Question 1

Find three additional polar representations of the point $\left( 5 , - \frac { \pi } { 3 } \right)$, given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$.

Accepted Answer

A)  $\left( 5 , \frac { 5 \pi } { 3 } \right) , \left( 5 , \frac { 2 \pi } { 3 } \right) , \left( - 5 , - \frac { 4 \pi } { 3 } \right)$ 
B)  $\left( 5 , \frac { 5 \pi } { 3 } \right) , \left( 5 , \frac { 2 \pi } { 3 } \right) , \left( 5 , - \frac { 4 \pi } { 3 } \right)$ 
C)  $\left( 5 , \frac { 5 \pi } { 3 } \right) , \left( - 5 , \frac { 2 \pi } { 3 } \right) , \left( - 5 , - \frac { 4 \pi } { 3 } \right)$ 
D)  $\left( - 5 , \frac { 5 \pi } { 3 } \right) , \left( - 5 , \frac { 2 \pi } { 3 } \right) , \left( 5 , - \frac { 4 \pi } { 3 } \right)$ 
E)  $\left( - 5 , \frac { 5 \pi } { 3 } \right) , \left( 5 , \frac { 2 \pi } { 3 } \right) , \left( - 5 , - \frac { 4 \pi } { 3 } \right)$ 
A)  $\left( 5 , \frac { 5 \pi } { 3 } \right) , \left( 5 , \frac { 2 \pi } { 3 } \right) , \left( - 5 , - \frac { 4 \pi } { 3 } \right)$ 
B)  $\left( 5 , \frac { 5 \pi } { 3 } \right) , \left( 5 , \frac { 2 \pi } { 3 } \right) , \left( 5 , - \frac { 4 \pi } { 3 } \right)$ 
C)  $\left( 5 , \frac { 5 \pi } { 3 } \right) , \left( - 5 , \frac { 2 \pi } { 3 } \right) , \left( - 5 , - \frac { 4 \pi } { 3 } \right)$ 
D)  $\left( - 5 , \frac { 5 \pi } { 3 } \right) , \left( - 5 , \frac { 2 \pi } { 3 } \right) , \left( 5 , - \frac { 4 \pi } { 3 } \right)$ 
E)  $\left( - 5 , \frac { 5 \pi } { 3 } \right) , \left( 5 , \frac { 2 \pi } { 3 } \right) , \left( - 5 , - \frac { 4 \pi } { 3 } \right)$

Question 2

Find two sets of polar coordinates with $0 \leq \theta < 2 \pi$ for the point $( - 2,0 )$, given in rectangular coordinates.

Accepted Answer

A)  $( 2,0 ) , ( - 2 , \pi )$ 
B)  $( \sqrt { 2 } , 0 ) , ( - \sqrt { 2 } , \pi )$ 
C)  $\left( - 2 , \frac { \pi } { 2 } \right) , \left( 2 , \frac { 3 \pi } { 2 } \right)$ 
D)  $( - 2,0 ) , ( 2 , \pi )$ 
E)  $\left( - \sqrt { 2 } , \frac { \pi } { 2 } \right) , \left( \sqrt { 2 } , \frac { 3 \pi } { 2 } \right)$ 
A)  $( 2,0 ) , ( - 2 , \pi )$ 
B)  $( \sqrt { 2 } , 0 ) , ( - \sqrt { 2 } , \pi )$ 
C)  $\left( - 2 , \frac { \pi } { 2 } \right) , \left( 2 , \frac { 3 \pi } { 2 } \right)$ 
D)  $( - 2,0 ) , ( 2 , \pi )$ 
E)  $\left( - \sqrt { 2 } , \frac { \pi } { 2 } \right) , \left( \sqrt { 2 } , \frac { 3 \pi } { 2 } \right)$

Question 3

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$2 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 0$$

Accepted Answer

A)  ellipse 
B)  hyperbola 
C)  parabola 
D)  circle 
A)  ellipse 
B)  hyperbola 
C)  parabola 
D)  circle

