Question 1

The graph of the following polar equation is a circle. Sketch its graph, and find its radius.
$r = 5 \sin ( \theta )$

Accepted Answer

A)

radius: 5 
B) 
 
radius: $2.5$ 
C) 
 
radius: 5 
D) 
 
radius: 5 
E) 
 
radius: 25 
A)

radius: 5 
B) 
 
radius: $2.5$ 
C) 
 
radius: 5 
D) 
 
radius: 5 
E) 
 
radius: 25

Question 2

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$3 y ^ { 2 } + 19 x ^ { 2 } + y - 115 = 0$$

Accepted Answer

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

Question 3

Which answer is a polar form of the given rectangular equation? $9 x y = 144$

Accepted Answer

A)  $r ^ { 2 } = 16 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 4 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 4 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 16 \sin \theta \cos \theta$ 
E)  $\theta = 16 \sin \theta \cos \theta$ 
A)  $r ^ { 2 } = 16 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 4 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 4 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 16 \sin \theta \cos \theta$ 
E)  $\theta = 16 \sin \theta \cos \theta$

Question 4

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 5

Find the vertices and asymptotes of the hyperbola.
$$9 y ^ { 2 } - 4 x ^ { 2 } = 36$$

Accepted Answer

A)  vertices: $( 0 , \pm 2 )$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
B)  vertices: $( 0 , \pm 2 )$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
C)  vertices: $( \pm 2,0 )$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
D)  vertices: $( \pm 2,0 )$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
E)  vertices: $( \pm 2,3 )$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
A)  vertices: $( 0 , \pm 2 )$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
B)  vertices: $( 0 , \pm 2 )$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
C)  vertices: $( \pm 2,0 )$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
D)  vertices: $( \pm 2,0 )$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
E)  vertices: $( \pm 2,3 )$ asymptote: $y = \pm \frac { 2 } { 3 } x$

Question 6

Use a graphing utility to graph the polar equation. Describe the resulting graph. $r = 4 \cos ( 7 \theta - 2 )$

Accepted Answer

A)  rose curve with 7 petals, rotated 2 radians 
B)  rose curve with 7 petals, rotated 4 radians 
C)  lemniscate rotated 4 radians 
D)  limaçon rotated 4 radians 
E)  cardioid rotated 2 radians 
A)  rose curve with 7 petals, rotated 2 radians 
B)  rose curve with 7 petals, rotated 4 radians 
C)  lemniscate rotated 4 radians 
D)  limaçon rotated 4 radians 
E)  cardioid rotated 2 radians

Question 7

Find the standard form of the parabola with the given characteristics.
directrix: $x = - 1$
vertex: $( 7 , - 1 )$

Accepted Answer

A)  $( x + 7 ) ^ { 2 } = 32 ( y - 1 )$ 
B)  $( x - 7 ) ^ { 2 } = 32 ( y + 1 )$ 
C)  $( x - 1 ) ^ { 2 } = 32 ( y + 7 )$ 
D)  $( y - 1 ) ^ { 2 } = 32 ( x + 7 )$ 
E)  $( y + 1 ) ^ { 2 } = 32 ( x - 7 )$ 
A)  $( x + 7 ) ^ { 2 } = 32 ( y - 1 )$ 
B)  $( x - 7 ) ^ { 2 } = 32 ( y + 1 )$ 
C)  $( x - 1 ) ^ { 2 } = 32 ( y + 7 )$ 
D)  $( y - 1 ) ^ { 2 } = 32 ( x + 7 )$ 
E)  $( y + 1 ) ^ { 2 } = 32 ( x - 7 )$

Question 8

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

Accepted Answer

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

Question 9

Rotate the axes to eliminate the $x y$-term in the following equation. $18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$

Accepted Answer

A)  $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$ 
B)  $6 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } + 6 \sqrt { 3 } x ^ { \prime } = 0$ 
C)  $6 \left( x ^ { \prime } \right) ^ { 2 } - 6 y ^ { \prime } = 0$ 
D)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { x ^ { \prime } } { 6 } = 0$ 
E)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } + \frac { y ^ { \prime } } { 12 } = 0$ 
A)  $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$ 
B)  $6 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } + 6 \sqrt { 3 } x ^ { \prime } = 0$ 
C)  $6 \left( x ^ { \prime } \right) ^ { 2 } - 6 y ^ { \prime } = 0$ 
D)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { x ^ { \prime } } { 6 } = 0$ 
E)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } + \frac { y ^ { \prime } } { 12 } = 0$

Question 10

Planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is $2 a$ (see figure). The polar equation of the orbit of a planet is given below, where $e$ is the eccentricity. If $a = 88.908 \times 10 ^ { 6 }$ miles and $e = 0.0260$, find the perihelion distance (the minimum distance from the sun to the planet). Round your answer to the nearest mile.

