{"id":1512,"date":"2015-11-12T18:35:28","date_gmt":"2015-11-12T18:35:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1512"},"modified":"2017-04-03T14:54:11","modified_gmt":"2017-04-03T14:54:11","slug":"find-the-equation-of-an-exponential-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/find-the-equation-of-an-exponential-function\/","title":{"raw":"Find the equation of an exponential function","rendered":"Find the equation of an exponential function"},"content":{"raw":"

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, then determine the constants a<\/em>\u00a0and b<\/em>, and evaluate the function.<\/p>\r\n\r\n

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How To: Given two data points, write an exponential model.<\/h3>\r\n
  1. If one of the data points has the form [latex]\\left(0,a\\right)[\/latex], then a<\/em>\u00a0is the initial value. Using a<\/em>, substitute the second point into the equation [latex]f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex], and solve for b<\/em>.<\/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{\\left(b\\right)}^{x}[\/latex]. Solve the resulting system of two equations in two unknowns to find a<\/em>\u00a0and b<\/em>.<\/li>\r\n\t
  3. Using the a<\/em>\u00a0and b<\/em>\u00a0found in the steps above, write the exponential function in the form [latex]f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex].<\/li>\r\n<\/ol><\/div>\r\n
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    Example 3: Writing an Exponential Model When the Initial Value Is Known<\/h3>\r\n

    In 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an algebraic function N<\/em>(t<\/em>) representing the population N<\/em>\u00a0of deer over time t<\/em>.<\/p>\r\n\r\n<\/div>\r\n

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    Solution<\/h3>\r\n

    We let our independent variable t<\/em>\u00a0be the number of years after 2006. Thus, the information given in the problem can be written as input-output pairs: (0, 80) and (6, 180). Notice that by choosing our input variable to be measured as years after 2006, we have given ourselves the initial value for the function, a\u00a0<\/em>= 80. We can now substitute the second point into the equation [latex]N\\left(t\\right)=80{b}^{t}[\/latex] to find b<\/em>:<\/p>\r\n\r\n

    [latex]\\begin{cases}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{cases}[\/latex]<\/div>\r\n

    NOTE:<\/strong> Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section.<\/em><\/p>\r\n

    The 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 gets large, the output will get increasingly larger, so much so that the model may not be useful in the long term.)<\/p>\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 \"Graph<\/span>\r\n

    Figure 3.\u00a0<\/strong>Graph showing the population of deer over time, [latex]N\\left(t\\right)=80{\\left(1.1447\\right)}^{t}[\/latex], t<\/em>\u00a0years after 2006<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    Try It 3<\/h3>\r\n

    A wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population N<\/em>\u00a0of wolves over time t<\/em>.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n

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    Example 4: Writing an Exponential Model When the Initial Value is Not Known<\/h3>\r\n

    Find an exponential function that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(2,1\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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    Solution<\/h3>\r\n

    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 a<\/em>\u00a0and b<\/em>.<\/p>\r\n\r\n