Deck 67: Graphs of Polar Equations

Question 1

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 5 \cos 2 \theta$$ ​

A)Symmetric with respect to the polar axis $$\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$$ ​
B)Symmetric with respect to the polar axis $$\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$$ ​
C)Symmetric with respect to the polar axis $$\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$$ ​
D)Symmetric with respect to the polar axis $$\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$$ ​
E)Symmetric with respect to the polar axis $$\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$$ ​

Answer 1

Symmetric with respect to the polar axis $$\begin{array} { c } | r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }\end{array}$$ ​

Question 2

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = 4 ( 1 - \cos \theta )$ ​

A)Symmetric with respect to polar axis $$\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$$
B)Symmetric with respect to polar axis $$\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$$
C)Symmetric with respect to polar axis $$\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$$
D)Symmetric with respect to polar axis $$\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$$ ​
E)Symmetric with respect to polar axis $$\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$$

Answer 2

Symmetric with respect to polar axis $$\begin{array} { l } | r | = 8 \text { when } \theta = \pi \\r = 0 \text { when } \theta = 0\end{array}$$

Question 3

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = 2$ ​

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $r = 2$
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B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $r = 2$
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C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $r = 2$
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D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $r = 2$
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E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $r = 2$
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Answer 3

Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $r = 2$
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Question 4

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 4 - 6 \cos \theta$$ ​

A)Symmetric with respect to the polar axis $| r | = 10$ when $\theta = \pi$
$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$ ​
B)Symmetric with respect to the polar axis ​ $| r | = 10$ when $\theta = \pi$
$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$
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C)Symmetric with respect to the polar axis ​ $$\begin{array} { l } | r | = 10 \text { when } \theta = \pi \\r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }\end{array}$$
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D)Symmetric with respect to the polar axis ​ $| r | = 10$ when $\theta = \pi$
$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$
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E)Symmetric with respect to the polar axis ​ $| r | = 10$ when $\theta = \pi$
$r = 0 \text { when } \cos \theta = \frac { 4 } { 6 }$
​

Question 5

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = 4 + 6 \sin \theta$ ​

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$ ​
B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$ ​
C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$ ​
D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$ ​
E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $| r | = 10$ when $\theta = \frac { \pi } { 2 }$
$r = 0$ when $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$ ​

Question 6

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 4 - 3 \sin \theta$$ ​

A)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​
B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​
C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​
D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​
E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​

Question 7

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = ( 1 + \sin \theta )$$ ​

A)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​
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B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​
C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​
D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​
E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 2 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​

Question 8

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 1$$ ​

A)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,polar axis,pole Circle with radius $$r = 1$$
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B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,polar axis,pole Circle with radius $$r = 1$$
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C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,polar axis,pole Circle with radius $$r = 1$$
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D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,polar axis,pole Circle with radius $$r = 1$$
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E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,polar axis,pole Circle with radius $$r = 1$$
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Question 9

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = \cos 3 \theta$$ ​

A)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$ ​
B)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$ ​
C)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$ ​
D)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$ ​
E)Symmetric with respect to the polar axis $| r | = 1$ when $\theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi$
$r = 0$ when $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$ ​

Question 10

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 4 \sin ( 2 \theta )$$ ​

A)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,the polar axis,and the pole $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$$ ​
B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,the polar axis,and the pole $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$$ ​
C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,the polar axis,and the pole $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$$ ​
D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,the polar axis,and the pole $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$$ ​
E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ ,the polar axis,and the pole $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi\end{array}$$ ​

Question 11

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = \frac { 3 \pi } { 7 }$ ​

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { 3 \pi } { 7 }$
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B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { 3 \pi } { 7 }$
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C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { 3 \pi } { 7 }$
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D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { 3 \pi } { 7 }$
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E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { 3 \pi } { 7 }$
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Question 12

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = \frac { \pi } { 4 }$ ​

A)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { \pi } { 4 }$
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B)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { \pi } { 4 }$
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C)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { \pi } { 4 }$
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D)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { \pi } { 4 }$
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E)Symmetric with respect to $\theta = \frac { \pi } { 2 }$ ,polar axis,pole Circle with radius $\frac { \pi } { 4 }$
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Question 13

