解:$(1)$∵直线$y = 3x - 11$经过点$P(1,b),$
∴$b = 3×1 - 11 = -8,$即点$P $的坐标为$(1,-8)。$
∵抛物线$y = ax^2$经过点$P(1,-8),$
∴$-8 = a×1^2,$解得$a = -8。$
∴$a = -8,$$b = -8$
$ (2)$抛物线$y = -8x^2$的顶点坐标为$(0,0),$对称轴为$y$轴
$ (3)$∵$a = -8,$
∴$a - k^2 = -8 - k^2 < 0。$
∴当$x < 0$时,二次函数$y = (a - k^2)x^2$中的$y$随$x$的增大而增大