2009 AMC 12A Problem 9

Attempt Problem 9 of the 2009 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Suppose that

f(x + 3) = 3x^2 + 7x + 4

and

f(x) = ax^2 + bx + c.

What is

a + b + c?

-1

0

1

2

3

a + b + c = f(1)

Write

1 = (-2) + 3

and use the given formula for

f(x + 3)

Note that

a + b + c = f(1).

Using

f(x + 3) = 3x^2 + 7x + 4

with

x = -2

gives

\begin{aligned} f(1) &= f(-2 + 3) \\ &= 3(-2)^2 + 7(-2) + 4 \\ &= 12 - 14 + 4 = 2. \end{aligned}

Thus, the correct answer is D.

Problem 9 in other years