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 4 - 5 \cos \theta$$ ​

A)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ ​
B)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ ​
C)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ ​
D)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ ​
E)Symmetric with respect to polar axis $| r | = 9$ when $\theta = \pi$
$r = 0 \text { when } \theta = \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ ​

Question 14

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 4 \sin 3 \theta$$ ​

A)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$$ ​
B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$$ ​
C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$$ ​
D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$$ ​
E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 4 \text { when } \theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 11 \pi } { 6 } \\r = 0 \text { when } \theta = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }\end{array}$$ ​

Question 15

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 3 + 7 \cos \theta$$ ​

A)Symmetric with respect to the polar axis $$| r | = 10 \text { when } \theta = 0$$ ​
B)Symmetric with respect to the polar axis $$| r | = 10 \text { when } \theta = 0$$ ​
C)Symmetric with respect to the polar axis $$| r | = 10 \text { when } \theta = 0$$ ​
D)Symmetric with respect to the polar axis $$| r | = 10 \text { when } \theta = 0$$ ​
E)Symmetric with respect to the polar axis $$| r | = 10 \text { when } \theta = 0$$ ​

Question 16

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = 5 \cos \theta$ ​

A)Symmetric with respect to polar axis Circle with radius 2.5
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B)Symmetric with respect to polar axis Circle with radius 2.5
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C)Symmetric with respect to polar axis Circle with radius 2.5
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D)Symmetric with respect to polar axis Circle with radius 2.5
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E)Symmetric with respect to polar axis Circle with radius 2.5
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Question 17

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$\gamma = ( 1 + \cos \theta )$$ ​

A)Symmetric with respect to polar axis $| r | = 2 \text { when } \theta = 0$
$r = 0$ when $\theta = \pi$ ​
B)Symmetric with respect to polar axis $| r | = 2 \text { when } \theta = 0$
$r = 0$ when $\theta = \pi$ ​
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C)Symmetric with respect to polar axis $| r | = 2 \text { when } \theta = 0$
$r = 0$ when $\theta = \pi$ ​
D)Symmetric with respect to polar axis $| r | = 2 \text { when } \theta = 0$
$r = 0$ when $\theta = \pi$ ​
E)Symmetric with respect to polar axis $| r | = 2 \text { when } \theta = 0$
$r = 0$ when $\theta = \pi$ ​

Question 18

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $r = 2 ( 1 - \sin \theta )$ ​

A)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$ ​
B)Symmetric with respect to $\frac { \pi } { 2 }$ $\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$ ​
C)Symmetric with respect to $\frac { \pi } { 2 }$ $$\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​
D)Symmetric with respect to $$\frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​
E)Symmetric with respect to $$\frac { \pi } { 2 }$$ $$\begin{array} { l } | r | = 4 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 2 }\end{array}$$ ​

Question 19

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = 3 - 5 \sin \theta$$ ​

A)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$$ ​
B)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$$ ​
C)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$$ ​
D)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$$ ​
E)Symmetric with respect to $$\theta = \frac { \pi } { 2 }$$ $$\begin{array} { c } | r | = 8 \text { when } \theta = \frac { 3 \pi } { 2 } \\r = 0 \text { when } \theta = \frac { \pi } { 6 } , \theta = \frac { 5 \pi } { 6 }\end{array}$$ ​

Question 20

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = \sec \theta$$ ​

A) $$x = 1 \Rightarrow \text { Line }$$
B) $$x = 1 \Rightarrow \text { Line }$$
C) $$x = 1 \Rightarrow \text { Line }$$
D) $$x = 1 \Rightarrow \text { Line }$$
E) $$x = 1 \Rightarrow \text { Line }$$

Question 21

Select the graph of the equation.​ $$r = - 4 \sec \theta$$ ​

A)​
B)​
C)​
D)​
E)​

Question 22

Select the graph of the equation.​ $$r = 1 - \sin \theta$$ ​

A)​
B)​
C)​
D)​
E)​

Question 23

Select the graph of the equation.​ $$r = 5 \sec \left( \theta - \frac { \pi } { 2 } \right)$$ ​

A)​
B)​
C)​
D)​
E)​

Question 24

Select the correct graph of the polar equation.Find an interval for $$\theta$$ for which the graph is traced only once.​ $$r = 4 - 8 \cos \theta$$ ​

