解:$(1)$由题意,设$y_{1} = a(x + 2)(a\neq 0),$$y - 1 = by_{1}(b\neq 0)$
$ $则$y = by_{1} + 1 = ab(x + 2)+1$
$ $又当$x = 0$时,$y = 7$
∴$2ab + 1 = 7,$解得$ab = 3$
∴$y$关于$x$的函数表达式为$y = 3(x + 2)+1 = 3x + 7$
$ (2)$由$(1)$得$y = 3x + 7$
∴当$y = -1$时,$3x + 7 = -1,$解得$x = -\frac 83$
则$x$的值为$-\frac 83$