Example 8 If the function $y=f(x+1)$ is an even function, is the graph of the function $y=f(x)$ a curve with axis symmetry? If so, write the equation of the axis of symmetry; if not, explain the reason.
Let $g(x)=f(1+x)$, then by the function $y=g(x)$ being an even function, we get $f(1+x)=f(1-x)$, so, the graph of the function $y=f(x)$ is a curve symmetric about an axis, with the equation of the axis of symmetry being $x=1$.

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