A)​ $$0 \leq \theta < 2 \pi$$
B)​ $$0 \leq \theta < 2 \pi$$
C)​ $$0 \leq \theta < 2 \pi$$
D)​ $$0 \leq \theta < 2 \pi$$
E)​ $$0 \leq \theta < 2 \pi$$

Question 25

Select the graph of $$r = 3 \cos \theta$$ over the interval.Describe the part of the graph obtained in this case.​ $$0 \leq \theta \leq \pi$$ ​

A)​ Upper half of circle
B)​ Upper half of circle
C)​ Upper half of circle
D)​ Upper half of circle
E)​ Upper half of circle

Question 26

Select the graph of $$r = 3 \cos \theta$$ over the interval.Describe the part of the graph obtained in this case.​ $$- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$ ​

A)​ Entire circle
B)​ Entire circle
C)​ Entire circle
D)​ Entire circle
E)​ Entire circle

Question 27

Select the graph of $$r = 3 \cos \theta$$ over the interval.Describe the part of the graph obtained in this case.​ $$\frac { \pi } { 2 } \leq \theta \leq \pi$$ ​

A)​ Lower half of circle
B)​ Lower half of circle
C)​ Lower half of circle
D)​ Lower half of circle
E)​ Lower half of circle

Question 28

Select the correct graph of the polar equation.Find an interval for $$\theta$$ for which the graph is traced only once.​ $$r = 5 + 4 \cos \theta$$ ​

A)​ $$0 \leq \theta < 2 \pi$$
B)​ $$0 \leq \theta < 2 \pi$$
C)​ $$0 \leq \theta < 2 \pi$$
D)​ $$0 \leq \theta < 2 \pi$$
E)​ $$0 \leq \theta < 2 \pi$$

Question 29

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = \frac { 8 } { 2 \sin \theta - 3 \cos \theta }$$ ​

A) $$y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$$ ​
B) $$y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$$ ​
C) $$y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$$ ​
D) $$y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$$ ​
E) $$y = \frac { 3 } { 2 } x + 4 \Rightarrow \text { Line }$$ ​

Question 30

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r ^ { 2 } = 4 \cos 2 \theta$$ ​

A)Symmetric with respect to the polar axis, $$\theta = \frac { \pi } { 2 }$$ ,and the pole
Lemniscate
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B)Symmetric with respect to the polar axis, $$\theta = \frac { \pi } { 2 }$$ ,and the pole
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C)Symmetric with respect to the polar axis, $$\theta = \frac { \pi } { 2 }$$ ,and the pole
Lemniscate
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D)Symmetric with respect to the polar axis, $$\theta = \frac { \pi } { 2 }$$ ,and the pole
Lemniscate
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E)Symmetric with respect to the polar axis, $$\theta = \frac { \pi } { 2 }$$ ,and the pole
Lemniscate
​

Question 31

Consider the equation $$r = \sin k \theta$$ .Select the correct graph of the equation for $$k = 2.5$$ . ​

A)​
B)​
C)​
D)​ ​
E)​

Question 32

Select the graph of the equation.​ $$r = 4 \sec \left( \theta + \frac { \pi } { 3 } \right)$$ ​

A)​
B)​
C)​
D)​
E)​

Question 33

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r ^ { 2 } = 9 \sin \theta$$ ​

A) $$\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$$ ​
B) $$\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$$ ​
C) $$\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$$ ​
D) $$\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$$ ​
E) $$\begin{array} { c } r = 3 \sqrt { \sin \theta } \\r = - 3 \sqrt { \sin \theta } \\0 \leq \theta \leq \pi\end{array}$$ ​

Question 34

Consider the equation $$r = 3 \sin k \varphi$$ .Select the correct graph of the equation for $$k = 1.5$$ . ​

A)​
B)​
C)​ ​
D)​
E)​

Question 35

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = \frac { 4 } { \sin \theta - 2 \cos \theta }$$ ​

A) $$y = 2 x + 4 \Rightarrow \text { Line }$$ ​
B) $$y = 2 x + 4 \Rightarrow \text { Line }$$ ​
C) $$y = 2 x + 4 \Rightarrow \text { Line }$$ ​
D) $$y = 2 x + 4 \Rightarrow \text { Line }$$ ​
E) $$y = 2 x + 4 \Rightarrow \text { Line }$$ ​

Question 36

Select the correct graph of the polar equation.Find an interval for $$\theta$$ for which the graph is traced only once.​ $$r = \cos \left( \frac { 3 \theta } { 2 } \right)$$ ​

