Reflecting over non-linear functions/curves

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Currently I am writing an IA (Internal Assessment) for the IB. It is a build up of deriving general rules to reflect over any line with equation $y=mx+b$. One step further that my teacher told me to take is researching if it is possible to reflect over non linear curves, for example, $x^2$. If the point $P=(0, -3)$ were reflected upon $y=x^2$ where would the reflected point $P'$ be and why. Is there a mathematical proof for this? Any ideas would be appreciated