{"id":643,"date":"2025-07-14T19:28:55","date_gmt":"2025-07-14T19:28:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=643"},"modified":"2026-01-29T20:42:02","modified_gmt":"2026-01-29T20:42:02","slug":"transformation-of-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/transformation-of-functions-learn-it-4\/","title":{"raw":"Transformation of Functions: Learn It 5","rendered":"Transformation of Functions: Learn It 5"},"content":{"raw":"

Identifying Horizontal Shifts<\/h2>\r\nWe just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift<\/strong>.\r\n\r\n
\r\n

horizontal shift<\/h3>\r\nA horizontal shift<\/strong> occurs when you add or subtract a constant value to the input [latex]x[\/latex] of the function [latex]f(x)[\/latex].\r\n\r\nThis shifts the graph of the function horizontally.\r\n
    \r\n \t
  • Rightward shift:<\/strong> If you subtract a constant [latex]c[\/latex] from [latex]x[\/latex] before applying the function [latex]f[\/latex], the graph of the function shifts to the right by [latex]c[\/latex] units.<\/li>\r\n<\/ul>\r\n

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

      \r\n \t
    • Leftward shift:<\/strong> If you add a constant [latex]c[\/latex] to [latex]x[\/latex] before applying the function [latex]f[\/latex], the graph of the function shifts to the left by [latex]c[\/latex] units.<\/li>\r\n<\/ul>\r\n

      [latex]h(x) = f(x+c)[\/latex]<\/p>\r\n \r\n\r\n<\/section>

      The image shows the graph of the cube root function [latex]f(x) = \\sqrt[3]{x}[\/latex] (solid blue line) and its horizontally shifted version [latex]f(x + 1)[\/latex] (dashed orange line).\"GraphOriginal Function [latex]f(x)[\/latex]<\/strong>\r\n
        \r\n \t
      • The solid blue curve represents the original function [latex]\\sqrt[3]{x}[\/latex].<\/li>\r\n \t
      • The function [latex]f(x)[\/latex] passes through the origin [latex](0,0)[\/latex] because [latex]\\sqrt[3]{0} = 0[\/latex].<\/li>\r\n<\/ul>\r\nHorizontally Shifted Function [latex]f(x+1)[\/latex]<\/strong>\r\n
          \r\n \t
        • The dashed orange curve represents the function\u00a0 [latex]f(x+1)\u00a0 = \\sqrt[3]{x+1}[\/latex].<\/li>\r\n \t
        • Each point on the graph of [latex]f(x+1)[\/latex] is exactly [latex]1[\/latex] unit to the left of the corresponding point on the graph of [latex]f(x)[\/latex].<\/li>\r\n \t
        • For example:\r\n
            \r\n \t
          • If [latex]x=0[\/latex], then [latex]\\sqrt[3]{0+1} =\u00a0 \\sqrt[3]{1} = 1[\/latex].<\/li>\r\n \t
          • If [latex]x=-2[\/latex], then [latex]\\sqrt[3]{-2+1} = \\sqrt[3]{-1} = -1[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nA horizontal shift involves moving the graph of a function left or right without altering its shape. In this case, adding [latex]1[\/latex] to the input of the function [latex]f(x) = \\sqrt[3]{x}[\/latex] results in a horizontal shift of the graph to the left<\/strong> by [latex]1[\/latex] unit.\r\n\r\n<\/section>
            The function [latex]G\\left(m\\right)[\/latex] gives the number of gallons of gas required to drive [latex]m[\/latex] miles. Interpret [latex]G\\left(m\\right)+10[\/latex] and [latex]G\\left(m+10\\right)[\/latex].[reveal-answer q=\"792859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792859\"][latex]G\\left(m\\right)+10[\/latex] can be interpreted as adding [latex]10[\/latex] to the output, gallons. This is the gas required to drive [latex]m[\/latex] miles, plus another [latex]10[\/latex] gallons of gas. The graph would indicate a vertical shift.[latex]G\\left(m+10\\right)[\/latex] can be interpreted as adding [latex]10 [\/latex]to the input, miles. So this is the number of gallons of gas required to drive [latex]10[\/latex] miles more than [latex]m[\/latex] miles. The graph would indicate a horizontal shift.[\/hidden-answer]<\/section>
            \r\n

            How To: Given a tabular function, create a new row to represent a horizontal shift.<\/strong><\/p>\r\n\r\n

              \r\n \t
            1. Identify the input row or column.<\/li>\r\n \t
            2. Determine the magnitude of the shift.<\/li>\r\n \t
            3. Add the shift to the value in each input cell.<\/li>\r\n<\/ol>\r\n<\/section>
              \r\n

              A function [latex]f\\left(x\\right)[\/latex] is given below. Create a table for the function [latex]g\\left(x\\right)=f\\left(x - 3\\right)[\/latex].<\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n
              [latex]x[\/latex]<\/strong><\/td>\r\n2<\/td>\r\n4<\/td>\r\n6<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
              [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n1<\/td>\r\n3<\/td>\r\n7<\/td>\r\n11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"797840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797840\"]\r\n

