Hyperbolas are relations that have asymptotes. When graphing rational functions you often produce a hyperbola. In this concept, hyperbolas will not be oriented in the same way as with rational functions, but the basic shape of a hyperbola will still be there.
Hyperbolas can be oriented so that they open side to side or up and down. One of the most common mistakes that you can make is to forget which way a given hyperbola should open. What are some strategies to help?
A hyperbola has two foci. For every point on the hyperbola, the difference of the distances to each foci is constant. This is what defines a hyperbola. The graphing form of a hyperbola that opens side to side is:
A hyperbola that opens up and down is:
Put the following hyperbola into graphing form and sketch it.
Solution: This relationship bridges the gap between ellipses which have eccentricity less than one and hyperbolas which have eccentricity greater than one. When eccentricity is equal to one, the shape is a parabola.
Square both sides and rearrange terms so that it is becomes a hyperbola in graphing form.
Concept Problem Revisited
Eccentricity is the ratio between the length of the focal radius and the length of the semi transverse axis. For hyperbolas, the eccentricity is greater than one.
A hyperbola is the collection of points that share a constant difference between the distances between two focus points.
1. Completely identify the components of the following conic.
2. Given the following graph, estimate the equation of the conic.
Shape: Hyperbola that opens vertically.
Center: (1, 3)
Foci: (1, 8), (1, -2)
Vertices: (1, 6), (1, 0)
Note that it is easiest to write the equations of the asymptotes in point-slope form using the center and the slope.