Accepted Answer

A)  $86,596,392$ miles 
B)  $91,219,608$ miles 
C)  $8,659,639$ miles 
D)  $9,121,961$ miles 
E)  $865,963,920$ miles 
A)  $86,596,392$ miles 
B)  $91,219,608$ miles 
C)  $8,659,639$ miles 
D)  $9,121,961$ miles 
E)  $865,963,920$ miles

Question 11

Find the standard form of the equation of the ellipse with the given characteristics.
vertices: $( 4 , - 6 ) , ( 4,10 ) \quad$ minor axis of length: 4

Accepted Answer

A)  $\frac { ( x + 4 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 64 } = 1$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 4 } + \frac { ( y - 2 ) ^ { 2 } } { 64 } = 1$ 
C)  $\frac { ( x - 4 ) ^ { 2 } } { 64 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 64 } + \frac { ( y - 4 ) ^ { 2 } } { 4 } = 1$ 
E)  $\frac { ( x + 2 ) ^ { 2 } } { 64 } + \frac { ( y + 4 ) ^ { 2 } } { 4 } = 1$ 
A)  $\frac { ( x + 4 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 64 } = 1$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 4 } + \frac { ( y - 2 ) ^ { 2 } } { 64 } = 1$ 
C)  $\frac { ( x - 4 ) ^ { 2 } } { 64 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 64 } + \frac { ( y - 4 ) ^ { 2 } } { 4 } = 1$ 
E)  $\frac { ( x + 2 ) ^ { 2 } } { 64 } + \frac { ( y + 4 ) ^ { 2 } } { 4 } = 1$

Question 12

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

Accepted Answer

A)  $r = \frac { 12 } { 4 - 3 \sin \theta }$ 
B)  $r = \frac { 12 } { 4 + 3 \cos \theta }$ 
C)  $r = \frac { 12 } { 4 - 3 \cos \theta }$ 
D)  $r = \frac { 12 } { 4 + 3 \sin \theta }$ 
E)  $r = \frac { 12 } { 3 - 5 \cos \theta }$ 
A)  $r = \frac { 12 } { 4 - 3 \sin \theta }$ 
B)  $r = \frac { 12 } { 4 + 3 \cos \theta }$ 
C)  $r = \frac { 12 } { 4 - 3 \cos \theta }$ 
D)  $r = \frac { 12 } { 4 + 3 \sin \theta }$ 
E)  $r = \frac { 12 } { 3 - 5 \cos \theta }$

Question 13

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 14

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

Accepted Answer

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

Question 15

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$y ^ { 2 } + 13 x + y - 79 = 0$

Accepted Answer

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

Question 16

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 17

Sketch the graph of the ellipse, using the latera recta.
$$4 x ^ { 2 } + 9 y ^ { 2 } = 36$$

Accepted Answer

The answer of Sketch the graph of the ellipse, using...

Question 18

Rotate the axes to eliminate the $x y$-term in the following equation and then write the equation in standard form.
$$3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$$

