What is the most direct way to derive an equation for a parabola from its x and y intercepts?

Question

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?

Answer 1

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.

Answer 2

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 $y = \frac{c}{ab}(-a)(-b) = c$.

Answer 3

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

$$y = c(x-a)(x-b)/(a*b)$$