11. The quadratic function $f(x)$ satisfies $f(x-2)=f(-x-2)$, and the y-intercept of its graph is 1, and the length of the segment intercepted on the x-axis is $2 \sqrt{2}, f(x)$ is given by $f(x)=$ $\qquad$ .
11. $f(x)=\frac{1}{2}(x+2)^{2}-1$.

The axis of symmetry is $x=-2$, and since $\left|x_{1}-x_{2}\right|=2 \sqrt{2}$, the parabola intersects the $x$-axis at the points $(-2-\sqrt{2}, 0)$ and $(-2+\sqrt{2}, 0)$. Let $f(x)=a(x+2+\sqrt{2})(x+2-\sqrt{2})$, and from $f(0)=1$ we get $a=\frac{1}{2}$.