9.5 HyperbolasPART 1

Hyperbola/Parabola Quiz: Friday

Conics Test: March 26

Definition of Hyperbola: A hyperbola is the set of points P(x,y) in a

plane such that the absolute value of the difference between the distances P to two fixed points in the plane, f1 and f2, called the foci, is constant.

P Q

F1 F2

What you need to know: A hyperbola has two axes of symmetry. One

axis contains the TRANSVERSE axis of the hyperbola, (a,0) to (-a,0), and the other axis contains the CONJUGATE axis, from (0,-b) to (0,b).

The endpoints of the TRANSVERSE are called vertices. The endpoints of the CONJUGATE are called co-vertices. The point in the VERY middle, is the center.

Standard Equation of a Hyperbola CENTERED AT THE ORIGIN (0,0)

2 2

2 21

x y

a b

Horizontal Vertical

In both cases: a²+b²=c². (it switched from the ellipse!!!!)

Length of the transverse is 2a and length of the conjugate is 2b

AND NOTE: Transverse is NOT ALWAYS longer than the CONJUGATE!!!

2 2

2 21

y x

a b

Example: Write the standard equation for the hyperbola

with vertices at (0,-4) and (0,4) and co-vertices at (-3,0) and (3,0). Then Sketch the graph.

Since the vertices lie along the y-axis, the equation is vertical.

We know that a=4 and b=3.

Example 1 Graph:

You try: Write the standard equation for the hyperbola

whose vertices are at V1(-5,0), and V2 (5,0) and whose co-vertices are at C1(0,-6) and C2(0,6). Graph it!

You Try Graph:

Asymptotes: Given a hyperbola centered at the origin, you can

find the asymptotes: Standard equation:

2 2

2 21

y x ay x

a b b

2 2

2 21

x y by x

a b a

Example 2: Find the equations for the asymptotes and

coordinates of the vertices for the following equation. Then sketch the graph.

2 2

116 36

y x

Example 2 Graph:

You Try: Find the equations of the asymptotes and the

coordinates of the vertices for the graph below. Then sketch the graph.

2 2

116 25

x y

You Try Graph:

Homework: Hyperbola WS #1

YOU HAVE TO PRACTICE!!!

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