Final answer:
The vertex of the parabola -16(y+1)=(x-2)² is (2, -1), the focus is (2, -5), and the directrix is the line y = 3.
Step-by-step explanation:
To identify the focus, vertex, and directrix from the given parabola equation -16(y+1)=(x-2)², we first rewrite it in standard form.
Given that the equation is in the format 4p(y-k)=(x-h)², where (h, k) is the vertex, we can determine that the vertex is (2, -1).
The coefficient of the (y+1) term is -16, so 4p = -16, giving us p = -4.
This means the focus, which lies p units from the vertex along the axis of symmetry, is at (2, -1+(-4)), which simplifies to (2, -5).
The directrix is a horizontal line (since the parabola opens up or down) that is located |p| units on the opposite side of the focus from the vertex.
Therefore, the directrix is y = -1-(-4) or y = 3.