{"id":1056,"date":"2025-07-22T00:04:53","date_gmt":"2025-07-22T00:04:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1056"},"modified":"2026-03-16T19:07:21","modified_gmt":"2026-03-16T19:07:21","slug":"exponential-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions-learn-it-4\/","title":{"raw":"Graphs of Exponential Functions: Learn It 4","rendered":"Graphs of Exponential Functions: Learn It 4"},"content":{"raw":"

Stretching, Compressing, or Reflecting an Exponential Function<\/h2>\r\nWhile horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch<\/strong> or compression<\/strong> occurs when we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|>0[\/latex].\r\n\r\n
For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the stretch, using [latex]a=3[\/latex], to get [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex].\r\n[caption id=\"attachment_5005\" align=\"aligncenter\" width=\"679\"]\"Two (a) [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] stretches the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of 3. (b) [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] compresses the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]\\frac{1}{3}[\/latex].[\/caption]<\/section>
\r\n

stretches and compressions of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)=a{b}^{x}[\/latex]\r\n
    \r\n \t
  • is stretched vertically by a factor of [latex]a[\/latex]if [latex]|a|>1[\/latex].<\/li>\r\n \t
  • is compressed vertically by a factor of [latex]a[\/latex]\u00a0if [latex]|a|<1[\/latex].<\/li>\r\n \t
  • has a\u00a0[latex]y[\/latex]-intercept is [latex]\\left(0,a\\right)[\/latex].<\/li>\r\n \t
  • has a horizontal asymptote of [latex]y=0[\/latex], range of [latex]\\left(0,\\infty \\right)[\/latex], and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are all unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/section>
    Exponential functions are stretched, compressed or reflected in the same manner you've used to transform other functions. Multipliers or negatives inside the function argument (in the exponent) affect horizontal transformations. Multipliers or negatives outside the function argument affect vertical transformations.<\/section>
    Sketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"418729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418729\"]Before graphing, identify the behavior and key points on the graph.\r\n
      \r\n \t
    • Since [latex]b=\\frac{1}{2}[\/latex] is between zero and one, the left tail of the graph will increase without bound as [latex]x[\/latex]\u00a0decreases, and the right tail will approach the [latex]x[\/latex]-axis as [latex]x[\/latex]\u00a0increases.<\/li>\r\n \t
    • Since [latex]a\u00a0= 4[\/latex], the graph of [latex]f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] will be stretched vertically by a factor of [latex]4[\/latex].<\/li>\r\n \t
    • Create a table of points:\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/strong><\/td>\r\n[latex]\u20133[\/latex]<\/td>\r\n[latex]\u20132[\/latex]<\/td>\r\n[latex]\u20131[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/strong><\/td>\r\n[latex]32[\/latex]<\/td>\r\n[latex]16[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]0.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
    • Plot the [latex]y[\/latex]-<\/em>intercept, [latex]\\left(0,4\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,8\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\r\n \t
    • Draw a smooth curve connecting the points.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(0,\\infty \\right)[\/latex], the horizontal asymptote is y\u00a0= 0.[\/caption][\/hidden-answer]<\/section>
      [ohm_question hide_question_numbers=1]321399[\/ohm_question]<\/section>\r\n

      Graphing Reflections<\/h3>\r\nIn addition to shifting, compressing, and stretching a graph, we can also reflect it about the [latex]x[\/latex]-axis or the [latex]y[\/latex]-axis. When we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by [latex]\u20131[\/latex], we get a reflection about the [latex]x[\/latex]-axis. When we multiply the input by[latex] \u20131[\/latex], we get a reflection about the [latex]y[\/latex]-axis.\r\n\r\n
      For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the two reflections alongside it. The reflection about the [latex]x[\/latex]-axis, [latex]g\\left(x\\right)={-2}^{x}[\/latex], and the reflection about the [latex]y[\/latex]-axis, [latex]h\\left(x\\right)={2}^{-x}[\/latex], are both shown below.\r\n[caption id=\"attachment_5009\" align=\"aligncenter\" width=\"676\"]\"Two (a) [latex]g\\left(x\\right)=-{2}^{x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the x-axis. (b) [latex]h\\left(x\\right)={2}^{-x}[\/latex] reflects the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] about the y-axis.[\/caption]<\/section>
      \r\n

