2006 AMC 10A 第 8 题

Question

抛物线

y = x^2 + bx + c

经过点

(2, 3)

和

(4, 3)

。求

c

。

A parabola with equation

y = x^2 + bx + c

passes through the points

(2, 3)

and

(4, 3).

What is

c?

2

5

7

10

11

两个点的

y

值相同，所以它们关于对称轴对称。

The two points have the same

y

-value, so they are symmetric about the vertex

对称轴为

x = 3

，因此

b = -6

。

The axis of symmetry is

x = 3,

giving

b = -6

代入两个点得到

3 = 4 + 2b + c

和

3 = 16 + 4b + c

。相减得

0 = 12 + 2b

，所以

b = -6

。

因此

c = 3 - 4 - 2(-6) = 11

。

所以正确答案是 E。

Substituting the points gives

3 = 4 + 2b + c

and

3 = 16 + 4b + c.

Subtracting yields

0 = 12 + 2b,

so

b = -6.

Then

c = 3 - 4 - 2(-6) = 11.

Thus, the correct answer is E.

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