Polar Coordinates and Graphing
P   r , 
O
 = directed angle
Polar Axis
Counterclockwise from polar axis to OP .
Plotting Points in the Polar
Coordinate System
 
r
,


   2,  lies
 3
The point
two units from the pole on the
terminal side of the angle    .
3



The point  r ,    3,  6  lies three units
from the pole on the terminal side of
the angle     .
6

The point  r ,    3,
 

the point  3,  6  .


11 
6 
coincides with
Multiple Representations of Points
In the polar coordinate system, each point does not have a unique
representation. In addition to  2 , we can use negative values
for r . Because r is a directed distance, the coordinates  r , 
and  r ,    represent the same point.
In general, the point r,  can be represented as
r,    r ,  2n 
where n is any integer.
or r,    r ,   2n  1  
Try plotting the following points in polar coordinates and find
three additional polar representations of the point:
 2 
1.  4,

3


5 

2.  5, 

3


7 

3.  3, 

6


 7 
4.   ,  
 8 6
Graphing a Polar Equation
r  4sin

0

6
r
0
2 2 3 4 2 3 2

3

2
2
3
5

6
7
6
3 11
2 6
2
0 2 4 2 0
circle
Using Symmetry to Sketch a
Polar Graph
Symmetry with respect to

the line   .
2
 r ,   
Symmetry with respect to
the Polar Axis.
 r , 
 

Symmetry with respect to
the Pole
 r , 
 r , 




 r ,  
 r , 
Tests for Symmetry
The graph of a polar equation is symmetric with respect to the
following if the given substitution yields an equivalent equation.
1. The line  

2
:
Replace  r ,  with  r , -   .
2. The polar axis:
Replace  r ,  with  r ,   .
3. The pole:
Replace  r ,  with  r ,  .
Using Symmetry to Sketch
Graph: r  3  2cos .
Replacing  r ,  by  r ,   produces
r  3  2cos   
 3  2cos  .
Thus, the graph is symmetric with respect to the polar axis, and you need
only plot points from 0 to .

0

6

3

2
2
3
5
6

Limaon
r
5
3 3
4
3
2
3 3
1
Symmetry Test Fails
Unfortunately the tests for symmetry can guarantee symmetry, but there are
polar curves that fail the test, yet still display symmetry. Let’s look at the graph
of r    2 . Try the symmetry tests. What happens?
Original
Equation
r    2
Replacement
r,  with  r ,   
New
Equation
r    3
 Not symmetric

about the line  = .
2
r    2
 r ,  with r ,  
r    2
 Not symmetric
about the polar axis.
r    2
 r ,  with  r , 
 r    2
 Not symmetric
about the pole.
All of the tests indicate that no symmetry exists. Now, let’s look at the graph.
You can see that the graph is symmetric with respect to

the line   .
2
r    2
Spiral of Archimedes
Quick Test for Symmetry
The graph of r  a  sin  is symmetric with respect to the line  

2
.
The graph of r  a  cos  is symmetric with respect to the polar axis.
Determine the effect of “a” on the graph of r  a sin .
Click on the graph below to open an interactive page.
Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Determine the effect of “a” on the graph of r  a cos .
Click on the graph below to open an interactive page.
Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Determine the effect of “a” and “b” on the graph of r  a  b sin .
Click on the graph below to open an interactive page.
Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Determine the effect of “a” on the graph of r  a sin b .
Click on the graph below to open an interactive page.
Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Determine the effect of “a” and “b’ on the graph of r  a cos b .
Click on the graph below to open an interactive page.
Please note the interactive page may open with the graph to the far left.
You may need to drag the graph slightly to the right.
Summary of Special Polar Graphs
Limacons:
a
1
b
Limacon with
inner loop
r  a  b cos 
r  a  b sin
 a  0, b  0 
a
1
b
Cardiod
(heart-shaped)
1
a
2
b
a
2
b
Dimpled
Limacon
Convex
Limacon
Rose Curves: b petals if b is odd
2b petals if b is even
b  2
r  a cos b
r  a cos b
r  a sin b
r  a sin b
Circles and Lemniscates
Circles
r  a cos 
Lemniscates
r  a sin
r 2  a2 sin2
r 2  a2 cos 2
Analyzing Polar Graphs
Analyze the basic features of r  3cos2 .
Type of Curve:
Rose Curve with 2b petals = 4 petals
Symmetry:
Polar axis, pole, and 
Maximum Value of | r |:
| r | = 3 when   0,  , , 3
2
Zeros of r:
r = 0 when  
You can use this same process to
Analyze any polar graph.
 3
, .
4 4

2

2
.
Analyze and Sketch the graph:
r  2  cos
r  1  sin
Type of curve:
Convex Limacon
Cardioid Limacon
Symmetry:
sym. @ polar axis
Maximum | r |
3,0 
sym. @ line  =
2
 3 
 2, 2 


Zeros of r
none

 
 0, 2 

