{"id":1067,"date":"2025-07-22T00:05:53","date_gmt":"2025-07-22T00:05:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1067"},"modified":"2026-03-17T16:06:16","modified_gmt":"2026-03-17T16:06:16","slug":"exponential-functions-learn-it-2-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-functions-learn-it-2-2\/","title":{"raw":"Exponential Functions: Learn It 2","rendered":"Exponential Functions: Learn It 2"},"content":{"raw":"

Finding Equations of Exponential Functions<\/h2>\r\nIn the previous examples, we were given an exponential function which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, determine the constants [latex]a[\/latex]\u00a0and [latex]b[\/latex], and evaluate the function.\r\n\r\n
How To: Given two data points, write an exponential model<\/strong>\r\n
    \r\n \t
  1. If one of the data points has the form [latex]\\left(0,a\\right)[\/latex], then [latex]a[\/latex]\u00a0is the initial value. Using [latex]a[\/latex], substitute the second point into the equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], and solve for [latex]b[\/latex].<\/li>\r\n \t
  2. If neither of the data points have the form [latex]\\left(0,a\\right)[\/latex], substitute both points into two equations with the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex]. Solve the resulting system of two equations to find [latex]a[\/latex] and [latex]b[\/latex].<\/li>\r\n \t
  3. Using the [latex]a[\/latex]\u00a0and [latex]b[\/latex] found in the steps above, write the exponential function in the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    When writing an exponential model from two data points, recall the processes you've learned to write other types of models from data points contained on the graphs of linear, power, polynomial, and rational functions. Each process was different, but each followed a fundamental characteristic of functions: that every point on the graph of a function satisfies the equation of the function. The process of writing an exponential model capitalizes on the same idea.<\/section>
    In 2006, [latex]80[\/latex] deer were introduced into a wildlife refuge. By 2012, the population had grown to [latex]180[\/latex] deer. The population was growing exponentially.\r\n[latex]\\\\[\/latex]\r\nWrite an algebraic function [latex]N(t)[\/latex] representing the population [latex]N[\/latex]\u00a0of deer over time [latex]t[\/latex].[reveal-answer q=\"910377\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"910377\"]We let our independent variable [latex]t[\/latex]\u00a0be the number of years after 2006. Thus, the information given in the problem can be written as input-output pairs: [latex](0, 80)[\/latex] and [latex](6, 180)[\/latex]. Notice that by choosing our input variable to be measured as years after 2006, we have given ourselves the initial value for the function, [latex]a\u00a0= 80[\/latex]. We can now substitute the second point into the equation [latex]N\\left(t\\right)=80{b}^{t}[\/latex] to find [latex]b[\/latex]:\r\n

    [latex]\\begin{array}{c}N\\left(t\\right)\\hfill & =80{b}^{t}\\hfill & \\hfill \\\\ 180\\hfill & =80{b}^{6}\\hfill & \\text{Substitute using point }\\left(6, 180\\right).\\hfill \\\\ \\frac{9}{4}\\hfill & ={b}^{6}\\hfill & \\text{Divide and write in lowest terms}.\\hfill \\\\ b\\hfill & ={\\left(\\frac{9}{4}\\right)}^{\\frac{1}{6}}\\hfill & \\text{Isolate }b\\text{ using properties of exponents}.\\hfill \\\\ b\\hfill & \\approx 1.1447 & \\text{Round to 4 decimal places}.\\hfill \\end{array}[\/latex]<\/p>\r\nNOTE:<\/strong> Unless otherwise stated, do not round any intermediate calculations. Round the final answer to four places for the remainder of this section.<\/em>\r\n\r\nThe exponential model for the population of deer is [latex]N\\left(t\\right)=80{\\left(1.1447\\right)}^{t}[\/latex]. Note that this exponential function models short-term growth. As the inputs get larger, the outputs will get increasingly larger resulting in the model not being useful in the long term due to extremely large output values.\r\n\r\nWe can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph below\u00a0passes through the initial points given in the problem, [latex]\\left(0,\\text{ 8}0\\right)[\/latex] and [latex]\\left(\\text{6},\\text{ 18}0\\right)[\/latex]. We can also see that the domain for the function is [latex]\\left[0,\\infty \\right)[\/latex] and the range for the function is [latex]\\left[80,\\infty \\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph showing the population of deer over time, [latex]N\\left(t\\right)=80{\\left(1.1447\\right)}^{t}[\/latex], t\u00a0years after 2006[\/caption]\r\n[\/hidden-answer]<\/section>

    Find an exponential function that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(2,1\\right)[\/latex].[reveal-answer q=\"904458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"904458\"]Because we don\u2019t have the initial value, we substitute both points into an equation of the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex] and then solve the system for [latex]a[\/latex]\u00a0and [latex]b[\/latex].\r\n
      \r\n \t
    • Substituting [latex]\\left(-2,6\\right)[\/latex] gives [latex]6=a{b}^{-2}[\/latex]<\/li>\r\n \t
    • Substituting [latex]\\left(2,1\\right)[\/latex] gives [latex]1=a{b}^{2}[\/latex]<\/li>\r\n<\/ul>\r\nUse the first equation to solve for [latex]a[\/latex]\u00a0in terms of [latex]b[\/latex]:\r\n

