MATH 4H
POLAR COORDINATES AND
GRAPHING HOMEWORK
NAME__________________________
DATE___________________________
Homework assignments are from the Merrill textbook.
HW # 62:
Ditto 9-1 – ALL (Packet p. 5)
HW # 63:
Ditto 9-3 – ALL (Packet p. 6)
HW # 64:
Ditto 9-2 – #1, 3, 4 (Packet p. 7)
HW # 65:
Ditto 9-2 – #5, 6 (Packet p. 7)
HW # 66:
Ditto 9-3 Enrichment Worksheet – ALL (Packet pp. 11-12)
HW # 67:
Ditto - Polar Computer WS – ALL (Packet p. 13)
HW # 68:
Ditto - Chapter 12 Test (Packet p. 14)
HW # 69:
p. 417 - # 2, 4, 7, 8, 10, 11, 14, 15, 20, 22, 26 - 30
HW # 70:
WS – Algebra 15-7
Study for Test!!!
p.1
1
2
3
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9-1: Practice Worksheet
Polar Coordinates
Graph and label the points that have the given polar coordinates.
1. a) (2.5, 0º)
b) (3, -135º)
c) (-1, -30º)
90º
120º
60º
150º
30º
0º
180º
210º
330º
240º
300º
270º
2.
5
Name four different pairs of polar coordinates that represent point A.
7.
8.
A
A
Graph each polar equation.
9. r = 3
10.
180
6
9-3: Practice Worksheet
Polar and Rectangular Coordinates
Find the polar coordinates of each point with the given rectangular coordinates.
1
√3
�
2
1. (0,4)
2. � , −
4. (4,0)
5. �−1, −√3�
2
3. �−
√3 1
, �
2 2
6. (2,2)
Find the rectangular coordinates of each point with the given polar coordinates.
7. (6, 120°)
𝜋𝜋
10. �4, �
6
8. (−4, 45°)
11. �0,
13𝜋𝜋
3
9. (3, 300°)
�
12. �3, −
Write each rectangular equation in polar form.
13. 𝑥𝑥 2 + 𝑦𝑦 2 = 9
14. 𝑦𝑦 = 3
15. 𝑥𝑥 2 − 𝑦𝑦 2 = 1
16. 𝑥𝑥 2 + 𝑦𝑦 2 − 2𝑦𝑦 = 0
Write each polar equation in rectangular form.
𝜋𝜋
17. 𝑟𝑟 = 4
18. 𝜃𝜃 = −
19. 𝑟𝑟 cos 𝜃𝜃 = 5
20. 𝑟𝑟 = −3 sec 𝜃𝜃
7
3
3𝜋𝜋
4
�
9-2: Practice Worksheet
Graphs of Polar Equations
Graph each polar equation. Identify the classical curve it represents.
1. 𝑟𝑟 = 2 + 2𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
2. 𝑟𝑟 = 3𝑠𝑠𝑠𝑠𝑠𝑠3𝜃𝜃
3. 𝑟𝑟 = 2 + 3𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
4. 𝑟𝑟 = 2 + 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
5. 𝑟𝑟 = 𝜃𝜃
6. 𝑟𝑟 2 = 5𝑐𝑐𝑐𝑐𝑐𝑐2𝜃𝜃
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Lab Worksheet
Objective: To graph polar equations, identify symmetries of graphs, and note the
relationships between graphs and coefficients.
I. Graph each of the following functions and sketch, on the axes below, the
graphs shown on the screen. Note: After sketching the graph on paper, erase
the graph from the screen.
𝑟 = 𝑠𝑖𝑛𝜃
𝑟 = 𝑠𝑖𝑛2𝜃
𝑟 = 𝑠𝑖𝑛3𝜃
𝑟 = 𝑠𝑖𝑛4𝜃
𝑟 = 𝑠𝑖𝑛5𝜃
𝑟 = 𝑠𝑖𝑛6𝜃
𝑟 = 𝑐𝑜𝑠𝜃
𝑟 = 𝑐𝑜𝑠2𝜃
𝑟 = 𝑐𝑜𝑠3𝜃
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𝑟 = 𝑐𝑜𝑠4𝜃
𝑟 = 𝑐𝑜𝑠5𝜃
𝑟 = 𝑐𝑜𝑠6𝜃
II. From your observations of the preceding graphs, complete the following table:
Function
Number of Petals Polar Axis Vertical Axis Symmetry
In Flower
Symmetry? Symmetry?
