Question 1

Test for symmetry with respect to $\theta = \pi / 2$, the polar axis, and the pole.
$$r = \frac { 6 } { 5 + \sin \theta }$$

Accepted Answer

A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$ 
A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$ C

Question 2

Consider the parametric equations $x = 5 \sqrt { t }$ and $y = 2 + 7 t$. Find the rectangular equation by eliminating the parameter.

Accepted Answer

A)  $y = \frac { 7 } { 25 } x ^ { 2 } + \frac { 2 } { 25 }$ 
B)  $y = 175 x ^ { 2 } + 2$ 
C)  $y = \frac { 1 } { 25 } x ^ { 2 } - \frac { 2 } { 7 }$ 
D)  $y = \frac { 7 } { 25 } x ^ { 2 } + 2$ 
E)  $y = \frac { 1 } { 175 } x ^ { 2 } + 14$ 
A)  $y = \frac { 7 } { 25 } x ^ { 2 } + \frac { 2 } { 25 }$ 
B)  $y = 175 x ^ { 2 } + 2$ 
C)  $y = \frac { 1 } { 25 } x ^ { 2 } - \frac { 2 } { 7 }$ 
D)  $y = \frac { 7 } { 25 } x ^ { 2 } + 2$ 
E)  $y = \frac { 1 } { 175 } x ^ { 2 } + 14$ D

Question 3

Which answer is a rectangular form of the given polar equation? $r = 10 \cos \theta$

Accepted Answer

A)  $( x - 5 ) ^ { 2 } + y ^ { 2 } = 25$ 
B)  $( x + 5 ) ^ { 2 } + y ^ { 2 } = 25$ 
C)  $( x + 5 ) ^ { 2 } + y ^ { 2 } = 5$ 
D)  $x ^ { 2 } + ( y + 5 ) ^ { 2 } = 25$ 
E)  $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 25$ 
A)  $( x - 5 ) ^ { 2 } + y ^ { 2 } = 25$ 
B)  $( x + 5 ) ^ { 2 } + y ^ { 2 } = 25$ 
C)  $( x + 5 ) ^ { 2 } + y ^ { 2 } = 5$ 
D)  $x ^ { 2 } + ( y + 5 ) ^ { 2 } = 25$ 
E)  $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 25$ A

Question 4

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ?
$y = x ^ { 2 } + 9$
$x = t + 6$

Accepted Answer

A) 
$\begin{array} { l } 
y = ( t + 6 ) ^ { 2 } - 9 \
x = t - 6
\end{array}$ 
B) 
$\begin{array} { l } 
y = ( t - 6 ) ^ { 2 } - 9 \
x = t - 6
\end{array}$ 
C) 
$\begin{array} { l } 
y = ( t - 6 ) ^ { 2 } + 9 \
x = 6 + t
\end{array}$ 
D) 
$\begin{array} { l } 
y = ( 6 + t ) ^ { 2 } + 9 \
x = 6 - t
\end{array}$ 
E) 
$y = ( 6 - t ) ^ { 2 } + 9$ 
A) 
$\begin{array} { l } 
y = ( t + 6 ) ^ { 2 } - 9 \
x = t - 6
\end{array}$ 
B) 
$\begin{array} { l } 
y = ( t - 6 ) ^ { 2 } - 9 \
x = t - 6
\end{array}$ 
C) 
$\begin{array} { l } 
y = ( t - 6 ) ^ { 2 } + 9 \
x = 6 + t
\end{array}$ 
D) 
$\begin{array} { l } 
y = ( 6 + t ) ^ { 2 } + 9 \
x = 6 - t
\end{array}$ 
E) 
$y = ( 6 - t ) ^ { 2 } + 9$

Question 5

Identify the center and radius of the circle below.
$$x ^ { 2 } + y ^ { 2 } + 6 x + 6 y - 5 = 0$$

Accepted Answer

A)  Center: $( 3,3 )$
Radius: $\sqrt { 11 }$ 
B)  Center: $( - 3 , - 3 )$
Radius: $\sqrt { 11 }$ 
C)  Center: $( 3,3 )$
Radius: $\sqrt { 23 }$ 
D)  Center: $( - 3 , - 3 )$
Radius: $\sqrt { 23 }$ 
E)  Center: $( - 6 , - 6 )$
Radius: $\sqrt { 77 }$ 
A)  Center: $( 3,3 )$
Radius: $\sqrt { 11 }$ 
B)  Center: $( - 3 , - 3 )$
Radius: $\sqrt { 11 }$ 
C)  Center: $( 3,3 )$
Radius: $\sqrt { 23 }$ 
D)  Center: $( - 3 , - 3 )$
Radius: $\sqrt { 23 }$ 
E)  Center: $( - 6 , - 6 )$
Radius: $\sqrt { 77 }$

Question 6

Convert the following polar equation to rectangular form.
$$\theta = \frac { 2 \pi } { 3 }$$

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 7

Find the vertex and focus of the parabola below.
$$y ^ { 2 } + 10 y - 8 x + 30 = 0$$