Accepted Answer

A)  $\left( x ^ { \prime } - 1 \right) ^ { 2 } = 3 \left( y ^ { \prime } - \frac { 4 } { 5 } \right)$ 
B)  $\left( x ^ { \prime } + 3 \right) ^ { 2 } = - 5 \left( y ^ { \prime } + \frac { 1 } { 3 } \right)$ 
C)  $\left( y ^ { \prime } + 1 \right) ^ { 2 } = - \left( x ^ { \prime } - \frac { 4 } { 5 } \right)$ 
D)  $\left( y ^ { \prime } - \frac { 4 } { 5 } \right) ^ { 2 } = 3 \left( x ^ { \prime } - 1 \right)$ 
E)  $\left( y ^ { \prime } + \frac { 4 } { 5 } \right) ^ { 2 } = x ^ { \prime } + \frac { 1 } { 3 }$ 
A)  $\left( x ^ { \prime } - 1 \right) ^ { 2 } = 3 \left( y ^ { \prime } - \frac { 4 } { 5 } \right)$ 
B)  $\left( x ^ { \prime } + 3 \right) ^ { 2 } = - 5 \left( y ^ { \prime } + \frac { 1 } { 3 } \right)$ 
C)  $\left( y ^ { \prime } + 1 \right) ^ { 2 } = - \left( x ^ { \prime } - \frac { 4 } { 5 } \right)$ 
D)  $\left( y ^ { \prime } - \frac { 4 } { 5 } \right) ^ { 2 } = 3 \left( x ^ { \prime } - 1 \right)$ 
E)  $\left( y ^ { \prime } + \frac { 4 } { 5 } \right) ^ { 2 } = x ^ { \prime } + \frac { 1 } { 3 }$

Question 19

Use a graphing utility to graph the rotated conic.
$$r = \frac { 1 } { 3 - 3 \sin ( \theta - 2 \pi / 3 ) }$$
Use either grid below for your graph, whichever is more convenient.

Accepted Answer

The answer of Use a graphing utility to graph the...

Question 20

Find the standard form of the equation of the ellipse with the given characteristics.
foci: $( - 2,6 ) , ( - 2,10 )$
endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$

Accepted Answer

A)  $\frac { ( x - 2 ) ^ { 2 } } { 77 } + \frac { ( y + 8 ) ^ { 2 } } { 81 } = 1$ 
B)  $\frac { ( x + 2 ) ^ { 2 } } { 77 } + \frac { ( y - 8 ) ^ { 2 } } { 81 } = 1$ 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y - 8 ) ^ { 2 } } { 77 } = 1$ 
D)  $\frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 77 } = 1$ 
E)  $\frac { ( x + 8 ) ^ { 2 } } { 81 } + \frac { ( y - 2 ) ^ { 2 } } { 77 } = 1$ 
A)  $\frac { ( x - 2 ) ^ { 2 } } { 77 } + \frac { ( y + 8 ) ^ { 2 } } { 81 } = 1$ 
B)  $\frac { ( x + 2 ) ^ { 2 } } { 77 } + \frac { ( y - 8 ) ^ { 2 } } { 81 } = 1$ 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 81 } + \frac { ( y - 8 ) ^ { 2 } } { 77 } = 1$ 
D)  $\frac { ( x - 8 ) ^ { 2 } } { 81 } + \frac { ( y + 2 ) ^ { 2 } } { 77 } = 1$ 
E)  $\frac { ( x + 8 ) ^ { 2 } } { 81 } + \frac { ( y - 2 ) ^ { 2 } } { 77 } = 1$

The graph of the following polar equation is a circle. Sketch its graph, and find its radius. $r = 5 \sin ( \theta )$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $3 y ^ { 2 } + 19 x ^ { 2 } + y - 115 = 0$

Which answer is a polar form of the given rectangular equation? $9 x y = 144$

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

Find the vertices and asymptotes of the hyperbola. $9 y ^ { 2 } - 4 x ^ { 2 } = 36$

Use a graphing utility to graph the polar equation. Describe the resulting graph. $r = 4 \cos ( 7 \theta - 2 )$

Find the standard form of the parabola with the given characteristics. directrix: $x = - 1$ vertex: $( 7 , - 1 )$

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

Rotate the axes to eliminate the $x y$ -term in the following equation. $18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$

Find the standard form of the equation of the ellipse with the given characteristics. vertices: $( 4 , - 6 ) , ( 4,10 ) \quad$ minor axis of length: 4

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

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 13 x + y - 79 = 0$

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

Sketch the graph of the ellipse, using the latera recta. $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ $Sketch the graph of the ellipse, using the latera recta. 4 x ^ { 2 } + 9 y ^ { 2 } = 36$

Rotate the axes to eliminate the $x y$ -term in the following equation and then write the equation in standard form. $3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$

Find the standard form of the equation of the ellipse with the given characteristics. foci: $( - 2,6 ) , ( - 2,10 )$ endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$

