{"id":1598,"date":"2024-05-30T00:20:15","date_gmt":"2024-05-30T00:20:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1598"},"modified":"2025-08-13T23:00:26","modified_gmt":"2025-08-13T23:00:26","slug":"transformations-of-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/transformations-of-functions-learn-it-2\/","title":{"raw":"Transformations of Functions: Learn It 2","rendered":"Transformations of Functions: Learn It 2"},"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).\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"386\"]\"Graph Graph of f(x) and f(x) + 1[\/caption]\r\n\r\nOriginal 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>
            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].\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<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"719880\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"719880\"]\r\n\r\nThe 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].\r\n

            [latex]\\begin{array}{c c} g\\left(5\\right) & = f\\left(5 - 3\\right)\\hfill \\\\ & =f\\left(2\\right)\\hfill \\\\ & =1\\hfill \\end{array}[\/latex]<\/p>\r\nWe continue with the other values to create this table.\r\n\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\nThe result is that the function [latex]f\\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.\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nThe graph below represents both of the functions. We can see the horizontal shift in each point.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the points from the previous table for f(x) and g(x)=f(x-3)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
            The graph below represents a transformation of the toolkit function [latex]f\\left(x\\right)={x}^{2}[\/latex]. Relate this new function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex], and then find a formula for [latex]g\\left(x\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of f(x)[\/caption]\r\n\r\n[reveal-answer q=\"937293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937293\"]Notice that the graph is identical in shape to the [latex]f\\left(x\\right)={x}^{2}[\/latex] function, but the [latex]x[\/latex]-<\/em>values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so\r\n

            [latex]g\\left(x\\right)=f\\left(x - 2\\right)[\/latex]<\/p>\r\nNotice how we must input the value [latex]x=2[\/latex] to get the output value [latex]y=0[\/latex]; the [latex]x[\/latex]-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the [latex]f\\left(x\\right)[\/latex] function to write a formula for [latex]g\\left(x\\right)[\/latex] by evaluating [latex]f\\left(x - 2\\right)[\/latex].\r\n

            [latex]\\begin{cases}f\\left(x\\right)={x}^{2}\\hfill \\\\ g\\left(x\\right)=f\\left(x - 2\\right)\\hfill \\\\ g\\left(x\\right)=f\\left(x - 2\\right)={\\left(x - 2\\right)}^{2}\\hfill \\end{cases}[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nTo determine whether the shift is [latex]+2[\/latex] or [latex]-2[\/latex] , consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function, [latex]f\\left(0\\right)=0[\/latex]. In our shifted function, [latex]g\\left(2\\right)=0[\/latex]. To obtain the output value of 0 from the function [latex]f[\/latex], we need to decide whether a plus or a minus sign will work to satisfy [latex]g\\left(2\\right)=f\\left(x - 2\\right)=f\\left(0\\right)=0[\/latex]. For this to work, we will need to subtract<\/em> 2 units from our input values.\r\n\r\n[\/hidden-answer]\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>
            [ohm2_question hide_question_numbers=1]19144[\/ohm2_question]<\/section>
            [ohm2_question hide_question_numbers=1]19145[\/ohm2_question]<\/section>
            [ohm2_question hide_question_numbers=1]19146[\/ohm2_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).<\/p>\n
                \"Graph
                Graph of f(x) and f(x) + 1<\/figcaption><\/figure>\n

                Original 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

                      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>\n\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