Writing an Exponential Model When the Initial Value Is Known

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 $N(t)=80b^t$ to find $b:$

NOTE: Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places for the remainder of this section.

The exponential model for the population of deer is $N(t)=80{\left(1.1447\right)}^t.$ (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.)

We can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph in Figure 3 passes through the initial points given in the problem, $\left(0,80\right)$ and $\left(\text{6},180\right).$ We can also see that the domain for the function is $[0,\infty ),$ and the range for the function is $[80,\infty ).$

Figure 3 Graph showing the population of deer over time, $N(t)=80{\left(1.1447\right)}^t,$ $t$ years after 2006