Problem:
A parabola with equation y=x2+bx+c passes through the points (2,3) and (4,3). What is c?
Answer Choices:
A. 2
B. 5
C. 7
D. 10
E. 11
Solution:
Substitute (2,3) and (4,3) into the equation to give
3=4+2b+c and 3=16+4b+c.
Subtracting corresponding terms in these equations gives 0=12+2b. So
b=−6 and c=3−4−2(−6)=(E)11
OR
The parabola is symmetric about the vertical line through its vertex, and the points (2,3) and (4,3) have the same y-coordinate. The vertex has x-coordinate (2+4)/2=3, so the equation has the form
y=(x−3)2+k
for some constant k. Since y=3 when x=4, we have 3=12+k and k=2. Consequently the constant term c is
(−3)2+k=9+2=(E)11
The problems on this page are the property of the MAA's American Mathematics Competitions