{"id":1058,"date":"2025-07-22T00:04:41","date_gmt":"2025-07-22T00:04:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1058"},"modified":"2026-03-16T18:56:03","modified_gmt":"2026-03-16T18:56:03","slug":"exponential-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions-learn-it-2\/","title":{"raw":"Graphs of Exponential Functions: Learn It 2","rendered":"Graphs of Exponential Functions: Learn It 2"},"content":{"raw":"

Graph exponential functions<\/h2>\r\n
\r\n

characteristics of the graph of the parent function [latex]f\\left(x\\right)={b}^{x}[\/latex]<\/h3>\r\nAn exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], [latex]b>0[\/latex], [latex]b\\ne 1[\/latex], has these characteristics:\r\n
    \r\n \t
  • one-to-one function<\/li>\r\n \t
  • The horizontal asymptote is [latex]y = 0[\/latex].<\/li>\r\n \t
  • The domain of [latex]f[\/latex] is all real numbers, [latex](-\\infty, \\infty)[\/latex].<\/li>\r\n \t
  • The range of [latex]f[\/latex] is all positive real numbers, [latex](0, \\infty)[\/latex].<\/li>\r\n \t
  • There is no [latex]x[\/latex]-intercept.<\/li>\r\n \t
  • The [latex]y[\/latex]-intercept is [latex]\\left(0,1\\right)[\/latex].<\/li>\r\n \t
  • The graph is\u00a0increasing if [latex]b \\gt 1[\/latex], which implies exponential growth.<\/li>\r\n \t
  • The graph decreasing if [latex]0 \\lt b \\lt 1[\/latex], which implies exponential decay.<\/li>\r\n<\/ul>\r\n<\/section>
    How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function<\/strong>\r\n
      \r\n \t
    1. Create a table of points.<\/li>\r\n \t
    2. Plot at least [latex]3[\/latex]\u00a0point from the table including the y<\/em>-intercept [latex]\\left(0,1\\right)[\/latex].<\/li>\r\n \t
    3. Draw a smooth curve through the points.<\/li>\r\n \t
    4. State the domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range, [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote, [latex]y=0[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
      When sketching the graph of an exponential function by plotting points, include a few input values left and right of zero as well as zero itself.\r\n[latex]\\\\[\/latex]\r\nWith few exceptions, such as functions that would be undefined at zero or negative input like the radical or (as you'll see soon) the logarithmic function, it is good practice to let the input equal [latex]-3, -2, -1, 0, 1, 2, \\text{ and } 3[\/latex] to get the idea of the shape of the graph.<\/section>
      \r\n
      \r\n\r\n[ohm_question hide_question_numbers=1]321385[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section><\/section>
      \r\n

      Exponential Growth vs. Decay<\/h3>\r\nThe graph of exponential growth is increasing like shown in the graph below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Notice that the graph gets close to the x-axis, but never touches it.[\/caption]\r\n

      The graph of the exponential decay function, [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] decreases, but with a similar (reflected) shape.<\/p>\r\n\"Graph\r\n

      The domain of [latex]g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}[\/latex] is all real numbers, the range is [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote is [latex]y=0[\/latex].<\/p>\r\n\r\n

      \r\n
      <\/div>\r\n<\/div>\r\n<\/section><\/section>
      \r\n
      \r\n
      Sketch a graph of [latex]f\\left(x\\right)={0.25}^{x}[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"410947\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"410947\"]Before graphing, identify the behavior and create a table of points for the graph.\r\n
        \r\n \t
      • Since [latex]b= 0.25[\/latex] is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote [latex]y= 0[\/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)={0.25}^{x}[\/latex]<\/strong><\/td>\r\n[latex]64[\/latex]<\/td>\r\n[latex]16[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]0.25[\/latex]<\/td>\r\n[latex]0.0625[\/latex]<\/td>\r\n[latex]0.015625[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
      • Plot the [latex]y[\/latex]-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].<\/li>\r\n<\/ul>\r\nDraw a smooth curve connecting the points.\r\n\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], and the horizontal asymptote is [latex]y=0[\/latex].[\/caption][\/hidden-answer]<\/section>
        \r\n
        [ohm_question hide_question_numbers=1]321384[\/ohm_question]<\/div>\r\n<\/section>
        Use a graphing utility to sketch a graph of [latex]f(x)={\\sqrt{2}(\\sqrt{2})}^{x}[\/latex]. State the domain and range.\r\n[reveal-answer q=\"334418\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"334418\"]\r\n

