解:
$ (1)$根据题意,得$\begin {cases}a - b + 1 = 6,①\\4a + 2b + 1 = 11,②\end {cases}$
$ ①×2 +②,$得$6a + 3 = 23,$
$ $解得$a = \frac {10}{3}.$
$ $将$a = \frac {10}{3}$代入$①,$得$\frac {10}{3} - b + 1 = 6,$
$ $解得$b = -\frac {5}{3}.$
$ (2)$由$(1)$可知,$y = \frac {10}{3}x^2 - \frac {5}{3}x + 1,$
$ $当$x = -3$时,$y = \frac {10}{3}×(-3)^2 - \frac {5}{3}×(-3) + 1 = 36.$