23. The parabola with equation $y=x^{2}$ is reflected in the line with equation $y=x+2$. Which of the following is the equation of the reflected parabola?
A $x=y^{2}+4 y+2$
B $x=y^{2}+4 y-2$
C $x=y^{2}-4 y+2$
D $x=y^{2}-4 y-2$
E $x=y^{2}+2$
Solution
C
Method 1
The parabola with equation $y=x^{2}$ meets the line with equation $y=x+2$ where $x^{2}=x+2$. This last equation may be rearranged as $x^{2}-x-2=0$ and then factorized as $(x+1)(x-2)=0$. So its solutions are -1 and 2. For $x=-1$, we have $y=1$ and for $x=2$ we have $y=4$. Therefore the parabola meets the line at the points with coordinates $(-1,1)$ and $(2,4)$ as shown in the figure on the right.

These two points remain where they are when they are reflected in the line. Therefore the reflected parabola also goes through these two points.
We can now test each equation given as an option to see if it is the equation of a curve which goes through the points $(-1,1)$ and $(2,4)$.
For example, because $-1 \neq 1^{2}+4 \times 1+2$ the coordinates of the point $(-1,1)$ do not satisfy the equation $x=y^{2}+4 y+2$. It follows that $x=y^{2}+4 y+2$ is not the equation of a curve which goes through $(-1,1)$. Therefore it is not the equation of the reflected parabola.
Because $-1=1^{2}-4 \times 1+2$ and $2=4^{2}-4 \times 4+2$, it follows that both the points $(-1,1)$ and $(2,4)$ lie on the curve with the equation $x=y^{2}-4 y+2$. It can be checked that these points do not lie on the curves given by any of the other equations.

Therefore, in the context of the SMC, we can conclude that the equation of the reflected parabola is $x=y^{2}-4 y+2$.
If we were not given the equations in the options we would need to calculate the equation of the reflected parabola. We adopt this approach in the second method.
Meтноd 2
We leave it to the reader to check that when the point $P$ with coordinates $(x, y)$ is reflected in the line with equation $y=x+2$, its image point $P^{\prime}$ has the coordinates $(y-2, x+2)$.
We put $x^{\prime}=y-2$ and $y^{\prime}=x+2$, so that $x=y^{\prime}-2$ and $y=x^{\prime}+2$

The point with coordinates $\left(x^{\prime}, y^{\prime}\right)$ lies on the image under reflection of the parabola if, and only if, the point $(x, y)$ lies on the parabola, that is, if, and only if, $y=x^{2}$. Therefore the condition that the point $\left(x^{\prime}, y^{\prime}\right)$ lies on the image of the parabola is expressed by the equation $\left(x^{\prime}+2\right)=\left(y^{\prime}-2\right)^{2}$.
On expansion this equation may be written as $x^{\prime}+2=y^{\prime 2}-4 y^{\prime}+4$. Rearranging this equation, and dropping the dashes, we deduce that the equation of the reflected parabola is
$$
x=y^{2}-4 y+2 \text {. }
$$