{"id":1604,"date":"2024-05-30T01:58:23","date_gmt":"2024-05-30T01:58:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1604"},"modified":"2025-08-13T23:03:38","modified_gmt":"2025-08-13T23:03:38","slug":"transformations-of-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/transformations-of-functions-learn-it-4\/","title":{"raw":"Transformations of Functions: Learn It 4","rendered":"Transformations of Functions: Learn It 4"},"content":{"raw":"

Graphing Functions Using Stretches and Compressions<\/h2>\r\nAdding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.\r\n

Vertical Stretches and Compressions<\/h3>\r\nWe can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.\r\n\r\nWhen we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch<\/strong>; if the constant is between 0 and 1, we get a vertical compression<\/strong>. The graph below\u00a0shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Vertical stretch and compression[\/caption]\r\n\r\n
\r\n

vertical stretches and compressions<\/h3>\r\nA vertical stretch or compression<\/strong> involves scaling the graph of a function [latex]f(x)[\/latex] by a constant factor [latex]a[\/latex].\r\n

[latex]g(x) = a \\cdot f(x)[\/latex]<\/p>\r\nThis transformation changes the output values of the function.\r\n

    \r\n \t
  • If [latex]a>1[\/latex]: The graph is stretched vertically.<\/li>\r\n \t
  • If [latex]0 < a < 1[\/latex]: The graph is compressed vertically.<\/li>\r\n \t
  • If [latex]a<0[\/latex]: A combination of vertical stretch\/compression and vertical reflection.<\/li>\r\n<\/ul>\r\n<\/section>
    How To: Given a function, graph its vertical stretch.<\/strong>\r\n
      \r\n \t
    1. Identify the value of [latex]a[\/latex].<\/li>\r\n \t
    2. Multiply all range values by [latex]a[\/latex].<\/li>\r\n \t
    3. If [latex]a>1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].\r\nIf [latex]{ 0 }<{ a }<{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].\r\nIf [latex]a<0[\/latex], the graph is either stretched or compressed and also reflected about the [latex]x[\/latex]-axis.<\/li>\r\n<\/ol>\r\n<\/section>
      A function [latex]P\\left(t\\right)[\/latex] models the number\u00a0of fruit flies in a population over time, and is graphed below.A scientist is comparing this population to another population, [latex]Q[\/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph to represent the growth of the population of fruit flies[\/caption]\r\n\r\n[reveal-answer q=\"951851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951851\"]Because the population is always twice as large, the new population\u2019s output values are always twice the original function\u2019s output values.If we choose four reference points, [latex](0, 1)[\/latex], [latex](3, 3)[\/latex], [latex](6, 2)[\/latex] and [latex](7, 0)[\/latex] we will multiply all of the outputs by [latex]2[\/latex].The following shows where the new points for the new graph will be located.\r\n\r\n
      [latex]\\begin{cases}\\left(0,\\text{ }1\\right)\\to \\left(0,\\text{ }2\\right)\\hfill \\\\ \\left(3,\\text{ }3\\right)\\to \\left(3,\\text{ }6\\right)\\hfill \\\\ \\left(6,\\text{ }2\\right)\\to \\left(6,\\text{ }4\\right)\\hfill \\\\ \\left(7,\\text{ }0\\right)\\to \\left(7,\\text{ }0\\right)\\hfill \\end{cases}[\/latex]<\/center>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the population function doubled[\/caption]\r\n\r\nSymbolically, the relationship is written as\r\n\r\n
      [latex]Q\\left(t\\right)=2P\\left(t\\right)[\/latex]<\/center>This means that for any input [latex]t[\/latex], the value of the function [latex]Q[\/latex] is twice the value of the function [latex]P[\/latex]. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t[\/latex], stay the same while the output values are twice as large as before.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      A function [latex]f[\/latex] is given in the table below. Create a table for the function [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex].\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=\"798923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"798923\"]\r\n\r\nThe formula [latex]g\\left(x\\right)=\\frac{1}{2}f\\left(x\\right)[\/latex] tells us that the output values of [latex]g[\/latex] are half of the output values of [latex]f[\/latex] with the same inputs. For example, we know that [latex]f\\left(4\\right)=3[\/latex]. Then:\r\n\r\n
      [latex]g\\left(4\\right)=\\frac{1}{2}\\cdot{f}(4) =\\frac{1}{2}\\cdot\\left(3\\right)=\\frac{3}{2}[\/latex]<\/center>We do the same for the other values to produce this table.\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]\\frac{1}{2}[\/latex]<\/td>\r\n[latex]\\frac{3}{2}[\/latex]<\/td>\r\n[latex]\\frac{7}{2}[\/latex]<\/td>\r\n[latex]\\frac{11}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[latex]\\\\[\/latex]\r\nAnalysis of the Solution<\/strong>\r\n\r\nThe result is that the function [latex]g\\left(x\\right)[\/latex] has been compressed vertically by [latex]\\frac{1}{2}[\/latex]. Each output value is divided in half, so the graph is half the original height.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      \r\n\r\n[caption id=\"attachment_1606\" align=\"alignright\" width=\"309\"]\"\" Graph of g(x) and f(x)[\/caption]\r\n\r\nThe graph shows two function: The toolkit function [latex]f(x) = x^3[\/latex] (green) and [latex]g(x)[\/latex] (red).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].[reveal-answer q=\"343479\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"343479\"]The red curve [latex]g(x)[\/latex] appears to be less steep compared to the green curve [latex]f(x)[\/latex]. This suggests a vertical compression.If [latex]g(x)[\/latex] is a vertical compression of [latex]f(x)[\/latex], we have: [latex]g(x) = a \\cdot f(x)[\/latex], where [latex]0 < a < 1[\/latex].To determine [latex]a[\/latex], it is helpful to look for a point on the graph that is relatively clear.\r\n
        \r\n \t
      • In this graph, it appears that [latex]g\\left(2\\right)=2[\/latex].<\/li>\r\n \t
      • With the basic cubic function at the same input, [latex]f\\left(2\\right)={2}^{3}=8[\/latex].<\/li>\r\n \t
      • Based on that, it appears that the outputs of [latex]g[\/latex] are [latex]\\frac{1}{4}[\/latex] the outputs of the function [latex]f[\/latex] because [latex]2=\\frac{1}{4} \\cdot 8[\/latex].<\/li>\r\n<\/ul>\r\nThus, [latex]g(x) = \\frac{1}{4} f(x) = \\frac{1}{4} x^3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]19152[\/ohm2_question]<\/section>\r\n

