Hyperbola: Definition, Formula & Examples

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  • 0:02 Hyperbolas in Real Life
  • 0:25 Definition & Standard Form
  • 2:40 Complete the Square
  • 4:12 Graphing Hyperbolas
  • 5:51 Lesson SUmmary
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Lesson Transcript
Instructor: David Karsner
Hyperbolas are a set of two similar curves called branches set on a plane governed by certain equations. They are, along with circles, parabolas and ellipses, conic sections, as you'll learn in this lesson.

Hyperbolas in Real Life

If you were to stand on the ground and throw a ball into the air, the ball would leave your hand reach a certain height and then fall back to the ground. The path that the ball would take would look like an arc. If this path were visible and you could somehow hold it up to a giant mirror, then the path and its reflection would create hyperbola. This lesson will give you the method in which one can take an equation of a hyperbola and find its center, vertices, and asymptotes and then graph it.

Definition and Standard Form

hyperbola from two cones and a plane

In this image we can see how a hyperbola is created from the intersection of a plane and two cones that meet on their tips. A hyperbola can open to the left and right or open up and down. A more formal definition of a hyperbola is a collection of all points, whose distances to two fixed points, called foci (plural for focus), is a constant difference. The equation of a hyperbola in standard form is:

((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1

The x and y are interchangeable and both give you an equation of an hyperbola. Let's look at some of its parts.

Parts of a Hyperbola

Center

Notice how the hyperbola has two lines of symmetry: one vertical and one horizontal. You can fold the hyperbola so that one half of it will lie completely on top of the other half. The folds will create two lines. The point at which the vertical fold and horizontal fold intersect is called the center of the hyperbola. Using the standard form of the hyperbola, the center is located at the point (h, k).

Vertices

The vertex of a parabola is the lowest point on a parabola if it is opening up and the highest point if it is opening down. The vertices of a hyperbola (which is composed of two parabolas) is the vertex of each branch of the hyperbola. To use the standard form of the hyperbola to find the vertices, you need to notice if the positive term is x^2 or y^2. If the positive term is x^2, the vertices are found at (h + a, k) and (h - a, k). If y^2 is the first term, the vertices are found at (h, k + b) and (h, k - b).

Asymptotes

The asymptotes of a hyperbola are two lines that lie between the two branches of the hyperbola. The intersection of the two lines occurs at the center of the hyperbola. The lines come very close to the hyperbola itself but don't cross it. If the equation has a positive x^2 term, then equations for the lines will be:

  • (y - k) = +b/a(x - h)
  • (y - k) = -b/a(x - h)

If the y^2 term is positive, the equations will be the same, except b/a is replaced with a/b.

Complete the Square

In the standard form of the equation, the values for a, b, g, and k are accessible. Knowing these values will make graphing the hyperbola possible. Converting the equation of a hyperbola so that it is in the standard form requires the ability to complete the square.

Example

Put the following equation into standard form:

4x^2 - y^2 - 24x + 4y + 28 = 0

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