Definition
A hyperbola is
the set of all
points such
that the
difference of
the distance
from two
given points
called foci is
constant
Definition
The parts
of a
hyperbola
are:
transverse
axis
Definition
The parts
of a
hyperbola
are:
conjugate axis
Definition
The parts
of a
hyperbola
are:
center
Definition
The parts
of a
hyperbola
are:
vertices
Definition
The parts
of a
hyperbola
are:
foci
Definition
The parts
of a
hyperbola
are:
the
asymptotes
Definition
The
distance
transverse
from
the
axis is 2a
center to
units
long
each vertex
is a units
2a
a
Definition
The
distance
The length
from the
of
the
center to the
conjugate
rectangle along
the
conjugate
axis is 2b
axis is b units
units
2b
b
Definition
The distance
from the
center to
each focus is
c units
where
c  a b
2
2
2
c
Sketch the graph of the hyperbola
2
2
x
y

1
25 36
What are the coordinates of the foci?
What are the coordinates of the vertices?
What are the equations of the
asymptotes?
6
y  x
5

61 ,0

6
y  x
5

61 ,0

How do get the hyperbola
into an up-down position?
switch x and y
identify vertices, foci,
asymptotes for:
2
2
y
x

1
25 36
5
y  x
6
0,
61

0, 
61

5
y  x
6
Definition
Standard equations:
( x  h) ( y  k )


1
2
2
a
b
2
2
( y  k ) ( x  h)


1
2
2
a
b
where (h,k) is the center
2
2
Definition
The equations of the
asymptotes are:
b
y  k   ( x  h)
a
for a hyperbola that
opens left & right
Definition
The equations of the
asymptotes are:
a
y  k   ( x  h)
b
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 where
2
2
2
c  a b
Summary
•If x term is positive,
hyperbola opens left & right
•If y term is positive,
hyperbola opens up & down
•a is always the positive
2
denominator
Example
Find the coordinates of the
center, foci, and vertices, and
the equations of the
asymptotes for the graph of :
4 x  y  24 x  4 y  28  0
2
2
then graph the hyperbola.
Hint: re-write in standard
form
Solution
( x  3) ( y  2)

1
1
4
Center: (-3,2)
2
2
Foci: (-3± 5 ,2)
Vertices: (-2,2), (-4,2)
Asymptotes: y  2  2( x  3)
Example
Find the coordinates of the
center, foci, and vertices, and
the equations of the
asymptotes for the graph of :
25 y  9 x  100 y  72 x  269  0
2
2
then graph the hyperbola.
Solution
( y  2) ( x  4)

1
9
25
2
2
Center: (-4,2)
Foci: (-4,2± 34 )
Vertices: (-4,-1), (-4,5)
3
Asymptotes: y  2   ( x  4)
5