Question

What is the equation of the parabola with focus (0,1/8) and vertex at the origin?

Answer 1

`y = 2x^2`

Explanation:

Please observe that the vertex, `(0,0)`, and the focus, `(0,1/8)`, are separated by a vertical distance of `1/8` in the positive direction; this means that the parabola opens upward. The vertex form of the equation for a parabola that opens upward is:

`y = a(x-h)^2+k" [1]"`

where `(h,k)` is the vertex.

Substitute the vertex, `(0,0)`, into equation [1]:

`y = a(x-0)^2+0`

Simplify:

`y = ax^2" [1.1]"`

A characteristic of the coefficient `a` is:

`a = 1/(4f)" [2]"`

where `f` is the signed distance from the vertex to the focus.

Substitute `f = 1/8` into equation [2]:

`a = 1/(4(1/8)`

`a = 2" [2.1]"`

Substitute equation [2.1] into equation [1.1]:

`y = 2x^2`