Loop – to – Loop
(No calculator!)
For each polar curve a) name the curve, b) determine the axis of symmetry, c) determine the
quadrangle values, and d) graph the curve. If you use a scale other than 1, be sure to indicate it on
the polar grid
1. r  2  2cos
_____________________ Name of Curve
r

r

r

_____________________ Axis of Symmetry
2. r  2  3cos
_____________________ Name of Curve
_____________________ Axis of Symmetry
3. r  2  2sin
_____________________ Name of Curve
_____________________ Axis of Symmetry
4. r  3  5cos
_____________________ Name of Curve
r

r

r

r

_____________________ Axis of Symmetry
5. r  5  2sin
_____________________ Name of Curve
_____________________ Axis of Symmetry
6. r  4  4cos
_____________________ Name of Curve
_____________________ Axis of Symmetry
7. r  2  3cos
_____________________ Name of Curve
_____________________ Axis of Symmetry
8. r  1  4sin
_____________________ Name of Curve
r

r

𝑟

_____________________ Axis of Symmetry
9. r  4  1sin
_____________________ Name of Curve
_____________________ Axis of Symmetry
10. r  3  3cos
_____________________ Name of Curve
_____________________ Axis of Symmetry
For each polar curve a) name the curve, b) determine the requested data, c) determine the
quadrangle values, and d) graph the curve. If you use a scale other than 1, be sure to indicate it on
the polar grid
11. r  2 sin 4
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ # of Petals
_____________________ Length of Petals
_____________________ Spacing between Petals
12. r  7 cos5
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ # of Petals
_____________________ Length of Petals
_____________________ Spacing between Petals
13. r  3sin
_____________________ Name of Curve
r

𝑟

_____________________ Axis of Symmetry
_____________________ Radius
_____________________ Center
14. r  6
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ Radius
_____________________ Center
15. r  4 cos 3
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ # of Petals
_____________________ Length of Petals
_____________________ Spacing between Petals
16. 𝑟 2 = 25 cos 2𝜃
_____________________ Name of Curve
r

r

𝑟

_____________________ Axis of Symmetry
_____________________ Length of Lobe
17. r  4sin
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ Radius
_____________________ Center
18. r  3sin 2
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ # of Petals
_____________________ Length of Petals
_____________________ Spacing between Petals
19. r  2cos
_____________________ Name of Curve
_____________________ Axis of Symmetry
_____________________ Radius
_____________________ Center
20. Why is it impossible to graph a six-petal rose using these equations?
Loop – to – Loop
(No calculator!)
For each polar curve a) name the curve, b) determine the axis of symmetry, c) determine the
quadrangle values, and d) graph the curve. If you use a scale other than 1, be sure to indicate it on
the polar grid
1. r  2  2cos
_____________________
Name of Curve
Cardioid
𝑟

4
0–axis
_____________________
Axis of Symmetry
2
0
2
2. r  2  3cos
Limaçon (loop)
_____________________
Name of Curve
0–axis
_____________________
Axis of Symmetry
𝑟

5
2
–1
0
3. r  2  2sin
Cardioid
_____________________
Name of Curve
𝜋
–axis
_____________________
Axis of Symmetry
2
r
2
4
2
0

4. r  3  5cos
Limaçon (loop)
_____________________
Name of Curve
0–axis
_____________________
Axis of Symmetry
r

5
3
–2
3
r-scale = 2
5. r  5  2sin
Limaçon (no loop)
_____________________
Name of Curve
3𝜋
–axis
_____________________
Axis of Symmetry
2
r

5
3
5
7
r-scale = 2
6. r  4  4cos
Cardioid
_____________________
Name of Curve
𝜋 –axis
_____________________
Axis of Symmetry
r

0
4
8
4
r-scale = 2
7. r  2  3cos
Limaçon (loop)
_____________________
Name of Curve
0–axis
_____________________
Axis of Symmetry
r
5
2
0
2

8. r  1  4sin
Limaçon (loop)
_____________________
Name of Curve
𝜋
–axis
_____________________
Axis of Symmetry
2
r

1
5
1
–3
9. r  4  1sin
Limaçon (no loop)
_____________________
Name of Curve
𝜋
–axis
_____________________
Axis of Symmetry
2
r

4
5
4
3
10. r  3  3cos
Cardioid
_____________________
Name of Curve
r

𝜋 –axis
_____________________
Axis of Symmetry
r-scale = 2
For each polar curve a) name the curve, b) determine the requested data, c) determine the
quadrangle values, and d) graph the curve. If you use a scale other than 1, be sure to indicate it on
the polar grid
11. r  2 sin 4
Rose
_____________________
Name of Curve
Where to start first
petal?
longest
radius = 2
Every
22.5
_____________________ Axis of Symmetry
so
8
_____________________
# of Petals
2 = 2 sin 4𝜃
1 = 1 sin 4𝜃
2
_____________________
Length of Petals
sine = 1 at 90
4𝜃 = 90°
45
_____________________
Spacing between Petals 𝜃 = 22.5°
12. r  7 cos5
Rose
_____________________
Name of Curve
Where to start first
petal?
longest radius = 7
Every 36
_____________________
Axis of Symmetry
so
7 = 7 cos 5𝜃
5
_____________________
# of Petals
1 = 1 sin 5𝜃
cosine = 1 at 0
7
_____________________
Length of Petals
5𝜃 = 0°
72
𝜃 = 0°
_____________________ Spacing between Petals
r-scale = 2
13. r  3sin
Circle
_____________________
Name of Curve
90
_____________________
Axis of Symmetry
1.5
_____________________
Radius
(1.5,90°)
_____________________
Center
r

0
3
3
3
14. r  6
Polar Circle
_____________________
Name of Curve
––
_____________________
Axis of Symmetry
6
_____________________
Radius
Pole
_____________________
Center
r

6
6
6
6
r-scale = 2
15. r  4 cos 3
Rose
_____________________
Name of Curve
Where to start first
petal?
longest radius = 4
Every 60
_____________________
Axis of Symmetry
so
4
=
4
cos 3𝜃
3
_____________________
# of Petals
1 = 1 sin 3𝜃
cosine
= 1 at 0
7
_____________________
Length of Petals
3𝜃 = 0°
120
𝜃 = 0°
_____________________ Spacing between Petals
16. 𝑟 2 = 25 cos 2𝜃
_____________________
Name of Curve
Lemniscate
0 –axis
_____________________
Axis of Symmetry
5
_____________________
Length of Lobe
r

5

5

17. r  4sin
Circle
_____________________
Name of Curve
270
_____________________
Axis of Symmetry
2
_____________________
Radius
(2,270°)
_____________________
Center
r

0
–4
0
4
18. r  3sin 2
Rose
_____________________
Name of Curve
Where to start first
petal?
longest radius = 3
Every 45
_____________________
Axis of Symmetry
so
3
=
3
sin 2𝜃
4
_____________________
# of Petals
1 = 1 sin 2𝜃
sine = 1 at 90
3
_____________________
Length of Petals
2𝜃 = 90°
90
_____________________ Spacing between Petals 𝜃 = 45°
19. r  2cos
Circle
_____________________
Name of Curve
180
_____________________
Axis of Symmetry
1
_____________________
Radius
(1,180°)
_____________________
Center
r

–2
0
2
0
20. Why is it impossible to graph a six-petal rose using these equations? Odd coefficients of theta
graph the exact number of petals but even coefficients of theta graph double the number of petals.
Since 6 is even you would get 12 petals not 6.