Accepted Answer

A)  Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 11 } { 8 } , 5 \right)$ 
B)  Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 21 } { 8 } , - 5 \right)$ 
C)  Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 69 } { 8 } , - 5 \right)$ 
D)  Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , 5 \right)$ 
E)  Vertex: $\left( - \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , - 5 \right)$ 
A)  Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 11 } { 8 } , 5 \right)$ 
B)  Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 21 } { 8 } , - 5 \right)$ 
C)  Vertex: $\left( \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 69 } { 8 } , - 5 \right)$ 
D)  Vertex: $\left( - \frac { 5 } { 8 } , 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , 5 \right)$ 
E)  Vertex: $\left( - \frac { 5 } { 8 } , - 5 \right)$
Focus: $\left( \frac { 59 } { 8 } , - 5 \right)$

Question 8

Find the standard form of the equation of the ellipse with the following characteristics.
foci: $( \pm 8,0 )$
major axis of length: 22

Accepted Answer

A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 57 } = 1$ 
B)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 121 } = 1$ 
D)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 64 } = 1$ 
E)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 420 } = 1$ 
A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 57 } = 1$ 
B)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 121 } = 1$ 
D)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 64 } = 1$ 
E)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 420 } = 1$

Question 9

Test for symmetry with respect to $\theta = \pi / 2$, the polar axis, and the pole.
$$r = \frac { 2 } { 4 + \sin \theta }$$

Accepted Answer

A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$ 
A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$

Question 10

Which set of parametric equations represents the following line or conic?
Use $x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right)$ and $y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right)$.
Line: passes through $( 7,6 )$ and $( - 4,3 )$

Accepted Answer

A) 
$\begin{array} { l } 
x = - 11 t + 7 \
y = - 3 t + 6
\end{array}$ 
B) 
$x = - 11 t + 7$
$y = 3 t + 6$ 
C) 
$x = - 11 t - 7$
$y = - 3 t - 6$ 
D) 
$x = - 3 t + 6$
$y = - 11 t + 7$ 
E) 
$\begin{array} { l } 
x = - 3 t - 6 \
y = - 11 t - 7
\end{array}$ 
A) 
$\begin{array} { l } 
x = - 11 t + 7 \
y = - 3 t + 6
\end{array}$ 
B) 
$x = - 11 t + 7$
$y = 3 t + 6$ 
C) 
$x = - 11 t - 7$
$y = - 3 t - 6$ 
D) 
$x = - 3 t + 6$
$y = - 11 t + 7$ 
E) 
$\begin{array} { l } 
x = - 3 t - 6 \
y = - 11 t - 7
\end{array}$

Question 11

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$3 y ^ { 2 } + 17 x ^ { 2 } + y - 103 = 0$$

Accepted Answer

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

Question 12

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Accepted Answer

A)  $r = \frac { 2 } { 5 + 2 \cos \theta }$ 
B)  $r = \frac { 4 } { 5 + 2 \cos \theta }$ 
C)  $r = \frac { 4 } { 5 - 2 \cos \theta }$ 
D)  $r = \frac { 4 } { 5 - 2 \sin \theta }$ 
E)  $r = \frac { 4 } { 5 + 2 \sin \theta }$ 
A)  $r = \frac { 2 } { 5 + 2 \cos \theta }$ 
B)  $r = \frac { 4 } { 5 + 2 \cos \theta }$ 
C)  $r = \frac { 4 } { 5 - 2 \cos \theta }$ 
D)  $r = \frac { 4 } { 5 - 2 \sin \theta }$ 
E)  $r = \frac { 4 } { 5 + 2 \sin \theta }$

Question 13

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 14

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 15

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)  $( y + 1 ) ^ { 2 } = 4 ( x - 2 )$

Question 16

Sketch the curve represented by the following parametric equations. Indicate the orientation of the curve.
$x = e ^ { t } , y = e ^ { - 2 t }$

Accepted Answer

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

Question 17

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$36 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 0$$

Accepted Answer

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

Question 18

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.
$$\theta = 50 ^ { \circ } , \quad v _ { 0 } = 104 \text { feet per second }$$

Accepted Answer

A)  maximum height $\approx 12$ feet range $\approx 171$ feet 
B)  maximum height $\approx 171$ feet range $\approx 12$ feet 
C)  maximum height $\approx 99$ feet range $\approx 333$ feet 
D)  maximum height $\approx 333$ feet $\quad$ range $\approx 99$ feet 
E)  maximum height $\approx 38$ feet range $\approx 8$ feet 
A)  maximum height $\approx 12$ feet range $\approx 171$ feet 
B)  maximum height $\approx 171$ feet range $\approx 12$ feet 
C)  maximum height $\approx 99$ feet range $\approx 333$ feet 
D)  maximum height $\approx 333$ feet $\quad$ range $\approx 99$ feet 
E)  maximum height $\approx 38$ feet range $\approx 8$ feet

Question 19

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

Accepted Answer

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

Question 20

Find the standard form of the equation of the hyperbola with the given characteristics.
vertices: $( 0 , \pm 4 )$ foci: $( 0 , \pm 8 )$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 64 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 64$ 
E)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 64$ 
A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 64 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 64$ 
E)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 64$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 6 } { 5 + \sin \theta }$

Consider the parametric equations $x = 5 \sqrt { t }$ and $y = 2 + 7 t$ . Find the rectangular equation by eliminating the parameter.

