Question 1

Find the graph of the following polar equation.
$$r = 3 + 2 \cos ( \theta )$$

Accepted Answer

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

Question 2

Which answer is a rectangular form of the given polar equation?
$r = 18 \cos \theta$

Accepted Answer

A)  $( x - 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
B)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
C)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 9$ 
D)  $x ^ { 2 } + ( y + 9 ) ^ { 2 } = 81$ 
E)  $x ^ { 2 } + ( y - 9 ) ^ { 2 } = 81$ 
A)  $( x - 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
B)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
C)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 9$ 
D)  $x ^ { 2 } + ( y + 9 ) ^ { 2 } = 81$ 
E)  $x ^ { 2 } + ( y - 9 ) ^ { 2 } = 81$

Question 3

Find the standard form of the equation of the ellipse below.
$$2 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 0$$

Accepted Answer

A)  $\frac { ( x - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } - \frac { ( y - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } - \frac { ( y + 6 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
A)  $\frac { ( x - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } - \frac { ( y - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } - \frac { ( y + 6 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$

Question 4

Solve the following system of quadratic equations algebraically by the method of substitution.
$$\left\{ \begin{array} { l } 
- 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \
- 4 x - 2 y = 0
\end{array} \right.$$

Accepted Answer

A)  $( \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
B)  $( - \sqrt { 5 } , 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
C)  $( - \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
D)  $( - \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 3 } , 2 \sqrt { 3 } )$ 
E)  $( - \sqrt { 3 } , 2 \sqrt { 3 } ) , ( \sqrt { 3 } , - 2 \sqrt { 3 } )$ 
A)  $( \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
B)  $( - \sqrt { 5 } , 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
C)  $( - \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
D)  $( - \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 3 } , 2 \sqrt { 3 } )$ 
E)  $( - \sqrt { 3 } , 2 \sqrt { 3 } ) , ( \sqrt { 3 } , - 2 \sqrt { 3 } )$

Question 5

Find the center, vertices, and foci of the ellipse below.
$$x ^ { 2 } + 8 y ^ { 2 } = 80$$

Accepted Answer

A)  Center: $( 0,0 )$
Vertices: $( 4 \sqrt { 5 } , 0 ) , ( - 4 \sqrt { 5 } , 0 )$
Foci: $( \sqrt { 70 } , 0 ) , ( - \sqrt { 70 } , 0 )$ 
B)  Center: $( 0,0 )$
Vertices: $( 0,10 ) , ( 0 , - 10 )$
Foci: $( 0,24 \sqrt { 11 } ) , ( 0 , - 24 \sqrt { 11 } )$ 
C)  Center: $( 80,10 )$
Vertices: $( 160,10 ) , ( - 160,10 )$
Foci: $( 80 + 24 \sqrt { 11 } , 10 ) , ( 80 - 24 \sqrt { 11 } , 10 )$ 
D)  Center: $( 80,10 )$
Vertices: $( 80,20 ) , ( 80 , - 20 )$
Foci: $( 80,10 + 24 \sqrt { 11 } ) , ( 80,10 - 24 \sqrt { 11 } )$ 
E)  Center: $( 0,0 )$
Vertices: $( 80,0 ) , ( - 80,0 )$
Foci: $( 6336,0 ) , ( - 6336,0 )$ 
A)  Center: $( 0,0 )$
Vertices: $( 4 \sqrt { 5 } , 0 ) , ( - 4 \sqrt { 5 } , 0 )$
Foci: $( \sqrt { 70 } , 0 ) , ( - \sqrt { 70 } , 0 )$ 
B)  Center: $( 0,0 )$
Vertices: $( 0,10 ) , ( 0 , - 10 )$
Foci: $( 0,24 \sqrt { 11 } ) , ( 0 , - 24 \sqrt { 11 } )$ 
C)  Center: $( 80,10 )$
Vertices: $( 160,10 ) , ( - 160,10 )$
Foci: $( 80 + 24 \sqrt { 11 } , 10 ) , ( 80 - 24 \sqrt { 11 } , 10 )$ 
D)  Center: $( 80,10 )$
Vertices: $( 80,20 ) , ( 80 , - 20 )$
Foci: $( 80,10 + 24 \sqrt { 11 } ) , ( 80,10 - 24 \sqrt { 11 } )$ 
E)  Center: $( 0,0 )$
Vertices: $( 80,0 ) , ( - 80,0 )$
Foci: $( 6336,0 ) , ( - 6336,0 )$