              The formula [latex]g\\left(x\\right)=f\\left(x - 3\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are the same as the output value of [latex]f[\/latex] when the input value is 3 less than the original value. For example, we know that [latex]f\\left(2\\right)=1[\/latex]. To get the same output from the function [latex]g[\/latex], we will need an input value that is 3 larger<\/em>. We input a value that is 3 larger for [latex]g\\left(x\\right)[\/latex] because the function takes 3 away before evaluating the function [latex]f[\/latex].<\/p>\r\n

              [latex]\\begin{align}g\\left(5\\right)&=f\\left(5 - 3\\right) \\\\ &=f\\left(2\\right) \\\\ &=1 \\end{align}[\/latex]<\/p>\r\n

              We continue with the other values to create this table.<\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n
              [latex]x[\/latex]<\/strong><\/td>\r\n5<\/td>\r\n7<\/td>\r\n9<\/td>\r\n11<\/td>\r\n<\/tr>\r\n
              [latex]x - 3[\/latex]<\/strong><\/td>\r\n2<\/td>\r\n4<\/td>\r\n6<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
              [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n1<\/td>\r\n3<\/td>\r\n7<\/td>\r\n11<\/td>\r\n<\/tr>\r\n
              [latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n1<\/td>\r\n3<\/td>\r\n7<\/td>\r\n11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

              The result is that the function [latex]g\\left(x\\right)[\/latex] has been shifted to the right by 3. Notice the output values for [latex]g\\left(x\\right)[\/latex] remain the same as the output values for [latex]f\\left(x\\right)[\/latex], but the corresponding input values, [latex]x[\/latex], have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.<\/p>\r\n\r\n

              Analysis of the Solution<\/h4>\r\nThe graph represents both of the functions. We can see the horizontal shift in each point.\r\n\r\n\"Graph\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>
              [ohm_question hide_question_numbers=1]317931[\/ohm_question]<\/section>
              [ohm_question hide_question_numbers=1]317932[\/ohm_question]<\/section>
              [ohm_question hide_question_numbers=1]317933[\/ohm_question]<\/section>","rendered":"

              Identifying Horizontal Shifts<\/h2>\n

              We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift<\/strong>.<\/p>\n

              \n

              horizontal shift<\/h3>\n

              A horizontal shift<\/strong> occurs when you add or subtract a constant value to the input [latex]x[\/latex] of the function [latex]f(x)[\/latex].<\/p>\n

              This shifts the graph of the function horizontally.<\/p>\n

                \n
              • Rightward shift:<\/strong> If you subtract a constant [latex]c[\/latex] from [latex]x[\/latex] before applying the function [latex]f[\/latex], the graph of the function shifts to the right by [latex]c[\/latex] units.<\/li>\n<\/ul>\n

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

                  \n
                • Leftward shift:<\/strong> If you add a constant [latex]c[\/latex] to [latex]x[\/latex] before applying the function [latex]f[\/latex], the graph of the function shifts to the left by [latex]c[\/latex] units.<\/li>\n<\/ul>\n

                  [latex]h(x) = f(x+c)[\/latex]<\/p>\n

                   <\/p>\n<\/section>\n

                  The image shows the graph of the cube root function [latex]f(x) = \\sqrt[3]{x}[\/latex] (solid blue line) and its horizontally shifted version [latex]f(x + 1)[\/latex] (dashed orange line).\"GraphOriginal Function [latex]f(x)[\/latex]<\/strong><\/p>\n
                    \n
                  • The solid blue curve represents the original function [latex]\\sqrt[3]{x}[\/latex].<\/li>\n
                  • The function [latex]f(x)[\/latex] passes through the origin [latex](0,0)[\/latex] because [latex]\\sqrt[3]{0} = 0[\/latex].<\/li>\n<\/ul>\n

                    Horizontally Shifted Function [latex]f(x+1)[\/latex]<\/strong><\/p>\n

                      \n
                    • The dashed orange curve represents the function\u00a0 [latex]f(x+1)\u00a0 = \\sqrt[3]{x+1}[\/latex].<\/li>\n
                    • Each point on the graph of [latex]f(x+1)[\/latex] is exactly [latex]1[\/latex] unit to the left of the corresponding point on the graph of [latex]f(x)[\/latex].<\/li>\n
                    • For example:\n
                        \n
                      • If [latex]x=0[\/latex], then [latex]\\sqrt[3]{0+1} =\u00a0 \\sqrt[3]{1} = 1[\/latex].<\/li>\n
                      • If [latex]x=-2[\/latex], then [latex]\\sqrt[3]{-2+1} = \\sqrt[3]{-1} = -1[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                        A horizontal shift involves moving the graph of a function left or right without altering its shape. In this case, adding [latex]1[\/latex] to the input of the function [latex]f(x) = \\sqrt[3]{x}[\/latex] results in a horizontal shift of the graph to the left<\/strong> by [latex]1[\/latex] unit.<\/p>\n<\/section>\n

                        The function [latex]G\\left(m\\right)[\/latex] gives the number of gallons of gas required to drive [latex]m[\/latex] miles. Interpret [latex]G\\left(m\\right)+10[\/latex] and [latex]G\\left(m+10\\right)[\/latex].<\/p>\n