Exponential Functions

An exponential function has the form $y=a{b}^x,$ where $a\ne 0$ and the base $b$ is a positive real number other than $1.$ Let's determine and interpret $a$ and $b$ of the given model. $$\begin{array}{c}y=2500(1.50{)}^t\end{array}$$ In this situation, $a$ represents the number of subscribers it received at the beginning of the model. Because $b$ is greater than $0$ and less than $1,$ it represents the growth factor of the number of referrals each year. To determine the annual percent increase, we should consider the exponential growth model. $$\begin{array}{c}y=a(1+r{)}^t\end{array}$$ In the model, $r$ is the percent increase written as a decimal and the quantity $1+r$ is the growth factor. Knowing that the growth factor is $1.50$ in our situation, we can find the percent increase. $$\begin{array}{c}1-r=1.50\Rightarrow r=0.50\text{ or }r=50\%\end{array}$$