Question

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 $y = (f + g)(x)?$

a parabola tangent to the $x$ -axis

a parabola not tangent to the $x$ -axis

a horizontal line

a non-horizontal line

the graph of a cubic function

Solution:

Write the parabola in vertex form $y=a(x-h)^2+k.$ Its reflection about the $x$ -axis is $y=-a(x-h)^2-k.$

Shifting in opposite directions gives $f(x)=a(x-h+5)^2+k$ and $g(x)=-a(x-h-5)^2-k.$

Adding, the squared terms cancel and $(f+g)(x)=20a(x-h),$ which is a non-horizontal line since $a\neq0.$

Thus, the correct answer is D.

2003 AMC 12A Problem 19

Solution:

Problem 19 in Other Years