        Horizontal Stretches and Compressions<\/h2>\r\nNow we consider changes to the inside of a function. When we multiply a function\u2019s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch<\/strong>; if the constant is greater than 1, we get a horizontal compression<\/strong> of the function.\r\n\r\nGiven a function [latex]y=f\\left(x\\right)[\/latex], the form [latex]y=f\\left(bx\\right)[\/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[\/latex].\u00a0The graph of [latex]y={\\left(0.5x\\right)}^{2}[\/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of 2. The graph of [latex]y={\\left(2x\\right)}^{2}[\/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[\/latex] by a factor of [latex]2[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the vertical stretch and compression of x^2[\/caption]\r\n\r\n
        \r\n

        horizontal stretches and compressions<\/h3>\r\nA horizontal stretch or compression<\/strong> involves scaling the graph of a function [latex]f(x)[\/latex] by a constant factor [latex]b[\/latex].\r\n

        [latex]g(x) = f(b \\cdot x)[\/latex]<\/p>\r\nThis transformation changes the input values of the function.\r\n

          \r\n \t
        • If [latex]b>1[\/latex]: The graph is compressed horizontally. The graph is compressed by [latex]\\dfrac{1}{b}[\/latex].<\/li>\r\n \t
        • If [latex]0 < b < 1[\/latex]: The graph is stretched horizontally. The graph is stretched by [latex]\\dfrac{1}{b}[\/latex].<\/li>\r\n \t
        • If [latex]b<0[\/latex]: A combination of horizontal stretch\/compression and horizontal reflection.<\/li>\r\n<\/ul>\r\n<\/section>
          How To: Given a description of a function, sketch a horizontal compression or stretch.\r\n<\/strong>\r\n
            \r\n \t
          1. Write a formula to represent the function.<\/li>\r\n \t
          2. Set [latex]g\\left(x\\right)=f\\left(bx\\right)[\/latex] where [latex]b{\\gt}1[\/latex] for a compression or [latex]0{\\lt}b{\\lt}1[\/latex] for a stretch.<\/li>\r\n<\/ol>\r\n<\/section>
            Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, [latex]R[\/latex], will progress in [latex]1[\/latex] hour the same amount as the original population does in [latex]2[\/latex] hours, and in [latex]2[\/latex] hours, it will progress as much as the original population does in [latex]4[\/latex] hours. Sketch a graph of this population.[reveal-answer q=\"855794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"855794\"]Symbolically, we could write\r\n

            [latex]\\begin{align}&R\\left(1\\right)=P\\left(2\\right), \\\\ &R\\left(2\\right)=P\\left(4\\right),\\text{ and in general,} \\\\ &R\\left(t\\right)=P\\left(2t\\right). \\end{align}[\/latex]<\/p>\r\nSee below\u00a0for a graphical comparison of the original population and the compressed population.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"976\"]\"Two (a) Original population graph (b) Compressed population graph[\/caption]\r\n[latex]\\\\[\/latex]\r\nAnalysis of the Solution<\/strong>\r\n[latex]\\\\[\/latex]\r\nNote that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.[\/hidden-answer]<\/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(\\frac{1}{2}x\\right)[\/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(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=\"261935\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261935\"]\r\n\r\nThe formula [latex]g\\left(x\\right)=f\\left(\\frac{1}{2}x\\right)[\/latex] tells us that the output values for [latex]g[\/latex] are the same as the output values for the function [latex]f[\/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\\left(2\\right)[\/latex] because [latex]g\\left(2\\right)=f\\left(\\frac{1}{2}\\cdot 2\\right)=f\\left(1\\right)[\/latex], and we do not have a value for [latex]f\\left(1\\right)[\/latex] in our table. Our input values to [latex]g[\/latex] will need to be twice as large to get inputs for [latex]f[\/latex] that we can evaluate. For example, we can determine [latex]g\\left(4\\right)\\text{.}[\/latex]\r\n

