Understanding Parabola Equations

This page introduces the fundamental components of parabolas and their equations. It covers both vertically and horizontally oriented parabolas, emphasizing the relationship between the equation form and the parabola's orientation.

For vertically oriented parabolas (y-axis symmetry), the standard form is x² = 4py, where p determines the distance from the vertex to the focus and directrix. The focus is located at (0, p) and the directrix at y = -p.

For horizontally oriented parabolas (x-axis symmetry), the equation takes the form y² = 4px. Here, the focus is at (p, 0) and the directrix is x = -p.

Vocabulary: Vertex - The point where a parabola changes direction, often the highest or lowest point.

Definition: Directrix - A line perpendicular to the axis of symmetry of a parabola, used in defining the parabola.

Highlight: The value of p in the equation determines whether the parabola opens upward/rightward (p > 0) or downward/leftward (p < 0).

The page also presents variations of these equations for parabolas with different orientations and openings, providing a comprehensive overview of parabola equation forms.

Deriving Parabola Equations from Graphs

This page demonstrates how to derive a parabola equation from a given graph or set of information. It presents an example of a horizontally oriented parabola.

The example shows a parabola with its vertex at (0, 0) and focus at $-2, 0$. The directrix is located at x = 2.

Example: For a parabola with focus at $-2, 0$ and directrix at x = 2, the equation is derived as y² = -8x.

The process of deriving the equation involves:

1. Identifying the orientation of the parabola (horizontal in this case).
2. Locating the focus and directrix.
3. Applying the general formula y² = 4px, where p is the distance from the vertex to the focus.
4. Calculating p as -2 (negative because the parabola opens to the left).

Highlight: The axis of symmetry for this parabola is y = 0, which is consistent with its horizontal orientation.

This example illustrates the practical application of parabola equation formulas and demonstrates how to use a parabola graph calculator conceptually.

Quadratics and parabolas are fundamental concepts in algebra, with wide-ranging applications in mathematics and physics. This guide provides a comprehensive overview of parabola equations, focusing on their key components and how to derive them from graphs.

• The guide covers different forms of parabola equations, including those with vertical and horizontal axes of symmetry.
• It explains crucial elements such as the vertex, focus, and directrix of parabolas.
• Examples are provided to illustrate how to write parabola equations from given information or graphs.
• The relationship between the focus, directrix, and the equation of a parabola is thoroughly explored.