jmc

The graph of a quadratic function has an axis of symmetry. Observing that $q(7)=q(8),$ $q(6)=q(9),$ and so on, we see that this graph's axis of symmetry is $x=7.5$, and so $q(15)=q(0)$.

The graph passes through the point $(0,-3)$, so $q(0)=-3$, which tells us that $q(15)=\boxed{-3}$ as well.