In Exercises 1–4, find the focus and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). y^2 = - 4x

Key Concepts

Parabola Definition

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. In algebra, parabolas can be represented by quadratic equations, typically in the form y^2 = 4px or x = 4py, where p is the distance from the vertex to the focus and the directrix.

Focus and Directrix

The focus of a parabola is a fixed point located inside the curve, while the directrix is a line perpendicular to the axis of symmetry of the parabola. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix, which is a defining property of parabolas.

Standard Form of Parabola

The standard form of a parabola can indicate its orientation and position. For the equation y^2 = -4x, it opens to the left, with the vertex at the origin (0,0). The coefficient -4 indicates that the distance from the vertex to the focus is 1 unit, and the directrix is a vertical line located at x = 1.