AMC 12 2003a Problem 19

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