      reflecting the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)=-{b}^{x}[\/latex]\r\n
        \r\n \t
      • reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the [latex]x[\/latex]-axis.<\/li>\r\n \t
      • has a [latex]y[\/latex]-intercept of [latex]\\left(0,-1\\right)[\/latex].<\/li>\r\n \t
      • has a range of [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\r\n \t
      • has a horizontal asymptote of [latex]y=0[\/latex] and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex]f\\left(x\\right)={b}^{-x}[\/latex]\r\n
          \r\n \t
        • reflects the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] about the [latex]y[\/latex]-axis.<\/li>\r\n \t
        • has a [latex]y[\/latex]-intercept of [latex]\\left(0,1\\right)[\/latex], a horizontal asymptote at [latex]y=0[\/latex], a range of [latex]\\left(0,\\infty \\right)[\/latex], and a domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/section>
          Find and graph the equation for a function, [latex]g\\left(x\\right)[\/latex], that reflects [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the [latex]x[\/latex]-axis. State its domain, range, and asymptote.[reveal-answer q=\"91748\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"91748\"]Since we want to reflect the parent function [latex]f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}[\/latex] about the [latex]x[\/latex]-<\/em>axis, we multiply [latex]f\\left(x\\right)[\/latex] by [latex]\u20131[\/latex] to get [latex]g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}[\/latex]. Next we create a table of points.\r\n<\/colgroup>\r\n\r\n\r\n\r\n
          [latex]x[\/latex]<\/strong><\/td>\r\n[latex]\u20133[\/latex]<\/td>\r\n[latex]\u20132[\/latex]<\/td>\r\n[latex]\u20131[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
          [latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\r\n[latex]\u201364[\/latex]<\/td>\r\n[latex]\u201316[\/latex]<\/td>\r\n[latex]\u20134[\/latex]<\/td>\r\n[latex]\u20131[\/latex]<\/td>\r\n[latex]\u20130.25[\/latex]<\/td>\r\n[latex]\u20130.0625[\/latex]<\/td>\r\n[latex]\u20130.0156[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlot the [latex]y[\/latex]-<\/em>intercept, [latex]\\left(0,-1\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(1,-0.25\\right)[\/latex].\r\n\r\nDraw a smooth curve connecting the points:\r\n\r\n[caption id=\"attachment_5007\" align=\"aligncenter\" width=\"306\"]\"Graph The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,0\\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/section>
          \r\n

          Summarizing Transformations of the Exponential Function<\/h3>\r\nNow that we have worked with each type of translation for the exponential function, we can summarize them\u00a0to arrive at the general equation for transforming exponential functions.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Transformations of the Parent Function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/th>\r\n<\/tr>\r\n
          Translation<\/th>\r\nForm<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
          Shift\r\n
            \r\n \t
          • Horizontally [latex]c[\/latex]\u00a0units to the left<\/li>\r\n \t
          • Vertically [latex]d[\/latex]\u00a0units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n
          		[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n
          Stretch and Compress\r\n
            \r\n \t
          • Stretch if [latex]|a| \\gt 1[\/latex]<\/li>\r\n \t
          • Compression if [latex]0 \\lt |a| \\lt 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n
          		[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n
          Reflect about the [latex]x[\/latex]-axis<\/td>\r\n[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n
          Reflect about the [latex]y[\/latex]-axis<\/td>\r\n[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\r\n<\/tr>\r\n
          General equation for all transformations<\/td>\r\n[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
          \r\n

