Write an equation of the parabola that has a focus at (4, 2) and a directrix at y 3.

Try watching this.

https://www.youtube.com/watch?v=gu-1uUqDxeQ

There are many fabulous you tube clip available.

The focus lies above the directrix....so this parabola turns upward

The vertex  lies midway between these and is given by  ( 4, (2 - 3) / 2 )  = (4, -1/2)

The focal distance,p, is the distance between the vertex and the focus....since the parabola turns upward this is positive and is given by  2 - (-1/2)  =  2 + 1/2  = 5/2

So....we have this form :

4p (y - k)  = (x - h)^2    where  the vertex is (h, k)  = (4, -1/2)  and p = 5/2

Filling in what we know, we have

4 (5/2) (y -  -1/2)  = (x - 4 ) ^2

10 ( y +1/2)  =  ( x - 4 )^2         divide both sides by 10

y +1/2  = (1/10) (x - 4)^2        subtract 1/2 from both sides

y  = (1/10) (x - 4)^2  - 1/2       this is in vertex form