9 – 2A
Polar Graphs
⎛ 3π ⎞
Example 1 Graph 2 ,
⎝ 4⎠
1. Place a radius 2 long on
the x axis at 0 radians
2. Rotate the radius
3π
radians.
4
3π
4
DO NOT include the radius
in the graph.
13π 12
π
0
23π 12
7π 6
11π 6
5π 4
3. Place a point at the terminal
end of the radius.
DO NOT include the radius
in the graph.
4. This is the graph of the
⎛ −2π ⎞
point 3,
⎝
3 ⎠
9 – 2A Polar Graphs
7π 4
4π
17π
3
12
19π
12
3π
2
5π
3
⎛ 4π ⎞
Graph 3 ,
⎝
3⎠
1. Place a radius 3 long on
the x axis at 0 radians
.
π
4
π /6
Example 2
−2π
radians
3
−2π
−8π
radians equals
3
12
π
3
π 12
11π 12
2. Rotate the radius
5π
12
5π 6
3. Place a point at the terminal
3π
end of the radius at
4
4. This is the graph of the
⎛ 3π ⎞
point 2,
⎝ 4⎠
2π
3
π
2
7π
12
3π
4
2π
3
7π
12
π
2
5π
12
π
3
5π 6
π
4
π /6
π 12
11π 12
π
0
13π 12
23π 12
7π 6
11π 6
5π 4
7π 4
4π
17π
3
12
Page 1 of 9
3π
2
19π
12
5π
3
©2013 Eitel
1. Place a radius 3 long on
the x axis at π radians
5π
2. Rotate the radius
radians
6
5π
10π
.
radians equals
6
12
3. Place a point at the terminal
11π
end of the radius at
6
DO NOT include the radius
in the graph.
5π ⎞
⎛
Example 3 Graph −3 ,
⎝
6⎠
π
7π
2π 12
2
3π 3
4
5π 6
5π
12
π
3
π
4
π /6
π 12
11π 12
π
0
13π 12
23π 12
7π 6
11π 6
5π 4
4. This is the graph of the
5π ⎞
⎛
point −3,
⎝
6⎠
7π 4
4π
17π
3
12
3π
2
19π
12
5π
3
5π ⎞
⎛
⎛ 11π ⎞
Note: The point −3 ,
is coterminal (lies at the same location) with the point 3 ,
⎝
⎝
6⎠
6 ⎠
13π ⎞
⎛
Example 4 Graph −2 ,
⎝
12 ⎠
1. Place a radius 2 long on
the x axis at π radians
2. Rotate the radius
.
13π
radians
12
3. Place a point at the terminal
π
end of the radius at
12
DO NOT include the radius
in the graph.
4. This is the graph of the
⎛ 13π ⎞
point −2 ,
⎝
12 ⎠
3π
4
2π
3
7π
12
π
2
5π
12
π
3
5π 6
π
4
π /6
π 12
11π 12
π
0
13π 12
23π 12
7π 6
11π 6
5π 4
7π 4
4π
17π
3
12
3π
2
19π
12
5π
3
13π ⎞
⎛
⎛ π⎞
Note: The point −2 ,
is coterminal (lies at the same location) with the point 2 ,
⎝
⎠
⎝ 12 ⎠
12
9 – 2A Polar Graphs
Page 2 of 9
©2013 Eitel
⎛ −5π ⎞
Example 5 Graph 2 ,
⎝ 12 ⎠
1. Place a radius 2 long on
the x axis at 0 radians
−5π
2. Rotate the radius
radians
12
−5π
19π
.
radians equals
12
12
3. Place a point at the terminal
19π
end of the radius at
12
DO NOT include the radius
in the graph.
3π
4
2π
3
7π
12
π
2
5π
12
π
3
π
4
5π 6
π /6
π 12
11π 12
π
0
13π 12
23π 12
7π 6
11π 6
5π 4
4. This is the graph of the
⎛ −5π ⎞
point 2,
⎝ 12 ⎠
7π 4
4π
17π
3
12
3π
2
19π
12
5π
3
⎛ −5π ⎞
⎛ 19π ⎞
Note: The point 2 ,
is coterminal (lies at the same location) with the point 2 ,
⎝ 12 ⎠
⎝ 12 ⎠
−5π ⎞
⎛
Example 6 Graph −3 ,
⎝
6 ⎠
1. Place a radius 3 long on
the x axis at −π radians
−5π
2. Rotate the radius
radians
6
−5π
−10π
.
radians equals
6
12
3. Place a point at the terminal
π
end of the radius at
6
DO NOT include the radius
in the graph.
4. This is the graph of the
⎛ −5π ⎞
point −3,
⎝
6 ⎠
3π
4
2π
3
7π
12
π
2
5π
12
π
3
5π 6
π
4
π /6
π 12
11π 12
π
0
13π 12
23π 12
7π 6
11π 6
5π 4
7π 4
4π
17π
3
12
3π
2
19π
12
5π
3
−5π ⎞
⎛
⎛ π⎞
Note: The point −3 ,
is coterminal (lies at the same location) with the point 3 ,
⎝
⎠
⎝ 6⎠
6
9 – 2A Polar Graphs
Page 3 of 9
©2013 Eitel
The point ( r , θ ) is coterminal with the points (r , θ + 2π k ) or (−r , θ + π + 2π k ) where k ∈ Z
3π
4
5π
E
6
7π
12
2π
3
π
2
5π
12
π
3
C
D
π
4
π
6
11π
12
π
12
B
π
F
13π
12
7π
6
23π
12
L
J
G
K
5π
4
0
A
H
4π
3
I
17π
12
3π
2
19π
12
5π
3
7π
4
11π
6
FInd the polar coordinates for each point shown on the graph. Use both + r and – r.
