A parabola y=ax^2+bx+c has vertex (4,2). If (2,0) is on the parabola, then abc equals

1986 AMC 12 Problem 13

Attempt Problem 13 of the 1986 AMC 12 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 1986 AMC 12 solutions, or check the answer key.

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13.

A parabola

y=ax^2+bx+c

has vertex

(4,2).

(2,0)

is on the parabola, then

abc

equals

-12

-6

0

6

12

Write the parabola in vertex form

y=a(x-4)^2+2

Use the given point to find

a,

then expand to read

b

and

c

Write the equation as

y=a(x-4)^2+2.

Substituting

(2,0)

gives

0=4a+2,

a=-\frac{1}{2}.

Expanding,

y=-\frac12x^2+4x-6.

Thus

b=4,

c=-6,

and

abc=(-\frac{1}{2})(4)(-6)=12.

Therefore the correct answer is E.