Example 5 Find the minimum value of $y=\sqrt{x^{2}+1}+\sqrt{(4-x)^{2}+4}$.
Solution: As shown in Figure 2, construct Rt $\triangle P A C$ and Rt $\triangle P B D$ such that $A C=1, B D=2, C P=x, P D=4-x$. Thus, the original problem is transformed into: finding a point $P$ on line $l$ such that the value of $P A + P B$ is minimized.

Since $y = P A + P B \geqslant A B$, the minimum value of $y = P A + P B$ is $A B$.

Draw $B E \parallel l$, intersecting the extension of $A C$ at point $E$. Then,
$$
\begin{array}{l}
A E = A C + C E \\
= A C + B D = 3, \\
B E = C D \\
= x + 4 - x = 4 .
\end{array}
$$

Therefore, $A B = 5$. Hence, the minimum value of $y$ is 5.