Problem:
The parabola with equation y=ax2+bx+c and vertex (h,k) is reflected about the line y=k. This results in the parabola with equation y=dx2+ex+f. Which of the following equals a+b+c+d+e+f ?
Answer Choices:
A. 2b
B. 2c
C. 2a+2b
D. 2h
E. 2k
Solution:
The equation of the first parabola can be written in the form
y=a(x−h)2+k=ax2−2axh+ah2+k
and the equation for the second (having the same shape and vertex, but opening in the opposite direction) can be written in the form
y=−a(x−h)2+k=−ax2+2axh−ah2+k
Hence,
a+b+c+d+e+f=a+(−2ah)+(ah2+k)+(−a)+(2ah)+(−ah2+k)=2k.
\section*{OR}
The reflection of a point (x,y) about the line y=k is (x,2k−y). Thus, the equation of the reflected parabola is
2k−y=ax2+bx+c, or equivalently, y=2k−(ax2+bx+c)
Hence a+b+c+d+e+f=2k.
The problems on this page are the property of the MAA's American Mathematics Competitions