Question 1

Find the vertex, the focus, and the directrix of the parabola.
-$$\mathrm { x } ^ { 2 } - 10 \mathrm { x } + 8 \mathrm { y } + 9 = 0$$

Accepted Answer

A)  Vertex: $\left( 5 , - \frac { 1 } { 2 } \right)$; Focus: $( 5 , - 6 )$; Directrix: $\mathrm { y } = - 10$ 
B)  Vertex: $\left( 5 , \frac { 15 } { 8 } \right)$; Focus: $( 5 , - 6 ) ;$ Directrix: $y = \frac { 17 } { 8 }$ 
C)  Vertex: $\left( 5 , \frac { 9 } { 2 } \right)$; Focus: $( 5,4 )$; Directrix: $y = 0$ 
D)  Vertex: $( 5,2 )$; Focus: $( 5,0 )$; Directrix: $y = 4$ 
A)  Vertex: $\left( 5 , - \frac { 1 } { 2 } \right)$; Focus: $( 5 , - 6 )$; Directrix: $\mathrm { y } = - 10$ 
B)  Vertex: $\left( 5 , \frac { 15 } { 8 } \right)$; Focus: $( 5 , - 6 ) ;$ Directrix: $y = \frac { 17 } { 8 }$ 
C)  Vertex: $\left( 5 , \frac { 9 } { 2 } \right)$; Focus: $( 5,4 )$; Directrix: $y = 0$ 
D)  Vertex: $( 5,2 )$; Focus: $( 5,0 )$; Directrix: $y = 4$ D

Question 2

Solve the problem.
-A rectangular board is 8 by 19 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the length of the string.

Accepted Answer

A)  8 
B)  19 
C)  16 
D)  38 
A)  8 
B)  19 
C)  16 
D)  38 B

Question 3

Match the polar equation with its graph.
-$$r = \frac { 4 } { 1 - \sin \theta }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
 
 4 
A) 
  
B) 
  
C) 
  
D) 
 
 4 D

Question 4

Find the eccentricity of the ellipse.
-$$23 x ^ { 2 } + y ^ { 2 } = 23$$

Accepted Answer

A)  $\frac { 4 \sqrt { 33 } } { 23 }$ 
B)  $\frac { \sqrt { 23 } } { 22 }$ 
C)  $\frac { 23 } { 4 \sqrt { 33 } }$ 
D)  $\frac { \sqrt { 506 } } { 23 }$ 
A)  $\frac { 4 \sqrt { 33 } } { 23 }$ 
B)  $\frac { \sqrt { 23 } } { 22 }$ 
C)  $\frac { 23 } { 4 \sqrt { 33 } }$ 
D)  $\frac { \sqrt { 506 } } { 23 }$

Question 5

Find the standard form of the equation of the parabola.
-Focus at $( 10,0 )$, directrix $x = - 10$

Accepted Answer

A)  $y = \frac { 1 } { 40 } x ^ { 2 }$ 
B)  $40 \mathrm { y } = \mathrm { x } ^ { 2 }$ 
C)  $x = \frac { 1 } { 40 } y ^ { 2 }$ 
D)  $y ^ { 2 } = 40 x$ 
A)  $y = \frac { 1 } { 40 } x ^ { 2 }$ 
B)  $40 \mathrm { y } = \mathrm { x } ^ { 2 }$ 
C)  $x = \frac { 1 } { 40 } y ^ { 2 }$ 
D)  $y ^ { 2 } = 40 x$

Question 6

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane.
-The hyperbola $x y = 36$

