Attachments
9_3Hyperbolas.ppt
9.3 Hyperbolas
Hyperbola: set of all points such that the difference of the
distances from any point to the foci is constant.
(0,b)
(0,-b)
Vertex (a,0)
Vertex (-a,0)
Asymptotes
This is an example of a horizontal transverse axis
(a, the biggest number, is under the x2 term with the minus
before the y)
Focus
Focus
The transverse axis is the line segment joining the
vertices(through the foci)
The midpoint of the transverse axis is the center of the
hyperbola..
Standard Equation of a Hyperbola (Center at Origin)
This is the equation
if the transverse axis is
horizontal.
The foci of the hyperbola lie on the major axis (the y-axis, or
the horizontal), c units from the center, where c2 = a2+ b2
(c, 0)
(c, 0)
(a, 0)
(a, 0)
(0, b)
(0, b)
The standard form of the Hyperbola with a center at (h, k) and a
horizontal axis is
The Hyperbola with a center at (h, k) and a horizontal axis has
the following characteristics
Standard Equation of a Hyperbola (Center at Origin)
This is the equation
if the transverse axis is
vertical.
The foci of the hyperbola lie on the major axis (the x-axis or
the vertical), c units from the center, where c2 = a2+ b2
(0, c)
(0, c)
(0, a)
(0, a)
(b, 0)
(b, 0)
The standard form of the Hyperbola with a center at (h, k) and a
vertical axis is
The Hyperbola with a center at (h, k) and a vertical axis has
the following characteristics
Hyperbola General Rules
x and y are both squared
Equation always equals 1
Equation is always minus(-)
a2 is always the first denominator
c2 = a2 + b2
c is the distance from the center to each foci on the major
axis
a is the distance from the center to each vertex on the major
axis
General Rules Continued
b is the distance from the center to each midpoint of the
rectangle used to draw the asymptotes. This distance runs
perpendicular to the distance (a).
Major axis has a length of 2a.
If x2 is first then the hyperbola is horizontal.
If y2 is first then the hyperbola is vertical.
Still More Rules
The center is in the middle of the 2 vertices and the 2
foci.
The vertices and the co-vertices are used to draw the rectangles
that form the asymptotes.
The vertices and the co-vertices are the midpoints of the
rectangle.
The co-vertices are not labeled on the hyperbola because they
are not actually part of the graph.
Ex 1: Write the equation in standard form of 4x2 16y2 = 64. Find
the foci and vertices of the hyperbola.
Get the equation in standard form (make it equal to 1):
4x2 16y2 = 64
64 64 64
Use c2 = a2 + b2 to find c.
c2 = 42 + 22
c2 = 16 + 4 = 20
c =
(c, 0)
(c,0)
(4,0)
(4, 0)
(0, 2)
(0,-2)
That means a = 4 b = 2
Vertices:
Foci:
Simplify...
x2 y2 = 116 4
Ex 2: Write an equation of the hyperbola whose foci are (0, 6)
and (0, 6) and whose vertices are (0, 4) and (0, 4). Its center is
(0, 0).
y2 x2 = 1a2 b2
Since the major axis is
vertical, the equation is
the following:
Since a = 4 and c = 6 , find b...
c2 = a2+ b2
62 = 42 + b2
36 = 16 + b2
20 = b2
The equation of the hyperbola:
y2 x2 = 116 20
(b, 0)
(b, 0)
(0, 4)
(0, 4)
(0, 6)
(0, 6)
How do you graph a hyperbola?
To graph a hyperbola, you need to know the center, the vertices,
the co-vertices, and the asymptotes...
Draw a rectangle using +a and +b as the sides...
(5, 0)
(5,0)
(4,0)
(4, 0)
(0, 3)
(0,-3)
a = 4 b = 3
The asymptotes intersect at the center of the hyperbola and pass
through the corners of a rectangle with corners (+ a, + b)
Ex 3: Graph the hyperbola x2 y2 = 116 9
c = 5
Draw the asymptotes (diagonals of rectangle)...
Draw the hyperbola...
Here are the equations of the asymptotes:
Horizontal Transverse Axis: y = + b xa
Vertical Transverse Axis: y = + a xb
Ex 4: Graph 4x2 9y2 = 36
Write in standard form (divide through by 36)
a = 3 b = 2 because x2 term is + transverse axis is horizontal
& vertices are (-3,0) & (3,0)
Draw a rectangle centered at the origin.
Draw asymptotes.
Draw hyperbola.
Examples
22
22
1
xy
ab
-=
22
22
1
yx
ab
-=
-
2
5
,
0
(
)
and
2
5
,
0
(
)
Vertices: (,)
hak
20
=
2
5
-
4
,
0
(
)
and
4
,
0
(
)
Foci: (,)
hck
Vertices: (,)
hka
22
22
()()
1
yx
a
h
k
b
--
-=
Foci: (0,)
c
Foci: (,)
hkc
22
22
()()
1
xy
a
k
h
b
--
-=
Vertices: (0,)
a
Vertices: (,0)
a
Foci: (,0)
c
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