Final answer:
The equation of the parabola with focus (4, 0) and directrix x = −4 is x = 8y². The parabola opens to the right, and the vertex is at the origin (0, 0).
Step-by-step explanation:
To identify the equation of a parabola with a given focus and directrix, one should be familiar with the definition and properties of a parabola. The focus of the parabola is a point F(h, k) and the directrix is a line y=k or x=h depending on the orientation of the parabola. A parabola is the set of all points that are equidistant from the focus and the directrix.
Given that the focus is F(4, 0) and the directrix is x = −4, the parabola opens sideways since the directrix is vertical, and it opens to the right because the focus has a positive x-coordinate. The distance between the focus and the directrix, which is the value of 4p, is 8 (as x-coordinate of the focus minus x-coordinate of the directrix equals 4 − (−4) = 8). Therefore, p = 2. The vertex V is at the midpoint between the focus and directrix; hence V(0, 0).
The standard form of the equation of a horizontal parabola with vertex at the origin is x = 4py2. In this case the equation becomes x = 4(2)y2 or x = 8y2.