Example 1 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 is $2 \sqrt{2}$. Find the analytical expression of $f(x)$.
(2002, Anhui Province High School Mathematics Competition)
Explanation: Utilizing the symmetry of the graph of a quadratic function, and combining it with the fact that the length of the segment it intercepts on the $x$-axis is $2 \sqrt{2}$, we know that the intersection points of $f(x)$ with the $x$-axis are $(2+\sqrt{2}, 0)$ and $(2-\sqrt{2}, 0)$. Therefore, we can use the zero-product form to solve.
Let $f(x)=a(x-2+\sqrt{2})(x-2-\sqrt{2})$.
Since the graph of the quadratic function passes through the point $(0,1)$, substituting into the undetermined form gives $a=\frac{1}{2}$. Thus,
$$
f(x)=\frac{1}{2}(x-2+\sqrt{2})(x-2-\sqrt{2}),
$$

which simplifies to $f(x)=\frac{1}{2} x^{2}-2 x+1$.