertically<\/li>\r\n \t
        2. horizontally<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"211400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211400\"]\r\n
            \r\n \t
          1. Reflecting the graph vertically means that each output value will be reflected over the horizontal [latex]t[\/latex]-<\/em>axis as shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"777\"]\"Graph Vertical reflection of the square root function[\/caption]\r\n\r\nTable of values<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            [latex]t[\/latex]<\/td>\r\n[latex]s(t) = \\sqrt{t}[\/latex]<\/td>\r\nReflected Function [latex]V(t)[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]1[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]4[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBecause each output value is the opposite of the original output value, we can write\r\n

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

          2. Reflecting horizontally means that each input value will be reflected over the vertical axis as shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"777\"]\"Graph Horizontal reflection of the square root function[\/caption]\r\n\r\nTable for [latex]s(t) = \\sqrt{t}[\/latex]<\/strong>\r\n\r\n\r\n\r\n\r\n
            [latex]t[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]s(t) = \\sqrt{t}[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable for reflected function\u00a0<\/strong>\r\n\r\n\r\n\r\n\r\n
            [latex]t[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]H(t)[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBecause each input value is the opposite of the original input value, we can write\r\n

            [latex]H\\left(t\\right)=s\\left(-t\\right)\\text{ or }H\\left(t\\right)=\\sqrt{-t}[\/latex]<\/p>\r\nNotice 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.<\/li>\r\n<\/ol>\r\nNote 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].[\/hidden-answer]\r\n\r\n<\/section>

            A function [latex]f\\left(x\\right)[\/latex] is given. Create a table for the functions below.\r\n
              \r\n \t
            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>\r\n<\/colgroup>\r\n\r\n\r\n\r\n
              [latex]x[\/latex]<\/strong><\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n
              [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"608272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"608272\"]\r\n
                \r\n \t
              1. For [latex]g\\left(x\\right)[\/latex], the negative sign outside the function indicates a vertical reflection, so the [latex]x[\/latex]-values stay the same and each output value will be the opposite of the original output value.\r\n<\/colgroup>\r\n\r\n\r\n\r\n
                [latex]x[\/latex]<\/strong><\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n
                [latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n[latex]\u20131[\/latex]<\/td>\r\n[latex]\u20133[\/latex]<\/td>\r\n[latex]\u20137[\/latex]<\/td>\r\n[latex]\u201311[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
              2. 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.\r\n<\/colgroup>\r\n\r\n\r\n\r\n
                [latex]x[\/latex]<\/strong><\/td>\r\n[latex]\u22122[\/latex]<\/td>\r\n[latex]\u22124[\/latex]<\/td>\r\n[latex]\u22126[\/latex]<\/td>\r\n[latex]\u22128[\/latex]<\/td>\r\n<\/tr>\r\n
                [latex]h\\left(x\\right)[\/latex] <\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
                [ohm2_question hide_question_numbers=1]19147[\/ohm2_question]<\/section>
                [ohm2_question hide_question_numbers=1]19148[\/ohm2_question]<\/section>
                [ohm2_question hide_question_numbers=1]19149[\/ohm2_question]<\/section>","rendered":"

                Graphing Functions Using Reflections about the Axes<\/h2>\n

                Another transformation that can be applied to a function is a reflection over the [latex]x[\/latex]– or [latex]y[\/latex]-axis. A vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis, while a horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis.<\/p>\n

                \"Graph
                Graph of the vertical and horizontal reflection of a function<\/figcaption><\/figure>\n
                Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the [latex]x[\/latex]-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the [latex]y[\/latex]-axis.<\/section>\n
                \n

                reflections<\/h3>\n

                A vertical reflection<\/strong> reflects a graph vertically across the [latex]x[\/latex]-axis. <\/span>This transformation changes the sign of the output values of [latex]f(x)[\/latex].<\/p>\n

                  \n
                • If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]x[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\n<\/ul>\n

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

                   <\/p>\n

                  A horizontal reflection<\/strong> reflects a graph horizontally across the [latex]y[\/latex]-axis. <\/span>This transformation changes the sign of the input values of [latex]f(x)[\/latex].<\/p>\n

                    \n
                  • If you reflect the graph of a function [latex]f(x)[\/latex] over the [latex]y[\/latex]-axis, the new function [latex]g(x)[\/latex] is given by:<\/li>\n<\/ul>\n

                    [latex]g(x) = f(-x)[\/latex]<\/p>\n<\/section>\n

                    How To: Given a function, reflect the graph both vertically and horizontally.<\/strong><\/p>\n
                      \n
                    1. Multiply all outputs by \u20131 for a vertical reflection. The new graph is a reflection of the original graph about the [latex]x[\/latex]-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 [latex]y[\/latex]-axis.<\/li>\n<\/ol>\n<\/section>\n
                      Reflect the graph of [latex]s\\left(t\\right)=\\sqrt{t}[\/latex]<\/p>\n
                        \n
                      1. vertically<\/li>\n
                      2. horizontally<\/li>\n<\/ol>\n