Question

What is the equation of the parabola with a focus at (0, 2) and vertex at (0,0)?

Answer 1

`y = 1/8x^2`

Explanation:

If the focus is above or below the vertex, then the vertex form of the equation of the parabola is:

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

If the focus is to the left or right the vertex, then the vertex form of the equation of the parabola is:

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

Our case uses equation [1] where we substitute 0 for both h and k:

`y = a(x-0)^2+0" [3]"`

The focal distance, f, from the vertex to the focus is:

`f = y_"focus"-y_"vertex"`

`f = 2-0`

`f = 2`

Compute the value of "a" using the following equation:

`a = 1/(4f)`

`a = 1/(4(2))`

`a = 1/8`

Substitute `a = 1/8` into equation [3]:

`y = 1/8(x-0)^2+0`

Simplify:

`y = 1/8x^2`