You have the correct answer.
I used GeoGebra to check. It has a very quick capability to graph the parabola and also show the focus and directrix.
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If you're curious and wanting to know the steps, then read on.
The general parabola can be written as
4p(y - k) = (x - h)^2
where p is the focal distance and (h,k) is the vertex.
The focal distance stretches from the focus to the vertex. It's exactly half the distance from focus to directrix.
In this case, the distance from the focus (3,0) to the directrix y = 4 is 4 units. So p = 4/2 = 2
The vertex is at the midpoint of the focus and the point (3,4). So the vertex is at (h,k) = (3,2)
Putting all this together leads to...
4p(y - k) = (x - h)^2
4*2(y - 2) = (x - 3)^2
8(y-2) = (x - 3)^2
y-2 = (1/8)(x - 3)^2
The only thing we need to change is the 1/8 needs to become negative so that the directrix is above the focus. This will make the parabola open downward forming a frown.
So that's how we get to y-2 = (-1/8)(x - 3)^2
Optionally, you can expand things out and get it into y = ax^2+bx+c form if you wanted.