      [latex]\\begin{array}{l}6=ab^{-2}\\\\\\frac{6}{b^{-2}}=a\\,\\,\\,\\,\\,\\,\\,\\,\\text{Divide.}\\\\a=6b^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\text{Use properties of exponents to rewrite the denominator.}\\end{array}[\/latex]<\/p>\r\nSubstitute [latex]a[\/latex]\u00a0in the second equation and solve for [latex]b[\/latex]:\r\n

      [latex]\\begin{array}{l}1=ab^{2}\\\\1=6b^{2}b^{2}=6b^{4}\\,\\,\\,\\,\\,\\text{Substitute }a.\\\\b=\\left(\\frac{1}{6}\\right)^{\\frac{1}{4}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Use properties of exponents to isolate }b.\\\\b\\approx0.6389\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Round 4 decimal places.}\\end{array}[\/latex]<\/p>\r\nUse the value of [latex]b[\/latex]\u00a0in the first equation to solve for the value of [latex]a[\/latex]:\r\n

      [latex]a=6b^{2}\\approx6\\left(0.6389\\right)^{2}\\approx2.4492[\/latex]<\/p>\r\nThus, the equation is [latex]f\\left(x\\right)=2.4492{\\left(0.6389\\right)}^{x}[\/latex].\r\n\r\nWe can graph our model to check our work. Notice that the graph below\u00a0passes through the initial points given in the problem, [latex]\\left(-2,\\text{ 6}\\right)[\/latex] and [latex]\\left(2,\\text{ 1}\\right)[\/latex]. The graph is an example of an exponential decay function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The graph of [latex]f\\left(x\\right)=2.4492{\\left(0.6389\\right)}^{x}[\/latex] models exponential decay.[\/caption][\/hidden-answer]<\/section>

      [ohm_question hide_question_numbers=1]321407[\/ohm_question]<\/section>
      Do two points always determine a unique exponential function?<\/strong>\r\n\r\n
      \r\n\r\nYes, provided the two points are either both above the [latex]x[\/latex]-axis or both below the [latex]x[\/latex]-axis and have different [latex]x[\/latex]-coordinates. But keep in mind that we also need to know that the graph is, in fact, an exponential function. Not every graph that looks exponential really is exponential. We need to know the graph is based on a model that shows the same percent growth with each unit increase in [latex]x[\/latex],\u00a0which in many real world cases involves time.\r\n\r\n<\/section>
      How To: Given the graph of an exponential function, write its equation<\/strong>\r\n
        \r\n \t
      1. First, identify two points on the graph. Choose the [latex]y[\/latex]-intercept as one of the two points whenever possible. Try to choose points that are as far apart as possible to reduce round-off error.<\/li>\r\n \t
      2. If one of the data points is the y-<\/em>intercept [latex]\\left(0,a\\right)[\/latex] , then [latex]a[\/latex]\u00a0is the initial value. Using [latex]a[\/latex], substitute the second point into the equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex] and solve for [latex]b[\/latex].<\/li>\r\n \t
      3. If neither of the data points have the form [latex]\\left(0,a\\right)[\/latex], substitute both points into two equations with the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex]. Solve the resulting system of two equations to find [latex]a[\/latex]\u00a0and [latex]b[\/latex].<\/li>\r\n \t
      4. Write the exponential function, [latex]f\\left(x\\right)=a{b}^{x}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
        Find an equation for the exponential function graphed below.\"Graph[reveal-answer q=\"440954\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"440954\"]We can choose the [latex]y[\/latex]-intercept of the graph, [latex]\\left(0,3\\right)[\/latex], as our first point. This gives us the initial value [latex]a=3[\/latex]. Next, choose a point on the curve some distance away from [latex]\\left(0,3\\right)[\/latex] that has integer coordinates. One such point is [latex]\\left(2,12\\right)[\/latex].\r\n

        [latex]\\begin{array}{llllll}y=a{b}^{x}& \\text{Write the general form of an exponential equation}. \\\\ y=3{b}^{x} & \\text{Substitute the initial value 3 for }a. \\\\ 12=3{b}^{2} & \\text{Substitute in 12 for }y\\text{ and 2 for }x. \\\\ 4={b}^{2} & \\text{Divide by 3}. \\\\ b=\\pm 2 & \\text{Take the square root}.\\end{array}[\/latex]<\/p>\r\nBecause we restrict ourselves to positive values of [latex]b[\/latex], we will use [latex]b\u00a0= 2[\/latex]. Substitute [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0into standard form to yield the equation [latex]f\\left(x\\right)=3{\\left(2\\right)}^{x}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

