Example 2 Find the minimum value of $y=\sqrt{x^{2}+1}+\sqrt{(4-x)^{2}+4}$.
Solution: Construct right triangles $\triangle P A C$ and $\triangle P B D$ such that $A C=1, B D=$ $2, P C=x, P D=4-x$, and transform the problem of finding the minimum value to: finding a point $P$ on $l$ such that the sum $P A+P B$ is minimized.

Take the symmetric point $A^{\prime}$ of $A$ with respect to $l$, then we have $y=P A+P B \geqslant A^{\prime} B=5$, i.e., $y_{\text {minimum }}=5$.