Accepted Answer

A)  $\mathrm { V } : ( 6,6 ) , ( - 6 , - 6 )$
$F : ( 6 \sqrt { 2 } , \sqrt { 2 } ) , ( - 6 \sqrt { 2 } , - \sqrt { 2 } )$ 
B)  $\mathrm { V } : ( 6 \sqrt { 2 } , 6 \sqrt { 2 } ) , ( - 6 \sqrt { 2 } , - 6 \sqrt { 2 } )$
F: $( 6,6 ) , ( - 6 , - 6 )$ 
C)  V: $( 6,6 ) , ( - 6 , - 6 )$
$F : ( 6 \sqrt { 2 } , 6 \sqrt { 2 } ) , ( - 6 \sqrt { 2 } , - 6 \sqrt { 2 } )$ 
D)  V: $( - 6,6 ) , ( 6 , - 6 )$
F: $( - 6 \sqrt { 2 } , 6 \sqrt { 2 } ) , ( 6 \sqrt { 2 } , - 6 \sqrt { 2 } )$ 
A)  $\mathrm { V } : ( 6,6 ) , ( - 6 , - 6 )$
$F : ( 6 \sqrt { 2 } , \sqrt { 2 } ) , ( - 6 \sqrt { 2 } , - \sqrt { 2 } )$ 
B)  $\mathrm { V } : ( 6 \sqrt { 2 } , 6 \sqrt { 2 } ) , ( - 6 \sqrt { 2 } , - 6 \sqrt { 2 } )$
F: $( 6,6 ) , ( - 6 , - 6 )$ 
C)  V: $( 6,6 ) , ( - 6 , - 6 )$
$F : ( 6 \sqrt { 2 } , 6 \sqrt { 2 } ) , ( - 6 \sqrt { 2 } , - 6 \sqrt { 2 } )$ 
D)  V: $( - 6,6 ) , ( 6 , - 6 )$
F: $( - 6 \sqrt { 2 } , 6 \sqrt { 2 } ) , ( 6 \sqrt { 2 } , - 6 \sqrt { 2 } )$

Question 7

Write an equation for the illustrated plane.
-

Accepted Answer

A)  $\mathrm { y } = 3$ 
B)  $x + z = 3$ 
C)  $x + z = 3$ and $y = 0$ 
D)  $x + z = - 3$ 
A)  $\mathrm { y } = 3$ 
B)  $x + z = 3$ 
C)  $x + z = 3$ and $y = 0$ 
D)  $x + z = - 3$

Question 8

Find a polar equation for the conic described.
-The hyperbola with a focus at the pole and transverse axis endpoints $\left( 4 , \frac { \pi } { 2 } \right)$ and $\left( - 8 , \frac { 3 \pi } { 2 } \right)$

Accepted Answer

A)  $r = \frac { 16 } { 1 + 3 \sin \theta }$ 
B)  $r = \frac { 16 } { 1 - 3 \sin \theta }$ 
C)  $r = \frac { 8 } { 2 + 6 \sin \theta }$ 
D)  $r = \frac { 16 } { 1 + 3 \cos \theta }$ 
A)  $r = \frac { 16 } { 1 + 3 \sin \theta }$ 
B)  $r = \frac { 16 } { 1 - 3 \sin \theta }$ 
C)  $r = \frac { 8 } { 2 + 6 \sin \theta }$ 
D)  $r = \frac { 16 } { 1 + 3 \cos \theta }$

Question 9

Determine the eccentricity, the type of conic, and the directrix.
-$$r = \frac { 3 } { 6 + 8 \cos \theta }$$

Accepted Answer

A)  Eccentricity $= \frac { 4 } { 3 } ;$ parabola; directrix $x = - \frac { 9 } { 4 }$ 
B)  Eccentricity $= \frac { 8 } { 3 } ;$ ellipse; directrix $y = - 4$ 
C)  Eccentricity $= \frac { 3 } { 4 }$; ellipse; directrix $x = - 4$ 
D)  Eccentricity $= \frac { 4 } { 3 } ;$ hyperbola; directrix $x = \frac { 3 } { 8 }$ 
A)  Eccentricity $= \frac { 4 } { 3 } ;$ parabola; directrix $x = - \frac { 9 } { 4 }$ 
B)  Eccentricity $= \frac { 8 } { 3 } ;$ ellipse; directrix $y = - 4$ 
C)  Eccentricity $= \frac { 3 } { 4 }$; ellipse; directrix $x = - 4$ 
D)  Eccentricity $= \frac { 4 } { 3 } ;$ hyperbola; directrix $x = \frac { 3 } { 8 }$

