Final answer:
The standard form of a parabola equation with a vertex (h, k) and a horizontal directrix y = d is typically (x - h)² = 4p(y - k) if the parabola opens upwards, but could also be (x - h)² = -4p(y - k) if it opens downwards. The sign of 4p depends on the direction the parabola opens. Option A is the correct answer.
Step-by-step explanation:
The question is asking for the standard form equation of a parabola that includes a vertex (h, k) and a horizontal directrix y = d. In the context of parabolas, p represents the distance from the vertex to the focus or from the vertex to the directrix, and it is positive if the parabola opens upwards and negative if it opens downwards.
Since the directrix is horizontal (parallel to the x-axis), the parabola opens either upward or downward, which means the equation will include (x - h)². The sign before 4p will determine whether the parabola opens upwards (positive if p > 0) or downwards (negative if p < 0). Considering the directrix is y = d, we know that the vertex form of the parabola's equation must have y - k in the expression because it is the y-value that is varying.
Therefore, the standard form of a parabola with vertex (h, k) and horizontal directrix y = d is either (x - h)² = 4p(y - k) if it opens upwards or (x - h)² = -4p(y - k) if it opens downwards. Since we do not have information on whether the parabola opens up or down, we cannot determine the sign of 4p, and both a) and c) could potentially be correct depending on the context. However, the typical standard form provided in textbooks and by convention when the directrix y=d and the parabola opens upwards is option a) (x - h)² = 4p(y - k).