2003 AMC 12A Problem 19

Problem 19 of 25HarderAlgebra Geometry

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)?

Show solution

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.