### Home > INT3 > Chapter 8 > Lesson 8.1.2 > Problem8-36

8-36.

The graph of an exponential function $y = ab^{x} + k$ passes through $\left(3, 7.5\right)$ and $\left(4, 6.25\right)$. It also has an asymptote at $y = 5$.

1. What is the equation of the function?

Substitute $5$ for $k$ and then use the points to write a system of equations.

Rewrite each equation by subtracting $5$ from both sides.

Solve the first equation for $a$.

Substitute for $a$ in the second equation.

Solve for $b$.

Substitute $1/2$ for $b$ to solve for $a$.

$y = 20\left(\frac{1}{2}\right)^x + 5$

$k = 5$
$7.5 = ab^{3} + 5$
$6.25 = ab^{4} + 5$

$ab^{3} = 2.5$
$ab^{4} = 1.25$

$a = \frac{2.5}{b^3}$

$\left( \frac{2.5}{b^3} \right) b^4 = 1.25$

$b = \frac{1}{2}$

$a = \frac{2.5}{(1/2)^3}$

$a = 20$

2. If the function also passes through $\left(8, w\right)$, what is the value of $w$?

Substitute $8$ for $x$ and $w$ for $y$.

$w = 20\left(\frac{1}{2}\right)^8 + 5$