31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola that opens to the right with directrix x = -4

Verified step by step guidance

1. Recall the definition of a parabola: it is the set of all points equidistant from the focus and the directrix.
2. Since the directrix is given as the vertical line \(x = -4\) and the parabola opens to the right, the axis of symmetry is horizontal along the x-axis.
3. The vertex is at the origin \((0,0)\), so the focus must be on the positive x-axis, at some point \((p,0)\), where \(p > 0\).
4. The distance from the vertex to the directrix is \(|p| = 4\), so the focus is at \((4,0)\).
5. Use the standard form of a parabola that opens right: \(y^2 = 4px\). Substitute \(p = 4\) to get the equation \(y^2 = 16x\).

Key Concepts

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric definition helps derive the equation of the parabola by relating distances from any point on the curve to the focus and directrix.

Orientation of Parabolas

The orientation of a parabola depends on the position of its focus and directrix. If the parabola opens right or left, its axis of symmetry is horizontal, and the equation involves x and y accordingly. For a parabola opening right, the directrix is vertical, and the equation typically has the form (y - k)^2 = 4p(x - h).

Vertex at the Origin and Directrix

When the vertex is at the origin, the parabola's equation simplifies since h = 0 and k = 0. Given the directrix x = -4, the focus lies on the opposite side of the vertex at x = 4, allowing calculation of the parameter p, which determines the distance from the vertex to the focus or directrix and shapes the parabola's equation.