2003 AMC 12A Problem 19

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

Write the parabola as

a(x-h)^2+k;

its reflection is

-a(x-h)^2-k

After the opposite

\pm5

shifts, add

f

and

g

and watch the squared terms cancel

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.

Problem 19 in other years