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.

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Suppose that $f(x + 3) = 3x^2 + 7x + 4$ and $f(x) = ax^2 + bx + c.$ What is $a + b + c?$

$-1$

$0$

$1$

$2$

$3$

Answer: D

Concepts:function substitution

Difficulty rating: 1350

Solution:

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.

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