Problem:
Let a and c be fixed positive numbers. For each real number t let (xt,yt) be the vertex of the parabola y=ax2+tx+c. If the set of vertices (xt,yt) for all real values of t is graphed in the plane, the graph is
Answer Choices:
A. a straight line
B. a parabola
C. part, but not all, of a parabola
D. one branch of a hyperbola
E. none of these
Solution:
The vertex of any parabola y=ax2+tx+c is on its axis of symmetry x=2a−t. Thus xi=2a−t and
yt=a(2a−t)2+t(2a−t)+c=c−4at2=c−axt2.
Thus every point (xt,yt) is on the parabola y=−ax2+c. Conversely, each point (x,y) on this parabola is of the form (xt,yt): just set t=−2ax.