{"id":1610,"date":"2024-05-30T02:41:05","date_gmt":"2024-05-30T02:41:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1610"},"modified":"2025-09-12T14:48:58","modified_gmt":"2025-09-12T14:48:58","slug":"transformations-of-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/transformations-of-functions-learn-it-5\/","title":{"raw":"Transformations of Functions: Learn It 5","rendered":"Transformations of Functions: Learn It 5"},"content":{"raw":"

Performing a Sequence of Transformations<\/h2>\r\nCombining transformations follows a specific order of operations similar to the mathematical order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When applying transformations to a function, the sequence ensures that each transformation is applied correctly and the resulting graph reflects the intended changes.\r\n\r\n
\r\n

order of transformations<\/h3>\r\nWhen transforming a function [latex]y = a \\cdot f(bx-c)+d[\/latex], the general order of transformation is as follows:\r\n
    \r\n \t
  1. Horizontal Shifts<\/strong> by [latex]c[\/latex] units.\r\n
      \r\n \t
    • [latex]f(x)[\/latex] is shifted to the right if [latex]c[\/latex] is positive<\/li>\r\n \t
    • [latex]f(x)[\/latex] is shifted to\u00a0the left if [latex]c[\/latex] is negative.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
    • Horizontal Stretches\/Compressions\u00a0<\/strong>by a factor of [latex]\\dfrac{1}{b}[\/latex].\r\n
        \r\n \t
      • [latex]f(x)[\/latex] is compressed horizontally if [latex]|b|>1[\/latex].<\/li>\r\n \t
      • [latex]f(x)[\/latex] is stretched horizontally if [latex]0<|b|<1[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
      • Reflections<\/strong>\r\n
          \r\n \t
        • Reflection across the y-axis if [latex]b[\/latex] is negative.<\/li>\r\n \t
        • Reflection across the x-axis if [latex]a[\/latex] is negative.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Vertical Stretches\/Compressions\u00a0<\/strong>by a factor of [latex]a[\/latex].\r\n
            \r\n \t
          • [latex]f(x)[\/latex] is stretched vertically if [latex]|a|>1[\/latex].<\/li>\r\n \t
          • [latex]f(x)[\/latex] is compressed vertically if [latex]0<|a|<1[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
          • Vertical Shifts<\/strong> by [latex]d[\/latex] units.\r\n
              \r\n \t
            • [latex]f(x)[\/latex] is shifted to the upward if [latex]d[\/latex] is positive<\/li>\r\n \t
            • [latex]f(x)[\/latex] is shifted to the downward if [latex]d[\/latex] is negative.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>
              \r\n
                \r\n \t
              • When combining vertical transformations written in the form [latex]af\\left(x\\right)+k[\/latex], first vertically stretch by [latex]a[\/latex] and then vertically shift by [latex]k[\/latex].<\/li>\r\n \t
              • When combining horizontal transformations written in the form [latex]f\\left(bx-h\\right)[\/latex], first horizontally shift by [latex]\\frac{h}{b}[\/latex] and then horizontally stretch by [latex]\\frac{1}{b}[\/latex].<\/li>\r\n \t
              • When combining horizontal transformations written in the form [latex]f\\left(b\\left(x-h\\right)\\right)[\/latex], first horizontally stretch by [latex]\\frac{1}{b}[\/latex] and then horizontally shift by [latex]h[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
                Given [latex]f(x)=|x|[\/latex], identify the transformations and graph the transformed function
                [latex]h(x)=f(x+1)-3 = |x+1|-3[\/latex]<\/center>Step-by-Step Transformations<\/strong>\r\n
