Chapter 4.1, Problem 32E

Solution

To write an exponential function of the form $y=a{b}^{x-h}+k$ whose graph has a y-intercept at 5 and asymptote of $y=2$.

One exponential function can be written as $y=3\cdot 2^x+2$.

Given information:

The graph of the function has y-intercept at 5 and asymptote of $y=2$.

Concept used:

An exponential growth model of the form $y=a{b}^{x-h}+k$ has an asymptote of $y=k$.

Calculation:

Since the function has an asymptote of $y=2$ , it gives, $k=2$.

The model can be written as $y=a{b}^{x-h}+2$.

Since the function has a y-intercept at 5, it passes through the point $\left(0,5\right)$.

Substitute $x=0$ and $y=5$ in the equation $y=a{b}^{x-h}+2$.

  $\begin{array}{c}5=a{b}^{0-h}+2\\ 5=a{b}^{-h}+2\\ a{b}^{-h}=5-2\\ a{b}^{-h}=3\\ \frac{a}{b^h}=3\end{array}$

Since there can be many exponential functions satisfying the given condition, consider $b=2$ and $h=0$.

When $b=2$ and $h=0$ , then, $\frac{a}{b^h}=3$ implies $a=3$.

Thus, one exponential function can be written as $y=3\cdot 2^x+2$.

Conclusion:

One exponential function can be written as $y=3\cdot 2^x+2$.