Parabola describing projectile motion. I am trying to create a function that will generate a parabola that describes projectile motion.

Here are my inputs:


The starting x-y coordinate of the throw
The initial x-y velocity vector.


I need an output for an equation of a parabola that fits this motion.
By Newton's second law $F = ma = m(0,-g)$, which es equivalent to the set of equations:
\begin{align}
\ddot x(t) &amp;= 0\\
\ddot y(t) &amp;= -g.
\end{align}
After integrating these, we get that:
\begin{align}
x(t) &amp;= x_0+v_{x}t\\
y(t) &amp;= y_0+v_yt-\frac{gt^2}{2},
\end{align}
where $(x_0,y_0)$ is the initial position of the object and $(v_x,v_y)$ its initial velocity. These equations define a parabola.

If you want to get an equation of the form $y = f(x)$, use that $t= \frac{x-x_0}{v_x}$, then:
$$
y = y_0+v_y\frac{x-x_0}{v_x} - \frac{g(x-x_0)^2}{2v_x^2},
$$
where $x\in[x_0,x_1]$ and $x_1$ is where the projectile reaches the ground again.
EDIT1:

Write down vertical acceleration equilibrium and integrate two times,remembering to include vertical component of projectile velocity $V$ shot at $ \alpha $ to x-axis as follows. 

The accelerations in x- and y-(vertical) directions are:

\begin{align}
\ddot x(t) &amp;= 0\\
\ddot y(t) &amp;= -g
\end{align}
First integration for velocity with components of initial $V$ components as  :
\begin{align}
\dot x(t) &amp;= V \cos \alpha \\
\dot y(t) &amp;= V \sin \alpha -g  \, t \\
\end{align}
Second integration is for position starting at origin with boundary condition ( x=0, y=0 ):
\begin{align}
\dot x(t) &amp;= V \cos \alpha \,t \\
\dot y(t) &amp;= V \sin \alpha \,t -g t^2/2 \,  \\
\end{align}

Now eliminate $t$ to find trajectory as a parabola reaching a maximum point before hitting ground.

$$ y = x \tan \alpha - g x^2 / 2 /(V \cos \alpha)^2 $$

It is possible to find maximum range and height using using above relations. Maximum range occurs at $ \alpha = 45^0 $ and of course maximum height $ V^2 / 2g $ when shot vertically up. 

It is an interesting fact I found that as $\alpha $ is varied, the envelope of all parabolas is another parabola.