In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Focus: (7, 0); Directrix: x = - 7

Key Concepts

Standard Form of a Parabola

The standard form of a parabola that opens horizontally is given by the equation (y - k)² = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus. This form allows for easy identification of the parabola's orientation and key features, such as the focus and directrix.

Focus and Directrix

The focus of a parabola is a fixed point from which distances to points on the parabola are measured, while the directrix is a line that is perpendicular to the axis of symmetry of the parabola. The parabola is defined as the set of points equidistant from the focus and the directrix, which is crucial for determining its equation.

Vertex of a Parabola

The vertex of a parabola is the point where it changes direction and is located midway between the focus and the directrix. For a parabola with a focus at (7, 0) and a directrix at x = -7, the vertex can be found by averaging the x-coordinates of the focus and the directrix, which is essential for writing the equation in standard form.