Question 4

Find the rectangular graph of the following polar equation.
$$\theta = \frac { 5 \pi } { 4 }$$

Accepted Answer

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

Question 5

Find a polar equation of the conic with the given characteristics and with one focus at the pole.
$\begin{array} { c c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \ \text { Hyperbola } & e = 5 & x = 2 \end{array}$

Accepted Answer

A)  $r = \frac { 10 } { 1 - 5 \cos \theta }$ 
B)  $r = \frac { 10 } { 1 + 5 \cos \theta }$ 
C)  $r = \frac { 2 } { 1 + 5 \cos \theta }$ 
D)  $r = \frac { 10 } { 1 + 5 \sin \theta }$ 
E)  $r = \frac { 10 } { 1 - 5 \sin \theta }$ 
A)  $r = \frac { 10 } { 1 - 5 \cos \theta }$ 
B)  $r = \frac { 10 } { 1 + 5 \cos \theta }$ 
C)  $r = \frac { 2 } { 1 + 5 \cos \theta }$ 
D)  $r = \frac { 10 } { 1 + 5 \sin \theta }$ 
E)  $r = \frac { 10 } { 1 - 5 \sin \theta }$

Question 6

Which set of parametric equations represents the following line or conic?
Use $x = h + a \sec \theta$ and $y = k + b \tan \theta$.
Hyperbola: vertices $\quad ( - 5,7 ) , ( - 3,7 )$
foci $( - 8,7 ) , ( 0,7 )$

Accepted Answer

A) 
$\begin{array} { l } 
x = 4 + \sqrt { 15 } \sec \theta \
y = - 7 + \tan \theta
\end{array}$ 
B) 
$\begin{array} { l } 
x = - 4 + \sqrt { 15 } \sec \theta \
y = 7 + \tan \theta
\end{array}$ 
C) 
$\begin{array} { l } 
x = 7 + \sqrt { 15 } \sec \theta \
y = - 4 + \tan \theta
\end{array}$ 
D) 
$\begin{array} { l } 
x = - 4 + \sec \theta \
y = 7 + \sqrt { 15 } \tan \theta
\end{array}$ 
E) 
$\begin{array} { l } 
x = 7 + \sec \theta \
y = - 4 + \sqrt { 15 } \tan \theta
\end{array}$ 
A) 
$\begin{array} { l } 
x = 4 + \sqrt { 15 } \sec \theta \
y = - 7 + \tan \theta
\end{array}$ 
B) 
$\begin{array} { l } 
x = - 4 + \sqrt { 15 } \sec \theta \
y = 7 + \tan \theta
\end{array}$ 
C) 
$\begin{array} { l } 
x = 7 + \sqrt { 15 } \sec \theta \
y = - 4 + \tan \theta
\end{array}$ 
D) 
$\begin{array} { l } 
x = - 4 + \sec \theta \
y = 7 + \sqrt { 15 } \tan \theta
\end{array}$ 
E) 
$\begin{array} { l } 
x = 7 + \sec \theta \
y = - 4 + \sqrt { 15 } \tan \theta
\end{array}$

Question 7

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)  $y = - \frac { \sqrt { 3 } } { 3 }$

Question 8

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$.
$$3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$$

Accepted Answer

A)  ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { 4 \sqrt { 3 } x + 33 }$ 
B)  ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
C)  parabola; $y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
D)  parabola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
E)  hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
A)  ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { 4 \sqrt { 3 } x + 33 }$ 
B)  ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
C)  parabola; $y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
D)  parabola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
E)  hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$

Question 9

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 10

Find the standard form of the equation of the ellipse with the following characteristics.
foci: $( \pm 4,0 ) \quad$ major axis of length: 14

Accepted Answer

A)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
B)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 16 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 16 } = 1$ 
E)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 180 } = 1$ 
A)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
B)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 16 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 16 } = 1$ 
E)  $\frac { x ^ { 2 } } { 196 } + \frac { y ^ { 2 } } { 180 } = 1$

