1984 AMC 12 Problem 22

Attempt Problem 22 of the 1984 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 1984 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

22.

Let

a

and

c

be fixed positive numbers. For each real number

t

let

(x_t,y_t)

be the vertex of the parabola

y=ax^2+tx+c.

If the set of vertices

(x_t,y_t)

for all real values of

t

is graphed in the plane, the graph is

a straight line

a parabola

part, but not all, of a parabola

one branch of a hyperbola

none of these

Write the vertex coordinates in terms of

t

Use

x_t=-\frac{t}{2a}

to eliminate

t

from

y_t

The vertex coordinates are

x_t=-\frac{t}{2a},\qquad y_t=c-\frac{t^2}{4a}.

Since

t=-2ax_t,

eliminating

t

gives

y_t=c-ax_t^2.

t

ranges over all reals, so does

x_t,

so the entire parabola is traced.

Therefore, the correct answer is B.

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