Question 6

Find the $y$-intercepts of the graph of the circle below.
$$( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$$

Accepted Answer

A)  $3 + 4 \sqrt { 2 } , 3 - 4 \sqrt { 2 }$ 
B)  $3 + 2 \sqrt { 10 } , 3 - 2 \sqrt { 10 }$ 
C)  $2 + 3 \sqrt { 5 } , 2 - 3 \sqrt { 5 }$ 
D)  $- 2 + 3 \sqrt { 3 } , - 2 - 3 \sqrt { 3 }$ 
E)  $- 2 + 3 \sqrt { 5 } , - 2 - 3 \sqrt { 5 }$ 
A)  $3 + 4 \sqrt { 2 } , 3 - 4 \sqrt { 2 }$ 
B)  $3 + 2 \sqrt { 10 } , 3 - 2 \sqrt { 10 }$ 
C)  $2 + 3 \sqrt { 5 } , 2 - 3 \sqrt { 5 }$ 
D)  $- 2 + 3 \sqrt { 3 } , - 2 - 3 \sqrt { 3 }$ 
E)  $- 2 + 3 \sqrt { 5 } , - 2 - 3 \sqrt { 5 }$

Question 7

Find the standard form of the equation of the hyperbola below.
$$8 x ^ { 2 } - y ^ { 2 } - 80 x - 16 y + 96 = 0$$

Accepted Answer

A)  $\frac { ( x - 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { ( y + 8 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } + \frac { ( y - 8 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( y + 5 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } + \frac { ( x - 8 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( y + 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { ( x - 8 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( y - 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { ( x + 8 ) ^ { 2 } } { ( 2 \sqrt { 2 } ) ^ { 2 } } = 1$ 
A)  $\frac { ( x - 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { ( y + 8 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } + \frac { ( y - 8 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( y + 5 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } + \frac { ( x - 8 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( y + 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { ( x - 8 ) ^ { 2 } } { ( 2 \sqrt { 10 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( y - 5 ) ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { ( x + 8 ) ^ { 2 } } { ( 2 \sqrt { 2 } ) ^ { 2 } } = 1$

Question 8

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$, given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$.

Accepted Answer

A)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
B)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
C)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
D)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
E)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
A)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
B)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
C)  $\left( - 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$ 
D)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( 5 , \frac { \pi } { 6 } \right) , \left( - 5 , - \frac { 11 \pi } { 6 } \right)$ 
E)  $\left( 5 , \frac { 7 \pi } { 6 } \right) , \left( - 5 , \frac { \pi } { 6 } \right) , \left( 5 , - \frac { 11 \pi } { 6 } \right)$

Question 9

Find the graph of the following polar equation.
$$r = 6 \sin ( 2 \theta )$$

Accepted Answer

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

Question 10

A projectile is launched from ground level at an angle of $\theta$ with the horizontal. The initial velocity is $v _ { 0 }$ feet per second and the path of the projectile is modeled by the parametric equations $x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 } \text {. }$
Use a graphing utility to graph the paths of a projectile launched from ground level with the values given for $\theta$ and $v _ { 0 }$. Use the graph to approximate the maximum height and range of the projectile to the nearest foot.
$\theta = 55 ^ { \circ } , \quad v _ { 0 } = 56 \text { feet per second }$

