Question 1

Find the center, vertices, and foci of the ellipse with the given equation.
-$$5 x ^ { 2 } + 7 y ^ { 2 } = 35$$

Accepted Answer

A)  Center: $( 0,0 )$; Vertices: $( - \sqrt { 7 } , 0 ) , ( - \sqrt { 7 } , 0 )$; Foci: $( - \sqrt { 2 } , 0 ) , ( \sqrt { 2 } , 0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( 0 , - 7 ) , ( 0,7 ) ;$ Foci: $( 0 , - 2 \sqrt { 6 } ) , ( 0,2 \sqrt { 6 } )$ 
C)  Center: $( 0,0 )$; Vertices: $( 0 , - \sqrt { 7 } ) , ( 0 , - \sqrt { 7 } )$; Foci: $( 0 , - \sqrt { 2 } ) , ( 0 , \sqrt { 2 } )$ 
D)  Center: $( 0,0 )$; Vertices: $( - 7,0 ) , ( 7,0 )$; Foci: $( - 2 \sqrt { 6 } , 0 ) , ( 2 \sqrt { 6 } , 0 )$ 
A)  Center: $( 0,0 )$; Vertices: $( - \sqrt { 7 } , 0 ) , ( - \sqrt { 7 } , 0 )$; Foci: $( - \sqrt { 2 } , 0 ) , ( \sqrt { 2 } , 0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( 0 , - 7 ) , ( 0,7 ) ;$ Foci: $( 0 , - 2 \sqrt { 6 } ) , ( 0,2 \sqrt { 6 } )$ 
C)  Center: $( 0,0 )$; Vertices: $( 0 , - \sqrt { 7 } ) , ( 0 , - \sqrt { 7 } )$; Foci: $( 0 , - \sqrt { 2 } ) , ( 0 , \sqrt { 2 } )$ 
D)  Center: $( 0,0 )$; Vertices: $( - 7,0 ) , ( 7,0 )$; Foci: $( - 2 \sqrt { 6 } , 0 ) , ( 2 \sqrt { 6 } , 0 )$

Question 2

Write an equation for the sphere with the given point as its center and the given number as its radius.
-$( 4,7,4 ) , \sqrt { \mathrm { r } }$

Accepted Answer

A)  $( x + 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = r ^ { 2 }$ 
B)  $( x - 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = r$ 
C)  $( x - 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = \sqrt { r }$ 
D)  $( x + 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = 2 r$ 
A)  $( x + 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = r ^ { 2 }$ 
B)  $( x - 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = r$ 
C)  $( x - 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = \sqrt { r }$ 
D)  $( x + 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = 2 r$

Question 3

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

Accepted Answer

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

Question 4

Match the polar equation with its graph.
-$$r = \frac { 3 } { 3 + 12 \cos \theta }$$

Accepted Answer

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

Question 5

Find a polar equation for the conic described.
-The ellipse with a focus at the pole and major axis endpoints $( 3,0 )$ and $( 1 , \pi )$

Accepted Answer

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

Question 6

Find the eccentricity of the ellipse.
-$$x ^ { 2 } + 6 y ^ { 2 } = 6$$

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 5 }$ 
B)  $\frac { \sqrt { 30 } } { 6 }$ 
C)  $\frac { 6 } { \sqrt { 35 } }$ 
D)  $\frac { \sqrt { 35 } } { 6 }$ 
A)  $\frac { \sqrt { 6 } } { 5 }$ 
B)  $\frac { \sqrt { 30 } } { 6 }$ 
C)  $\frac { 6 } { \sqrt { 35 } }$ 
D)  $\frac { \sqrt { 35 } } { 6 }$

Question 7

Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Center $( - 2,4 )$, focus $( - 2,11 )$, vertex $( - 2,6 )$

Accepted Answer

A)  $\frac { ( y - 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 53 } = 1$ 
B)  $\frac { ( y + 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$ 
C)  $\frac { ( y + 4 ) ^ { 2 } } { 4 } + \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$ 
D)  $\frac { ( y - 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$ 
A)  $\frac { ( y - 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 53 } = 1$ 
B)  $\frac { ( y + 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$ 
C)  $\frac { ( y + 4 ) ^ { 2 } } { 4 } + \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$ 
D)  $\frac { ( y - 4 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 45 } = 1$

Question 8

Find the standard form of the equation of the parabola.
-Vertex at $( 9,8 )$, opens downward, focal width $= 2$

