Is it possible to find parabola that passes through two points and its vertex above first point on given height and located between points?

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I'm trying to calculate body trajectory that launched by some impulse.

I have two points: start position and finish position which can be located on different heights (points is not symmetric). I know the height that body will reach, so parabola's vertex located that height higher of first point. Vertex located between points but we don't know it's X coordinate (if points on same height it will be in the middle between them).

Is it possible to find parabola's equation in this case?