Question 10

Match the polar equation with its graph.
-$$r = \frac { 10 } { 4 - 4 \sin \theta }$$

Accepted Answer

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

Question 11

Graph the parabola.
-$$x = \frac { 1 } { 6 } ( y - 4 ) ^ { 2 } + 4$$

Accepted Answer

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

Question 12

Find the center, vertices, and foci of the ellipse with the given equation.
-$$\frac { ( x + 5 ) ^ { 2 } } { 100 } + \frac { ( y - 1 ) ^ { 2 } } { 64 } = 1$$

Accepted Answer

A)  Center: $( - 5,1 )$; Vertices: $( 1 , - 15 ) , ( 1,5 ) ;$ Foci: $( 1 , - 11 ) , ( 1,1 )$ 
B)  Center: $( - 5,1 )$; Vertices: $( 1 , - 15 ) , ( 1,5 )$; Foci: $( 1 , - 13 ) , ( 1,3 )$ 
C)  Center: $( - 5,1 )$; Vertices: $( - 15,1 ) , ( 5,1 )$; Foci: $( - 13,1 ) , ( 3,1 )$ 
D)  Center: $( - 5,1 )$; Vertices: $( - 15,1 ) , ( 5,1 )$; Foci: $( - 11,1 ) , ( 1,1 )$ 
A)  Center: $( - 5,1 )$; Vertices: $( 1 , - 15 ) , ( 1,5 ) ;$ Foci: $( 1 , - 11 ) , ( 1,1 )$ 
B)  Center: $( - 5,1 )$; Vertices: $( 1 , - 15 ) , ( 1,5 )$; Foci: $( 1 , - 13 ) , ( 1,3 )$ 
C)  Center: $( - 5,1 )$; Vertices: $( - 15,1 ) , ( 5,1 )$; Foci: $( - 13,1 ) , ( 3,1 )$ 
D)  Center: $( - 5,1 )$; Vertices: $( - 15,1 ) , ( 5,1 )$; Foci: $( - 11,1 ) , ( 1,1 )$

Question 13

Graph the parabola.
-$$x = \frac { 1 } { 4 } y ^ { 2 }$$

Accepted Answer

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

Question 14

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no
cross-product term.
-$$2 x ^ { 2 } + 2 \sqrt { 3 } x y + 4 y ^ { 2 } - 8 = 0$$

Accepted Answer

A)  $\frac { \mathrm { u } ^ { 2 } } { 8 } + \frac { 5 \mathrm { v } ^ { 2 } } { 8 } = 1$ 
B)  $\frac { 5 \mathrm { u } ^ { 2 } } { 8 } + \frac { \mathrm { v } ^ { 2 } } { 8 } = 1$ 
C)  $\frac { 2 u ^ { 2 } } { 8 } + \frac { v ^ { 2 } } { 8 } = 1$ 
D)  $\frac { 3 \mathrm { u } ^ { 2 } } { 16 } + \frac { \mathrm { v } ^ { 2 } } { 16 } = 1$ 
A)  $\frac { \mathrm { u } ^ { 2 } } { 8 } + \frac { 5 \mathrm { v } ^ { 2 } } { 8 } = 1$ 
B)  $\frac { 5 \mathrm { u } ^ { 2 } } { 8 } + \frac { \mathrm { v } ^ { 2 } } { 8 } = 1$ 
C)  $\frac { 2 u ^ { 2 } } { 8 } + \frac { v ^ { 2 } } { 8 } = 1$ 
D)  $\frac { 3 \mathrm { u } ^ { 2 } } { 16 } + \frac { \mathrm { v } ^ { 2 } } { 16 } = 1$

Question 15

Solve the problem.
-A comet follows the hyperbolic path described by $\frac { x ^ { 2 } } { 21 } - \frac { y ^ { 2 } } { 20 } = 1$, where $x$ and $y$ are in millions. If the sun is the fc of the path, how close to the sun is the vertex of the path?

