6.5 Graphs of Polar Equations

I. General Form

A.) Graphs of Polar Functions- An infinite collection of rectangular coordinates (x, y) can be represented by an equation in terms of x and/or y.

Collections of polar coordinates can be represented in a similar fashion, where

sinor

cos

r a b c

r a b c

4

3

2

1

1

2

3

4

4 2 2 4

On your TI-83+, change your MODE to POLAR. Set your window to [0,2π];[-5,5]; [-5,5] and graph

3sinr

Start (0,0)

This direction

B.) Ex. 1- Try a few of these.

1.) 2cosr

Make a table!!!

r r0

6

4

3

2

23

34

56

7

6

54

43

32

53

74

116

23

2

1

0

1

2

3

23

2

10

1

2

3

2 1 1 2

2

1.5

1

0.5

0.5

1

1.5

2

4

3

2

1

1

2

3

4

4 2 2 4

2.) 4sin 2r

4

3

2

1

1

2

3

4

2 2 4

r r0

6

4

3

2

23

34

56

7

6

54

43

32

53

74

116

02 3

4

2 3

0

2 3

4

2 3

02 3

4

2 30

2 3

4

2 3

4

3

2

1

1

2

3

2 2 4

II. Analyzing Polar EquationsA.) Characteristics of a Polar:

(Much the same as the characteristics of a rectangular equation.)

DomainRange ( values)ContinuitySymmetryBoundednessMax -valuesAsymptotesPetals

r

r

B.) Symmetry Tests -

TEST REPLACE WITH

x-axis

y-axis

Origin

, or , r r ,r

,r

,r

, or , r r

, or , r r

C.) Ex. 2- Determine the symmetry for

x-axis:

y-axis:

Origin:

2sin .r

2sin 2sinr

sin 2sin2sin

rr

NO!

YES!

2sin2sin

rr

NO!

D.) Ex. 3 - Analyze

Domain: ,

2sin .r

Range: [-2, 2]Continuity: YesSymmetry: axisy Boundedness: BDD

Max -values: 2rAsymptotes:NonePetals: None

E.) Ex. 4 – Use the graph from example 1 to analyze

Domain: ,

4sin 2 .r

Range: [-4, 4]Continuity: YesSymmetry: , ,originy x

Boundedness: BDDMax -values: 4rAsymptotes:NonePetals: 4

4

3

2

1

1

2

3

2 2 4

F.) Ex. 5 – Use your graphing calculator to analyze the following polar equations:

ROW 1 -

3sin 4r ROW 2 -

5cos 3r ROW 3 -

2sin 3r

ROW 4 -

4cos 4r ROW 5 -

6cos 6r

4

3

2

1

1

2

3

2 2 4

A.) Def. – A ROSE CURVE is any polar equation in

the form of where n is

an integer greater than 1.

III. Rose Curves

sin or cosr a n r a n

If n is odd, there are n petals.

If n is even, there are 2n petals.

B.) For all rose curves . sin & cosr a n r a n

Domain: ,

Range: [- , ]a aContinuity: YesSym: - even : all three

- odd: cos - -axis sin - -axis

nn x

yBound.: BDDMax -values: r aAsym.: NonePetals: odd even 2

n nn n

MORE EXCITEMENT TO COME TOMORROW!!!!!

A.) Any polar equation in the form of

is called

a LIMAÇON (“leemasahn” or “snail”) CURVE.

IV. Limaçon Curves

sin or cosr a b r a b

B.) Ex. 6- Analyze 2 3sinr

Domain: , Range: [-1, 5]Continuity: Yes

Symmetry: axisy Boundedness: BDDMax -values: 5rAsymptotes: None

Petals: None

1.5

1

0.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3 2 1 1 2 3

C.) In general-

Domain: ,

Range: [ , ]Continuity: YesSymmetry: sin : -axis cos : -axisBoundedness: BDDMax -values: Asymptotes: NonePetals: None

a b a b

r a b yr a b x

r a b

sincos

r a br a b

A.) Any polar equation in the form of or is called a LEMNISCATE CURVE.

V. Lemniscate Curves 2 2 sin 2r a

2 2 cos 2r a

B.) Ex. 7- Analyze 2 4sin 2 on 0, 2r

3Domain: 0, ,2 2

Range: [-2, 2]Continuity: YesSym.: origin

Boundedness: BDDMax -values: 2rAsymptotes: NonePetals: 2

1.5

1

0.5

0.5

1

1.5

2 1 1 2

C.) In general-

Too Difficult!!!

2

2

sincos

r a br a b

Each one will have a different domain

A.) The polar equation is called THE SPIRAL OF ARCHIMEDES.

VI. The Spiral of Archimedes

r

on 0,r B.) Ex. 8- Analyze

4

3

2

1

1

2

3

4

5

2 2 4 6

Domain: 0,

Range: 0,Continuity: YesSym.: None

Bound.: BelowMax -values: NonerAsym.: None

Petals: None