9. Given the function $f(x)=a x^{2}-c(a, c$ are real numbers). If $-4 \leqslant f(1) \leqslant-1,-1 \leqslant f(2) \leqslant 2$, then the maximum value of $f(8)$ is $\qquad$ .
9.122.

Since $f(1)=a-c, f(2)=4a-c$, therefore, $a=\frac{1}{3}(f(2)-f(1)), c=\frac{1}{3}(f(2)-4f(1))$.
Thus, $f(8)=64a-c=21f(2)-20f(1)$.
Hence $-1=21 \times(-1)-20 \times(-1) \leqslant f(8)$ $\leqslant 21 \times 2-20 \times(-4)=122$.

When $f(x)=2x^{2}-6$, it satisfies the given conditions, and $f(8)=122$. Therefore, the maximum value of $f(8)$ is 122.