Exponential Functions

First, we write an exponential function to model the given situation, thus, recall its general form. $$\begin{array}{c}y=a{b}^x\end{array}$$ In this formula, $a$ is the initial value and $b=1+r,$ where $r$ is the rate of change. If the function represents growth then $r>0.$ Conversely, if it represents decay, then $r<0.$

Writing the Equation

To write the equation, we first need to define the variables. Let $y$ be the population, and $x$ the number of years after that have passed. In this case, the initial value is a population of $120000.$ Since the population increases $1.2\%$ each year, we have that $r=0.012.$ $$\begin{array}{c}y=120000[1+0.012{]}^x\\ \Updownarrow \\ y=120000(1.012{)}^x\end{array}$$

Finding the Amount After the Specified Time

Now, we want to find the population after $15$ years. To do so, we will substitute $15$ for $x$ in our model equation. We found that, after $15$ years, the population will be approximately $143512.$