To Pole?
𝑠𝑖𝑛2𝜃
𝑠𝑖𝑛3𝜃
𝑠𝑖𝑛4𝜃
𝑠𝑖𝑛5𝜃
𝑠𝑖𝑛6𝜃
𝑐𝑜𝑠2𝜃
𝑐𝑜𝑠3𝜃
𝑐𝑜𝑠4𝜃
𝑐𝑜𝑠5𝜃
𝑐𝑜𝑠6𝜃
III. Summary:
Given 𝑟 = sin 𝑛𝜃 or 𝑟 = cos 𝑛𝜃,
(a) How many petals are on the graph when 𝑛 is even? _____________
(b) How many petals are on the graph when 𝑛 is odd? _____________
(c) Which functions are symmetric with respect to the polar axis? _______
(d) Which functions are symmetric with respect to the vertical axis? _____
(e) ) Which functions are symmetric with respect to the pole? __________
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Complete Analysis
Ex.: 𝑟 = 2(1 + 𝑐𝑜𝑠𝜃)
(a) Identify:
(c) Intercepts:
(b) Symmetry:
(d) Extent:
(e) Tangents at pole:
(f) Table.
(g) Graph.
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9-3: Enrichment Worksheet
Polar Roses
The polar equation 𝒓𝒓 = 𝒂𝒂 𝒔𝒔𝒔𝒔𝒔𝒔 𝒏𝒏𝜽𝜽 graphs as a rose.
When 𝒏𝒏 = 𝟏𝟏, the rose is a circle – a flower with one leaf.
Sketch the graphs of these roses.
1. 𝑟𝑟 = 2 sin 2𝜃𝜃
2. 𝑟𝑟 = −2𝑠𝑠𝑠𝑠𝑠𝑠3𝜃𝜃
3. 𝑟𝑟 = −2 sin 4𝜃𝜃
4. 𝑟𝑟 = 2 sin 5𝜃𝜃
{OVER}
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5. The graph of the equation 𝑟𝑟 = 𝑎𝑎 sin 𝑛𝑛𝜃𝜃 is a rose. Use your results from Exercises 1-4 to
complete the conjectures.
a. The length of a petal is ________ units.
b. If 𝑛𝑛 is an odd integer, the number of petals is __________.
c. If 𝑛𝑛 is an even integer, the number of petals is __________.
6. Write 𝑟𝑟 = 2 sin 2𝜃𝜃 in rectangular form.
7. The total area, 𝐴𝐴, of the three petals in the three petal rose 𝑟𝑟 = 𝑎𝑎 sin 3𝜃𝜃 is given by
1
1
𝐴𝐴 = 𝑎𝑎2 𝜋𝜋. For a four petal rose, the area is 𝐴𝐴 = 𝑎𝑎2 𝜋𝜋.
4
a. Find the area of a four petal rose with 𝑎𝑎 = 6.
2
b. Write the equation of a three petal rose with area 36𝜋𝜋.
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Chapter 9 - Computer
Graphs of Polar Equations
You can graph polar equations on your TI Calculator.
Example 1: Graph 𝑟𝑟 = 2 cos 3𝜃𝜃
• Add a graph page.
• Press Menu – Graph Entry/Edit – Polar
• Enter 2 cos 3𝜃𝜃 next to 𝑟𝑟1(𝑥𝑥) =.
• Be sure to check your domain.
The default is 0 ≤ 𝜃𝜃 ≤ 6.28.
• Zoom Square to get a “pretty” picture.
Example 2: Graph 𝑟𝑟 2 = 4 cos 2𝜃𝜃
• Add a graph page.
