Problem:
A parabola y=ax2+bx+c has vertex (4,2). If (2,0) is on the parabola, then abc equals
Answer Choices:
A. −12
B. −6
C. 0
D. 6
E. 12
Solution:
Since (4,2) is the vertex, x=4 is the axis of symmetry. Since (2,0) is on the parabola, by symmetry so is (6,0). In other words, 2 and 6 are roots of ax2+bx+c=0. Thus
y=ax2+bx+c=a(x−2)(x−6)
Substituting (4,2) we obtain
2=a(2)(−2)=−4a
So a=−1/2. Thus y=−21x2+4x−6 and abc=(−21)(4)(−6)=12.
OR
That (2,0) is on the graph means that 0=4a+2b+c. That (4,2) is on the graph means 2=16a+4b+c. That x=4 is the axis of symmetry means 4=−b/2a. Thus we have 3 equations in the three unknowns a,b,c, which we may now solve.