Question

The vertex of a parabola is `(1, 3)` and another point on the parabola is `(- 1, - 1)`. What is another point also on the parabola?

Answer 1

Assuming the parabola has a vertical axis of symmetry
`(3,-1)` is also on the parabola.

Explanation:

With a vertical axis of symmetry, if the vertex is at `(1,3)`
then the axis of symmetry is `x=1`.

If we think of the points on the parabola as being `(x,f(x))`
This means that for any value `k`,
`color(white)("XXX")f(k-1)=f(k+1)`

We are told `(-1,-1)` is on the parabola.

`color(white)("XXX")-1=f(-1) = f(2-1) = f(2+1) = f(3)`

Therefore
`color(white)("XXX")(3,f(3)) = (3,-1)` is on the parabola.

Note: you could also use the general vertex form:
`color(white)("XXX")y=m(x-a)^2+b`
with the given data values to determine the complete equation for this parabola as:
`color(white)("XXX")y=2(x-1)^2+3`
and then establish as many points as you would like on this curve.