10.2 Graphing Polar EquationsDay 2

Yesterday, we graphed polar equations using “brute force” – making tables of values. But this is very inefficient! We can bypass having to make all these separate calculations by learning some rules.  

Symmetry – Tests for Symmetry on Polar Graphs

If the following substitution is made and the equation is equivalent to the original equation, then the graph has the indicated symmetry.    

WRT the Pole (Origin)

Replace r with –r

WRT the Polar Axis (x-axis) 

Replace θ with –θ

WRT the line (y-axis) 

Replace θ by θ – πor (r, θ) with (–r, –θ)

2

EX 1: Identify the kind(s) of symmetry each polar graph possesses.

A)

Pole Polar Axis       B)

Pole Polar Axis

  

5 10cosr

2

2 36cos 2r

2

5 10cos 5 10cos( ) 5 10cos( )

5 10cos 5 10cos

rr r

r r

2 2 2

2 2 2

( ) 36cos 2 36cos 2( ) ( ) 36cos 2( )

36cos 2 36cos 2 36cos 2

r rr

r r r

 SPIRAL Also called the “Spiral of Archimedes” No special rules!  Typical Graph:  

 

r k

CIRCLES There are three forms for a circle.

Center of circle at _______

Radius = _______

 Typical Graph:  

Contains the _______

Tangent to _______

Center on _______

Diameter = _______

Radius = _______

If a > 0, circle is ____ of pole

If a < 0, circle is ____ of pole

 Typical Graph:

Contains the _______

Tangent to _______

Center on _______

Diameter = _______

Radius = _______

If a > 0, circle is ____ of pole

If a < 0, circle is ____ of pole

 Typical Graph:

r k cosr a sinr a pole

kpole

2

polar axisa

EW

polepolar axis

2 a

NS

2a

2a

EX 2:   Radius:______  Center On: Polar Axis /  Circle is N S E W of the pole   

 

6cosr

2

3

LIMAÇONS French for “snail.”

OR (oriented on polar axis) (oriented on )

cos sinr a b r a b 2

Limaçon with Inner Loop

 When or a < b

Diameter = _______

Inner Loop = _______

Cardioid (heart-shaped)

 When or a = b

Diameter

= _______ = ______

Dimpled Limaçon 

When

Larger = _______

Smaller = _______

Convex Limaçon 

When or a ≥ 2b

Larger = _______

Smaller = _______

For all the “bumps,” they hit the polar axis or (whichever is the opposite of where it is oriented) at _______

Typical Graph: 

Typical Graph: Typical Graph: Typical Graph: 

1ab 1a

b 1 2ab 2a

b

2

b a

b aa b 2 a

a b

a b

a b

a b

±a

smaller

larger2a

a a

smaller largerinnerloop

diam

a

a

          

EX 3:  

Type:__________________ On: Polar Axis /  Lengths:_________________ _________________________

EX 4:  

Type:__________________ On: Polar Axis /  Lengths:_________________ _________________________

EX 5:  

Type:__________________ On: Polar Axis /  Lengths:_________________ _________________________

2 4cosr 2 2sinr 4 2cosr

Limaçon with Inner Loop

2 2

2

Diam: 6

Inner Loop: 2

Cardioid

Diam: 4

Convex Limaçon

Larger: 6

Smaller: 2

Oriented Oriented Oriented

ROSES These look like flowers…we call each loop a “petal.”

Length of each petal = ______If b is even, there are _______ petals.If b is odd, there are _______ petals.

(*Since the values from 0 to 2π give us the points, having b be an odd number, the values actually repeat themselves and overlap the already existing values so we do not get double the

number of petals like we do with b being even.)

First peak is at _______

Peaks are ______ radians apart(n is number of petals)

Typical Graphs:

First peak is at _______

Peaks are ______ radians apart(n is number of petals)

cosr a b sinr a b

θ = 02n

2b

2n

2bb

a

  

              

EX 6:

Length of Petals:_______ Number of Petals:_______ First Peak at:_______

Each Petal _______ rad apart 

EX 7:  Length of Petals:_______ Number of Petals:_______ First Peak at:_______

Each Petal _______ rad apart

5cos 2r 4sin 3r

54

θ = 0

2

43

23

6

LEMNISCATE These look like “figure eights.”

Oriented on _______________ 

Oriented on _______________

Maximum distance out is ________

Typical Graph: Typical Graph:

2 2 cos 2r a 2 2 sin 2r a

polar axis 4

2a a

EX 8:  Oriented On: Polar Axis /  Maximum Distance:_______   

EX 9:  Oriented On: Polar Axis /  Maximum Distance:_______

2 4cos 2r 2 4sin 2r

4 4

2 2

Ex 10: Transform the rectangular equation into a polar equation and graph.

2 2

2 2

2

6 0

6

6 sin

6sin

x y y

x y y

r r

r

CircleRadius 3Center onN of pole

2

Ex 11: Determine an equation of the polar graph.

A) B)

Equation:____________________Why?_______________________________________________________________________________

Equation:____________________Why?_______________________________________________________________________________

r = 3cos 2θpetal graph w/ 4 petalspeak on polar axispetal length is 3

Dimpled Limiçon 2 on

larger = 8 smaller = 2bump hits at 5 a = 5, b = 3

r = 5 + 3sin θ

2θ cos

a = 3

Homework

#1003 Pg 501 # 1–17 odd, 21, 23, 24, 27, 29, 31, 37, 41, 43, 44–47, 49–53 odd