Consider a quadratic function Q(y)=Ay^2+By+C. If Q(1)=0, Q(2)=0 and Q(4)=-6, find C the constant term in Q.?

Question

Consider a quadratic function Q(y)=Ay^2+By+C. If Q(1)=0, Q(2)=0 and Q(4)=-6, find C the constant term in Q.

C=_____

Answer 1

C=6

Answer 2

due to symmetry of quadratic function, we are able to find the symmetry axis, since q(1)=0 and q(2)=0 y=1.5 is the axis of symmetrie. Furthermore, we know that the tangent line at the vertex or top is zero so the derivative in y=1.5 must be equal to zero

f'(y) = 2ay+b=0 ==> f'(1.5) =3a+b=0 ==> 3a=-b substitute -3a for b

f(1)=0 thus a*1^2-3a*1+c=0 ==> c=2a  substitute 2a for c

f(4)=a*16-3a*4+2a=-6 ==> 16a-12a+2a=-6 ==> 6a=-6 a=-1, b=3, c=-2

check your answer by using these values and using the points