Accepted Answer

A)  maximum height $\approx 49$ feet range $\approx 4$ feet 
B)  maximum height $\approx 4$ feet range $\approx 49$ feet 
C)  maximum height $\approx 33$ feet range $\approx 92$ feet 
D)  maximum height $\approx 92$ feet range $\approx 33$ feet 
E)  maximum height $\approx 43$ feet $\quad$ range $\approx 8$ feet 
A)  maximum height $\approx 49$ feet range $\approx 4$ feet 
B)  maximum height $\approx 4$ feet range $\approx 49$ feet 
C)  maximum height $\approx 33$ feet range $\approx 92$ feet 
D)  maximum height $\approx 92$ feet range $\approx 33$ feet 
E)  maximum height $\approx 43$ feet $\quad$ range $\approx 8$ feet

Question 11

Match the graph with its equation.

Accepted Answer

A)  $x ^ { 2 } = 4 y$ 
B)  $x ^ { 2 } = \frac { y } { 4 }$ 
C)  $y ^ { 2 } = \frac { x } { 4 }$ 
D)  $y ^ { 2 } = 4 x$ 
E)  $( y + 1 ) ^ { 2 } = 4 ( x - 2 )$ 
A)  $x ^ { 2 } = 4 y$ 
B)  $x ^ { 2 } = \frac { y } { 4 }$ 
C)  $y ^ { 2 } = \frac { x } { 4 }$ 
D)  $y ^ { 2 } = 4 x$ 
E)  $( y + 1 ) ^ { 2 } = 4 ( x - 2 )$

Question 12

Which answer is a polar form of the given rectangular equation?
$$16 x y = 144$$

Accepted Answer

A)  $r ^ { 2 } = 9 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 3 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 3 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 9 \sin \theta \cos \theta$ 
E)  $\theta = 9 \sin \theta \cos \theta$ 
A)  $r ^ { 2 } = 9 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 3 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 3 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 9 \sin \theta \cos \theta$ 
E)  $\theta = 9 \sin \theta \cos \theta$

Question 13

Identify the center and radius of the circle below.
$$x ^ { 2 } + y ^ { 2 } + 10 x + 10 y - 5 = 0$$

Accepted Answer

A)  Center: $( 5,5 )$
Radius: $\sqrt { 15 }$ 
B)  Center: $( - 5 , - 5 )$
Radius: $\sqrt { 15 }$ 
C)  Center: $( 5,5 )$
Radius: $\sqrt { 55 }$ 
D)  Center: $( - 5 , - 5 )$
Radius: $\sqrt { 55 }$ 
E)  Center: $( - 10 , - 10 )$
Radius: $\sqrt { 205 }$ 
A)  Center: $( 5,5 )$
Radius: $\sqrt { 15 }$ 
B)  Center: $( - 5 , - 5 )$
Radius: $\sqrt { 15 }$ 
C)  Center: $( 5,5 )$
Radius: $\sqrt { 55 }$ 
D)  Center: $( - 5 , - 5 )$
Radius: $\sqrt { 55 }$ 
E)  Center: $( - 10 , - 10 )$
Radius: $\sqrt { 205 }$

Question 14

Which answer is a rectangular form of the given polar equation?
$$r = \frac { 5 } { 8 + \sin \theta }$$

Accepted Answer

A)  $63 x ^ { 2 } - 64 y ^ { 2 } - 10 x + 25 = 0$ 
B)  $65 x ^ { 2 } + 64 y ^ { 2 } - 10 x + 25 = 0$ 
C)  $63 x ^ { 2 } + 64 y ^ { 2 } + 10 x - 25 = 0$ 
D)  $64 x ^ { 2 } + 63 y ^ { 2 } + 10 y - 25 = 0$ 
E)  $64 x ^ { 2 } + 65 y ^ { 2 } - 10 y + 25 = 0$ 
A)  $63 x ^ { 2 } - 64 y ^ { 2 } - 10 x + 25 = 0$ 
B)  $65 x ^ { 2 } + 64 y ^ { 2 } - 10 x + 25 = 0$ 
C)  $63 x ^ { 2 } + 64 y ^ { 2 } + 10 x - 25 = 0$ 
D)  $64 x ^ { 2 } + 63 y ^ { 2 } + 10 y - 25 = 0$ 
E)  $64 x ^ { 2 } + 65 y ^ { 2 } - 10 y + 25 = 0$

Question 15

Match the graph to a set of parametric equations.

