Example 1 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 Question)
Solve As shown in Figure 30-3, let the coordinates of point $B$ be $\left(y_{1}^{2}-4, y_{1}\right)$, and point $C\left(y^{2}-4, y\right)$. Clearly, $y_{1}^{2}-4 \neq 0$, so $k_{A B}=\frac{y_{1}-2}{y_{1}^{2}-4}=\frac{1}{y_{1}+2}$.
Since $A B \perp B C$, we have $k_{B C}=-\left(y_{1}+2\right)$, thus
$$
\left\{\begin{array}{l}
y-y_{1}=-\left(y_{1}+2\right)\left[x-\left(y_{1}^{2}-4\right)\right], \\
y^{2}=x+4 .
\end{array}\right.
$$

Eliminating $x$, noting that $y \neq y_{1}$, we get $\left(2+y_{1}\right)\left(y+y_{1}\right)+1=0$, which simplifies to $y_{1}^{2}+(2+y) y_{1}+(2 y+1)=0$.
By $\Delta \geqslant 0$, solving yields $y \leqslant 0$ or $y \geqslant 4$.
When $y=0$, the coordinates of point $B$ are $(-3,-1)$; when $y=4$, the coordinates of point $B$ are $(5,-3)$, both satisfying the conditions. Therefore, the range of the ordinate $y$ of point $C$ is $(-\infty, 0] \cup[4,+\infty)$.