Final answer:
The equation of the parabola with vertex (3, 0) and directrix x = 0.25 is not represented by any of the given options. The correct form is derived from (y - 0)² = 11(x - 3) for a parabola that opens to the right, which simplifies to y = √[11(x - 3)].
Step-by-step explanation:
The question involves finding the equation of a parabola given the vertex and the directrix. We know that if the directrix is a vertical line (like x = constant), the parabola must open horizontally. Since the vertex is at (3, 0) and the directrix is x = 0.25, the parabola opens to the right, away from the directrix.
The general equation for a horizontally opening parabola with vertex (h, k) is (y - k)² = 4p(x - h), where p is the distance from the vertex to the directrix. Here, since the vertex is (3, 0) and the directrix is x = 0.25, p = 3 - 0.25 = 2.75. Plugging these values into the equation, we get (y - 0)² = 4(2.75)(x - 3), which simplifies to y² = 11(x - 3). This equation needs to be in y = ax² + bx + c form, so rewrite it as y = √[11(x - 3)] because we are dealing with the top half of the parabola.
Therefore, none of the options a), b), c), or d) are correct for the given vertex and directrix.