Question 1

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

Accepted Answer

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 $$\begin{array} { c } 
| r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \
r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }
\end{array}$$ ​   
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 $$\begin{array} { c } 
| r | = 5 \text { when } \theta = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 } \
r = 0 \text { when } \theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }
\end{array}$$ ​

Question 2

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

Accepted Answer

A) ​   $$0 \leq \theta < \pi$$ 
B) ​   $$0 \leq \theta < \pi$$ 
C) ​   $$0 \leq \theta < \pi$$ 
D) ​   $$0 \leq \theta < \pi$$ 
E) ​   $$0 \leq \theta < \pi$$ 
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 3

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

Accepted Answer

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) ​   $$0 \leq \theta < 4 \pi$$ 
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 4

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

Accepted Answer

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) ​   $$0 \leq \theta < 4 \pi$$ 
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 5

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

Accepted Answer

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 $\theta = \frac { 7 \pi } { 6 } , \theta = \frac { 11 \pi } { 6 }$ ​   
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 equation.​ $$r = 2 - \sin \left( \theta - \frac { \pi } { 4 } \right)$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 7

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

Accepted Answer

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) ​   $$\begin{array} { l } 
\theta _ { \min } = 0 \
\theta _ { \max } = 2 \pi \
\theta _ { s t e p } = \pi / 24 \
X _ { \min } = - 18 \
X _ { \max } = 18 \
X _ { s c l } = 3 \
Y _ { \min } = - 18 \
Y _ { \max } = 18 \
Y _ { s c l } = 3
\end{array}$$ 
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 8

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

Accepted Answer

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 $\theta = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 }$ ​   
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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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) ​   $$0 \leq \theta < 2 \pi$$ 
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 11

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

Accepted Answer

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 
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 12

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

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   ​ 
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   ​ 
E) ​

Question 13

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

Accepted Answer

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 $$\begin{array} { l } 
| r | = 8 \text { when } \theta = \pi \
r = 0 \text { when } \theta = 0
\end{array}$$   
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 $$\begin{array} { l } 
| r | = 8 \text { when } \theta = \pi \
r = 0 \text { when } \theta = 0
\end{array}$$

Question 14

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

Accepted Answer

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)  $$y = 2 x + 4 \Rightarrow \text { Line }$$ ​   
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 15

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

Accepted Answer

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 
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 16

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

Accepted Answer

A) ​   
B) ​   
C) ​   ​ 
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   ​ 
D) ​   
E) ​

Question 17

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

Accepted Answer

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) ​   $$0 \leq \theta < 2 \pi$$ 
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 18

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

Accepted Answer

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 $\theta = \pi$ ​
​   
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 $\theta = \pi$ ​   
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 $\theta = \pi$ ​
​   
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 19

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

Accepted Answer

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) ​   $$\begin{array} { l } 
\theta _ { \min } = 0 \
\theta _ { \max } = 2 \pi \
\theta _ { s t e p } = \pi / 24 \
X _ { \min } = - 5 \
X _ { \max } = 5 \
X _ { s c l } = 1 \
Y _ { \min } = - 5 \
Y _ { \max } = 5 \
Y _ { s c l } = 1
\end{array}$$ 
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 20

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

Accepted Answer

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 $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​   
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 $$\theta = \frac { \pi } { 2 }$$ $$| r | = 7 \text { when } \theta = \frac { 3 \pi } { 2 }$$ ​

Exam 67: Graphs of Polar Equations

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ r=5cos⁡2θr = 5 \cos 2 \thetar=5cos2θ ​

Select the correct graph of the polar equation.Find an interval for θ\thetaθ for which the graph is traced only once.​ r2=4sin⁡2θr ^ { 2 } = 4 \sin 2 \thetar2=4sin2θ ​

Select the correct graph of the polar equation.Find an interval for θ\thetaθ for which the graph is traced only once.​ r=4sin⁡(5θ2)r = 4 \sin \left( \frac { 5 \theta } { 2 } \right)r=4sin(25θ​) ​

Select the correct graph of the polar equation.Find an interval for θ\thetaθ for which the graph is traced only once.​ r=cos⁡(3θ2)r = \cos \left( \frac { 3 \theta } { 2 } \right)r=cos(23θ​) ​

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

Select the graph of the equation.​ r=2−sin⁡(θ−π4)r = 2 - \sin \left( \theta - \frac { \pi } { 4 } \right)r=2−sin(θ−4π​) ​

Select the correct graph of the polar equation.Describe your viewing window.​ r=9π2r = \frac { 9 \pi } { 2 }r=29π​ ​

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ r=cos⁡3θr = \cos 3 \thetar=cos3θ ​

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ r2=4cos⁡2θr ^ { 2 } = 4 \cos 2 \thetar2=4cos2θ ​

Select the correct graph of the polar equation.Find an interval for θ\thetaθ for which the graph is traced only once.​ r=5+4cos⁡θr = 5 + 4 \cos \thetar=5+4cosθ ​

Select the graph of r=3cos⁡θr = 3 \cos \thetar=3cosθ over the interval.Describe the part of the graph obtained in this case.​ π2≤θ≤π\frac { \pi } { 2 } \leq \theta \leq \pi2π​≤θ≤π ​

Consider the equation r=sin⁡kθr = \sin k \thetar=sinkθ .Select the correct graph of the equation for k=2.5k = 2.5k=2.5 . ​

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

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ r=4sin⁡θ−2cos⁡θr = \frac { 4 } { \sin \theta - 2 \cos \theta }r=sinθ−2cosθ4​ ​

Select the graph of r=4cos⁡θr = 4 \cos \thetar=4cosθ over the interval.Describe the part of the graph obtained in this case.​ π4≤θ≤3π4\frac { \pi } { 4 } \leq \theta \leq \frac { 3 \pi } { 4 }4π​≤θ≤43π​ ​

Consider the equation r=3sin⁡kφr = 3 \sin k \varphir=3sinkφ .Select the correct graph of the equation for k=1.5k = 1.5k=1.5 . ​

Select the correct graph of the polar equation.Find an interval for θ\thetaθ for which the graph is traced only once.​ r=4−8cos⁡θr = 4 - 8 \cos \thetar=4−8cosθ ​

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

Select the correct graph of the polar equation.Describe your viewing window.​ r=74r = \frac { 7 } { 4 }r=47​ ​

Select the graph of the polar equation using symmetry,zeros,maximum r-values,and any other additional points.​ r=4−3sin⁡θr = 4 - 3 \sin \thetar=4−3sinθ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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