Question 1

Which of the following polar equations represents a vertical line in the polar coordinate system?

Accepted Answer

$$	heta = \frac{\pi}{2}$$

Question 2

Given the polar equation $$r = 3 \cos 	heta$$, what is the center and radius of the corresponding circle?

Accepted Answer

Center at $$(1.5, 0)$$, radius $$1.5$$

Question 3

For the polar equation $$r = 2 + 4 \sin 	heta$$, what is the general shape of the graph?

Accepted Answer

Limacon with an inner loop

Question 4

If $$r = 5 \sin 3	heta$$, how many petals does the graph have and what is the length of each petal?

Accepted Answer

3 petals, each of length $$5$$

Question 5

Which of the following equations represents a lemniscate symmetric about the polar axis?

Accepted Answer

$$r^2 = 9$$ \cos 2	heta$$

Question 6

A polar equation $$r = 4 - 2 \cos 	heta is $$given. What is the value of $$\frac{|a|}{|b|}$$ and what is the shape of the graph?

Accepted Answer

$$\frac{4}{2} = 2$$, limacon with a dimple

Question 7

Which of the following is true about the graph of $$r = a \sin n	heta$$ when $$n is $$even?

Accepted Answer

The graph has $$2n$$ petals

Question 8

For the equation $$r = 2 \sin 	heta$$, what is the radius and location of the circle?

Accepted Answer

Radius $$1$$, center at $$(0, 1)$$

Question 9

Which of the following equations will produce a graph that passes through the pole?

Accepted Answer

$$r = 2 + 2 \sin 	heta$$

Question 10

Given $$r = 3 \cos 2	heta$$, what is the length of each loop of the lemniscate?

Accepted Answer

$$3$$

Question 11

A real-world application: A lighthouse rotates its beam in a pattern described by $$r = 5 \sin 2	heta. $$How many bright spots (maxima) will be observed in one full rotation?

Accepted Answer

4

Question 12

Which of the following is the correct symmetry for the graph of $$r = a \cos n	heta$$ when $$n is $$odd?

Accepted Answer

Symmetric about the polar axis

Question 13

For the equation $$r = 2 + 2 \cos 	heta$$, what is the general shape and does it pass through the pole?

Accepted Answer

Cardioid, passes through the pole

Question 14

If $$r = 4 \sin 	heta$$ describes the cross-section of a circular fountain, what is the radius and where is the center located?

Accepted Answer

Radius $$2$$, center at $$(0, 2)$$

Question 15

Which equation represents a limacon with no inner loop and no dimple?

Accepted Answer

$$r = 2 + 2 \sin 	heta$$

Question 16

A rose curve $$r = 6 \cos 4	heta is $$drawn. How many petals does it have and what is the length of each petal?

Accepted Answer

8 petals, each of length $$6$$

Question 17

Which of the following is the correct equation for a line passing through the pole at an angle $$\alpha$$ with the polar axis?

Accepted Answer

$$	heta = \alpha$$

Question 18

For the equation $$r = 2 - 3 \sin 	heta$$, what is the value of $$\frac{|a|}{|b|}$$ and what is the general shape?

Accepted Answer

$$\frac{2}{3} \u003c 1$$, limacon with an inner loop

Question 19

A medical imaging device scans in a pattern described by $$r^2 = 16$$ \sin 2	heta$$. What is the length of each loop and how many loops are present?

Accepted Answer

Length $$4$$, 2 loops

Graphing Polar Equations: Circles, Limacons, Roses, and Lemniscates

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