Question

The axis of symmetry of a parabola is `x=6`, and one point on the parabola is `(2, 8)`. What is another point on the parabola?

Answer 1

There could be infinite parabolas, but along with `(2,8)` we have `(10,8)` too on all parabolas as `x=6` is axis of symmetry.

Explanation:

Given axis of symmetry `x=6`, the equation of parabola is of the form `y=a(x-6)^2+k`

As it passes through `(2,8)`, we have

`8=a(2-6)^2+k` or `16a=8-k` or `a=(8-k)/16=1/2-k/16`

Hence, we could have a series of parabolas using different values of `k`. Let us choose `k=-8,24,40`, which gives `a=1,-1,-2` and three equations are

`y=(x-6)^2-8`,

`y=-(x-6)^2+24` and `y=-2(x-6)^2+40`

and parabolas appear as (graph not drawn to scale) follows.

graph{(y-(x-6)^2+8)(y+2(x-6)^2-40)(y+(x-6)^2-24)=0 [0, 15, -15.84, 24.16]}

Note that it also passes through `(10,8)` as `x=6` is axis of symmetry.