jmc

Given that $(8,8)$ is on the graph of $y=\frac{1}{4}f\left(\frac{1}{2}x\right)$, we can substitute $8$ for both $x$ and $y$ in that equation to obtain $$8=\frac{1}{4}f\left(\frac{1}{2}\cdot 8\right).$$We can rewrite this information as $$32=f(4),$$which tells us that $(4,32)$ must be on the graph of $y=f(x)$. The sum of coordinates of this point is $\boxed{36}$.