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 at a distance p from the vertex along the axis of symmetry, while the directrix is a line perpendicular to this axis, also at a distance p from the vertex but in the opposite direction. Together, they define the parabola's shape and orientation.

Graphing Parabolas

To graph a parabola, one must identify its vertex, focus, and directrix. The equation y^2 = 4x indicates a horizontal parabola that opens to the right, with the vertex at the origin (0,0). Understanding how to plot these elements helps in visualizing the parabola and matching it to given graphs.