                  \r\n \t
                • Original Function: The original function is [latex]f(x)=|x|[\/latex].<\/li>\r\n \t
                • Transformations:\r\n
                    \r\n \t
                  1. Horizontal Shift:<\/strong> The term [latex]|x+1|[\/latex]indicates a horizontal shift to the left by [latex]1[\/latex] unit. This is because [latex]x+1 = 0[\/latex] when [latex]x=-1[\/latex]so the entire graph of [latex]f(x)=|x|[\/latex] is moved [latex]1[\/latex] unit to the left.<\/li>\r\n \t
                  2. Vertical Shift:<\/strong> The term [latex]-3[\/latex] indicates a vertical shift downward by [latex]3[\/latex] units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\nGraph<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n
                    Original Graph\r\n[latex]y=|x|[\/latex]\r\n<\/strong><\/td>\r\nLeft by [latex]1[\/latex] unit\r\n[latex]y=|x+1|[\/latex]\r\n<\/strong><\/td>\r\nDownward by [latex]3[\/latex] units\r\n[latex]y = |x+1|-3[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
                    [latex](0,0)[\/latex]<\/td>\r\n[latex](-1,0)[\/latex]<\/td>\r\n[latex](-1,-3)[\/latex]<\/td>\r\n<\/tr>\r\n
                    [latex](1,1)[\/latex]<\/td>\r\n[latex](0,1)[\/latex]<\/td>\r\n[latex](0,-2)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of an absolute function and how it was transformed[\/caption]\r\n\r\n<\/section>
                    \r\n\r\n[caption id=\"\" align=\"alignright\" width=\"372\"]\"Graph Graph of a half-circle[\/caption]\r\n\r\nUse the given graph of [latex]f(x)[\/latex] to draw the transformed function:\r\n\r\n
                    [latex]g(x)=f(\\frac{1}{2}x+1)-3[\/latex]<\/center>[reveal-answer q=\"840785\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"840785\"]The original function [latex]f(x)[\/latex] has key points: [latex](-2,0)[\/latex], [latex](0,2)[\/latex], and [latex](2,0)[\/latex].\r\n[latex]\\\\[\/latex]\r\nStep-by-Step Transformations<\/strong>\r\n
                      \r\n \t
                    1. Horizontal Shift Left by [latex]1[\/latex] unit<\/li>\r\n \t
                    2. Horizontal Stretch by a factor of [latex]2[\/latex] (Note: this impact the [latex]x[\/latex]-values)<\/li>\r\n \t
                    3. Vertical Shift Down by [latex]3[\/latex] units<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                      Original Point<\/strong><\/td>\r\nLeft by [latex]1[\/latex] unit<\/strong><\/td>\r\nHorizontal Stretch by a factor of [latex]2[\/latex]<\/strong><\/td>\r\nDown by [latex]3[\/latex] units<\/strong><\/td>\r\n<\/tr>\r\n
                      [latex](-2,0)[\/latex]<\/td>\r\n[latex](-3,0)[\/latex]<\/td>\r\n[latex](-6,0)[\/latex]<\/td>\r\n[latex](-6,-3)[\/latex]<\/td>\r\n<\/tr>\r\n
                      [latex](0,2)[\/latex]<\/td>\r\n[latex](-1,2)[\/latex]<\/td>\r\n[latex](-2,2)[\/latex]<\/td>\r\n[latex](-2,-1)[\/latex]<\/td>\r\n<\/tr>\r\n
                      [latex](2,0)[\/latex]<\/td>\r\n[latex](1,0)[\/latex]<\/td>\r\n[latex](2,0)[\/latex]<\/td>\r\n[latex](2,-3)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"396\"]\"Graph Graph of a vertically stretched and translated half-circle[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
                      Write a formula for the graph shown below, which is a transformation of the toolkit square root function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of h(x)[\/caption]\r\n\r\n[reveal-answer q=\"639112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"639112\"]The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as\r\n

