Final answer:
The vertex of the parabola is (-4, 2), the focus is (-4, -4), and the directrix is y = 8.
Step-by-step explanation:
To determine the vertex, focus, and directrix of the equation x²+8x-12y+64=0, we need to convert it into the standard form of a parabola equation, which is (x-h)² = 4p(y-k). In this equation, (h,k) represents the vertex of the parabola and p represents the distance between the vertex and the focus and directrix.
By completing the square, we can rewrite the equation as (x+4)²-12(y-2) = 0. Comparing this with the standard form, we can determine that the vertex is at (-4, 2).
Since the coefficient of (y-2) is negative, the parabola opens downwards. Therefore, the focus will be below the vertex. Using the formula p = -c, where c is the coefficient of (y-2), we find that p = -(-12/2) = 6.
Since the parabola is opening downwards, the focus will be at (-4, 2-6) = (-4, -4) and the directrix will be the line y = 2+6 = 8.