How do i find the equation of a parabola given 2 points and the axis of symmetry, but no vertex?

Question

The points I have are (6,0) and (18,0), and the Axis is x=12. but that is it.

Answer 1

The points (6,0) and (18,0) are the solutions so can write the equation in factored form (x-6)(x-18)=0

y= x^2 -24x +108

Answer 2

There are many parabolas that meet this criteria. Here are a few:

To find one, you can write the vertex form with what you know:

y = a(x - 12)² + k

Then you can pick a "a" value, for instance, let's pick a = 1. The equation is then:

y = (x - 12)² + k. Now, plug in a point like (6, 0) and solve for "k"

0 = (6 - 12)² + k

0 = (-6)² + k

0 = 36 + k

k = -36

So the equation is:

y = (x - 12)² - 36

(this is the red curve I have above)