Accepted Answer

A)  $4.6$ million 
B)  41 million 
C)  $6.4$ million 
D)  $1.8$ million 
A)  $4.6$ million 
B)  41 million 
C)  $6.4$ million 
D)  $1.8$ million

Question 16

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term.
-$$x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$$

Accepted Answer

A)  $x = \frac { 1 } { 2 } \mathrm { u } - \frac { \sqrt { 3 } } { 2 } \mathrm { v }$ and $\mathrm { y } = \frac { \sqrt { 3 } } { 2 } \mathrm { u } + \frac { 1 } { 2 } \mathrm { v }$ 
B)  $x = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$ 
C)  $x = - v$ and $y = u$ 
D)  $x = \frac { \mathrm { u } } { \sqrt { 2 } } - \frac { \mathrm { v } } { \sqrt { 2 } }$ and $\mathrm { y } = \frac { \mathrm { u } } { \sqrt { 2 } } + \frac { \mathrm { v } } { \sqrt { 2 } }$ 
A)  $x = \frac { 1 } { 2 } \mathrm { u } - \frac { \sqrt { 3 } } { 2 } \mathrm { v }$ and $\mathrm { y } = \frac { \sqrt { 3 } } { 2 } \mathrm { u } + \frac { 1 } { 2 } \mathrm { v }$ 
B)  $x = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$ 
C)  $x = - v$ and $y = u$ 
D)  $x = \frac { \mathrm { u } } { \sqrt { 2 } } - \frac { \mathrm { v } } { \sqrt { 2 } }$ and $\mathrm { y } = \frac { \mathrm { u } } { \sqrt { 2 } } + \frac { \mathrm { v } } { \sqrt { 2 } }$

Question 17

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term.
-$$4 x ^ { 2 } + 6 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$$

Accepted Answer

A)  $x = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ 
B)  $x = \frac { 1 } { 2 } u - \frac { \sqrt { 3 } } { 2 } v$ and $y = \frac { \sqrt { 3 } } { 2 } u + \frac { 1 } { 2 } v$ 
C)  $x = - v$ and $y = u$ 
D)  $x = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$ 
A)  $x = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ 
B)  $x = \frac { 1 } { 2 } u - \frac { \sqrt { 3 } } { 2 } v$ and $y = \frac { \sqrt { 3 } } { 2 } u + \frac { 1 } { 2 } v$ 
C)  $x = - v$ and $y = u$ 
D)  $x = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$

Question 18

Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Vertices at $( \pm 3,0 )$, foci at $( \pm 9,0 )$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 72 } - \frac { y ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 72 } = 1$ 
A)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 72 } - \frac { y ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 72 } = 1$

Question 19

Find the eccentricity of the hyperbola.
-$$x ^ { 2 } - 16 y ^ { 2 } = 1$$

Accepted Answer

A)  4 
B)  $\frac { \sqrt { 17 } } { 4 }$ 
C)  $\sqrt { 17 }$ 
D)  17 
A)  4 
B)  $\frac { \sqrt { 17 } } { 4 }$ 
C)  $\sqrt { 17 }$ 
D)  17

Question 20

Find the eccentricity of the hyperbola.
-$$8 x ^ { 2 } - 9 y ^ { 2 } = 72$$

Accepted Answer

A)  $\frac { \sqrt { 13 } } { 3 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 17 } } { 3 }$ 
A)  $\frac { \sqrt { 13 } } { 3 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 17 } } { 3 }$

Find the vertex, the focus, and the directrix of the parabola. - $\mathrm { x } ^ { 2 } - 10 \mathrm { x } + 8 \mathrm { y } + 9 = 0$

Solve the problem. -A rectangular board is 8 by 19 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the length of the string.

