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.

   Step 1(a):

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

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

   Step 2(a):

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

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

   Step 3(a):

   Solve the first equation for $a$.

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

   Step 4(a):

   Substitute for $a$ in the second equation.

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

   Step 5(a):

   Solve for $b$.

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

   Step 6(a):

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

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

   $a = 20$

   Answer (a):

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

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

   Step (b):

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

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