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
