Question

what is the directrix and focus of the equation 1/16(y+4)^2=x-3

Answer 1

Perform a substitution
x' = x-3
y' = y+4
and you are left with the equatiom
y'^2 = 16x'

This is the equation of a parabola with axis along the line y = -4 and vertex at (x',y') = (0,0)
which is (x,y) = (3,-4)

The factor 16 tells you where the focus and directrix are.
16 = 4p,
where p is the distance from the vertex to the focus and directrix.

That puts the focus at (x,y) = (7,-4)
The directrix is the vertical line x = -1