{"id":641,"date":"2025-07-14T19:28:32","date_gmt":"2025-07-14T19:28:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=641"},"modified":"2026-01-29T20:31:37","modified_gmt":"2026-01-29T20:31:37","slug":"transformation-of-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/transformation-of-functions-learn-it-3\/","title":{"raw":"Transformation of Functions: Learn It 4","rendered":"Transformation of Functions: Learn It 4"},"content":{"raw":"

Identifying Vertical Shifts<\/h2>\r\nOne simple kind of transformation involves shifting the entire graph of a function up and down, known as a vertical shift.\u00a0<\/strong>\r\n\r\n
\r\n

vertical shift<\/h3>\r\nA vertical shift<\/strong> occurs when you add or subtract a constant value to the function [latex]f(x)[\/latex].\r\n\r\nThis shifts the graph of the function vertically without changing its shape.\r\n
    \r\n \t
  • Upward shift:<\/strong> If you add a constant [latex]c[\/latex] to the function [latex]f(x)[\/latex], the graph of the function shifts upward 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
    • Downward shift:<\/strong> If you subtract a constant [latex]c[\/latex] to the function [latex]f(x)[\/latex], the graph of the function shifts downward by [latex]c[\/latex] units.<\/li>\r\n<\/ul>\r\n

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

      The image below shows the graph of a function [latex]f(x)[\/latex] (solid blue line) and its vertically 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\nVertically 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 higher than 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 = 0 + 1\u00a0 = 1[\/latex].<\/li>\r\n \t
          • If [latex]x=1[\/latex], then [latex]\\sqrt[3]{1} +1 = 1 + 1\u00a0 = 2[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nA vertical shift involves moving the graph of a function up or down without altering its shape. In this case, adding [latex]1[\/latex] to the function [latex]f(x) = \\sqrt[3]{x}[\/latex] results in a vertical shift of the graph upward by [latex]1[\/latex] unit.\r\n\r\n<\/section>
            \r\n

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

              \r\n \t
            1. Identify the output row or column.<\/li>\r\n \t
            2. Determine the magnitude<\/strong> of the shift.<\/li>\r\n \t
            3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/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\\right)-3[\/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=\"371465\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371465\"]\r\n

              The formula [latex]g\\left(x\\right)=f\\left(x\\right)-3[\/latex] tells us that we can find the output values of [latex]g[\/latex] by subtracting 3 from the output values of [latex]f[\/latex]. For example:<\/p>\r\n

              [latex]\\begin{align}f\\left(2\\right)&=1 && \\text{Given} \\\\[1.5mm] g\\left(x\\right)&=f\\left(x\\right)-3 && \\text{Given transformation} \\\\[1.5mm] g\\left(2\\right)&=f\\left(2\\right)-3 \\\\& =1 - 3 \\\\ &=-2 \\end{align}[\/latex]<\/p>\r\n

              Subtracting 3 from each [latex]f\\left(x\\right)[\/latex] value, we can complete a table of values for [latex]g\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/colgroup>\r\n\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
              [latex]g\\left(x\\right)[\/latex] <\/strong><\/td>\r\n\u22122<\/td>\r\n0<\/td>\r\n4<\/td>\r\n8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
              \r\n
              \r\n
              \r\n

              Analysis of the Solution<\/h4>\r\n

              As with the earlier vertical shift, notice the input values stay the same and only the output values change.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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

              The function [latex]h\\left(t\\right)=-4.9{t}^{2}+30t[\/latex] gives the height [latex]h[\/latex] of a ball (in meters) thrown upward from the ground after [latex]t[\/latex] seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function [latex]b\\left(t\\right)[\/latex] to [latex]h\\left(t\\right)[\/latex], and then find a formula for [latex]b\\left(t\\right)[\/latex].<\/p>\r\n

              [latex]b\\left(t\\right)=h\\left(t\\right)+10=-4.9{t}^{2}+30t+10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>

              [ohm_question hide_question_numbers=1]317927[\/ohm_question]<\/section>
              [ohm_question hide_question_numbers=1]317930[\/ohm_question]<\/section>","rendered":"

              Identifying Vertical Shifts<\/h2>\n

              One simple kind of transformation involves shifting the entire graph of a function up and down, known as a vertical shift.\u00a0<\/strong><\/p>\n

              \n

              vertical shift<\/h3>\n

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

              This shifts the graph of the function vertically without changing its shape.<\/p>\n

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

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

                  \n
                • Downward shift:<\/strong> If you subtract a constant [latex]c[\/latex] to the function [latex]f(x)[\/latex], the graph of the function shifts downward by [latex]c[\/latex] units.<\/li>\n<\/ul>\n

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

                  The image below shows the graph of a function [latex]f(x)[\/latex] (solid blue line) and its vertically 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

                    Vertically 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 higher than 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 = 0 + 1\u00a0 = 1[\/latex].<\/li>\n
                      • If [latex]x=1[\/latex], then [latex]\\sqrt[3]{1} +1 = 1 + 1\u00a0 = 2[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

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

                        \n

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

                          \n
                        1. Identify the output row or column.<\/li>\n
                        2. Determine the magnitude<\/strong> of the shift.<\/li>\n
                        3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.<\/li>\n<\/ol>\n<\/section>\n
                          \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\\right)-3[\/latex].<\/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]f\\left(x\\right)[\/latex] <\/strong><\/td>\n1<\/td>\n3<\/td>\n7<\/td>\n11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n