jmc

We substitute these two points into the given equation to solve for $c$. Plugging in $(-1,-11)$, we get $-11=(-1{)}^2-1b+c\Rightarrow -b+c=-12$. Plugging in $(3,17)$, we get $17=3^2+3b+c\Rightarrow 3b+c=8$. In summary, we have the two equations $$\begin{array}{rl}-b+c& =-12\\ 3b+c& =8\end{array}$$ Multiplying the first equation by 3, we have $-3b+3c=-36$. Adding the second equation to this last one, we have $(-3b+3c)+(3b+c)=-36+8\Rightarrow c=\boxed{-7}$.

The parabola is graphed below: