2001 AMC 12 Problem 13

Problem 13 of 25IntermediateGeometry

The parabola with equation

y = ax^2 + bx + c

and vertex

(h, k)

is reflected about the line

y = k.

This results in the parabola with equation

y = dx^2 + ex + f.

Which of the following equals

a + b + c + d + e + f?

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Solution

The value

a + b + c

is the first parabola at

x = 1,

and

d + e + f

is the reflected parabola at

x = 1.

Reflecting the curve about

y = k

replaces each height

y

by

2k - y.

So at

x = 1

the two heights sum to

(a + b + c) + (d + e + f) = 2k.

Thus, the correct answer is E.

More practice

Concepts: parabola · transformation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.