With a given point, find the curve that intersects zero.

Question

Background

For a pseudo-trajectory of a cannon, I am to find, with a given point maximum $(M_x,M_y)$, the equation of the curve that intersects the origin and the point $(2M_x,0)$. I have the equation $f(x)=m\left(x-M_x\right)^2+M_y$ that will have its maximum at the point $(M_x,M_y)$, but I am thus far unable to find an $m$ such that $f(0)=0$ and $f(2M_x)=0$ (restating the required intersects). All I know is that $m<0$ must hold, or else the equations maximum would be $\infty$.

Question

What might variable $m$ be and how might I find it?

Graph

Answer 1

If the parabola intersects the point $(2M_x,0)$ then the following equation must hold $$0 =m(2M_x-M_x)^2+M_y=mM_x^2+M_y$$ which implies that $$m = -\frac{M_y}{M_x^2}$$