Question

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

Answer 1

Two points are not sufficient to specify a quadratic equation.

Explanation:

Consider a quadratic equation in factored form.

`y=a(x-r_1)(x-r_2)`

If we specify `r_1` and `r_2`, then we know exactly two points on this parabola, namely `(r_1, 0)`, and `(r_2, 0)`. 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.