Accepted Answer

A)  $( x - 9 ) ^ { 2 } = - 2 ( y - 8 )$ 
B)  $( x - 9 ) ^ { 2 } = - 0.5 ( y - 8 )$ 
C)  $( x - 9 ) ^ { 2 } = 2 ( y - 8 )$ 
D)  $( y - 8 ) ^ { 2 } = 2 ( x - 9 )$ 
A)  $( x - 9 ) ^ { 2 } = - 2 ( y - 8 )$ 
B)  $( x - 9 ) ^ { 2 } = - 0.5 ( y - 8 )$ 
C)  $( x - 9 ) ^ { 2 } = 2 ( y - 8 )$ 
D)  $( y - 8 ) ^ { 2 } = 2 ( x - 9 )$

Question 9

Evaluate the expression.
-$$r = \langle 7,6 , - 5 \rangle , v = \langle 4,2,4 \rangle , w = \langle - 4,5,1 \rangle$$
$\mathbf { r } + \mathbf { v }$

Accepted Answer

A)  $\langle 11,8 , - 1 \rangle$ 
B)  $\langle - 28 , - 12,20 \rangle$ 
C)  $\langle - 11 , - 8,1 \rangle$ 
D)  $\langle 28,12 , - 20 \rangle$ 
A)  $\langle 11,8 , - 1 \rangle$ 
B)  $\langle - 28 , - 12,20 \rangle$ 
C)  $\langle - 11 , - 8,1 \rangle$ 
D)  $\langle 28,12 , - 20 \rangle$

Question 10

Find an equation in standard form for the ellipse that satisfies the given conditions.
-An ellipse with major axis from $( - 2 , - 2 )$ to $( 10 , - 2 )$; minor axis from $( 4 , - 4 )$ to $( 4,0 )$

Accepted Answer

A)  $\frac { ( x - 4 ) ^ { 2 } } { 36 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$ 
B)  $\frac { ( x - 6 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$ 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 36 } + \frac { ( y - 4 ) ^ { 2 } } { 4 } = 1$ 
D)  $\frac { ( x + 4 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$ 
A)  $\frac { ( x - 4 ) ^ { 2 } } { 36 } + \frac { ( y + 2 ) ^ { 2 } } { 4 } = 1$ 
B)  $\frac { ( x - 6 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$ 
C)  $\frac { ( x + 2 ) ^ { 2 } } { 36 } + \frac { ( y - 4 ) ^ { 2 } } { 4 } = 1$ 
D)  $\frac { ( x + 4 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$

Question 11

Find an equation that matches the parabola's graph.
-

Accepted Answer

A)  $y = x ^ { 2 }$ 
B)  $x = y ^ { 2 }$ 
C)  $y = - x ^ { 2 }$ 
D)  $x = - y ^ { 2 }$ 
A)  $y = x ^ { 2 }$ 
B)  $x = y ^ { 2 }$ 
C)  $y = - x ^ { 2 }$ 
D)  $x = - y ^ { 2 }$

Question 12

Find the vertices and foci of the hyperbola.
-$$\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 1$$

Accepted Answer

A)  Vertices: $( 8,2 ) , ( 0,2 )$; Foci: $( 9,2 ) , ( - 1,2 )$ 
B)  Vertices: $( 2,8 ) , ( 2,0 )$; Foci: $( 2,9 ) , ( 2 , - 1 )$ 
C)  Vertices: $( 7,2 ) , ( 1,2 )$; Foci: $( 1,2 ) , ( 7,2 )$ 
D)  Vertices: $( 2,7 ) , ( 2,1 )$; Foci: $( 2,1 ) , ( 2,7 )$ 
A)  Vertices: $( 8,2 ) , ( 0,2 )$; Foci: $( 9,2 ) , ( - 1,2 )$ 
B)  Vertices: $( 2,8 ) , ( 2,0 )$; Foci: $( 2,9 ) , ( 2 , - 1 )$ 
C)  Vertices: $( 7,2 ) , ( 1,2 )$; Foci: $( 1,2 ) , ( 7,2 )$ 
D)  Vertices: $( 2,7 ) , ( 2,1 )$; Foci: $( 2,1 ) , ( 2,7 )$

Question 13

Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Foci at $( \pm 9,0 )$, transverse axis with length 6

