March 27th copyright2009merrydavidson
HYPERBOLA’S• A hyperbola looks sort of like two
mirrored parabolas.• The two "halves" being called
"branches". • A hyperbola has two foci and two
vertices.• Hyperbola’s also have asymptotes.
HYPERBOLA’SDefinition:
The set of all points for which the difference in the distance to two fixed points (called the foci) is constant.
Horizontal Hyperbola
d2
d1
d1-d2= constant
Horizontal Hyperbola
d2d1
d1-d2= constant
Horizontal Hyperbola
d2d1
d1-d2= constant
Horizontal Hyperbola
c = center to focus
b = center to boxa = center to vertex
“x” term is positive
Major/Transverse axis goes through vertices.
Minor/Conjugate axis
ab
c
asymptote
HORIZONTAL hyperbola
Vertical Hyperbola
c = center to focusb = center to box
a = center to vertex
“y” term is positive
Major/Transverse axis goes through vertices.
Minor/Conjugate axis b
a
c
asymptote
Vertical Hyperbola
Horizontal Hyperbola Vertical Hyperbola
c2 = a2 + b2
SUMMARY
MAJOR axis does NOT mean longer.
Ex 1:Horizontal/Vertical? Center:a2= a=
b2= b=
c
19)3(
25)1( 22
yx
(-1,3)25 5
9 3 c2 = a2 + b2
2 25 9c
34 5.8c 5.8
Let’s draw what we know so far…
Draw in the box.
Ex 1: 19)3(
25)1( 22
yx
Draw in the asymptotes.Draw in the branches.
Place “c” on the graph.F1: F2: ( 1 34,3) ( 1 34,3)
Ex 2:
Horizontal/Vertical? Center:
a2= a=
b2= b= c =
2 24( 1) 25 100y x
(0,1)
25 5
4 2
c2 = a2 + b2
2 25 4c 29 5.4c 5.4
Let’s draw what we know so far…
2 2( 1) 125 4y x
Insert a “4” on your notes!
Draw in the box.
Ex 2:
Draw in the asymptotes.Draw in the branches.
Place “c” on the graph.F1: F2:
2 2( 1) 125 4y x
(0,1 29) (0,1 29)
Ex 3: Find the center, foci, and the equation of the asymptotes.
2 22 21
4 16x y
Center:Major axis: horizontal or vertical
a = Verticesb =c = Foci
4x2 – y2 - 16x - 4y -4 = 0
(2,-2)
24
(4,-2),(0,-2)
(4 2 5)20 2 5
Ex 4: Find the equation of the hyperbola if the vertices are at (5,0) and (-5,0) and b2 = 49.
Draw in what you know.
Where is the vertex?(0,0)
Major axis: H or V?
Fill in the equation.2 2
125 49x y
Ex 5.
Write the equation for the hyperbola graphed on your notes.
HW: WS 10-5