Question 11

Find the vertex and focus of the parabola.
$$x ^ { 2 } - 8 y = 0$$

Accepted Answer

A)  vertex: $( 2,0 ) \quad$ focus: $( 0,0 )$ 
B)  vertex: $( - 2,0 )$ focus: $( 0,0 )$ 
C)  vertex: $( 0,0 ) \quad$ focus: $( 2,0 )$ 
D)  vertex: $( 0,0 ) \quad$ focus: $( 0,2 )$ 
E)  vertex: $( 0,0 ) \quad$ focus: $( 0 , - 2 )$ 
A)  vertex: $( 2,0 ) \quad$ focus: $( 0,0 )$ 
B)  vertex: $( - 2,0 )$ focus: $( 0,0 )$ 
C)  vertex: $( 0,0 ) \quad$ focus: $( 2,0 )$ 
D)  vertex: $( 0,0 ) \quad$ focus: $( 0,2 )$ 
E)  vertex: $( 0,0 ) \quad$ focus: $( 0 , - 2 )$

Question 12

Find the center and foci of the ellipse.
$\frac { ( x - 9 ) ^ { 2 } } { 32 } + \frac { ( y + 5 ) ^ { 2 } } { 36 } = 1$

Accepted Answer

A)  center: $( - 9,5 ) \quad$ foci: $( - 9,3 ) , ( - 9,7 )$ 
B)  center: $( 9 , - 5 ) \quad$ foci: $( 9 , - 7 ) , ( 9 , - 3 )$ 
C)  center: $( 9 , - 5 ) \quad$ foci: $( 7 , - 5 ) , ( 11 , - 5 )$ 
D)  center: $( - 9,5 ) \quad$ foci: $( - 11 , - 5 ) , ( - 7 , - 5 )$ 
E)  center: $( 9 , - 5 ) \quad$ foci: $( - 11,5 ) , ( - 7,5 )$ 
A)  center: $( - 9,5 ) \quad$ foci: $( - 9,3 ) , ( - 9,7 )$ 
B)  center: $( 9 , - 5 ) \quad$ foci: $( 9 , - 7 ) , ( 9 , - 3 )$ 
C)  center: $( 9 , - 5 ) \quad$ foci: $( 7 , - 5 ) , ( 11 , - 5 )$ 
D)  center: $( - 9,5 ) \quad$ foci: $( - 11 , - 5 ) , ( - 7 , - 5 )$ 
E)  center: $( 9 , - 5 ) \quad$ foci: $( - 11,5 ) , ( - 7,5 )$

Question 13

Find the graph of the following polar equation.
$$r = - 3 \sin ( 3 \theta )$$

Accepted Answer

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

Question 14

Find the eccentricity of the following ellipse. Round your answer to two decimals.
$$4 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 0$$

Accepted Answer

A)  $1.25$ 
B)  $1.50$ 
C)  $329.06$ 
D)  $0.75$ 
E)  $0.67$ 
A)  $1.25$ 
B)  $1.50$ 
C)  $329.06$ 
D)  $0.75$ 
E)  $0.67$

Question 15

Find an interval for $\theta$ for which the graph is traced only once.
$$r = 2 \sin \left( \frac { 3 \theta } { 2 } \right)$$

Accepted Answer

A) 
$0 \leq \theta < \frac { \pi } { 3 }$ 
B) 
$0 \leq \theta < \pi$ 
C) 
$0 \leq \theta < 4 \pi$ 
D) 
$0 \leq \theta < \frac { 4 \pi } { 3 }$ 
E) 
$- 2 \pi \leq \theta < 2 \pi$ 
A) 
$0 \leq \theta < \frac { \pi } { 3 }$ 
B) 
$0 \leq \theta < \pi$ 
C) 
$0 \leq \theta < 4 \pi$ 
D) 
$0 \leq \theta < \frac { 4 \pi } { 3 }$ 
E) 
$- 2 \pi \leq \theta < 2 \pi$

Question 16

Match the graph with its equation.

