DefinitionA hyperbola is the set of all points such that the difference of the distance from two given points called foci is constant

DefinitionThe parts of a hyperbola are:

transverse axis

DefinitionThe parts of a hyperbola are:

conjugate axis

DefinitionThe parts of a hyperbola are:

center

DefinitionThe parts of a hyperbola are:

vertices

DefinitionThe parts of a hyperbola are:

foci

DefinitionThe parts of a hyperbola are:

the asymptotes

DefinitionThe distance from the center to each vertex is a units

a

The transverse axis is 2a units long

2a

DefinitionThe distance from the center to the rectangle along the conjugate axis is b units

b

2b

The length of the conjugate axis is 2b units

DefinitionThe distance from the center to each focus is c units where

c

2 2 2c a b

Sketch the graph of the hyperbola

What are the coordinates of the foci?What are the coordinates of the vertices?What are the equations of the asymptotes?

2 2

125 36

x y

65

y x 65

y x

61,0 61,0

How do get the hyperbola into an up-down position?

switch x and y

2 2

125 36

y x

identify vertices, foci,asymptotes for:

0, 61

0, 61

56

y x56

y x

Definition

2 2

2 2

( ) ( )1

x h y k

a b

2 2

2 2

( ) ( )1

y k x h

a b

where (h,k) is the center

Standard equations:

DefinitionThe equations of the asymptotes are:

( )b

y k x ha

for a hyperbola that opens left & right

DefinitionThe equations of the asymptotes are:

( )a

y k x hb

for a hyperbola that opens up & down

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 where2 2 2c a b

Summary

•If x term is positive, hyperbola opens left & right•If y term is positive, hyperbola opens up & down•a2 is always the positive denominator

Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes for the graph of :

2 24 24 4 28 0x y x y then graph the hyperbola.Hint: re-write in standard form

Example

Solution2 2( 3) ( 2)

11 4

x y

Center: (-3,2)

Foci: (-3± ,2)5

Vertices: (-2,2), (-4,2)Asymptotes: 2 2( 3)y x

ExampleFind the coordinates of the center, foci, and vertices, and the equations of the asymptotes for the graph of :

2 225 9 100 72 269 0y x y x

then graph the hyperbola.

Solution2 2( 2) ( 4)

19 25

y x

Center: (-4,2)

Foci: (-4,2± )34Vertices: (-4,-1), (-4,5)

Asymptotes:3

2 ( 4)5

y x