Accepted Answer

A)  Witch of Agnesi:
$x = 2 \cot \theta$
$y = 2 \sin ^ { 2 } \theta$ 
B)  Lissajous curve:
$x = 3 \cos \theta$
$y = \sin 2 \theta$ 
C)  Involute of circle
$\begin{array} { l } 
x = \cos \theta + \theta \sin \theta \
y = \sin \theta - \theta \cos \theta
\end{array}$ 
D)  Evolute of ellipse:
$x = 3 \cos ^ { 3 } \theta$
$y=5 \sin ^{3} \theta$ 
E)  $\begin{array}{l}
x=\cot \theta \
y=6 \sin \theta \cos \theta
\end{array}$ 
A)  Witch of Agnesi:
$x = 2 \cot \theta$
$y = 2 \sin ^ { 2 } \theta$ 
B)  Lissajous curve:
$x = 3 \cos \theta$
$y = \sin 2 \theta$ 
C)  Involute of circle
$\begin{array} { l } 
x = \cos \theta + \theta \sin \theta \
y = \sin \theta - \theta \cos \theta
\end{array}$ 
D)  Evolute of ellipse:
$x = 3 \cos ^ { 3 } \theta$
$y=5 \sin ^{3} \theta$ 
E)  $\begin{array}{l}
x=\cot \theta \
y=6 \sin \theta \cos \theta
\end{array}$

Question 16

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ?
$y = x ^ { 2 } + 2$

Accepted Answer

A) 
$x = t + 6$
$y = ( t + 6 ) ^ { 2 } - 2$ 
B) 
$x = t - 6$
$y = ( t - 6 ) ^ { 2 } - 2$ 
C) 
$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } + 2
\end{array}$ 
D) 
$x = 6 + t$
$y = ( 6 + t ) ^ { 2 } + 2$ 
E) 
$\begin{array} { l } 
x = 6 - t \
y = ( 6 - t ) ^ { 2 } + 2
\end{array}$ 
A) 
$x = t + 6$
$y = ( t + 6 ) ^ { 2 } - 2$ 
B) 
$x = t - 6$
$y = ( t - 6 ) ^ { 2 } - 2$ 
C) 
$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } + 2
\end{array}$ 
D) 
$x = 6 + t$
$y = ( 6 + t ) ^ { 2 } + 2$ 
E) 
$\begin{array} { l } 
x = 6 - t \
y = ( 6 - t ) ^ { 2 } + 2
\end{array}$

Question 17

Find the graph of the following polar equation.
$
r=2 \sin (2 \theta)
$

Accepted Answer

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

Question 18

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$.
$$r = \sqrt { 3 } + 2 \cos \theta$$

Accepted Answer

A)  zeros: $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
B)  zeros: $\frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 }$ 
C)  zeros: $\frac { 5 \pi } { 3 } , \frac { 7 \pi } { 3 }$ 
D)  zeros: $\frac { \pi } { 3 } , \frac { 11 \pi } { 3 }$ 
E)  zeros: $\frac { 5 \pi } { 2 } , \frac { 7 \pi } { 2 }$ 
A)  zeros: $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
B)  zeros: $\frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 }$ 
C)  zeros: $\frac { 5 \pi } { 3 } , \frac { 7 \pi } { 3 }$ 
D)  zeros: $\frac { \pi } { 3 } , \frac { 11 \pi } { 3 }$ 
E)  zeros: $\frac { 5 \pi } { 2 } , \frac { 7 \pi } { 2 }$

Question 19

Find the $y$-intercepts of the graph of the circle below.
$$( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$$

