Colleen Beaudoin
February, 2009


Review: The geometric definition relies on a
cone and a plane intersecting it
Algebraic definition: a set of points in
the plane such that the difference of the
distances from two fixed points, called
foci, remains constant.

y




A


d1 
d2

x






f1










f2


d2
d1






foci


B


From each
point in the
plane, the
difference of
the distances to
the foci is a
constant.
Example:
Point A: d1-d2 = c
Point B: d1-d2 = c
Center

y

Transverse Axis







x






f1








f2







Conjugate Axis







foci
Vertices
Algebraic Definition of a hyperbola: a set of points in
the plane such that the difference of the distances
from two fixed points, called foci, remains constant.


How is the definition similar to that of an ellipse?
How is it different?





Both variables are squared.
Equation:
Compare the equations of ellipses and
hyperbolas.
What makes the hyperbola different from the
parabola?
What makes the hyperbola different from a
circle?
2
2
2
( x - h) ( y - k )
( y - k ) ( x - h)


1
or

2
2
2
2
a
b
b
a
Procedure to graph:
1. Put in standard form (above): x squared term - y
squared term = 1
2. Determine if the hyperbola is opening vertically
or horizontally. (If x is first, it’s horizontal. If y
is first, it’s vertical.)
3. Plot the center (h,k)
4. Plot the endpoints of the horizontal axis by
moving “a” units left and right from the center.
2
2
2
2
( x - h) ( y - k )
( y - k ) ( x - h)


1
or

2
2
2
2
a
b
b
a
2
To graph:
5. Plot the endpoints of the vertical axis by
moving “b” units up and down from the
center.
Note: The line segment that contains the vertices of
the hyperbola is known as the transverse axis. The
other axis is the conjugate axis.
6. Draw a rectangle such that each of the axis
endpoints is the midpoint of a side.
2
2
2
( x - h) ( y - k )
( y - k ) ( x - h)

 1 or

2
2
2
2
a
b
b
a
2
To graph:
7. Sketch the diagonals of the rectangle and
extend them outside of the rectangle. (These
are the asymptotes of the hyperbola.)
8. Draw each branch of the hyperbola – Be sure to
go through the vertex of each (the endpoint of
the transverse axis) and approach the
asymptotes.
2
2
2
( x - h) ( y - k )
( y - k ) ( x - h)


1
or

2
2
2
2
a
b
b
a
2
To graph:
9. Use the following formula to help locate the foci:
c2 = a2 + b2
Move “c” units left and right form the center if the
transverse axis is horizontal
OR Move “c” units up and down form the center if
the transverse axis is vertical
Label the points f1 and f2 for the two foci.
Note: It is not necessary to plot the foci to graph the hyperbola, but it is common
practice to locate them.
The equation of each asymptote can be found by
using the point-slope formula. Use the center
as “the point” and slope can be found by
counting on the graph (from the point to the
corner of the rectangle).
Or the following formulas can be used:
With horizontal transverse axis:
b
b
y = k + (x - h) and y = k - (x - h)
a
a
With vertical transverse axis:
a
a
y = k + (x - h) and y = k - (x - h)
b
b
2
2
y
x

1
64 36
1. Put in standard form.
Done
2. Determine if the hyperbola is opening vertically or
horizontally.
Vertically because “y” is first.
3. Identify the center.
(0,0)
4. Identify the endpoints of the horizontal axis.
(6,0) and (-6,0)
5. Identify the endpoints of the vertical axis.
(0,8) and (0,-8)
Which pair of endpoints are the vertices?
(0,8) and (0,-8)
2
2
y
x

1
64 36
6. Draw a rectangle such that each of the axis
endpoints is the midpoint of a side.
7. Sketch the asymptotes of the hyperbola.
8. Draw each branch of the hyperbola – Be sure to
go through the vertex of each (the endpoint of
the transverse axis) and approach the
asymptotes.
2
2
y
x

1
64 36
9. Locate the foci.
(0,10) and (0,-10)
10. Find the equations of the asymptotes.
4
4
y  x and y 
x
3
3
2
2
y
x

1
64 36
















        
















Transverse axis
y
Conjugate axis
Center
x
                 
Asymptotes
( x  3) ( y  2)

1
36
16
2
2
1. Put in standard form.
Done
2. Determine if the hyperbola is opening vertically or
horizontally.
Horizontally because “x” is first.
3. Identify the center.
(3,-2)
4. Identify the endpoints of the horizontal axis.
(-3,-2) and (9,-2)
5. Identify the endpoints of the vertical axis.
(3,2) and (3,-6)
Which pair of endpoints are the vertices?
(-3,-2) and (9,-2)
( x  3) ( y  2)

1
36
16
2
2
6. Draw a rectangle such that each of the axis
endpoints is the midpoint of a side.
7. Sketch the asymptotes of the hyperbola.
8. Draw each branch of the hyperbola – Be sure to
go through the vertex of each (the endpoint of
the transverse axis) and approach the
asymptotes.
( x  3) ( y  2)

1
36
16
2
9. Locate the foci.
(3+2√13,-2) and (3-2√13,-2)
10. Find the equations of the asymptotes.
4
4
y  x and y 
x
3
3
2
( x  3) ( y  2)

1
36
16
2
2

y










          














x








       
Exp. 3: Write the equation in
standard form and graph: y 2  4 x 2  64
1. Put in standard form.
y 2 x2
 1
64 16
y 2 x2
 1
64 16

y













         













x


 
 

        
Write the equation for a hyperbola with xintercepts at 5 and -5 and foci (6,0) and (-6,0).
1)
2)
3)
4)
5)
6)
How can you tell if the graph of an equation
will be a line, parabola, circle, ellipse, or
hyperbola?
What’s the standard form of a hyperbola?
What’s the standard form of an ellipse?
What’s the standard form of a parabola?
What’s the standard form of a circle?
How are the various equations similar and
different?