A hyperbola is discontinious, meaning it has gaps on the graphs. It can be defined as the locus of all points in a plane such that the absolute value of the differences of the distances from two foci is constant. The Locus is the various points that form a curve.
The graph of a parabola is two disconnected branches that approach two asymptotes.
The Center of a hyperbola is the midpoint of the segment with endpoints at the foci.
The Vertices are the intersection of this segment, of the center, and each branch of the curve.
In a hyperbola, just like in the ellipse, the foci are inside the parabola.
Axes of Symmetry:
- Transverse axis: the length of 2a. It connects the vertices.
- Conjugate axis: the length of 2b. It is perpendicular to transverse.
2a is the absolute value of the differences between the distances from any point to the foci. And it is represented by the following formulas:
In a Horizontal Hyperbola:
|PF1 – PF2| = 2a
Where “PF” equals the point focus.
In a Vertical Hyperbola:
|PF1 – PF2| = 2b
2a is defined as the distance from vertex to vertex.
2b is the distance from
2c is defined as the distance from one focus to the other focus.
Therefore, c units is the distance from the focus to the center; a untis is the distance from the vertex to the center; and b units is the the length of the hyperbola.
Equations for Horizontal and Vertical Hyperbola – General Form
In hyperbolas a will be the one that goes first, not the greatest one in value.
How to Graph Hyperbolas
To graph, you use your a, b and c values. If you know the formulas for each one of the points (vertices, foci, etc.), find the points, and then graph them. However, it is easier to do so with the a, b and c values and you don’t have to memorize anything. You just have to understand that a is the distance from vertez to center, that b is the length and c is the distance from foci to center. This can be seen in the following image.
Want somebody to help you review? Want to see more specific cases where you can analyze the problem? Or want somebody to guide you through different examples?
The Conic Club 