Match the polar equation with its graph. - $r = \frac { 4 } { 1 - \sin \theta }$

Find the eccentricity of the ellipse. - $23 x ^ { 2 } + y ^ { 2 } = 23$

Find the standard form of the equation of the parabola. -Focus at $( 10,0 )$ , directrix $x = - 10$

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. -The hyperbola $x y = 36$

Write an equation for the illustrated plane. -

Find a polar equation for the conic described. -The hyperbola with a focus at the pole and transverse axis endpoints $\left( 4 , \frac { \pi } { 2 } \right)$ and $\left( - 8 , \frac { 3 \pi } { 2 } \right)$

Determine the eccentricity, the type of conic, and the directrix. - $r = \frac { 3 } { 6 + 8 \cos \theta }$

Match the polar equation with its graph. - $r = \frac { 10 } { 4 - 4 \sin \theta }$

Graph the parabola. - $x = \frac { 1 } { 6 } ( y - 4 ) ^ { 2 } + 4$ $Graph the parabola. - x = \frac { 1 } { 6 } ( y - 4 ) ^ { 2 } + 4$

Find the center, vertices, and foci of the ellipse with the given equation. - $\frac { ( x + 5 ) ^ { 2 } } { 100 } + \frac { ( y - 1 ) ^ { 2 } } { 64 } = 1$

Graph the parabola. - $x = \frac { 1 } { 4 } y ^ { 2 }$ $Graph the parabola. - x = \frac { 1 } { 4 } y ^ { 2 }$

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no cross-product term. - $2 x ^ { 2 } + 2 \sqrt { 3 } x y + 4 y ^ { 2 } - 8 = 0$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $4 x ^ { 2 } + 6 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Vertices at $( \pm 3,0 )$ , foci at $( \pm 9,0 )$

Find the eccentricity of the hyperbola. - $x ^ { 2 } - 16 y ^ { 2 } = 1$

Find the eccentricity of the hyperbola. - $8 x ^ { 2 } - 9 y ^ { 2 } = 72$

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 8: Analytic Geometry in Two and Three Dimensions

Find the vertex, the focus, and the directrix of the parabola. - x2−10x+8y+9=0\mathrm { x } ^ { 2 } - 10 \mathrm { x } + 8 \mathrm { y } + 9 = 0x2−10x+8y+9=0

Solve the problem. -A rectangular board is 8 by 19 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the length of the string.

Match the polar equation with its graph. - r=41−sin⁡θr = \frac { 4 } { 1 - \sin \theta }r=1−sinθ4​

Find the eccentricity of the ellipse. - 23x2+y2=2323 x ^ { 2 } + y ^ { 2 } = 2323x2+y2=23

Find the standard form of the equation of the parabola. -Focus at (10,0)( 10,0 )(10,0) , directrix x=−10x = - 10x=−10

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. -The hyperbola xy=36x y = 36xy=36

Write an equation for the illustrated plane. -

Find a polar equation for the conic described. -The hyperbola with a focus at the pole and transverse axis endpoints (4,π2)\left( 4 , \frac { \pi } { 2 } \right)(4,2π​) and (−8,3π2)\left( - 8 , \frac { 3 \pi } { 2 } \right)(−8,23π​)

Determine the eccentricity, the type of conic, and the directrix. - r=36+8cos⁡θr = \frac { 3 } { 6 + 8 \cos \theta }r=6+8cosθ3​

Match the polar equation with its graph. - r=104−4sin⁡θr = \frac { 10 } { 4 - 4 \sin \theta }r=4−4sinθ10​

Graph the parabola. - x=16(y−4)2+4x = \frac { 1 } { 6 } ( y - 4 ) ^ { 2 } + 4x=61​(y−4)2+4

