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?

   Step 1 (a):

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

   Step 2 (a):

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

   Step 3 (a):

   Solve the first equation for $a$.

   Step 4 (a):

   Substitute for $a$ in the second equation.

   Step 5 (a):

   Solve for $b$.

   Step 6 (a):

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

   Answer (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 $w$?

   Step (b):

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

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