Given that $ (8,8) $ is on the graph of $ y=\frac 14f\left(\frac 12x\right) $, we can substitute $ 8 $ for both $ x $ and $ y $ in that equation to obtain $$8 = \frac14f\left(\frac 12\cdot 8\right).$$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} $.