13. 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 $C$.

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$$
\begin{array}{l}
\text { Let } B\left(b^{2}-4, b\right), C\left(c^{2}-4, c\right) \text {, then } A B \perp B C \Rightarrow \overrightarrow{B A} \cdot \overrightarrow{B C}=0 \\
\Rightarrow\left(b^{2}-4\right)\left(b^{2}-c^{2}\right)+(b-2)(b-c)=(b-2)(b-c)[(b+2)(b+c)+1]=0 \\
\Rightarrow(b+2)(b+c)+1=b^{2}+(c+2) b+2 c+1=0 .
\end{array}
$$

Thus, $\Delta=(c+2)^{2}-4(2 c+1)=c^{2}-4 c \geqslant 0 \Rightarrow c \in(-\infty, 0] \cup[4,+\infty)$.
Therefore, the range of the y-coordinate of point $C$ is $(-\infty, 0] \cup[4,+\infty)$.