• Press Menu – Graph Entry/Edit – Polar
• Enter √4 cos 2𝜃𝜃 next to 𝑟𝑟1(𝑥𝑥) =.
• Enter −√4 cos 2𝜃𝜃 next to 𝑟𝑟2(𝑥𝑥) =.
• Zoom Square to get a “pretty” picture.
Use your graphing calculator to graph each polar equation. Adjust the window to see a
complete curve. Identify the classical curve it represents.
1. 𝑟𝑟 = 2 + 2 sin 𝜃𝜃
2. 𝑟𝑟 = 1 + 2 cos 𝜃𝜃
3. 𝑟𝑟 = 6 cos 6𝜃𝜃
4. 𝑟𝑟 2 = 8 sin 2𝜃𝜃
5. 𝑟𝑟 = 3 + 3 cos 𝜃𝜃
6. 𝑟𝑟 = 2𝜃𝜃
7. 𝑟𝑟 = 0.5𝜃𝜃
8. 𝑟𝑟 = 2 + 4 cos 𝜃𝜃
9. 𝑟𝑟 2 = 12 cos 2𝜃𝜃
10. 𝑟𝑟 = 3𝜃𝜃
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Chapter 12 Test
1. Find another pair of polar coordinates that represents �2,
−2𝜋𝜋 ≤ 𝜃𝜃 ≤ 0.
4𝜋𝜋
3
�, where 𝑟𝑟 > 0 and
2. Graph 𝑚𝑚∠𝜃𝜃 = 45°.
3. Find polar coordinates for the point named by rectangular coordinates �−√2, √2�.
4. Find rectangular coordinates for the point named by the polar coordinates �−3,
5. Find the distance between (3, 30°) and (4, −60°).
2𝜋𝜋
3
�.
6. Change 𝑟𝑟 = 3 cos 𝜃𝜃 from polar to rectangular form. Identify the curve.
7. Change 𝑥𝑥 2 = 4 − 𝑦𝑦 from rectangular to polar form.
𝜋𝜋
8. Find the polar equation of a circle with center �2, � and radius 4.
4
9. Sketch the graph of 𝑟𝑟 = 2 + 2 sin 𝜃𝜃.
10. Sketch the graph of 𝑟𝑟 2 = 9 cos 2𝜃𝜃.
𝜋𝜋
11. Find the intercepts of the graph of 𝑟𝑟 = 4 cos �𝜃𝜃 − �.
4
12. Find the symmetry, intercepts, extent, tangents at the pole, and number of leaves for the
graph of 𝑟𝑟 = 2 sin 4𝜃𝜃.
13. For the graph of the Limacon 𝑟𝑟 = 𝑎𝑎 + 𝑏𝑏 cos 𝜃𝜃 to have a loop, what must be true about 𝑎𝑎
and 𝑏𝑏.
14. Sketch the graph of 𝑟𝑟 = 2 − 3 cos 𝜃𝜃.
15. Sketch the graph of 𝑟𝑟 = 2𝜃𝜃.
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Chapter 9 Mid-Chapter Test
Name four different pairs of polar coordinates that represent point 𝑨𝑨.
𝜋𝜋
2. 𝐴𝐴 �2, �
1. 𝐴𝐴(25, −30°)
3
Graph each point or equation.
3. �1.5,
5𝜋𝜋
4
4. 𝑟𝑟 = √10
�
6. 𝑟𝑟 = 2 sin 2𝜃𝜃
5. 𝜃𝜃 = −270°
{OVER}
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Identify the classical curve represented by the graph of each equation.
7. 𝑟𝑟 = 1 + 2 cos 𝜃𝜃
8. 𝑟𝑟 = 4𝜃𝜃
Find the polar coordinates of each point with the given rectangular coordinates..
10. �−1, −√3�
9. �−2, −2√3�
Find the rectangular coordinates of each point with the given polar coordinates..
𝜋𝜋
12. �−2, − �
11. (3, 150°)
3
13. Write 𝑦𝑦 = −3 in polar form.
14. Write 𝑟𝑟 = −4 csc 𝜃𝜃 in rectangular form.
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