2001 AMC 12 Problem 13

Below is the professionally curated solution for Problem 13 of the 2001 AMC 12, 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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Concepts:parabola transformation

Difficulty rating: 1600

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

$2b$

$2c$

$2a + 2b$

$2h$

$2k$

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.

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