Let $h, h_M: \mathbb {F} _q^n\to \mathbb {F} _q$ be quadratic forms with $h$ not degenerate. Fix $k\in \mathbb {F} _q$ and set $C_n(k,h)_{\mathbb {F} _q}:= \{h(x_1,\dots ,x_n)=k\}\subset \mathbb {F}_q^n$. We compute (in many cases) the image of $h_{M|C_n(k,h)_{\mathbb {F} _q}}$. This question is related to a question on the numerical range of matrices over a finite field.