2001 AMC 12 Problem 13

Attempt Problem 13 of the 2001 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 12 solutions, or check the answer key.

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13.

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?

2b

2c

2a + 2b

2h

2k

Note that

a + b + c

is the first parabola evaluated at

x = 1,

and

d + e + f

is the second at

x = 1

Reflecting about

y = k

sends the value

y

at each

x

2k - y

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

2k - y.

So at

x = 1

the two heights sum to

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

Thus, the correct answer is E.

Problem 13 in other years