Accepted Answer

A) 
$\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 4 } = 1$ 
B) 
$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ 
C) 
$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ 
D) 
$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ 
E) 
$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ 
A) 
$\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 4 } = 1$ 
B) 
$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ 
C) 
$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ 
D) 
$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ 
E) 
$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$

Question 17

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
$x = \left( v _ { 0 } \cos \theta \right) t$ and $y = \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$.
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.
$\theta = 55 ^ { \circ } , \quad v _ { 0 } = 56$ 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)  $\quad$ maximum height $\approx 43$ feet 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)  $\quad$ maximum height $\approx 43$ feet range $\approx 8$ feet

Question 18

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = 5 + 4 \cos ( \theta )$$

Accepted Answer

A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries. 
A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries.

Question 19

Find the center and foci of the following ellipse.
$$4 x ^ { 2 } + 8 y ^ { 2 } - 16 x + 32 y + 32 = 0$$

Accepted Answer

A)  Center: $( - 2,2 )$
Foci: $( - 2 + \sqrt { 2 } , 2 ) , ( - 2 - \sqrt { 2 } , 2 )$ 
B)  Center: $( 2 , - 2 )$
Foci: $( 2 + \sqrt { 2 } , - 2 ) , ( 2 - \sqrt { 2 } , - 2 )$ 
C)  Center: $( 2 , - 2 )$
Foci: $( 2 , - 2 + \sqrt { 2 } ) , ( 2 , - 2 - \sqrt { 2 } )$ 
D)  Center: $( - 2,2 )$
Foci: $( - 2,2 + \sqrt { 2 } ) , ( - 2,2 - \sqrt { 2 } )$ 
E)  Center: $( - 2,2 )$
Foci: $( - 2 + \sqrt { 6 } , 2 ) , ( - 2 - \sqrt { 6 } , - 2 )$ 
A)  Center: $( - 2,2 )$
Foci: $( - 2 + \sqrt { 2 } , 2 ) , ( - 2 - \sqrt { 2 } , 2 )$ 
B)  Center: $( 2 , - 2 )$
Foci: $( 2 + \sqrt { 2 } , - 2 ) , ( 2 - \sqrt { 2 } , - 2 )$ 
C)  Center: $( 2 , - 2 )$
Foci: $( 2 , - 2 + \sqrt { 2 } ) , ( 2 , - 2 - \sqrt { 2 } )$ 
D)  Center: $( - 2,2 )$
Foci: $( - 2,2 + \sqrt { 2 } ) , ( - 2,2 - \sqrt { 2 } )$ 
E)  Center: $( - 2,2 )$
Foci: $( - 2 + \sqrt { 6 } , 2 ) , ( - 2 - \sqrt { 6 } , - 2 )$

Question 20

Match the graph with its equation.

Accepted Answer

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

Find three additional polar representations of the point $\left( 5 , - \frac { \pi } { 3 } \right)$ , given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$ .

Find two sets of polar coordinates with $0 \leq \theta < 2 \pi$ for the point $( - 2,0 )$ , given in rectangular coordinates.

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $2 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 0$

Find the rectangular graph of the following polar equation. $\theta = \frac { 5 \pi } { 4 }$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Hyperbola e=5 x=2

Which set of parametric equations represents the following line or conic? Use $x = h + a \sec \theta$ and $y = k + b \tan \theta$ . Hyperbola: vertices $\quad ( - 5,7 ) , ( - 3,7 )$ foci $( - 8,7 ) , ( 0,7 )$

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

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$ . $3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$

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 standard form of the equation of the ellipse with the following characteristics. foci: $( \pm 4,0 ) \quad$ major axis of length: 14

Find the vertex and focus of the parabola. $x ^ { 2 } - 8 y = 0$

Find the center and foci of the ellipse. $\frac { ( x - 9 ) ^ { 2 } } { 32 } + \frac { ( y + 5 ) ^ { 2 } } { 36 } = 1$

Find the graph of the following polar equation. $r = - 3 \sin ( 3 \theta )$

Find the eccentricity of the following ellipse. Round your answer to two decimals. $4 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 0$

Find an interval for $\theta$ for which the graph is traced only once. $r = 2 \sin \left( \frac { 3 \theta } { 2 } \right)$

Match the graph with its equation.

