Download Hyperbolas

Hyperbolas
Topic 7.5
Definitions

Hyperbola: set of all points where the
absolute value of the difference of the
distance from any point on the hyperbola to
the foci is contant
15
10
5
-20
-10
10
-5
-10
-15
20
Two Standard Equations

2
2
(
y

k
)
(
x

h
)
Vertical Hyperbola:

1
2
2
a
b

Foci: (h, - c  k ) and (h, c  k )
** To Find C, c  a  b **
2
2
(h, -a  k ) and ( h, a  k )

Vertices:

a
Asymptote Slopes: 
b
2
Two Standard Equations
( x  h) ( y  k )
Horizontal Hyperbola:

1
2
2
a
b
2


2
Foci: (-c  h, k ) and (c  h, k )
** To Find C, c  a  b **
2
2

Vertices: (-a  h, k ) and (a  h, k )

b
Asymptote Slopes: 
a
2
Writing in Standard Form
1.
Complete the square for both the xterms and y-terms and move the
constant to the other side of the
equation
The first term will be the positive term
between x2 and y2
2.
Divide all terms by the constant
Example: x2  9 y 2  4 x  54 y  113  0
(x  4 x)  (9 y  54 y)  113 Group terms
2
2
(x  4 x  __)  9( y  6 y  __)  113 Complete the square
2
2
(x  4 x  4)  9( y  6 y  9)  113  4  (9)(9)
2
2
(x  2)2  9( y  3)2  36
(x  2)2 9( y  3)2 36


36
36
36
(x  2) ( y  3)

1
36
4
2
2
Simplify
Don’t forget to the
negative nine!
Divide by Constant
Graphing the hyperbola
1.
2.
3.
4.
5.
6.
Put equation in standard form
Graph the center (h, k)
Graph the foci (look at the equation to determine
your direction)
Graph the vertices
Graph the asymptotes (start at the center and
use “Rise over Run”)
Draw “U” shapes that go through the vertices and
stay in between the asymptotes
Example:
(x  2) ( y  3)

1
36
4
2
2
6
1) Graph Center
4
(-4,3)
2) Graph Foci
(8,3)
(-4.32,3 )
(8.32,3)
(2,3)
2
3) Graph Vertices
4) Graph Asymptotes
5) Graph Hyperbola
-5
5
10
You Try! Write the following equation
in standard form, then graph it.
 x2  4 y 2  6 x  16 y  29  0
4( y  4 y  __)  ( x  6 x  __)  29  4(__)  __
2
2
4( y  4 y  4)  ( x  6 x  9)  29  16  9
4( y  (2))2  ( x  (3))2  36
2
2
( y  (2)) ( x  (3))

 1
9
36
2
2
6
4
Center: (-3,-2)
Foci: (-3, -8.71) & (-3, 4.71)
Vertices: (-3,-8) and (-3,4)
1
Asymptotes: 
2
2
-10
-5
5
-2
-4
-6
-8
-10