1984 AHSME Problem 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+bx+c. If the set of the vertices (x_t, y_t) for all real values of t is graphed on the plane, the graph is

\mathrm{(A) \ } \text{a straight line} \qquad \mathrm{(B) \ } \text{a parabola} \qquad \mathrm{(C) \ } \text{part, but not all, of a parabola} \qquad \mathrm{(D) \ } \text{one branch of a hyperbola} \qquad \mathrm{(E) \ } \text{none of these}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.