1. Lesson 9.3 Hyperbolas

2. Hyperbola
Set of all points where the difference between the distances to two fixed points (foci) is a positive constant.
20 cm
12 cm
7cm
15 cm
12 cm
4cm
10cm
2cm
Focus
Focus

3. Other parts of a hyperbola:
Transversal axis can run vertically
Axis (transverse)
Center
Vertices
Foci

4. Equation of a Hyperbola Similar to ellipse
Similarities/Differences:
Subtraction between x2 and y2 terms
Variable above a determines the direction of axis
a is still the distance from center to a vertex
c is still the distance from center to a focus
c is now larger so…
Horizontal axis
Vertical axis

5. Example
Find the standard form of the equation with foci (-1, 2) and (5, 2) and vertices (0, 2) and (4, 2).

6. Asymptotes Question: What is b in an hyperbola?
It is still a distance from the center in the opposite direction of the axis
How it applies to a parabola has to do with a new part unique to hyperbolas.
Asymptotes
Lines that bound the hyperbola
Pass through the diagonals of a rectangle with dimensions 2a and 2b

7. Asymptotes (h, k + b) (h - a, k) (h, k) (h + a, k) (h, k - b)
Conjugate axis
(h, k + b)
(h - a, k)
(h, k)
(h + a, k)
(h, k - b)
b is the distance from center to edge of rectangle along conjugate axis

8. Equations for Asymptotes
Horizontal hyperbola (transverse axis)
Vertical hyperbola (transverse axis)
b is up/down – a is left/right
a is up/down – b is left/right
slope is rise over run →
slope is rise over run →

9. Example
Sketch the graph of 4x2 – y2 = 16, include the asymptotes.

10. Example
Find the standard form of the hyperbola with vertices (3, -5), (3, 1) and asymptotes

11. Eccentricity of an Hyperbola
where e > 1 (since c is larger than a)
large e → flatter curve
e close to 1 → more curved - pointed
Problem Set 9.3