1. A (4 , 0 ) or (−4 , π )
2. B
7π ⎞
⎛ π⎞
⎛
3,
or −3 ,
⎝ 6⎠
⎝
6 ⎠
3. C
7π ⎞
⎛ π⎞
⎛
4,
or −4 ,
⎝
⎝
3⎠
3 ⎠
4. D
19π ⎞
⎛ 7π ⎞
⎛
3,
or −3 ,
⎝ 12 ⎠
⎝
12 ⎠
5. E
7π ⎞
⎛ 3π ⎞
⎛
4,
or −4 ,
⎝
⎝
4 ⎠
4 ⎠
6. F (2 , π ) or (−2 , 0 )
7. G
13π ⎞
⎛ 7π ⎞
⎛
4,
or −4 ,
⎝
⎝
6 ⎠
6 ⎠
8. H
10π ⎞
⎛ 4π ⎞
⎛
3,
or −3 ,
⎝
⎝
3 ⎠
3 ⎠
9. I
2π ⎞
⎛ 5π ⎞
⎛
10. J 1 ,
or −1 ,
⎝
⎠
⎝
3
3 ⎠
9 – 2A Polar Graphs
11. K
3π ⎞
⎛ 7π ⎞
⎛
3,
or −3 ,
⎝
⎠
⎝
4
4 ⎠
Page 4 of 9
π⎞
⎛ 3π ⎞
⎛
4,
or −4 ,
⎝
⎝
2 ⎠
2⎠
11π ⎞
⎛ 23π ⎞
⎛
12. L 2 ,
or −2 ,
⎝
⎠
⎝
12
12 ⎠
©2013 Eitel
Example 1 Graph r = 3 cos( θ )
Use a calculator. Round off values to 1 decimal place.
θ
0
π 12
π 6
π 4
π 3
5π 12
π 2
R
3
2.9
2.6
2.1
1.5
.8
0
θ
π
R
–3
13π 12 7π 6
–2.9
–2.6
5π
6
7π 12 2π 3
– .8
–1.5
3π 4
5π 6 11π 12
–2.1
–2.6
5π 4
4π 3 17π 12 3π 2 19π 12 5π 3
7π 4 11π 6 23π 12
–2.1
–1.5
2.1
3π
4
2π
3
–.8
0
r = 3 cos( θ )
π
5π
7π
2
12
12
.8
π
3
1.5
2.6
2.9
π
4
π
6
11π
12
π
12
π
0
13π
12
7π
6
9 – 2A Polar Graphs
–2.9
23π
12
5π
4
4π
3
17π
12
3π
2
19π
12
Page 5 of 9
5π
3
7π
4
11π
6
©2013 Eitel
Example 2 Graph r = 2 − 2sin( θ )
Use a calculator. Round off values to 1 decimal place.
θ
0
π 12
π 6
π 4
π 3
5π 12
π 2
R
2
1.5
1.0
.59
.27
.07
0
θ
π
R
2
13π 12 7π 6
2.5
3
5π
6
5π 4
.07
3π 4
.27
.59
1.0
1.5
7π 4 11π 6 23π 12
3.7
3.4
2π
3
3.9
4
.3.9
r = 2 − 2sin( θ )
π
5π
7π
2
12
12
π
3
3.7
3
2.5
π
4
π
6
11π
12
π
12
π
0
13π
12
7π
6
9 – 2A Polar Graphs
5π 6 11π 12
4π 3 17π 12 3π 2 19π 12 5π 3
3.4
3π
4
7π 12 2π 3
23π
12
5π
4
4π
3
17π
12
3π
2
19π
12
Page 6 of 9
5π
3
7π
4
11π
6
©2013 Eitel
Example 3 Graph r2 = 16cos ( 2θ )
Use a calculator. Round off values to 1 decimal place.
r2 = 16cos ( 2θ ) is equal to r = ± 4 cos ( 2θ ) Put 4 cos ( 2θ ) in the calculator and add the ± sign to
the answer. cos ( 2θ ) is negative at π 3 so 4 cos ( 2θ ) is undefined. I listed the answer as UND.
θ
0
π 12
π 6
R
4
± 3.7 ± 2.8
θ
π
13π 12 7π 6
R
4
± 3.7 ± 2.8
π 4
π 3
5π 12
π 2
7π 12 2π 3
0
UND UND UND UND UND
5π 4
4π 3 17π 12 3π 2 19π 12 5π 3
0
UND UND
0
3π 4
5π 6 11π 12
0
± 2.8 ± 3.7
7π 4 11π 6 23π 12
UND UND
0
± 2.8 ± 3.7
(± 3.7,π /12) means graph 2 points on at (+ 3.7,π /12) which will be in the first quadrant and one at
(− 3.7,π /12 ) which will be in the third quadrant
5π
6
3π
4
2π
3
r2 = 16 cos ( 2θ )
π
5π
7π
2
12
12
π
3
π
4
π
6
11π
12
π
12
π
0
13π
12
7π
6
9 – 2A Polar Graphs
23π
12
5π
4
4π
3
17π
12
3π
2
19π
12
Page 7 of 9
5π
3
7π
4
11π
6
©2013 Eitel
Circle
r = 2cos(θ )
DImpled Lemacon
r = 3 + 2sin(θ)
Cardioid
r = 2 + 2cos(θ)
Lemacon with inner loop
r = 2 + 3cos(θ)
3 petal Rose
r = 2sin(3θ )
8 petal Rose
r = 2cos(4θ )
9 – 2A Polar Graphs
Page 8 of 9
©2013 Eitel
Lemniscate
r2 = 4 sin(2θ)
9 – 2A Polar Graphs
Page 9 of 9
©2013 Eitel