Q. 89 - Precalculus Enhanced with Graphing Utilities

Question

Find an exponential function with horizontal asymptote $y=2$ whose graph contains the points $(0,3)$ and $(1,5)$.

Step-by-Step Solution

Answer

The exponential function is $y=3^x+2$.

Step 1. Use the given information.

Consider the given horizontal asymptote $y=2$.

The two points on the exponential function are $(0,3)$ and $(1,5)$.

If the horizontal asymptote is $y=0$, then the exponential function is $y=a^x$.

If the horizontal asymptote is $y=2$, then the exponential function is $y=a^x+2$.

Step 2. Finding exponential function.

On the point $(1,5)$, the exponential function is

$$\begin{gathered} 5=a^1+2 \\ 5=a+2 \\ a=5-2 \\ a=3 \end{gathered}$$

So the exponential function is $y=3^x+2$.