Question

Using the vertex form, how do you solve for the variable a, with the points (3,1) the vertex and(5,9)?

Answer 1

The answer depends upon what you intend by the variable `a`

If the vertex is `(hatx,haty) = (3,1)`
and another point on the parabola is `(x,y) =(5,9)`

Then the vertex form can be written
`color(white)("XXXXX")` `y = m(x-hatx)^2+haty`
which, with `(x,y)` set to `(5,9)`, becomes
`color(white)("XXXXX")`9 = m(5-3)^2+1#

`8 = 2m`

`m =4)`

and the vertex form is

`y = 4(x-3)^2+1`

Option 1: (less likely option, but possible)
The vertex form is sometimes written as
`color(white)("XXXXX")y = m(x-a)^2+b`
in which case
`color(white)("XXXXX")a=3`

Option 2:
The generalized standard form of a parabola is usually written as
`color(white)("XXXXX")y=ax^2+bx+c`
in which case
`color(white)("XXXXX")a = 4`