Reflect the parabola $y=x^{2}$ over the point $(1 ; 1)$. What is the equation of the reflection?
I. solution. Reflection does not change the position of the axis of the parabola, and the vertex of the reflected parabola is the reflection of the vertex of $y=x^{2}$, which is the origin, so its coordinates are $(2 ; 2)$.

The equation of the reflected parabola can thus be written in the form

$$
y=a(x-2)^{2}+2
$$

Using the fact that the reflected parabola also passes through the point $(1 ; 1)$, the value of the parameter $a$ can be determined:

$$
1=a(1-2)^{2}+2, \quad \text { from which } \quad a=-1
$$

Thus, the equation of the reflection is $y=-x^{2}+4 x-2$.

Remark. In the solution, we used the geometry of the parabola, specifically how the characteristic parts of the parabola (axes, vertex) appear in the equation of the figure. The following solution uses only the information contained in the equation of a curve.

II. solution. The point $P(x ; y)$ is clearly on the reflected parabola if and only if its reflection $P^{\prime}\left(x^{\prime} ; y^{\prime}\right)$ on the point $(1 ; 1)$ is on the original parabola, $y=x^{2}$.

The reflection of $P(x ; y)$ on the point $(1 ; 1)$ is the point $P^{\prime}(2-x ; 2-y)$, which lies on the curve $y=x^{2}$ if and only if substituting its coordinates into the equation yields an equality:

$$
2-y=(2-x)^{2}
$$

We have thus found that $P(x ; y)$ lies on the reflected parabola if and only if its coordinates satisfy (1), which is therefore the equation of the reflection. Simplifying,

$$
y=2-(2-x)^{2}=-x^{2}+4 x-2
$$

follows.

Remark. The method of the second solution can be applied in much more general circumstances. If the equation of a figure $M$ is $E(x ; y)=0$ and the image of $M$ under the transformation $\Phi$ is $\Phi(M)$, then the equation of $\Phi(M)$ is $E\left(\Phi_{x}^{-1}(x ; y) ; \Phi_{y}^{-1}(x ; y)\right)=0$, where $\left(\Phi_{x}^{-1}(x ; y) ; \Phi_{y}^{-1}(x ; y)\right)$ is the image of the point $P(x ; y)$ under the inverse of the transformation $\Phi$, $\Phi^{-1}$. This is a consequence of the obvious statement that the point $P$ lies on $\Phi(M)$ if and only if the point $\Phi^{-1}(P)$ lies on $M$.