You can move them, but you'll get a different parabola. The parabola I posted has it's specific focus and directrix. As does any parabola.
You can't just move the focus and directrix around and keep the same parabola.
Let's find the equatio of a parabola with vertex at V(-4,2) and directrix at y=5.
See, the distance from the vertex to the directrix(line y=5) is 3 units. 5-2=3.
So, the focus is F(-4,-1). Because 2-3=-1.
We can use
We know h and k, the coordinates of the vertex. So,
We want to express it as
There she is.
Parabolas are used a lot in real world applications. One area is in highway construction. Vertical curves on roads(hills and valleys) are laid out using the principles of parabolas.