A)​ $$0 \leq \theta < 4 \pi$$
B)​ $$0 \leq \theta < 4 \pi$$
C)​ $$0 \leq \theta < 4 \pi$$
D)​ $$0 \leq \theta < 4 \pi$$
E)​ $$0 \leq \theta < 4 \pi$$

Question 37

Select the graph of $$r = 4 \cos \theta$$ over the interval.Describe the part of the graph obtained in this case.​ $$\frac { \pi } { 4 } \leq \theta \leq \frac { 3 \pi } { 4 }$$ ​

A)​ Left half of circle
B)​ Left half of circle
C)​ Left half of circle
D)​ Left half of circle
E)​ Left half of circle

Question 38

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ $$r = \csc \theta$$ ​

A) $$y = 1 \Rightarrow \text { Line }$$
B) $$y = 1 \Rightarrow \text { Line }$$
C) $$y = 1 \Rightarrow \text { Line }$$
D) $$y = 1 \Rightarrow \text { Line }$$
E) $$y = 1 \Rightarrow \text { Line }$$

Question 39

Select the graph of the equation.​ $$r = 2 - \sin \left( \theta - \frac { \pi } { 4 } \right)$$ ​

A)​
B)​
C)​
D)​
E)​

Question 40

Select the graph of the equation.​ $$r = 2 \sec \left( \theta - \frac { \pi } { 4 } \right)$$ ​

A)​
B)​ ​
C)​
D)​
E)​

Question 41

Select the correct graph of the polar equation.Find an interval for $$\theta$$ for which the graph is traced only once.​ $$r ^ { 2 } = \frac { 4 } { \theta }$$ ​

A)​ $$0 < \theta < \infty$$
B)​ $$0 < \theta < \infty$$
C)​ $$0 < \theta < \infty$$
D)​ $$0 < \theta < \infty$$
E)​ $$0 < \theta < \infty$$

Question 42

Select the correct graph of the polar equation.Describe your viewing window.​ $$r = \frac { 7 } { 4 }$$ ​

A)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
B)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
C)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
D)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
E)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$

Question 43

Select the correct graph of the polar equation.Describe your viewing window.​ $$r = \frac { 9 \pi } { 2 }$$ ​

A)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 18 \\X _ { \max } = 18 \\X _ { s c l } = 3 \\Y _ { \min } = - 18 \\Y _ { \max } = 18 \\Y _ { s c l } = 3\end{array}$$
B)​ $\begin{array}{l}\theta_{\min }=0 \\\theta_{\max }=2 \pi \\\theta_{s t e p}=\pi / 24 \\X_{\min }=-18 \\X_{\max }=18 \\X_{s c l}=3 \\Y_{\min }=-18 \\Y_{\max }=18 \\Y_{s c l}=3\end{array}$
C)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 18 \\X _ { \max } = 18 \\X _ { s c l } = 3 \\Y _ { \min } = - 18 \\Y _ { \max } = 18 \\Y _ { s c l } = 3\end{array}$$
D)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 18 \\X _ { \max } = 18 \\X _ { s c l } = 3 \\Y _ { \min } = - 18 \\Y _ { \max } = 18 \\Y _ { s c l } = 3\end{array}$$
E)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 18 \\X _ { \max } = 18 \\X _ { s c l } = 3 \\Y _ { \min } = - 18 \\Y _ { \max } = 18 \\Y _ { s c l } = 3\end{array}$$

Question 44

Select the correct graph of the polar equation.Describe your viewing window.​ $$r = 8 \sin \theta \cos ^ { 2 } \theta$$ ​

A)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 3 \\X _ { \max } = 3 \\X _ { s c l } = 1 \\Y _ { \min } = - 3 \\Y _ { \max } = 3 \\Y _ { s c l } = 1\end{array}$$
B)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 3 \\X _ { \max } = 3 \\X _ { s c l } = 1 \\Y _ { \min } = - 3 \\Y _ { \max } = 3 \\Y _ { s c l } = 1\end{array}$$
C)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 3 \\X _ { \max } = 3 \\X _ { s c l } = 1 \\Y _ { \min } = - 3 \\Y _ { \max } = 3 \\Y _ { s c l } = 1\end{array}$$
D)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 3 \\X _ { \max } = 3 \\X _ { s c l } = 1 \\Y _ { \min } = - 3 \\Y _ { \max } = 3 \\Y _ { s c l } = 1\end{array}$$
E)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 3 \\X _ { \max } = 3 \\X _ { s c l } = 1 \\Y _ { \min } = - 3 \\Y _ { \max } = 3 \\Y _ { s c l } = 1\end{array}$$

