Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. y^2 = 8x

Verified step by step guidance

1. Rewrite the given equation y^2 = 8x in standard form for a parabola. The standard form for a parabola that opens horizontally is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix.
2. Compare the given equation y^2 = 8x with the standard form. Notice that h = 0, k = 0, and 4p = 8. Solve for p by dividing 8 by 4, which gives p = 2.
3. Identify the vertex of the parabola. Since h = 0 and k = 0, the vertex is at (0, 0).
4. Determine the focus of the parabola. Since the parabola opens to the right (positive x-direction) and p = 2, the focus is located at (h + p, k), which is (0 + 2, 0) or (2, 0).
5. Find the directrix of the parabola. The directrix is a vertical line located at x = h - p. Substituting h = 0 and p = 2, the directrix is x = -2.

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 quadratic equations, typically in the form y^2 = 4px or x = 4py, where p is the distance from the vertex to the focus. Understanding the standard form of a parabola is essential for identifying its key features.

Vertex, Focus, and Directrix

The vertex of a parabola is the point where it changes direction, while the focus is a fixed point inside the parabola that determines its shape. The directrix is a line perpendicular to the axis of symmetry of the parabola, equidistant from the vertex as the focus. These elements are crucial for graphing the parabola and understanding its geometric properties.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, focus, and directrix, and understanding its orientation (opening direction). For the equation y^2 = 8x, the parabola opens to the right, and knowing how to derive and plot these key points allows for an accurate representation of the curve. Familiarity with transformations and symmetry also aids in graphing.