        \r\n
        \r\n
        [ohm_question hide_question_numbers=1]321409[\/ohm_question]<\/section>
        [ohm_question hide_question_numbers=1]321408[\/ohm_question]<\/section>
        \r\n

        How To: Given two points on the curve of an exponential function, use a graphing calculator to find the equation.<\/strong><\/p>\r\n\r\n

          \r\n \t
        1. Press [STAT].<\/strong><\/li>\r\n \t
        2. Clear any existing entries in columns L1<\/strong> or L2.<\/strong><\/li>\r\n \t
        3. In L1<\/strong>, enter the x<\/em>-coordinates given.<\/li>\r\n \t
        4. In L2<\/strong>, enter the corresponding y<\/em>-coordinates.<\/li>\r\n \t
        5. Press [STAT] <\/strong>again. Cursor right to CALC<\/strong>, scroll down to ExpReg (Exponential Regression)<\/strong>, and press [ENTER].<\/strong><\/li>\r\n \t
        6. The screen displays the values of a<\/em> and b<\/em> in the exponential equation [latex]y=a\\cdot {b}^{x}[\/latex].<\/li>\r\n<\/ol>\r\nHow To: Given two points on the curve of an exponential function, use Desmos to find the equation.<\/strong>\r\n
            \r\n \t
          1. Click the [ + ] icon in the top left corner and choose \u201cTable.\u201d<\/li>\r\n \t
          2. In the x\u2081 column, enter the x-coordinates of the given points.<\/li>\r\n \t
          3. In the y\u2081 column, enter the corresponding y-coordinates.<\/li>\r\n \t
          4. Click below the table to enter a new expression.<\/li>\r\n \t
          5. Type the model:\r\n[latex]y_1 \\sim a \\cdot b^{x_1}[\/latex]<\/li>\r\n \t
          6. Desmos will display values for [latex]a[\/latex] and [latex]b[\/latex] that best fit the data in the form\r\n[latex]y = a \\cdot b^x[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n
            \r\n
            \r\n

            Use a graphing calculator to find the exponential equation that includes the points [latex]\\left(2,24.8\\right)[\/latex] and [latex]\\left(5,198.4\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"455667\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455667\"]\r\n

            Follow the guidelines above. First press [STAT]<\/strong>, [EDIT]<\/strong>, [1: Edit\u2026], <\/strong>and clear the lists L1<\/strong> and L2<\/strong>. Next, in the L1<\/strong> column, enter the x<\/em>-coordinates, 2 and 5. Do the same in the L2<\/strong> column for the y<\/em>-coordinates, 24.8 and 198.4.<\/p>\r\n

            Now press [STAT]<\/strong>, [CALC]<\/strong>, [0: ExpReg] <\/strong>and press [ENTER]<\/strong>. The values a\u00a0<\/em>= 6.2 and b\u00a0<\/em>= 2 will be displayed. The exponential equation is [latex]y=6.2\\cdot {2}^{x}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

            [ohm_question hide_question_numbers=1]321410[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section>","rendered":"

            Finding Equations of Exponential Functions<\/h2>\n

            In the previous examples, we were given an exponential function which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, determine the constants [latex]a[\/latex]\u00a0and [latex]b[\/latex], and evaluate the function.<\/p>\n

            How To: Given two data points, write an exponential model<\/strong><\/p>\n
              \n
            1. If one of the data points has the form [latex]\\left(0,a\\right)[\/latex], then [latex]a[\/latex]\u00a0is the initial value. Using [latex]a[\/latex], substitute the second point into the equation [latex]f\\left(x\\right)=a{b}^{x}[\/latex], and solve for [latex]b[\/latex].<\/li>\n
            2. If neither of the data points have the form [latex]\\left(0,a\\right)[\/latex], substitute both points into two equations with the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex]. Solve the resulting system of two equations to find [latex]a[\/latex] and [latex]b[\/latex].<\/li>\n
            3. Using the [latex]a[\/latex]\u00a0and [latex]b[\/latex] found in the steps above, write the exponential function in the form [latex]f\\left(x\\right)=a{b}^{x}[\/latex].<\/li>\n<\/ol>\n<\/section>\n
              When writing an exponential model from two data points, recall the processes you’ve learned to write other types of models from data points contained on the graphs of linear, power, polynomial, and rational functions. Each process was different, but each followed a fundamental characteristic of functions: that every point on the graph of a function satisfies the equation of the function. The process of writing an exponential model capitalizes on the same idea.<\/section>\n
              In 2006, [latex]80[\/latex] deer were introduced into a wildlife refuge. By 2012, the population had grown to [latex]180[\/latex] deer. The population was growing exponentially.
              \n[latex]\\\\[\/latex]
              \nWrite an algebraic function [latex]N(t)[\/latex] representing the population [latex]N[\/latex]\u00a0of deer over time [latex]t[\/latex].<\/p>\n