Polar Graphs
Calc AB- Section10.5
To define polar coordinates we first fix an origin called the ______
POLE
and an ____________
INITIAL RAY from O.
Then each point P can be located by assigning to it a
r ,   where r
POLAR coordinate pair ______
________
DISTANCE from ____
O to ____
P and  gives
gives the directed ___________
ANGLE from the initial ray to ray OP.
the directed ________
UNIQUE
Polar coordinates are NOT __________.
P
r ,  
r
POLE O

INITIAL RAY
x
Example 1: Graph P  2,  6. Find another to describe the point
P using a negative angle.
P

O
11

6
6
Example 2: Find all polar coordinates of the point P  2,  6
 

 2,  2k 
 6

7


 2 k 
 2,
6


x
Try these! Graph the points below
7 

 


2
,

3
,

2
.
5
,




A. 
B. 
C. 
6 
2
 3


Graphs of the Form
Equation:
r a
Equation:    0
r  a and   0
Graph: A ________
_______, a ,
CIRCLE with RADIUS
and centered at __
O
LINE through O making an
Graph: A ______
ANGLE    0 with the initial ray
______________
Example 3: Graph the sets of points whose polar
coordinates satisfy the following conditions:
a) 1  r  2 and 0   

2
b) 3  r  2 and  
2
5
 
d)
3
6
(no restrictions on r)

4
c) r  0 and  

4
Relating Polar and Rectangular Coordinates
Ray
x = _________
r cos 
r

cos   x
r
sin   y
r
x
y
x
Initial ray
r sin 
y = _________
2
2
2
x

y

r
__________
 y
  tan  
x
*
1
*Adjust your answer
if angle is in Q2 or 3
Example 4: Find the polar equation for the circle x   y  3  9
2
x2  y 2  6 y  9  9
r  6r sin   0
2
r  r  6sin    0
r 0
r  6sin   0
r  6sin 
2
Example 5: Replace the following polar equations by equivalent
Cartesian equations and identify their graphs.
r cos  4
a)
c)
x  4
VERTICAL LINE at x   4
r 2  4r cos 
b)
x2  y2  4x
x  4x  y  0
2
x
2
2
 4 x  4 y 2  0  4
x  2 
2
 y2  4
CIRCLE CENTERED at 2,0  RADIUS 2
4
r
2 cos   sin 
r  2 cos   sin    4
2 r cos   r sin   4
2x  y  4
y  2x  4
LINE with
SLOPE 2
Y-INTERCEPT -4