Reflect a curve about a point

Question

What is the equation of reflection of the curve $$ y=5x^{2}-7x+2 $$ about the pont (3,-3)?

The answer is explained as follows;

To get the reflection about origin , the x and y co-ordinates have to be inter-changed. Since reflection is about (3,-3), Replace x with x-3 and y with y+3.

So answer is $$ x=5y^{2}+23y+29 $$

When I plot the two curves, It doesn't look like they are reflections of one another. May be my understanding of this is wrong. Is the answer correct? If so, how?

Answer 1

Let's use the following steps:

1. shift the coordinate system so that the point of reflection becomes the origin
2. reflect around the origin. The reflection of a point $(x,y)$ equals the point $(-x,-y)$.
3. shift our coordinate system back to its original position.

Applying to your question:

1. Let $y' = y+3$ and $x' = x-3$. The point of reflection is the origin in our new coordinate system. Our original curve now has equation $$y' -3 = 5(x'+3)^2 - 7(x'+3) +2.$$
2. Reflect around the origin. Replace each $x'$ by $-x'$ and each $y'$ by $-y'$. The reflected curve has equation $$-y' -3 = 5(-x'+3)^2 - 7(-x'+3) +2.$$
3. Go back to the original coordinates. The curve we are looking for has equation $$-(y+3) -3 = 5(-(x-3) +3)^2 - 7(-(x-3)+3) + 2.$$ Now rewrite until you obtain an equation of the form $y = \ldots$.

Answer 2

I'll solve it without coordinate interchanging.

This curve is parabola, and reflected curve is parabola, so trivially. And the vertex of this curve is $(\frac7{10}, -\frac9{20})$, and focus of this curve is $(\frac7{10}, -\frac25)$.

So vertex of reflected curve is $(\frac{53}{10}, -\frac{111}{20})$, and focus is $(\frac{53}{10}, -\frac{28}{5})$.

So, the equation of curve must be $y=-5x^2+53x-146$.