Accepted Answer

A)  $5 + 4 \sqrt { 2 } , 5 - 4 \sqrt { 2 }$ 
B)  $5 + 2 \sqrt { 10 } , 5 - 2 \sqrt { 10 }$ 
C)  $2 + \sqrt { 61 } , 2 - \sqrt { 61 }$ 
D)  $- 2 + \sqrt { 11 } , - 2 - \sqrt { 11 }$ 
E)  $- 2 + \sqrt { 61 } , - 2 - \sqrt { 61 }$ 
A)  $5 + 4 \sqrt { 2 } , 5 - 4 \sqrt { 2 }$ 
B)  $5 + 2 \sqrt { 10 } , 5 - 2 \sqrt { 10 }$ 
C)  $2 + \sqrt { 61 } , 2 - \sqrt { 61 }$ 
D)  $- 2 + \sqrt { 11 } , - 2 - \sqrt { 11 }$ 
E)  $- 2 + \sqrt { 61 } , - 2 - \sqrt { 61 }$

Question 20

Convert the following polar equation to rectangular form.
$\theta = - \frac { 5 \pi } { 3 }$

Accepted Answer

A)  $y = \sqrt { 3 }$ 
B)  $y = \frac { \sqrt { 3 } } { 3 }$ 
C)  $y = - \sqrt { 3 }$ 
D)  $y = 3$ 
E)  $y = - \frac { \sqrt { 3 } } { 3 }$ 
A)  $y = \sqrt { 3 }$ 
B)  $y = \frac { \sqrt { 3 } } { 3 }$ 
C)  $y = - \sqrt { 3 }$ 
D)  $y = 3$ 
E)  $y = - \frac { \sqrt { 3 } } { 3 }$

Find the graph of the following polar equation. $r = 3 + 2 \cos ( \theta )$

Which answer is a rectangular form of the given polar equation? $r = 18 \cos \theta$

Find the standard form of the equation of the ellipse below. $2 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 0$

Solve the following system of quadratic equations algebraically by the method of substitution. $\left\{ \begin{array} { l } - 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \\- 4 x - 2 y = 0\end{array} \right.$

Find the center, vertices, and foci of the ellipse below. $x ^ { 2 } + 8 y ^ { 2 } = 80$

Find the $y$ -intercepts of the graph of the circle below. $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$

Find the standard form of the equation of the hyperbola below. $8 x ^ { 2 } - y ^ { 2 } - 80 x - 16 y + 96 = 0$

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$ , given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$ .

Find the graph of the following polar equation. $r = 6 \sin ( 2 \theta )$

Match the graph with its equation.

Which answer is a polar form of the given rectangular equation? $16 x y = 144$

Identify the center and radius of the circle below. $x ^ { 2 } + y ^ { 2 } + 10 x + 10 y - 5 = 0$

Which answer is a rectangular form of the given polar equation? $r = \frac { 5 } { 8 + \sin \theta }$

Match the graph to a set of parametric equations.

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 2$

Find the graph of the following polar equation. $r=2 \sin (2 \theta)$

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$ . $r = \sqrt { 3 } + 2 \cos \theta$

Find the $y$ -intercepts of the graph of the circle below. $( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$

Convert the following polar equation to rectangular form. $\theta = - \frac { 5 \pi } { 3 }$

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Exam 9: Topics in Analytic Geometry

Find the graph of the following polar equation. r=3+2cos⁡(θ)r = 3 + 2 \cos ( \theta )r=3+2cos(θ)

Which answer is a rectangular form of the given polar equation? r=18cos⁡θr = 18 \cos \thetar=18cosθ

Find the standard form of the equation of the ellipse below. 2x2+8y2+24x−64y+16=02 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 02x2+8y2+24x−64y+16=0

Solve the following system of quadratic equations algebraically by the method of substitution. {−4x2−3y2+48=0−4x−2y=0\left\{ \begin{array} { l } - 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \\- 4 x - 2 y = 0\end{array} \right.{−4x2−3y2+48=0−4x−2y=0​