                      [latex]h\\left(x\\right)=f\\left(x - 1\\right)+2[\/latex]<\/p>\r\nUsing the formula for the square root function, we can write\r\n

                      [latex]h\\left(x\\right)=\\sqrt{x - 1}+2[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nNote that this transformation has changed the domain and range of the function. This new graph has domain [latex]\\left[1,\\infty \\right)[\/latex] and range [latex]\\left[2,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                      A common model for learning has an equation similar to [latex]k\\left(t\\right)=-{2}^{-t}+1[\/latex], where [latex]k[\/latex] is the percentage of mastery that can be achieved after [latex]t[\/latex] practice sessions. This is a transformation of the function [latex]f\\left(t\\right)={2}^{t}[\/latex] shown below. Sketch a graph of [latex]k\\left(t\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of f(t)[\/caption]\r\n\r\n[reveal-answer q=\"533018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"533018\"]This equation combines three transformations into one equation.\r\n
                        \r\n \t
                      • A horizontal reflection: [latex]f\\left(-t\\right)={2}^{-t}[\/latex]<\/li>\r\n \t
                      • A vertical reflection: [latex]-f\\left(-t\\right)=-{2}^{-t}[\/latex]<\/li>\r\n \t
                      • A vertical shift: [latex]-f\\left(-t\\right)+1=-{2}^{-t}+1[\/latex]<\/li>\r\n<\/ul>\r\nWe can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points [latex](0, 1)[\/latex] and [latex](1, 2)[\/latex].\r\n
                          \r\n \t
                        1. First, we apply a horizontal reflection: [latex](0, 1) (\u20131, 2)[\/latex].<\/li>\r\n \t
                        2. Then, we apply a vertical reflection: [latex](0, \u22121) (-1, \u20132)[\/latex].<\/li>\r\n \t
                        3. Finally, we apply a vertical shift: [latex](0, 0) (-1, 1)[\/latex].<\/li>\r\n<\/ol>\r\nThis means that the original points, [latex](0,1)[\/latex] and [latex](1,2)[\/latex] become [latex](0,0)[\/latex] and [latex](1,1)[\/latex] after we apply the transformations.\r\n\r\nIn the graphs below, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Graphs Graphs of all the transformations[\/caption]\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nAs a model for learning, this function would be limited to a domain of [latex]t\\ge 0[\/latex], with corresponding range [latex]\\left[0,1\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
                          Given the table below\u00a0for the function [latex]f\\left(x\\right)[\/latex], create a table of values for the function [latex]g\\left(x\\right)=2f\\left(3x\\right)+1[\/latex].\r\n\r\n\r\n\r\n\r\n
                          [latex]x[\/latex]<\/strong><\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]12[\/latex]<\/td>\r\n[latex]18[\/latex]<\/td>\r\n[latex]24[\/latex]<\/td>\r\n<\/tr>\r\n
                          [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n[latex]10[\/latex]<\/td>\r\n[latex]14[\/latex]<\/td>\r\n[latex]15[\/latex]<\/td>\r\n[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"669282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"669282\"]\r\n\r\nThere are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, [latex]f\\left(3x\\right)[\/latex] is a horizontal compression by [latex]\\frac{1}{3}[\/latex], which means we multiply each [latex]x\\text{-}[\/latex] value by [latex]\\frac{1}{3}[\/latex].\r\n\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(3x\\right)[\/latex] <\/strong><\/td>\r\n[latex]10[\/latex]<\/td>\r\n[latex]14[\/latex]<\/td>\r\n[latex]15[\/latex]<\/td>\r\n[latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLooking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation.\r\n\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]2f\\left(3x\\right)[\/latex] <\/strong><\/td>\r\n[latex]20[\/latex]<\/td>\r\n[latex]28[\/latex]<\/td>\r\n[latex]30[\/latex]<\/td>\r\n[latex]34[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFinally, we can apply the vertical shift, which will add 1 to all the output values.\r\n\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)=2f\\left(3x\\right)+1[\/latex]<\/strong><\/td>\r\n[latex]21[\/latex]<\/td>\r\n[latex]29[\/latex]<\/td>\r\n[latex]31[\/latex]<\/td>\r\n[latex]35[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>
                          [ohm2_question hide_question_numbers=1]19154[\/ohm2_question]<\/section>
                          [ohm2_question hide_question_numbers=1]19153[\/ohm2_question]<\/section>","rendered":"

                          Performing a Sequence of Transformations<\/h2>\n

                          Combining transformations follows a specific order of operations similar to the mathematical order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When applying transformations to a function, the sequence ensures that each transformation is applied correctly and the resulting graph reflects the intended changes.<\/p>\n

                          \n

                          order of transformations<\/h3>\n

                          When transforming a function [latex]y = a \\cdot f(bx-c)+d[\/latex], the general order of transformation is as follows:<\/p>\n

