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 the standard form of a parabola. The standard form for a parabola that opens horizontally is \( (y - k)^2 = 4p(x - h) \).
2. Identify the values of \( h \), \( k \), and \( p \) by comparing \( y^2 = 8x \) with \( (y - k)^2 = 4p(x - h) \). Here, \( h = 0 \), \( k = 0 \), and \( 4p = 8 \).
3. Solve for \( p \) using the equation \( 4p = 8 \). This gives \( p = 2 \).
4. Determine the vertex of the parabola, which is at \( (h, k) = (0, 0) \).
5. Find the focus and directrix: The focus is \( (h + p, k) = (2, 0) \) and the directrix is the line \( x = h - p = -2 \).

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, parabolas can be represented by quadratic equations, typically in the form y^2 = 4px or x = ay^2. The orientation of the parabola (opening left, right, up, or down) depends on the equation's structure.

Vertex of a Parabola

The vertex of a parabola is the point where it changes direction, representing either the maximum or minimum value of the quadratic function. For the equation y^2 = 8x, the vertex is located at the origin (0,0), which is the point of symmetry for the parabola. Understanding the vertex is crucial for graphing the parabola accurately.

Focus and Directrix

The focus and directrix are key components that define a parabola's shape and position. The focus is a fixed point inside the parabola where all reflected lines converge, while the directrix is a line perpendicular to the axis of symmetry. For the equation y^2 = 8x, the focus is at (2,0) and the directrix is the line x = -2, which helps in accurately sketching the parabola.