In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. y^2 - 6x = 0

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, it can be represented by a quadratic equation in the form y^2 = 4px or x = ay^2, where 'p' is the distance from the vertex to the focus and the directrix. Understanding the standard forms of parabolas is essential for identifying their properties.

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. These elements are crucial for defining the parabola's shape and orientation, and they help in graphing the parabola accurately.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, focus, and directrix, and understanding its orientation (opening direction). For the equation y^2 - 6x = 0, rewriting it in standard form reveals its vertex and allows for the identification of the focus and directrix. This process is vital for visualizing the parabola and understanding its geometric properties.