Problem:
The parabola y=ax2+bx+c has vertex (p,p) and y-intercept (0,−p), where pî€ =0. What is b ?
Answer Choices:
A. −p
B. 0
C. 2
D. 4
E. p
Solution:
A parabola with the given equation and with vertex (p,p) must have equation y=a(x−p)2+p. Because the y-intercept is (0,−p) and pî€ =0, it follows that a=−2/p. Thus
y=−p2​(x2−2px+p2)+p=−p2​x2+4x−p
so b=4​.
The problems on this page are the property of the MAA's American Mathematics Competitions