Find the center, vertices, and foci of the ellipse below. x2+8y2=80x ^ { 2 } + 8 y ^ { 2 } = 80x2+8y2=80

Find the yyy -intercepts of the graph of the circle below. (x−3)2+(y+2)2=36( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36(x−3)2+(y+2)2=36

Find the standard form of the equation of the hyperbola below. 8x2−y2−80x−16y+96=08 x ^ { 2 } - y ^ { 2 } - 80 x - 16 y + 96 = 08x2−y2−80x−16y+96=0

Find three additional polar representations of the point (−5,−5π6)\left( - 5 , - \frac { 5 \pi } { 6 } \right)(−5,−65π​) , given in polar coordinates, using −2π<θ<2π- 2 \pi < \theta < 2 \pi−2π<θ<2π .

Find the graph of the following polar equation. r=6sin⁡(2θ)r = 6 \sin ( 2 \theta )r=6sin(2θ)

Match the graph with its equation.

Which answer is a polar form of the given rectangular equation? 16xy=14416 x y = 14416xy=144

Identify the center and radius of the circle below. x2+y2+10x+10y−5=0x ^ { 2 } + y ^ { 2 } + 10 x + 10 y - 5 = 0x2+y2+10x+10y−5=0

Which answer is a rectangular form of the given polar equation? r=58+sin⁡θr = \frac { 5 } { 8 + \sin \theta }r=8+sinθ5​

Match the graph to a set of parametric equations.

Which set of parametric equations represents the graph of the following rectangular equation using t=6−xt = 6 - xt=6−x ? y=x2+2y = x ^ { 2 } + 2y=x2+2

Find the graph of the following polar equation. r=2sin⁡(2θ)r=2 \sin (2 \theta)r=2sin(2θ)

Find any zeros of rrr on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . r=3+2cos⁡θr = \sqrt { 3 } + 2 \cos \thetar=3​+2cosθ

Find the yyy -intercepts of the graph of the circle below. (x−5)2+(y+2)2=36( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36(x−5)2+(y+2)2=36

Convert the following polar equation to rectangular form. θ=−5π3\theta = - \frac { 5 \pi } { 3 }θ=−35π​

Functions and Their Graphs

Polynomial and Rational Functions

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Trigonometric Functions

Analytic Trigonometry

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Analytic Geometry in Three Dimensions

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Find the graph of the following polar equation. $r = 3 + 2 \cos ( \theta )$

Which answer is a rectangular form of the given polar equation? $r = 18 \cos \theta$

Find the standard form of the equation of the ellipse below. $2 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 0$

Solve the following system of quadratic equations algebraically by the method of substitution. $\left\{ \begin{array} { l } - 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \\- 4 x - 2 y = 0\end{array} \right.$

Find the center, vertices, and foci of the ellipse below. $x ^ { 2 } + 8 y ^ { 2 } = 80$

Find the $y$ -intercepts of the graph of the circle below. $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$

Find the standard form of the equation of the hyperbola below. $8 x ^ { 2 } - y ^ { 2 } - 80 x - 16 y + 96 = 0$

Find three additional polar representations of the point $\left( - 5 , - \frac { 5 \pi } { 6 } \right)$ , given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$ .

Find the graph of the following polar equation. $r = 6 \sin ( 2 \theta )$

Which answer is a polar form of the given rectangular equation? $16 x y = 144$

Identify the center and radius of the circle below. $x ^ { 2 } + y ^ { 2 } + 10 x + 10 y - 5 = 0$

Which answer is a rectangular form of the given polar equation? $r = \frac { 5 } { 8 + \sin \theta }$

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 2$

Find the graph of the following polar equation. $r=2 \sin (2 \theta)$

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$ . $r = \sqrt { 3 } + 2 \cos \theta$

Find the $y$ -intercepts of the graph of the circle below. $( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$

Convert the following polar equation to rectangular form. $\theta = - \frac { 5 \pi } { 3 }$