AMC12 2009 A

AMC12 2009 A · Q9

AMC12 2009 A · Q9. It mainly tests Functions basics.

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

假设 $f(x+3) = 3x^2 + 7x + 4$ 且 $f(x) = ax^2 + bx + c$。$a+b+c$ 是多少？

(A) -1 -1

(B) 0 0

(D) 2 2

(E) 3 3

Answer

Correct choice: (D)

正确答案：（D）

Solution

As $f(x)=ax^2 + bx + c$, we have $f(1)=a\cdot 1^2 + b\cdot 1 + c = a+b+c$. To compute $f(1)$, set $x=-2$ in the first formula. We get $f(1) = f(-2+3) = 3(-2)^2 + 7(-2) + 4 = 12 - 14 + 4 = \boxed{2}$.

因为 $f(x)=ax^2 + bx + c$，所以 $f(1)=a\cdot 1^2 + b\cdot 1 + c = a+b+c$。为计算 $f(1)$，在第一个式子中令 $x=-2$。得到 $f(1) = f(-2+3) = 3(-2)^2 + 7(-2) + 4 = 12 - 14 + 4 = \boxed{2}$。

Topics

AL.010 Functions basics