Example: Writing an Exponential Model When the Initial Value Is Known

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(t) representing the population N of deer over time t.

Answer: We let our independent variable t be 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 = 80. We can now substitute the second point into the equation [latex]N\left(t\right)=80{b}^{t}[/latex] to find b:

[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]

NOTE: Unless otherwise stated, do not round any intermediate calculations. Round the final answer to four places for the remainder of this section. 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 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. We can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph below passes 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].

Example: Writing an Exponential Model When the Initial Value is Not Known

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

Answer: Because we don’t 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 and b.

• Substituting [latex]\left(-2,6\right)[/latex] gives [latex]6=a{b}^{-2}[/latex]
• Substituting [latex]\left(2,1\right)[/latex] gives [latex]1=a{b}^{2}[/latex]

Use the first equation to solve for a in terms of b:

[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]

Substitute a in the second equation and solve for b:

[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]

Use the value of b in the first equation to solve for the value of a:

[latex]a=6b^{2}\approx6\left(0.6389\right)^{2}\approx2.4492[/latex]

Thus, the equation is [latex]f\left(x\right)=2.4492{\left(0.6389\right)}^{x}[/latex]. We can graph our model to check our work. Notice that the graph below passes 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.

Example: Writing an Exponential Function Given Its Graph

Find an equation for the exponential function graphed below.

Answer: We can choose the y-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].

[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]

Because we restrict ourselves to positive values of b, we will use b = 2. Substitute a and b into standard form to yield the equation [latex]f\left(x\right)=3{\left(2\right)}^{x}[/latex].

Example: Calculating Continuous Growth

A person invested $1,000 in an account earning a nominal interest rate of 10% per year compounded continuously. How much was in the account at the end of one year?

Answer: Since the account is growing in value, this is a continuous compounding problem with growth rate r = 0.10. The initial investment was $1,000, so P = 1000. We use the continuous compounding formula to find the value after t = 1 year:

[latex]\begin{array}{c}A\left(t\right)\hfill & =P{e}^{rt}\hfill & \text{Use the continuous compounding formula}.\hfill \\ \hfill & =1000{\left(e\right)}^{0.1} & \text{Substitute known values for }P, r,\text{ and }t.\hfill \\ \hfill & \approx 1105.17\hfill & \text{Use a calculator to approximate}.\hfill \end{array}[/latex]

The account is worth $1,105.17 after one year.

Example: Calculating Continuous Decay

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-[latex]222[/latex] decay to in [latex]3[/latex] days?

Answer: Since the substance is decaying, the rate, [latex]17.3%[/latex], is negative. So, [latex]r=-0.173[/latex]. The initial amount of radon-[latex]222[/latex] was [latex]100[/latex] mg, so [latex]a= 100[/latex]. We use the continuous decay formula to find the value after [latex]t= 3[/latex] days:

[latex]\begin{array}{c}A\left(t\right)\hfill & =a{e}^{rt}\hfill & \text{Use the continuous growth formula}.\hfill \\ \hfill & =100{e}^{-0.173\left(3\right)} & \text{Substitute known values for }a, r,\text{ and }t.\hfill \\ \hfill & \approx 59.5115\hfill & \text{Use a calculator to approximate}.\hfill \end{array}[/latex]

So [latex]59.5115[/latex] mg of radon-[latex]222[/latex] will remain.