2006 AMC 12B Problem 12

Attempt Problem 12 of the 2006 AMC 12B 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 2006 AMC 12B solutions, or check the answer key.

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12.

The parabola

y = ax^2 + bx + c

has vertex

(p, p)

and

y

-intercept

(0, -p),

where

p \neq 0.

What is

b?

-p

0

2

4

p

Write the parabola in vertex form

y = a(x - p)^2 + p

Use the

y

-intercept to find

a

in terms of

p,

then read off the coefficient of

x

The vertex form is

y = a(x - p)^2 + p.

x = 0,

y = ap^2 + p = -p,

ap^2 = -2p

and

a = -\dfrac{2}{p}.

Expanding,

y = a x^2 - 2ap\, x + ap^2 + p,

b = -2ap = -2\left(-\dfrac{2}{p}\right)p = 4.

Thus, the correct answer is D.

Problem 12 in other years