# Reflect a curve about a point

analytic geometryfunctionsreflection

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?

2. reflect around the origin. The reflection of a point $$(x,y)$$ equals the point $$(-x,-y)$$.
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 $$y = \ldots$$.