Exercise 43 Page 428

Recall that an exponential function is characterized for having consecutive $y\text{-values}$ differing by a common factor if the corresponding $x\text{-values}$ are separated by equal intervals. We can consider a population of bacteria that doubles every hour. The table below shows the population size at the specified time.

As we can see, after every $1$ hour that passes the population is multiplied by $2.$ This makes an exponential function the perfect model if we represent the time with the $x\text{-variable}$ and the population size with $y.$ Now, we know that our function is of the form shown below. We can evaluate it at two different points to find the values for $a$ and $b.$ For example, we can try with $(0,50).$ Now that we know $a=50,$ we can use this and substitute another point. Let's use $(1,100).$ Now that we know that $a=50$ and $b=2,$ we can write our function. The graph of this function is shown below.

As we can see, the behavior of the graph is clearly nonlinear. This can also be seen in the data set, since a linear function is characterized by having a constant difference between consecutive pairs of $y\text{-values}$ corresponding to equal intervals of $x,$ which is not the case here.