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = 5 + 4 \cos ( \theta )$

Find the center and foci of the following ellipse. $4 x ^ { 2 } + 8 y ^ { 2 } - 16 x + 32 y + 32 = 0$

Match the graph with its equation.

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Exam 9: Topics in Analytic Geometry

Find three additional polar representations of the point (5,−π3)\left( 5 , - \frac { \pi } { 3 } \right)(5,−3π​) , given in polar coordinates, using −2π<θ<2π- 2 \pi < \theta < 2 \pi−2π<θ<2π .

Find two sets of polar coordinates with 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π for the point (−2,0)( - 2,0 )(−2,0) , given in rectangular coordinates.

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 2x2+5y2+8x−2y=02 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 02x2+5y2+8x−2y=0

Find the rectangular graph of the following polar equation. θ=5π4\theta = \frac { 5 \pi } { 4 }θ=45π​

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Hyperbola e=5 x=2

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

Use the discriminant to classify the graph; then use the quadratic formula to solve for yyy . 3x2−23xy+y2+103x−6y−24=03 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 03x2−23​xy+y2+103​x−6y−24=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 standard form of the equation of the ellipse with the following characteristics. foci: (±4,0)( \pm 4,0 ) \quad(±4,0) major axis of length: 14

Find the vertex and focus of the parabola. x2−8y=0x ^ { 2 } - 8 y = 0x2−8y=0

Find the center and foci of the ellipse. (x−9)232+(y+5)236=1\frac { ( x - 9 ) ^ { 2 } } { 32 } + \frac { ( y + 5 ) ^ { 2 } } { 36 } = 132(x−9)2​+36(y+5)2​=1

Find the graph of the following polar equation. r=−3sin⁡(3θ)r = - 3 \sin ( 3 \theta )r=−3sin(3θ)

Find the eccentricity of the following ellipse. Round your answer to two decimals. 4x2+9y2−24x+18y−36=04 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 04x2+9y2−24x+18y−36=0

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

Match the graph with its equation.

Test the graph of the following equation for symmetry with respect to θ=π2\theta = \frac { \pi } { 2 }θ=2π​ , the polar axis, and the pole. r=5+4cos⁡(θ)r = 5 + 4 \cos ( \theta )r=5+4cos(θ)

Find the center and foci of the following ellipse. 4x2+8y2−16x+32y+32=04 x ^ { 2 } + 8 y ^ { 2 } - 16 x + 32 y + 32 = 04x2+8y2−16x+32y+32=0

Match the graph with its equation.

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Find three additional polar representations of the point $\left( 5 , - \frac { \pi } { 3 } \right)$ , given in polar coordinates, using $- 2 \pi < \theta < 2 \pi$ .

Find two sets of polar coordinates with $0 \leq \theta < 2 \pi$ for the point $( - 2,0 )$ , given in rectangular coordinates.

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $2 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 0$

Find the rectangular graph of the following polar equation. $\theta = \frac { 5 \pi } { 4 }$

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

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$ . $3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$

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 standard form of the equation of the ellipse with the following characteristics. foci: $( \pm 4,0 ) \quad$ major axis of length: 14

Find the vertex and focus of the parabola. $x ^ { 2 } - 8 y = 0$

Find the center and foci of the ellipse. $\frac { ( x - 9 ) ^ { 2 } } { 32 } + \frac { ( y + 5 ) ^ { 2 } } { 36 } = 1$

Find the graph of the following polar equation. $r = - 3 \sin ( 3 \theta )$

Find the eccentricity of the following ellipse. Round your answer to two decimals. $4 x ^ { 2 } + 9 y ^ { 2 } - 24 x + 18 y - 36 = 0$

Find an interval for $\theta$ for which the graph is traced only once. $r = 2 \sin \left( \frac { 3 \theta } { 2 } \right)$

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = 5 + 4 \cos ( \theta )$

Find the center and foci of the following ellipse. $4 x ^ { 2 } + 8 y ^ { 2 } - 16 x + 32 y + 32 = 0$