  ### Home > CCA2 > Chapter 7 > Lesson 7.2.3 > Problem7-150

7-150.

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. Find the equation of the function.

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

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

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

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

Solve the first equation for $a$.

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

Substitute for $a$ in the second equation.

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

Solve for $b$.

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

$\text{Substitute }\frac{1}{2}\text{ for }b \text{ to solve for }a.$

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

$a = 20$

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

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

Substitute $8\ \text{for}\ x$ and $w\ \text{for}\ y$.

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