1. Hyperbolas
Chapter 8 Section 5

2. Objective
You will be able to differentiate between Horizontal and Vertical Hyperbolas and then be able to sketch a graphical representation of the equation.

3. Vocabulary
Hyperbolas: the set of all points in a plane such that the Absolute Value of the difference of the distances from two fixed points is constant.
Foci (Plural of Focus): the two fixed points inside each hyperbolic section used to generate the graph.

4. Vocabulary
Center: midpoint of the distance between the 2 vertices of the hyperbola.
Tranverse Axis: segment of length 2a whose endpoints are the vertices of the hyperbola. (~Major Axis)
Conjugate Axis: segment of length 2b that is perpendicular to the transverse axis of the hyperbola at the center. (~Minor Axis)
Asymptote: Lines which graph cannot touch or cross.

5. Hyperbola This gives the Pythagorean equation c2 = a2 + b2. a = Leg
The hyperbolic equation is a difference of 2 squares so the Pythagorean is a sum.
a = Leg
(Distance from center to transverse vertex)
b = Leg
(Distance from center to conjugate vertex)
c = Hypotenuse (distance from
center to a focus.)

7. Equations

8. Table of Equations and important information
Equations of Hyperbolass centered at the Origin
Standard Form
Direction of Major Axis
Horizontal
Vertical
Foci
(h + c, k), (h - c, k)
(h, k + c), (h, k - c)
Transverse Vertices
(h + a, k), (h - a, k)
(h, k + a), (h, k - a)
Conjugate Co-Vertices
(h, k + b), (h, k - b)
(h + b, k), (h - b, k)
Equations of Hyperbolas centered at (h, k)
Asymptotes

9. Quick Clues
The Denominator does not matter. The graph is Horizontal if the x2 is positive and the graph is Vertical if the y2 is positive.