Example 8 Given point $A(0,2)$ and two points $B, C$ on the parabola $y^{2}=x+4$, such that $A B \perp B C$. Find the range of the y-coordinate of point $C$.
(2002 National High School League Problem)
As shown in Figure 8-7, we can set $B\left(m^{2}-4, m\right), C\left(x_{0}^{2}-4, x_{0}\right)$. Then, from $k_{A B} \cdot k_{B C^{C}}=-1$, we have
$$
\frac{m-2}{4-m^{2}} \cdot \frac{m-x_{0}}{m^{2}-x_{0}^{2}}=-1
$$

Rearranging and simplifying, we get: $m^{2}+\left(x_{0}+2\right) m+2 x_{0}+1=0$.
Clearly, the equation in terms of $m$ has solutions, i.e., $\Delta \geqslant 0,\left(x_{0}+2\right)^{2}-4\left(2 x_{0}+1\right) \geqslant 0$. Solving this, we get $x_{0} \geqslant 4$ or $x_{0} \leqslant 0$.