Hi, is it possible to find the (best fit) equation of a/any Parabola given *two* points and the following additional constraints:

i) The Axis is parallel to the y-axis ( vertical )

ii) The Axis lies between the two points

iii) The vertex is higher than both points

iv) The vertex is a maxima, or the Parabola opens downwards.

v) One point can be the origin.

This is basically for the path of a projectile under gravity that originates from a start point and must hit a target point, following a parabolic path.

So for the equation: y = a(x - b)^2 + c, I understand that I need three equations to solve all constants, which normally requires three points.

Is it possible to find ( or approximate ) the third equation given the above constraints somehow?

Is there possibly a (simple'ish) iterative technique? ( Unfortunately speed is a factor as this is part of program's run-time algorithm ).

Many thanks for any help!