        Before graphing, identify the behavior and create a table of points for the graph.<\/p>\r\n\r\n

          \r\n \t
        • Since [latex]b= \\sqrt{2}[\/latex], which is greater than one, we know the function is increasing, and we can verify this by creating a table of values. The left tail of the graph will get really close to the x-axis and the right tail will increase without bound.<\/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)=\\sqrt{2}{(\\sqrt{2})}^{x}[\/latex]<\/strong><\/td>\r\n[latex]0.5[\/latex]<\/td>\r\n[latex]0.71[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]1.41[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]2.83[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
        • Plot the [latex]y[\/latex]-intercept, [latex]\\left(0,1.41\\right)[\/latex], along with two other points. We can use [latex]\\left(-1,1\\right)[\/latex] and [latex]\\left(1,2\\right)[\/latex].<\/li>\r\n<\/ul>\r\n

          Draw a smooth curve connecting the points.<\/p>\r\n\"Screen\r\n\r\nThe domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]; the range is [latex]\\left(0,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section>","rendered":"

          Graph exponential functions<\/h2>\n
          \n

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

          An exponential function with the form [latex]f\\left(x\\right)={b}^{x}[\/latex], [latex]b>0[\/latex], [latex]b\\ne 1[\/latex], has these characteristics:<\/p>\n

            \n
          • one-to-one function<\/li>\n
          • The horizontal asymptote is [latex]y = 0[\/latex].<\/li>\n
          • The domain of [latex]f[\/latex] is all real numbers, [latex](-\\infty, \\infty)[\/latex].<\/li>\n
          • The range of [latex]f[\/latex] is all positive real numbers, [latex](0, \\infty)[\/latex].<\/li>\n
          • There is no [latex]x[\/latex]-intercept.<\/li>\n
          • The [latex]y[\/latex]-intercept is [latex]\\left(0,1\\right)[\/latex].<\/li>\n
          • The graph is\u00a0increasing if [latex]b \\gt 1[\/latex], which implies exponential growth.<\/li>\n
          • The graph decreasing if [latex]0 \\lt b \\lt 1[\/latex], which implies exponential decay.<\/li>\n<\/ul>\n<\/section>\n
            How To: Given an exponential function of the form [latex]f\\left(x\\right)={b}^{x}[\/latex], graph the function<\/strong><\/p>\n
              \n
            1. Create a table of points.<\/li>\n
            2. Plot at least [latex]3[\/latex]\u00a0point from the table including the y<\/em>-intercept [latex]\\left(0,1\\right)[\/latex].<\/li>\n
            3. Draw a smooth curve through the points.<\/li>\n
            4. State the domain, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], the range, [latex]\\left(0,\\infty \\right)[\/latex], and the horizontal asymptote, [latex]y=0[\/latex].<\/li>\n<\/ol>\n<\/section>\n
              When sketching the graph of an exponential function by plotting points, include a few input values left and right of zero as well as zero itself.
              \n[latex]\\\\[\/latex]
              \nWith few exceptions, such as functions that would be undefined at zero or negative input like the radical or (as you’ll see soon) the logarithmic function, it is good practice to let the input equal [latex]-3, -2, -1, 0, 1, 2, \\text{ and } 3[\/latex] to get the idea of the shape of the graph.<\/section>\n
              \n
              \n
              \n