How do I find the equation of a parabola given 2 points

Question

i was given (4,0) and (28,6) and nothing else

Answer 1

There're four standard parabolas, using either point as a vertex, and then having the parabola open upwards, downwards, to the right or to the left. There's an infinite number of other parabolas, but the most simple might be an upward opening parabola with (4,0) as the vertex. All the equations will be quadratics

standard equation for an upward opening parabola is y-h = a(x-k) where (k,h) is the vertex. Let (4,0) be the vertex, k=0, h=4. Then y=a(x-4)^2. Now, use the other point to calculate "a"

plug (28,6) into the equation, where 28=x and 6=y

6=a(28-4)^2

6=a(24)^2

a = 6/24(24) =1/4(24) = 1/96

the parabola equation is:

y=(1/96)(x-4)^2 or

96y=x^2-8x+16

You can do the same approach using (28,6) as the vertex for a downward opening parabola, that makes a<0

Or use x-k = a(y-h)^2 with a rightward opening parabola and (4,0) as the vertex or

(28,6) as the vertex with a leftward opening parabola.

If neither point is a vertex, then there are an infinite number of other equations for a parabola passing through those 2 points