2003 AMC 12A Problems/Problem 19

Problem

A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$ -axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$ , respectively. Which of the following describes the graph of $y=(f+g)x$ ?

$\textbf{(A)}\ \text{a parabola tangent to the }x\text{-axis}$ $\textbf{(B)}\ \text{a parabola not tangent to the }x\text{-axis}\qquad\textbf{(C)}\ \text{a horizontal line}$ $\textbf{(D)}\ \text{a non-horizontal line}\qquad\textbf{(E)}\ \text{the graph of a cubic function}$

Solution

If we take the parabola $ax^2 + bx + c$ and reflect it over the x - axis, we have the parabola $-ax^2 - bx - c$ . Without loss of generality, let us say that the parabola is translated 5 units to the left, and the reflection to the right. Then:

$\begin{align*} f(x) &= a(x+5)^2 + b(x+5) + c = ax^2 + (10a+b)x + 25a + 5b + c \\ g(x) &= -a(x-5)^2 - b(x-5) - c = -ax^2 + 10ax -bx - 25a + 5b - c \end{align*}$

Adding them up produces:

\[(f + g)(x) &= ax^2 + (10a+b)x + 25a + 5b + c - ax^2 + 10ax -bx - 25a + 5b - c &= 20ax + 10b\] (Error compiling LaTeX. Unknown error_msg)

This is a line with slope $20a$ . Since $a$ cannot be $0$ (because $ax^2 + bx + c$ would be a line) we end up with $\boxed{\textbf{(D)} \text{ a non-horizontal line }}$

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2003 AMC 12A Problems/Problem 19

Problem

Solution