equation for a parabola?

Question

what is the equation or a parabola with vertex (0,0) and focus of (0,4)?

Answer 1

Let (x,y) be a point on the parabola.
By the definition of a parabola,
x^2+(y-4)^2 = (y+4)^2
Simplify,
x^2-8y+16 = 8y+16
y = x^2/16
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Method II Using formula
y = x^2/(4p) = x^2/16, p is the distance from the vertex to the focus point.

Answer 2

What is the equation or a parabola with vertex (0,0) and focus of (0,4)?

The directed distance from the vertex to the focus is p.
p = 4 - 0 = 4
4p = 16

The formula of a parabola in vertex form is:

4p(y - k) = (x - h)²
16(y - 0) = (x - 0)²
16y = x²
y = x²/16

Answer 3

General form of parabola:
y=4p(x-h)^2+k, Since p is directrix which is 4 then
y=16x^2