          transformations of exponential functions<\/h3>\r\nA transformation of an exponential function has the form\r\n\r\n[latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex], where the parent function, [latex]y={b}^{x}[\/latex], [latex]b>1[\/latex], is\r\n
            \r\n \t
          • shifted horizontally [latex]c[\/latex]\u00a0units to the left.<\/li>\r\n \t
          • stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a| > 0[\/latex].<\/li>\r\n \t
          • compressed vertically by a factor of [latex]|a|[\/latex] if [latex]0 < |a| < 1[\/latex].<\/li>\r\n \t
          • shifted vertically [latex]d[\/latex]\u00a0units.<\/li>\r\n \t
          • reflected about the [latex]x[\/latex]-<\/em>axis when [latex]a\u00a0< 0[\/latex].<\/li>\r\n<\/ul>\r\nNote the order of the shifts, transformations, and reflections follow the order of operations.\r\n\r\n<\/section>
            Write the equation for the function described below. Give the horizontal asymptote, domain, and range.\r\n
              \r\n \t
            • [latex]f\\left(x\\right)={e}^{x}[\/latex] is vertically stretched by a factor of [latex]2[\/latex], reflected across the [latex]y[\/latex]-axis, and then shifted up [latex]4[\/latex]\u00a0units.<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"290621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"290621\"]\r\n\r\nWe want to find an equation of the general form [latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex]. We use the description provided to find [latex]a, b, c[\/latex], and [latex]d[\/latex].\r\n
                \r\n \t
              • We are given the parent function [latex]f\\left(x\\right)={e}^{x}[\/latex], so [latex]b\u00a0= e[\/latex].<\/li>\r\n \t
              • The function is stretched by a factor of [latex]2[\/latex], so [latex]a\u00a0= 2[\/latex].<\/li>\r\n \t
              • The function is reflected about the [latex]y[\/latex]-axis. We replace [latex]x[\/latex]\u00a0with [latex]\u2013x[\/latex]\u00a0to get: [latex]{e}^{-x}[\/latex].<\/li>\r\n \t
              • The graph is shifted vertically [latex]4[\/latex] units, so [latex]d\u00a0= 4[\/latex].<\/li>\r\n<\/ul>\r\nSubstituting in the general form, we get:\r\n

                [latex]\\begin{array}{llll}f\\left(x\\right)\\hfill & =a{b}^{x+c}+d\\hfill \\\\ \\hfill & =2{e}^{-x+0}+4\\hfill \\\\ \\hfill & =2{e}^{-x}+4\\hfill \\end{array}[\/latex]<\/p>\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(4,\\infty \\right)[\/latex]; the horizontal asymptote is [latex]y=4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                [ohm_question hide_question_numbers=1]321400[\/ohm_question]<\/section>
                [ohm_question hide_question_numbers=1]321401[\/ohm_question]<\/section><\/section>","rendered":"

                Stretching, Compressing, or Reflecting an Exponential Function<\/h2>\n

                While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch<\/strong> or compression<\/strong> occurs when we multiply the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex] by a constant [latex]|a|>0[\/latex].<\/p>\n

                For example, if we begin by graphing the parent function [latex]f\\left(x\\right)={2}^{x}[\/latex], we can then graph the stretch, using [latex]a=3[\/latex], to get [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] and the compression, using [latex]a=\\frac{1}{3}[\/latex], to get [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex].<\/p>\n
                \"Two
                (a) [latex]g\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex] stretches the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of 3. (b) [latex]h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}[\/latex] compresses the graph of [latex]f\\left(x\\right)={2}^{x}[\/latex] vertically by a factor of [latex]\\frac{1}{3}[\/latex].<\/figcaption><\/figure>\n<\/section>\n
                \n

                stretches and compressions of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\n

                The function [latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/p>\n

                  \n
                • is stretched vertically by a factor of [latex]a[\/latex]if [latex]|a|>1[\/latex].<\/li>\n
                • is compressed vertically by a factor of [latex]a[\/latex]\u00a0if [latex]|a|<1[\/latex].<\/li>\n
                • has a\u00a0[latex]y[\/latex]-intercept is [latex]\\left(0,a\\right)[\/latex].<\/li>\n
                • has a horizontal asymptote of [latex]y=0[\/latex], range of [latex]\\left(0,\\infty \\right)[\/latex], and domain of [latex]\\left(-\\infty ,\\infty \\right)[\/latex] which are all unchanged from the parent function.<\/li>\n<\/ul>\n<\/section>\n
                  Exponential functions are stretched, compressed or reflected in the same manner you’ve used to transform other functions. Multipliers or negatives inside the function argument (in the exponent) affect horizontal transformations. Multipliers or negatives outside the function argument affect vertical transformations.<\/section>\n
                  Sketch a graph of [latex]f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]. State the domain, range, and asymptote.<\/p>\n