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

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Limits and an Introduction to Calculus

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Exam 9: Topics in Analytic Geometry

The graph of the following polar equation is a circle. Sketch its graph, and find its radius. r=5sin⁡(θ)r = 5 \sin ( \theta )r=5sin(θ)

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 3y2+19x2+y−115=03 y ^ { 2 } + 19 x ^ { 2 } + y - 115 = 03y2+19x2+y−115=0

Which answer is a polar form of the given rectangular equation? 9xy=1449 x y = 1449xy=144

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

Find the vertices and asymptotes of the hyperbola. 9y2−4x2=369 y ^ { 2 } - 4 x ^ { 2 } = 369y2−4x2=36

Use a graphing utility to graph the polar equation. Describe the resulting graph. r=4cos⁡(7θ−2)r = 4 \cos ( 7 \theta - 2 )r=4cos(7θ−2)

Find the standard form of the parabola with the given characteristics. directrix: x=−1x = - 1x=−1 vertex: (7,−1)( 7 , - 1 )(7,−1)

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

Rotate the axes to eliminate the xyx yxy -term in the following equation. 18x2−123xy+6y2+12x+123y=018 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 018x2−123​xy+6y2+12x+123​y=0

Find the standard form of the equation of the ellipse with the given characteristics. vertices: (4,−6),(4,10)( 4 , - 6 ) , ( 4,10 ) \quad(4,−6),(4,10) minor axis of length: 4

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

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 three additional polar representations of the point (5,−π2)\left( 5 , - \frac { \pi } { 2 } \right)(5,−2π​) , given in polar coordinates, using −2π<θ<2π- 2 \pi < \theta < 2 \pi−2π<θ<2π .

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. y2+13x+y−79=0y ^ { 2 } + 13 x + y - 79 = 0y2+13x+y−79=0

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

Sketch the graph of the ellipse, using the latera recta. 4x2+9y2=364 x ^ { 2 } + 9 y ^ { 2 } = 364x2+9y2=36

Rotate the axes to eliminate the xyx yxy -term in the following equation and then write the equation in standard form. 3x2−12xy+12y2+155y+3=03 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 03x2−12xy+12y2+155​y+3=0

Use a graphing utility to graph the rotated conic. r=13−3sin⁡(θ−2π/3)r = \frac { 1 } { 3 - 3 \sin ( \theta - 2 \pi / 3 ) }r=3−3sin(θ−2π/3)1​ Use either grid below for your graph, whichever is more convenient.

Find the standard form of the equation of the ellipse with the given characteristics. foci: (−2,6),(−2,10)( - 2,6 ) , ( - 2,10 )(−2,6),(−2,10) endpoints of the major axis: (−2,−1),(−2,17)( - 2 , - 1 ) , ( - 2,17 )(−2,−1),(−2,17)

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

The graph of the following polar equation is a circle. Sketch its graph, and find its radius. $r = 5 \sin ( \theta )$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $3 y ^ { 2 } + 19 x ^ { 2 } + y - 115 = 0$

Which answer is a polar form of the given rectangular equation? $9 x y = 144$

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

Find the vertices and asymptotes of the hyperbola. $9 y ^ { 2 } - 4 x ^ { 2 } = 36$

Use a graphing utility to graph the polar equation. Describe the resulting graph. $r = 4 \cos ( 7 \theta - 2 )$

Find the standard form of the parabola with the given characteristics. directrix: $x = - 1$ vertex: $( 7 , - 1 )$

Rotate the axes to eliminate the $x y$ -term in the following equation. $18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$

Find the standard form of the equation of the ellipse with the given characteristics. vertices: $( 4 , - 6 ) , ( 4,10 ) \quad$ minor axis of length: 4

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 13 x + y - 79 = 0$

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

Sketch the graph of the ellipse, using the latera recta. $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ $Sketch the graph of the ellipse, using the latera recta. 4 x ^ { 2 } + 9 y ^ { 2 } = 36$

Rotate the axes to eliminate the $x y$ -term in the following equation and then write the equation in standard form. $3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$

Find the standard form of the equation of the ellipse with the given characteristics. foci: $( - 2,6 ) , ( - 2,10 )$ endpoints of the major axis: $( - 2 , - 1 ) , ( - 2,17 )$