Question 45

Select the correct graph of the polar equation.Describe your viewing window.​ $$r = \frac { 7 } { 4 }$$ ​

A)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
B)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
C)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
D)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$
E)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 5 \\X _ { \max } = 5 \\X _ { s c l } = 1 \\Y _ { \min } = - 5 \\Y _ { \max } = 5 \\Y _ { s c l } = 1\end{array}$$

Question 46

Select the correct graph of the polar equation.Find an interval for $$\theta$$ for which the graph is traced only once.​ $$r = 4 \sin \left( \frac { 5 \theta } { 2 } \right)$$ ​

A)​ $$0 \leq \theta < 4 \pi$$
B)​ $$0 \leq \theta < 4 \pi$$
C)​ $$0 \leq \theta < 4 \pi$$
D)​ $$0 \leq \theta < 4 \pi$$
E)​ $$0 \leq \theta < 4 \pi$$

Question 47

Select the correct graph of the polar equation.Find an interval for $$\theta$$ for which the graph is traced only once.​ $$r ^ { 2 } = 4 \sin 2 \theta$$ ​

A)​ $$0 \leq \theta < \pi$$
B)​ $$0 \leq \theta < \pi$$
C)​ $$0 \leq \theta < \pi$$
D)​ $$0 \leq \theta < \pi$$
E)​ $$0 \leq \theta < \pi$$

Question 48

Select the correct graph of the polar equation.Describe your viewing window.​ $$r = 2 \csc \theta + 5$$ ​

A)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 12 \\X _ { \max } = 12 \\X _ { s c l } = 2 \\Y _ { \min } = - 12 \\Y _ { \max } = 12 \\Y _ { s c l } = 2\end{array}$$
B)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 12 \\X _ { \max } = 12 \\X _ { s c l } = 2 \\Y _ { \min } = - 12 \\Y _ { \max } = 12 \\Y _ { s c l } = 2\end{array}$$
C)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 12 \\X _ { \max } = 12 \\X _ { s c l } = 2 \\Y _ { \min } = - 12 \\Y _ { \max } = 12 \\Y _ { s c l } = 2\end{array}$$
D)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 12 \\X _ { \max } = 12 \\X _ { s c l } = 2 \\Y _ { \min } = - 12 \\Y _ { \max } = 12 \\Y _ { s c l } = 2\end{array}$$
E)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 12 \\X _ { \max } = 12 \\X _ { s c l } = 2 \\Y _ { \min } = - 12 \\Y _ { \max } = 12 \\Y _ { s c l } = 2\end{array}$$

Question 49

Select the correct graph of the polar equation.Describe your viewing window.​ $$r = \frac { \pi } { 4 }$$ ​

A)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 2 \\X _ { \max } = 2 \\X _ { s c l } = 0.5 \\Y _ { \min } = - 2 \\Y _ { \max } = 2 \\Y _ { s c l } = 0.5\end{array}$$
B)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 2 \\X _ { \max } = 2 \\X _ { s c l } = 0.5 \\Y _ { \min } = - 2 \\Y _ { \max } = 2 \\Y _ { s c l } = 0.5\end{array}$$
C)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 2 \\X _ { \max } = 2 \\X _ { s c l } = 0.5 \\Y _ { \min } = - 2 \\Y _ { \max } = 2 \\Y _ { s c l } = 0.5\end{array}$$
D)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 2 \\X _ { \max } = 2 \\X _ { s c l } = 0.5 \\Y _ { \min } = - 2 \\Y _ { \max } = 2 \\Y _ { s c l } = 0.5\end{array}$$
E)​ $$\begin{array} { l } \theta _ { \min } = 0 \\\theta _ { \max } = 2 \pi \\\theta _ { s t e p } = \pi / 24 \\X _ { \min } = - 2 \\X _ { \max } = 2 \\X _ { s c l } = 0.5 \\Y _ { \min } = - 2 \\Y _ { \max } = 2 \\Y _ { s c l } = 0.5\end{array}$$