13. Let the quadratic function $f(x)$ satisfy $f(x+2)=f(-x+2)$, and its graph intersects the $y$-axis at the point $(0,1)$, and the segment it intercepts on the $x$-axis has a length of $2 \sqrt{2}$. Find the analytical expression of $f(x)$.
Three, 13.
$f(x)$ is symmetric about $x=2$, let $f(x)=a(x-2)^{2}+b$. According to the problem, $4 a+b=1$. Suppose the graph of $f(x)$ intersects the $x$-axis at $x_{1}$ and $x_{2}$, then $\left|x_{1}-x_{2}\right|=2 \sqrt{2}$, i.e., $2 \sqrt{-\frac{b}{a}}=2 \sqrt{2}$. Solving this, we get $\left\{\begin{array}{l}a=\frac{1}{2}, \\ b=-1 .\end{array}\right.$ Therefore, $f(x)=\frac{1}{2}(x-2)^{2}-1$.