Parabola properties assumptions

Question

I am trying to model projectile trajectory but I'm having some trouble. I didn't realise parabolas are this complicated...

I have some assumptions that I would like to be clarified.

1. If I specify a starting point and an ending point for a throw, the apex of the throw must be higher than the starting and ending positions in order for the path to be modeled as a parabola.
2. As long as the apex is higher than the starting and ending positions' height, the apex can be anywhere between the starting and ending positions as long as it lies on the same plane as them.

Are these assumptions correct?

EDIT: This diagram shows that in my case, the maxima doesnt have to be perfectly in between the start and ending positions. It is true for the red points but not true for the green points.

Answer 1

Your assumptions are half correct. Here's why:

1. The apex can never be lower than the higher point of the starting and endpoints, The best you can get is, the highest point of the two (start or end) itself will become the apex.
2. You can always define a vertical parabola $y=ax^2+bx+c$ passing through $3$ given points. Just substitute the points' coordinates in the equation and solve for $a,b,c$. However, no $2$ points should have the same ordinate and all the three points cannot have the same abscissa as the projectile will have to have $0$ horizontal velocity in the first case, and infinite in the latter.

Note: After an answer is acquired, confirm that it is concave downwards. If not, the parabola for projectile motion cannot be defined with the given points.

Hope this helps.