Accepted Answer

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

Question 14

Find an equation in standard form for the ellipse that satisfies the given conditions.
-Vertices at $( \pm 12,0 )$ and foci at $( \pm 6,0 )$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 11,664 } + \frac { y ^ { 2 } } { 12 } = 1$ 
B)  $\frac { x ^ { 2 } } { 144 } + \frac { y } { 108 } = 1$ 
C)  $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 108 } = 1$ 
D)  $\frac { x } { 12 } + \frac { y ^ { 2 } } { 108 } = 2$ 
A)  $\frac { x ^ { 2 } } { 11,664 } + \frac { y ^ { 2 } } { 12 } = 1$ 
B)  $\frac { x ^ { 2 } } { 144 } + \frac { y } { 108 } = 1$ 
C)  $\frac { x ^ { 2 } } { 144 } + \frac { y ^ { 2 } } { 108 } = 1$ 
D)  $\frac { x } { 12 } + \frac { y ^ { 2 } } { 108 } = 2$

Question 15

Find the vertices and foci of the hyperbola.
-$$\frac { ( x + 5 ) ^ { 2 } } { 64 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$$

Accepted Answer

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

Question 16

Convert the equation to Cartesian form.
-$$r = \frac { 3 } { 5 - 6 \cos \theta }$$

Accepted Answer

A)  $11 x ^ { 2 } + 18 x - 25 y ^ { 2 } + 9 = 0$ 
B)  $11 x ^ { 2 } - 36 x + 25 y ^ { 2 } + 9 = 0$ 
C)  $11 x ^ { 2 } + 36 x - 25 y ^ { 2 } + 9 = 0$ 
D)  $11 x ^ { 2 } + 18 x + 25 y ^ { 2 } + 9 = 0$ 
A)  $11 x ^ { 2 } + 18 x - 25 y ^ { 2 } + 9 = 0$ 
B)  $11 x ^ { 2 } - 36 x + 25 y ^ { 2 } + 9 = 0$ 
C)  $11 x ^ { 2 } + 36 x - 25 y ^ { 2 } + 9 = 0$ 
D)  $11 x ^ { 2 } + 18 x + 25 y ^ { 2 } + 9 = 0$

Question 17

Find the vertex, focus, directrix, and focal width of the parabola.
-$$( y + 7 ) ^ { 2 } = 12 ( x + 3 )$$

Accepted Answer

A)  Vertex: $( - 7 , - 3 )$; Focus: $( - 7,0 )$; Directrix: $y = - 6$; Focal width: 3 
B)  Vertex: $( - 7 , - 3 )$; Focus: $( - 7,9$; Directrix: $y = - 15$; Focal width: 12 
C)  Vertex: $( - 3 , - 7 )$; Focus: $( 0 , - 7 )$; Directrix: $x = - 6$; Focal width: 12 
D)  Vertex: $( - 3 , - 7 )$; Focus: $( 9 , - 7 )$; Directrix: $x = - 15$; Focal width: 12 
A)  Vertex: $( - 7 , - 3 )$; Focus: $( - 7,0 )$; Directrix: $y = - 6$; Focal width: 3 
B)  Vertex: $( - 7 , - 3 )$; Focus: $( - 7,9$; Directrix: $y = - 15$; Focal width: 12 
C)  Vertex: $( - 3 , - 7 )$; Focus: $( 0 , - 7 )$; Directrix: $x = - 6$; Focal width: 12 
D)  Vertex: $( - 3 , - 7 )$; Focus: $( 9 , - 7 )$; Directrix: $x = - 15$; Focal width: 12

Question 18

Find an equation that matches the parabola's graph.
-

Accepted Answer

A)  $y = - \frac { 1 } { 3 } x ^ { 2 }$ 
B)  $y = \frac { 1 } { 3 } x ^ { 2 }$ 
C)  $x = \frac { 1 } { 3 } y ^ { 2 }$ 
D)  $x = - \frac { 1 } { 3 } y ^ { 2 }$ 
A)  $y = - \frac { 1 } { 3 } x ^ { 2 }$ 
B)  $y = \frac { 1 } { 3 } x ^ { 2 }$ 
C)  $x = \frac { 1 } { 3 } y ^ { 2 }$ 
D)  $x = - \frac { 1 } { 3 } y ^ { 2 }$

Question 19

Find the center, vertices, and foci of the ellipse with the given equation.
-$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$

