Given a parabola with equation $y=ax^2+bx+c$ which has as a tip the point $(3, 1)$ and passes through $(2, 0)$. Find out the product $abc$

Question

Given a parabola with equation $y=ax^2+bx+c$ which has as a tip the point $(3, 1)$ and passes through $(2, 0)$. Find out the product $abc$.

I can't solve the question. I am trying to solve it with the use of analytic geometry on parabolas, but it isn't working out for me. I have that $x=-\frac{b}{2a}$ is a tip. Hence $y=\frac{b^2}{4a}-\frac{b^2}{2a}+c$. I don't know how to finish it off from here. I assume we need to get an expression of the form of $abc=...$ but I haven't managed to get that. Could you please explain to me how to solve it?

Answer 1

Hint:

1. Prove that if $y=ax^2+bx+c$, then it's "tip point" is at $x=-\dfrac{b}{2a}$.

2. Prove that if a curve passes through a point, then it satisfies the equation of the curve.

Answer 2

Since there is no $xy$ term in the equation, the parabola has axes parallel to coordinate axes. Specifically, it is just the parabola $y=Ax^2$ translated (no rotation).

The new vertex is $(3,1)$ instead of $(0,0)$. Hence the equation of parabola is $$(y-1)=A(x-3)^2$$

$A$ can be obtained by using that $(2,0)$ lies on parabola whence $a,b,c$ can be found.