Final answer:
The equation of the parabola with a focus of (-5, 0) and a directrix of y = 8 is (x + 5)² = -16(y - 4) in standard form, indicating that the parabola opens downwards.
Step-by-step explanation:
To write the equation of a parabola given a focus of (-5, 0) and a directrix of y = 8, one must understand that a parabola is a set of points equidistant from the focus and the directrix. The vertex (h, k) of the parabola lies midway between the focus and directrix. Since the directrix is y = 8 and the focus has a y-coordinate of 0, the y-coordinate of the vertex will be (8 + 0)/2 = 4. The x-coordinate of the vertex will be the same as the focus, -5, so the vertex of the parabola is (-5, 4).
The standard form of a vertical parabola is (x - h)² = 4p(y - k), where (h, k) is the vertex and 'p' is the distance from the vertex to the focus or directrix. Since the distance from the vertex to the focus and directrix is 4 (since the focus is at y = 0 and the directrix is at y = 8), we have p = 4. Therefore, the equation is (x + 5)² = 16(y - 4).
This is the equation of the parabola in standard form. Note that the directrix y = 8 is above the vertex, indicating that the parabola opens downwards. Hence, the standard form of the parabola is (x + 5)² = -16(y - 4), where the negative sign indicates the downward opening of the parabola.