What is the most direct way to derive an equation for a parabola from its x and y intercepts? I have a pair of points at my disposal.  One of these points represents the parabola's maximum y-value, which always occurs at x=0.  I also have a point which represents the parabola's x-intercept(s).  Given this information, is there a way to rapidly derive the formula for this parabolic curve?  My issue is that I need to generate this equation directly in computer software, but all the standard-formula definitions for a parabolic curve use its Vertex, not its intercepts.  Is there some standard form of equation into which these intercepts can be 'plugged in' in order to produce a working relation?  If not, what is the most computationally direct way to solve this problem?
 A: I'll assume you meant you know the $x$-intercepts and maximum height.
If you have any parabola with $x$-intercepts $a,b$, $a\neq b$, and maximum height $c$, then you can write it as $$y=k(x-a)(x-b)$$ where $$k=-c\left(\frac{4}{(a-b)^2}\right).$$
(Notice that the value $c$ must be positive)
If $a=b$ we actually can't specify $k$ without more information.  
A: the equation would look like this
$$ y = k(x-a)(x-b)$$
now we have to figure out what k is. We know what the maximum value is, call it c, and that it's x value is 0. Therefor we can plug this into the equation so that we get the following
$$c = k(-a)(-b)$$
    $$c = kab$$
therefor
     $k = c/(ab)$
therefor, your equation is
$$y = c(x-a)(x-b)/(a*b)$$
A: To answer question found in the title: "... an equation for a parabola from its $x$ and $y$ intercepts", the correct equation is:
$$y = \frac{c}{ab}(x-a)(x-b),$$
where $a, b$ are the $x$-intercepts and $c$ is the $y$-intercept.  We can prove this is correct by noting that $y = 0$ when $x=a$ or $x=b$ is substituted, and when $x=0$, we have $y = \frac{c}{ab}(-a)(-b) = c$.