                            \n
                          1. Horizontal Shifts<\/strong> by [latex]c[\/latex] units.\n
                              \n
                            • [latex]f(x)[\/latex] is shifted to the right if [latex]c[\/latex] is positive<\/li>\n
                            • [latex]f(x)[\/latex] is shifted to\u00a0the left if [latex]c[\/latex] is negative.<\/li>\n<\/ul>\n<\/li>\n
                            • Horizontal Stretches\/Compressions\u00a0<\/strong>by a factor of [latex]\\dfrac{1}{b}[\/latex].\n
                                \n
                              • [latex]f(x)[\/latex] is compressed horizontally if [latex]|b|>1[\/latex].<\/li>\n
                              • [latex]f(x)[\/latex] is stretched horizontally if [latex]0<|b|<1[\/latex].<\/li>\n<\/ul>\n<\/li>\n
                              • Reflections<\/strong>\n
                                  \n
                                • Reflection across the y-axis if [latex]b[\/latex] is negative.<\/li>\n
                                • Reflection across the x-axis if [latex]a[\/latex] is negative.<\/li>\n<\/ul>\n<\/li>\n
                                • Vertical Stretches\/Compressions\u00a0<\/strong>by a factor of [latex]a[\/latex].\n
                                    \n
                                  • [latex]f(x)[\/latex] is stretched vertically if [latex]|a|>1[\/latex].<\/li>\n
                                  • [latex]f(x)[\/latex] is compressed vertically if [latex]0<|a|<1[\/latex].<\/li>\n<\/ul>\n<\/li>\n
                                  • Vertical Shifts<\/strong> by [latex]d[\/latex] units.\n
                                      \n
                                    • [latex]f(x)[\/latex] is shifted to the upward if [latex]d[\/latex] is positive<\/li>\n
                                    • [latex]f(x)[\/latex] is shifted to the downward if [latex]d[\/latex] is negative.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n
                                      \n
                                        \n
                                      • When combining vertical transformations written in the form [latex]af\\left(x\\right)+k[\/latex], first vertically stretch by [latex]a[\/latex] and then vertically shift by [latex]k[\/latex].<\/li>\n
                                      • When combining horizontal transformations written in the form [latex]f\\left(bx-h\\right)[\/latex], first horizontally shift by [latex]\\frac{h}{b}[\/latex] and then horizontally stretch by [latex]\\frac{1}{b}[\/latex].<\/li>\n
                                      • When combining horizontal transformations written in the form [latex]f\\left(b\\left(x-h\\right)\\right)[\/latex], first horizontally stretch by [latex]\\frac{1}{b}[\/latex] and then horizontally shift by [latex]h[\/latex].<\/li>\n<\/ul>\n<\/section>\n
                                        Given [latex]f(x)=|x|[\/latex], identify the transformations and graph the transformed function<\/p>\n
                                        [latex]h(x)=f(x+1)-3 = |x+1|-3[\/latex]<\/div>\n

                                        Step-by-Step Transformations<\/strong><\/p>\n

                                          \n
                                        • Original Function: The original function is [latex]f(x)=|x|[\/latex].<\/li>\n
                                        • Transformations:\n
                                            \n
                                          1. Horizontal Shift:<\/strong> The term [latex]|x+1|[\/latex]indicates a horizontal shift to the left by [latex]1[\/latex] unit. This is because [latex]x+1 = 0[\/latex] when [latex]x=-1[\/latex]so the entire graph of [latex]f(x)=|x|[\/latex] is moved [latex]1[\/latex] unit to the left.<\/li>\n
                                          2. Vertical Shift:<\/strong> The term [latex]-3[\/latex] indicates a vertical shift downward by [latex]3[\/latex] units.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n

                                            Graph<\/strong><\/p>\n\n\n\n\n\n
                                            Original Graph
                                            \n[latex]y=|x|[\/latex]
                                            \n<\/strong><\/td>\n                                            		Left by [latex]1[\/latex] unit
                                            \n[latex]y=|x+1|[\/latex]
                                            \n<\/strong><\/td>\n                                            		Downward by [latex]3[\/latex] units
                                            \n[latex]y = |x+1|-3[\/latex]<\/strong><\/td>\n<\/tr>\n
                                            [latex](0,0)[\/latex]<\/td>\n[latex](-1,0)[\/latex]<\/td>\n[latex](-1,-3)[\/latex]<\/td>\n<\/tr>\n
                                            [latex](1,1)[\/latex]<\/td>\n[latex](0,1)[\/latex]<\/td>\n[latex](0,-2)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                            \"Graph
                                            Graph of an absolute function and how it was transformed<\/figcaption><\/figure>\n<\/section>\n
                                            \n
                                            \"Graph
                                            Graph of a half-circle<\/figcaption><\/figure>\n

                                            Use the given graph of [latex]f(x)[\/latex] to draw the transformed function:<\/p>\n

                                            [latex]g(x)=f(\\frac{1}{2}x+1)-3[\/latex]<\/div>\n