            [latex]g\\left(4\\right)=f\\left(\\frac{1}{2}\\cdot 4\\right)=f\\left(2\\right)=1[\/latex]<\/p>\r\nWe do the same for the other values to produce the table below.\r\n\r\n\r\n\r\n\r\n
            [latex]x[\/latex]<\/strong><\/td>\r\n4<\/td>\r\n8<\/td>\r\n12<\/td>\r\n16<\/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[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Graph Graph of the previous table[\/caption]\r\n\r\nThis figure shows the graphs of both of these sets of points.\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nBecause each input value has been doubled, the result is that the function [latex]g\\left(x\\right)[\/latex] has been stretched horizontally by a factor of 2.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section> \r\n\r\n
            \r\n\r\n[caption id=\"\" align=\"alignright\" width=\"487\"]\"Graph Graph of f(x) being vertically compressed to g(x)[\/caption]\r\n\r\nRelate the function [latex]g\\left(x\\right)[\/latex] to [latex]f\\left(x\\right)[\/latex].[reveal-answer q=\"623190\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"623190\"]The orange graph\u00a0 [latex]g(x)[\/latex] appears to be a horizontally compressed version of the blue graph of [latex]f(x)[\/latex].\r\n[latex]\\\\[\/latex]\r\nIf [latex]g(x)[\/latex] is a horizontal compression of [latex]f(x)[\/latex], we have: [latex]g(x) = f(b \\cdot x)[\/latex], where [latex]b > 1[\/latex]. The graph is compressed by [latex]\\dfrac{1}{b}[\/latex].To determine [latex]b[\/latex], it is helpful to look for a point on the graph that is relatively clear.\r\n
              \r\n \t
            • In the compressed graph [latex]g(x)[\/latex], the end point is [latex](2, 4)[\/latex].<\/li>\r\n \t
            • The end point of [latex]f(x)[\/latex] is [latex](6,4)[\/latex].<\/li>\r\n \t
            • We can see that the [latex]x[\/latex]-values have been compressed by [latex]\\frac{1}{3}[\/latex], because [latex]2=\\frac{1}{3} \\cdot 6[\/latex].<\/li>\r\n \t
            • This means that [latex]\\dfrac{1}{b} = \\dfrac{1}{3}[\/latex], which means [latex]b = 3[\/latex].<\/li>\r\n<\/ul>\r\nThus, [latex]g(x)=f(3x)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
              [ohm2_question hide_question_numbers=1]19150[\/ohm2_question]<\/section>","rendered":"

              Graphing Functions Using Stretches and Compressions<\/h2>\n

              Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.<\/p>\n

              Vertical Stretches and Compressions<\/h3>\n

              We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.<\/p>\n

              When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch<\/strong>; if the constant is between 0 and 1, we get a vertical compression<\/strong>. The graph below\u00a0shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.<\/p>\n

              \"Graph
              Vertical stretch and compression<\/figcaption><\/figure>\n
              \n

              vertical stretches and compressions<\/h3>\n

              A vertical stretch or compression<\/strong> involves scaling the graph of a function [latex]f(x)[\/latex] by a constant factor [latex]a[\/latex].<\/p>\n

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

              This transformation changes the output values of the function.<\/p>\n

                \n
              • If [latex]a>1[\/latex]: The graph is stretched vertically.<\/li>\n
              • If [latex]0 < a < 1[\/latex]: The graph is compressed vertically.<\/li>\n
              • If [latex]a<0[\/latex]: A combination of vertical stretch\/compression and vertical reflection.<\/li>\n<\/ul>\n<\/section>\n
                How To: Given a function, graph its vertical stretch.<\/strong><\/p>\n
                  \n
                1. Identify the value of [latex]a[\/latex].<\/li>\n
                2. Multiply all range values by [latex]a[\/latex].<\/li>\n
                3. If [latex]a>1[\/latex], the graph is stretched by a factor of [latex]a[\/latex].
                  \nIf [latex]{ 0 }<{ a }<{ 1 }[\/latex], the graph is compressed by a factor of [latex]a[\/latex].\nIf [latex]a<0[\/latex], the graph is either stretched or compressed and also reflected about the [latex]x[\/latex]-axis.<\/li>\n<\/ol>\n<\/section>\n
                  A function [latex]P\\left(t\\right)[\/latex] models the number\u00a0of fruit flies in a population over time, and is graphed below.A scientist is comparing this population to another population, [latex]Q[\/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.<\/p>\n
                  \"Graph
                  Graph to represent the growth of the population of fruit flies<\/figcaption><\/figure>\n