Accepted Answer

A)  Center: $( 0,0 )$; Vertices: $( - 5,0 ) , ( 5,0 )$; Foci: $( - 3,0 ) , ( 3,0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( 0 , - 5 ) , ( 0,5 )$; Foci: $( 0 , - 3 ) , ( 0,3 )$ 
C)  Center: $( 0,0 )$; Vertices: $( - 5,0 ) , ( 5,0 )$; Foci: $( - 4,0 ) , ( 4,0 )$ 
D)  Center: $( 0,0 )$; Vertices: $( 0 , - 5 ) , ( 0,5 )$; Foci: $( 0 , - 4 ) , ( 0,4 )$ 
A)  Center: $( 0,0 )$; Vertices: $( - 5,0 ) , ( 5,0 )$; Foci: $( - 3,0 ) , ( 3,0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( 0 , - 5 ) , ( 0,5 )$; Foci: $( 0 , - 3 ) , ( 0,3 )$ 
C)  Center: $( 0,0 )$; Vertices: $( - 5,0 ) , ( 5,0 )$; Foci: $( - 4,0 ) , ( 4,0 )$ 
D)  Center: $( 0,0 )$; Vertices: $( 0 , - 5 ) , ( 0,5 )$; Foci: $( 0 , - 4 ) , ( 0,4 )$

Question 20

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$5 x ^ { 2 } - 7 x y - 2 y ^ { 2 } + 14 = 0$$

Accepted Answer

A)  Ellipse 
B)  Parabola 
C)  Hyperbola 
A)  Ellipse 
B)  Parabola 
C)  Hyperbola

Find the center, vertices, and foci of the ellipse with the given equation. - $5 x ^ { 2 } + 7 y ^ { 2 } = 35$

Write an equation for the sphere with the given point as its center and the given number as its radius. - $( 4,7,4 ) , \sqrt { \mathrm { r } }$

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

Match the polar equation with its graph. - $r = \frac { 3 } { 3 + 12 \cos \theta }$

Find a polar equation for the conic described. -The ellipse with a focus at the pole and major axis endpoints $( 3,0 )$ and $( 1 , \pi )$

Find the eccentricity of the ellipse. - $x ^ { 2 } + 6 y ^ { 2 } = 6$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Center $( - 2,4 )$ , focus $( - 2,11 )$ , vertex $( - 2,6 )$

Find the standard form of the equation of the parabola. -Vertex at $( 9,8 )$ , opens downward, focal width $= 2$

Evaluate the expression. - $r = \langle 7,6 , - 5 \rangle , v = \langle 4,2,4 \rangle , w = \langle - 4,5,1 \rangle$ $\mathbf { r } + \mathbf { v }$

Find an equation in standard form for the ellipse that satisfies the given conditions. -An ellipse with major axis from $( - 2 , - 2 )$ to $( 10 , - 2 )$ ; minor axis from $( 4 , - 4 )$ to $( 4,0 )$

Find an equation that matches the parabola's graph. -

Find the vertices and foci of the hyperbola. - $\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 1$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Foci at $( \pm 9,0 )$ , transverse axis with length 6

Find an equation in standard form for the ellipse that satisfies the given conditions. -Vertices at $( \pm 12,0 )$ and foci at $( \pm 6,0 )$

Find the vertices and foci of the hyperbola. - $\frac { ( x + 5 ) ^ { 2 } } { 64 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$

Convert the equation to Cartesian form. - $r = \frac { 3 } { 5 - 6 \cos \theta }$

Find the vertex, focus, directrix, and focal width of the parabola. - $( y + 7 ) ^ { 2 } = 12 ( x + 3 )$

Find an equation that matches the parabola's graph. -

Find the center, vertices, and foci of the ellipse with the given equation. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $5 x ^ { 2 } - 7 x y - 2 y ^ { 2 } + 14 = 0$

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 center, vertices, and foci of the ellipse with the given equation. - 5x2+7y2=355 x ^ { 2 } + 7 y ^ { 2 } = 355x2+7y2=35

Write an equation for the sphere with the given point as its center and the given number as its radius. - (4,7,4),r( 4,7,4 ) , \sqrt { \mathrm { r } }(4,7,4),r​

Find the eccentricity of the hyperbola. - 16x2−y2=116 x ^ { 2 } - y ^ { 2 } = 116x2−y2=1

Match the polar equation with its graph. - r=33+12cos⁡θr = \frac { 3 } { 3 + 12 \cos \theta }r=3+12cosθ3​

Find a polar equation for the conic described. -The ellipse with a focus at the pole and major axis endpoints (3,0)( 3,0 )(3,0) and (1,π)( 1 , \pi )(1,π)

Find the eccentricity of the ellipse. - x2+6y2=6x ^ { 2 } + 6 y ^ { 2 } = 6x2+6y2=6

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Center (−2,4)( - 2,4 )(−2,4) , focus (−2,11)( - 2,11 )(−2,11) , vertex (−2,6)( - 2,6 )(−2,6)

Find the standard form of the equation of the parabola. -Vertex at (9,8)( 9,8 )(9,8) , opens downward, focal width =2= 2=2