Which answer is a rectangular form of the given polar equation? $r = 10 \cos \theta$

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 9$ $x = t + 6$

Identify the center and radius of the circle below. $x ^ { 2 } + y ^ { 2 } + 6 x + 6 y - 5 = 0$

Convert the following polar equation to rectangular form. $\theta = \frac { 2 \pi } { 3 }$

Find the vertex and focus of the parabola below. $y ^ { 2 } + 10 y - 8 x + 30 = 0$

Find the standard form of the equation of the ellipse with the following characteristics. foci: $( \pm 8,0 )$ major axis of length: 22

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 2 } { 4 + \sin \theta }$

Which set of parametric equations represents the following line or conic? Use $x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right)$ and $y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right)$ . Line: passes through $( 7,6 )$ and $( - 4,3 )$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $3 y ^ { 2 } + 17 x ^ { 2 } + y - 103 = 0$

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

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.

Match the graph with its equation.

Sketch the curve represented by the following parametric equations. Indicate the orientation of the curve. $x = e ^ { t } , y = e ^ { - 2 t }$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $36 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 0$

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

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: $( 0 , \pm 4 )$ foci: $( 0 , \pm 8 )$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Exam 9: Topics in Analytic Geometry

Test for symmetry with respect to θ=π/2\theta = \pi / 2θ=π/2 , the polar axis, and the pole. r=65+sin⁡θr = \frac { 6 } { 5 + \sin \theta }r=5+sinθ6​

Consider the parametric equations x=5tx = 5 \sqrt { t }x=5t​ and y=2+7ty = 2 + 7 ty=2+7t . Find the rectangular equation by eliminating the parameter.

Which answer is a rectangular form of the given polar equation? r=10cos⁡θr = 10 \cos \thetar=10cosθ

Which set of parametric equations represents the graph of the following rectangular equation using t=6−xt = 6 - xt=6−x ? y=x2+9y = x ^ { 2 } + 9y=x2+9 x=t+6x = t + 6x=t+6

Identify the center and radius of the circle below. x2+y2+6x+6y−5=0x ^ { 2 } + y ^ { 2 } + 6 x + 6 y - 5 = 0x2+y2+6x+6y−5=0

Convert the following polar equation to rectangular form. θ=2π3\theta = \frac { 2 \pi } { 3 }θ=32π​

Find the vertex and focus of the parabola below. y2+10y−8x+30=0y ^ { 2 } + 10 y - 8 x + 30 = 0y2+10y−8x+30=0

Find the standard form of the equation of the ellipse with the following characteristics. foci: (±8,0)( \pm 8,0 )(±8,0) major axis of length: 22

Test for symmetry with respect to θ=π/2\theta = \pi / 2θ=π/2 , the polar axis, and the pole. r=24+sin⁡θr = \frac { 2 } { 4 + \sin \theta }r=4+sinθ2​

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 3y2+17x2+y−103=03 y ^ { 2 } + 17 x ^ { 2 } + y - 103 = 03y2+17x2+y−103=0

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

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.

Match the graph with its equation.

Sketch the curve represented by the following parametric equations. Indicate the orientation of the curve. x=et,y=e−2tx = e ^ { t } , y = e ^ { - 2 t }x=et,y=e−2t

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 36x2+16y2+144x+20y−335=036 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 036x2+16y2+144x+20y−335=0

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

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (0,±4)( 0 , \pm 4 )(0,±4) foci: (0,±8)( 0 , \pm 8 )(0,±8)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 6 } { 5 + \sin \theta }$

Consider the parametric equations $x = 5 \sqrt { t }$ and $y = 2 + 7 t$ . Find the rectangular equation by eliminating the parameter.

Which answer is a rectangular form of the given polar equation? $r = 10 \cos \theta$

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 9$ $x = t + 6$

Identify the center and radius of the circle below. $x ^ { 2 } + y ^ { 2 } + 6 x + 6 y - 5 = 0$

Convert the following polar equation to rectangular form. $\theta = \frac { 2 \pi } { 3 }$

Find the vertex and focus of the parabola below. $y ^ { 2 } + 10 y - 8 x + 30 = 0$

Find the standard form of the equation of the ellipse with the following characteristics. foci: $( \pm 8,0 )$ major axis of length: 22

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 2 } { 4 + \sin \theta }$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $3 y ^ { 2 } + 17 x ^ { 2 } + y - 103 = 0$

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.

Sketch the curve represented by the following parametric equations. Indicate the orientation of the curve. $x = e ^ { t } , y = e ^ { - 2 t }$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $36 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 0$

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

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: $( 0 , \pm 4 )$ foci: $( 0 , \pm 8 )$