Find the center, vertices, and foci of the ellipse with the given equation. - (x+5)2100+(y−1)264=1\frac { ( x + 5 ) ^ { 2 } } { 100 } + \frac { ( y - 1 ) ^ { 2 } } { 64 } = 1100(x+5)2​+64(y−1)2​=1

Graph the parabola. - x=14y2x = \frac { 1 } { 4 } y ^ { 2 }x=41​y2

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no cross-product term. - 2x2+23xy+4y2−8=02 x ^ { 2 } + 2 \sqrt { 3 } x y + 4 y ^ { 2 } - 8 = 02x2+23​xy+4y2−8=0

Solve the problem. -A comet follows the hyperbolic path described by x221−y220=1\frac { x ^ { 2 } } { 21 } - \frac { y ^ { 2 } } { 20 } = 121x2​−20y2​=1 , where xxx and yyy are in millions. If the sun is the fc of the path, how close to the sun is the vertex of the path?

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - x2+2xy+y2−8x+8y=0x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0x2+2xy+y2−8x+8y=0

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - 4x2+6xy+4y2−8x+8y=04 x ^ { 2 } + 6 x y + 4 y ^ { 2 } - 8 x + 8 y = 04x2+6xy+4y2−8x+8y=0

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Vertices at (±3,0)( \pm 3,0 )(±3,0) , foci at (±9,0)( \pm 9,0 )(±9,0)

Find the eccentricity of the hyperbola. - x2−16y2=1x ^ { 2 } - 16 y ^ { 2 } = 1x2−16y2=1

Find the eccentricity of the hyperbola. - 8x2−9y2=728 x ^ { 2 } - 9 y ^ { 2 } = 728x2−9y2=72

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Discrete Mathematics

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Find the vertex, the focus, and the directrix of the parabola. - $\mathrm { x } ^ { 2 } - 10 \mathrm { x } + 8 \mathrm { y } + 9 = 0$

Match the polar equation with its graph. - $r = \frac { 4 } { 1 - \sin \theta }$

Find the eccentricity of the ellipse. - $23 x ^ { 2 } + y ^ { 2 } = 23$

Find the standard form of the equation of the parabola. -Focus at $( 10,0 )$ , directrix $x = - 10$

Find the vertices and foci of the conic section without axis rotation by analyzing the graph geometrically in the xy-plane. -The hyperbola $x y = 36$

Find a polar equation for the conic described. -The hyperbola with a focus at the pole and transverse axis endpoints $\left( 4 , \frac { \pi } { 2 } \right)$ and $\left( - 8 , \frac { 3 \pi } { 2 } \right)$

Determine the eccentricity, the type of conic, and the directrix. - $r = \frac { 3 } { 6 + 8 \cos \theta }$

Match the polar equation with its graph. - $r = \frac { 10 } { 4 - 4 \sin \theta }$

Graph the parabola. - $x = \frac { 1 } { 6 } ( y - 4 ) ^ { 2 } + 4$ $Graph the parabola. - x = \frac { 1 } { 6 } ( y - 4 ) ^ { 2 } + 4$

Find the center, vertices, and foci of the ellipse with the given equation. - $\frac { ( x + 5 ) ^ { 2 } } { 100 } + \frac { ( y - 1 ) ^ { 2 } } { 64 } = 1$

Graph the parabola. - $x = \frac { 1 } { 4 } y ^ { 2 }$ $Graph the parabola. - x = \frac { 1 } { 4 } y ^ { 2 }$

Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no cross-product term. - $2 x ^ { 2 } + 2 \sqrt { 3 } x y + 4 y ^ { 2 } - 8 = 0$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. - $4 x ^ { 2 } + 6 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Vertices at $( \pm 3,0 )$ , foci at $( \pm 9,0 )$

Find the eccentricity of the hyperbola. - $x ^ { 2 } - 16 y ^ { 2 } = 1$

Find the eccentricity of the hyperbola. - $8 x ^ { 2 } - 9 y ^ { 2 } = 72$