Evaluate the expression. - r=⟨7,6,−5⟩,v=⟨4,2,4⟩,w=⟨−4,5,1⟩r = \langle 7,6 , - 5 \rangle , v = \langle 4,2,4 \rangle , w = \langle - 4,5,1 \rangler=⟨7,6,−5⟩,v=⟨4,2,4⟩,w=⟨−4,5,1⟩ r+v\mathbf { r } + \mathbf { v }r+v

Find an equation in standard form for the ellipse that satisfies the given conditions. -An ellipse with major axis from (−2,−2)( - 2 , - 2 )(−2,−2) to (10,−2)( 10 , - 2 )(10,−2) ; minor axis from (4,−4)( 4 , - 4 )(4,−4) to (4,0)( 4,0 )(4,0)

Find an equation that matches the parabola's graph. -

Find the vertices and foci of the hyperbola. - (y−4)216−(x−2)29=1\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 116(y−4)2​−9(x−2)2​=1

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Foci at (±9,0)( \pm 9,0 )(±9,0) , transverse axis with length 6

Find an equation in standard form for the ellipse that satisfies the given conditions. -Vertices at (±12,0)( \pm 12,0 )(±12,0) and foci at (±6,0)( \pm 6,0 )(±6,0)

Find the vertices and foci of the hyperbola. - (x+5)264−(y−3)236=1\frac { ( x + 5 ) ^ { 2 } } { 64 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 164(x+5)2​−36(y−3)2​=1

Convert the equation to Cartesian form. - r=35−6cos⁡θr = \frac { 3 } { 5 - 6 \cos \theta }r=5−6cosθ3​

Find the vertex, focus, directrix, and focal width of the parabola. - (y+7)2=12(x+3)( y + 7 ) ^ { 2 } = 12 ( x + 3 )(y+7)2=12(x+3)

Find an equation that matches the parabola's graph. -

Find the center, vertices, and foci of the ellipse with the given equation. - x225+y29=1\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 125x2​+9y2​=1

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - 5x2−7xy−2y2+14=05 x ^ { 2 } - 7 x y - 2 y ^ { 2 } + 14 = 05x2−7xy−2y2+14=0

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 center, vertices, and foci of the ellipse with the given equation. - $5 x ^ { 2 } + 7 y ^ { 2 } = 35$

Write an equation for the sphere with the given point as its center and the given number as its radius. - $( 4,7,4 ) , \sqrt { \mathrm { r } }$

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

Match the polar equation with its graph. - $r = \frac { 3 } { 3 + 12 \cos \theta }$

Find a polar equation for the conic described. -The ellipse with a focus at the pole and major axis endpoints $( 3,0 )$ and $( 1 , \pi )$

Find the eccentricity of the ellipse. - $x ^ { 2 } + 6 y ^ { 2 } = 6$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Center $( - 2,4 )$ , focus $( - 2,11 )$ , vertex $( - 2,6 )$

Find the standard form of the equation of the parabola. -Vertex at $( 9,8 )$ , opens downward, focal width $= 2$

Evaluate the expression. - $r = \langle 7,6 , - 5 \rangle , v = \langle 4,2,4 \rangle , w = \langle - 4,5,1 \rangle$ $\mathbf { r } + \mathbf { v }$

Find an equation in standard form for the ellipse that satisfies the given conditions. -An ellipse with major axis from $( - 2 , - 2 )$ to $( 10 , - 2 )$ ; minor axis from $( 4 , - 4 )$ to $( 4,0 )$

Find the vertices and foci of the hyperbola. - $\frac { ( y - 4 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 9 } = 1$

Find an equation in standard form for the hyperbola that satisfies the given conditions. -Foci at $( \pm 9,0 )$ , transverse axis with length 6

Find an equation in standard form for the ellipse that satisfies the given conditions. -Vertices at $( \pm 12,0 )$ and foci at $( \pm 6,0 )$

Find the vertices and foci of the hyperbola. - $\frac { ( x + 5 ) ^ { 2 } } { 64 } - \frac { ( y - 3 ) ^ { 2 } } { 36 } = 1$

Convert the equation to Cartesian form. - $r = \frac { 3 } { 5 - 6 \cos \theta }$

Find the vertex, focus, directrix, and focal width of the parabola. - $( y + 7 ) ^ { 2 } = 12 ( x + 3 )$

Find the center, vertices, and foci of the ellipse with the given equation. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $5 x ^ { 2 } - 7 x y - 2 y ^ { 2 } + 14 = 0$