2001 AMC 12 Problems/Problem 13

Problem

The parabola with equation $p(x) = ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$ . This results in the parabola with equation $q(x) = dx^2+ex+f$ . Which of the following equals $a+b+c+d+e+f$ ?

$(\mathrm{A})\ 2b \qquad (\mathrm{B})\ 2c \qquad (\mathrm{C})\ 2a+2b \qquad (\mathrm{D})\ 2h \qquad (\mathrm{E})\ 2k$

Solution

We write $p(x)$ as $a(x-h)^2+k$ (this is possible for any parabola). Then the reflection of $p(x)$ is $q(x) = -a(x-h)^2+k$ . Then we find $p(x) + q(x) = 2k$ . Since $p(1) = a+b+c$ and $q(1) = d+e+f$ , we have $a+b+c+d+e+f = 2k$ , so the answer is $\mathrm{E}$ .

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2001 AMC 12 Problems/Problem 13

Problem

Solution