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

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

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

$\frac{2}{3} < 1$, 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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