Answer
$
f(t)=250.252\cdot (1.028)^t
$
Work Step by Step
We are given that $f(-8)= 200$ and $f(30)= 580$. The ratio method gives:
$$
\begin{aligned}
\frac{a b^{30}}{a b^{-8}} & =\frac{f(30)}{f(-8)} \\
b^{38} & =\frac{580}{200} \\
b & =\left(\frac{580}{200}\right)^{1 / 38} \approx 1.0284
\end{aligned}
$$
$$
\begin{aligned}
a(1.0284)^{-8} & =200 \\
a & =\frac{200}{(1.0284)^{-8}} \approx 250.252
\end{aligned}
$$
It follows that
$$
f(t)=250.252\cdot (1.028)^t
$$
