9.5A Graph Hyperbolas
Algebra II
Hyperbolas
• Like an ellipse but instead of the sum of distances it
is the difference
• A hyperbola is the set of all points P such that the
differences from P to two fixed points, called foci, is
constant
• The line thru the foci intersects the hyperbola @ two
points (the vertices)
• The line segment joining the vertices is the
transverse axis, and it’s midpoint is the center of the
hyperbola.
• Has 2 branches and 2 asymptotes
• The asymptotes contain the diagonals of a rectangle
centered at the hyperbolas center
Asymptotes
Vertex (-a,0)
(0,b)
Vertex (a,0)
Focus
Focus
(0,-b)
This is an example of a horizontal transverse axis
Vertical transverse axis
2
2
y
x
 2 1
2
a
b
Standard Form of Hyperbola w/ center @
origin
Transverse
Equation
Asymptotes Vertices
Axis
Horizontal
x2 y2
 2 1
2
a
b
y 2 x2
Vertical


1
2
2
a
b
y=+/- (b/a)x
(+/-a,o)
y=+/- (a/b)x
(0,+/-a)
Foci lie on transverse axis, c units from the center c2 = a2+b2
Ex.
1)Graph
2
2
9 y  16 x  144
• Write in standard form (divide through by 144)
• a=4 b=3
• transverse axis is vertical & vertices are
(0,4) & (0, -4) Plot other pts from b value
(3,0) , (-3,0) to make rectangle
• Draw a rectangle centered at the origin.
• Draw asymptotes.
• Draw hyperbola with foci.
• Graph
Ex. 2 Graph
x  4y  9
2
2
9.5B Write Equations of
Hyperbolas
Algebra II
Ex. 1)Write the equation of a hyperbola
with foci (0,-3) & (0,3) and vertices (0,-2) &
(0,2).
• Vertical because foci & vertices lie on the y-axis
• Center @ origin because f & v are equidistant from the
origin
• Since c=3 & a=2, c2 = b2 + a2
2
2
•
9 = b2 + 4
y
x
 1
•
5 = b2
4
5
•
+/-√5 = b
Standard Form of Hyperbola w/ center @
origin
Transverse
Equation
Asymptotes Vertices
Axis
Horizontal
x2 y2
 2 1
2
a
b
y 2 x2
Vertical


1
2
2
a
b
y=+/- (b/a)x
(+/-a,o)
y=+/- (a/b)x
(0,+/-a)
Foci lie on transverse axis, c units from the center c2 = a2+b2
Ex. 2 Write an equation of the hyperbola
with the given foci & vertices
• Foci:   3 6 , 0,  3 6 , 0
• Vertices: (-2,0), (2,0)
Assignment