How do you write a quadratic equation when given two points?

Apr 21, 2018

Two points are not sufficient to specify a quadratic equation.

Explanation:

Consider a quadratic equation in factored form.

$y = a \left(x - {r}_{1}\right) \left(x - {r}_{2}\right)$

If we specify ${r}_{1}$ and ${r}_{2}$, then we know exactly two points on this parabola, namely $\left({r}_{1} , 0\right)$, and $\left({r}_{2} , 0\right)$. But there are an infinite number of parabolas that contain these two points because we can make the $a$ coefficient any real number.

We need a THIRD point to fix the parabola.

One special circumstance exists, though. Suppose the two points we are given are (1, 4) and (6, 4). Because these points lie on a vertical line, there would be NO parabola that could contain both of these points. In general, if the two points have the same $y$-value, the two points cannot be on the same parabola.