1. Ch 4: The Hyperbola Objectives:
Use the standard and general forms of the equation of a hyperbola
Graph hyperbolas
©2003 Roy L. Gover

2. Applications
Hyperbolic
Telescope
The Hyperbola:
Sonic
Booms

3. Definition
A hyperbola is the set of all points such that the difference of the distance from two given points called foci is constant

4. Definition
The graph of a hyperbola may look like this…

5. Definition
…or like this:

6. Definition
The parts of a hyperbola are:
transverse axis

7. Definition
The parts of a hyperbola are:
conjugate axis

8. Definition
The parts of a hyperbola are:
center

9. Definition
The parts of a hyperbola are:
vertices

10. Definition
The parts of a hyperbola are:
foci

11. Definition
The parts of a hyperbola are:
the rectangle

12. Definition
The parts of a hyperbola are:
the asymptotes

13. Definition a The transverse axis is 2a units long
The distance from the center to each vertex is a units a

14. Definition
The distance from the center to the rectangle along the conjugate axis is b units 2b
The length of the conjugate axis is 2b units b

15. Definition
The distance from the center to each focus is c units where c

16. Try This
Name the indicated parts of the hyperbola

17. Definition
The distance from the center up to the vertex or down to the vertex is a units a
For the “up & down” hyperbola, the transverse axis is vertical and the conjugate axis is horizontal
Transverse Axis

18. Definition
Standard equations:
where (h,k) is the center

19. Summary Vertices and foci are always on the transverse axis
Distance from the center to each vertex is a units
Distance from center to each focus is c units where

20. a2 is always the first denominator
Summary
If x term is first, hyperbola opens left & right
If y term is first, hyperbola opens up & down
a2 is always the first denominator

21. Example
Find the equation of the hyperbola that has foci at (2,5) and (-4,5) and a transverse axis 4 units long.

22. Try This
Find the equation of the hyperbola that has foci at (4,0) and (-4,0) and the vertices are at (1,0) and (-1,0). Hint: first sketch the graph.

23. Definition The equations of the asymptotes are:
for a hyperbola that opens left & right

24. Definition The equations of the asymptotes are:
for a hyperbola that opens up & down

25. Example
Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes for the graph of :
then graph the hyperbola.

26. Try This
Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes for the graph of :
then graph the hyperbola.
Hint: re-write in standard form

27. Solution Center: (-3,2) Foci: (-3± ,2) Vertices: (-2,2), (-4,2)
Asymptotes: