Question 1

Graph the parabola.
-$12 y=x^{2}$

Accepted Answer

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

Question 2

Find the vertex, focus, directrix, and focal width of the parabola.
-$$- \frac { 1 } { 40 } x ^ { 2 } = y$$

Accepted Answer

A)  Vertex: $( 0,0 )$; Focus: $( 0 , - 10 )$; Directrix: $y = 10$; Focal width: 40 
B)  Vertex: $( 0,0 )$; Focus: $( 0,10 )$; Directrix: y $= - 10$; Focal width: 10 
C)  Vertex: $( 0,0 )$; Focus: $( - 20,0 )$; Directrix: $x = 10$; Focal width: 160 
D)  Vertex: $( 0,0 )$; Focus: $( 0 , - 10 )$; Directrix: $y = 10$; Focal width: 160 
A)  Vertex: $( 0,0 )$; Focus: $( 0 , - 10 )$; Directrix: $y = 10$; Focal width: 40 
B)  Vertex: $( 0,0 )$; Focus: $( 0,10 )$; Directrix: y $= - 10$; Focal width: 10 
C)  Vertex: $( 0,0 )$; Focus: $( - 20,0 )$; Directrix: $x = 10$; Focal width: 160 
D)  Vertex: $( 0,0 )$; Focus: $( 0 , - 10 )$; Directrix: $y = 10$; Focal width: 160

Question 3

Determine the eccentricity, the type of conic, and the directrix.
-$$r = \frac { 5 } { 2 + 8 \sin \theta }$$

Accepted Answer

A)  Eccentricity $= \frac { 8 } { 5 }$; ellipse; directrix $y = - 20$ 
B)  Eccentricity $= 4$; ellipse; directrix $y = - \frac { 5 } { 4 }$ 
C)  Eccentricity $= 4$; hyperbola; directrix $y = \frac { 5 } { 8 }$ 
D)  Eccentricity $= \frac { 1 } { 4 } ;$ ellipse; directrix $\mathrm { y } = - 20$ 
A)  Eccentricity $= \frac { 8 } { 5 }$; ellipse; directrix $y = - 20$ 
B)  Eccentricity $= 4$; ellipse; directrix $y = - \frac { 5 } { 4 }$ 
C)  Eccentricity $= 4$; hyperbola; directrix $y = \frac { 5 } { 8 }$ 
D)  Eccentricity $= \frac { 1 } { 4 } ;$ ellipse; directrix $\mathrm { y } = - 20$

Question 4

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$5 x ^ { 2 } + 5 x y - 6 y ^ { 2 } + 3 x - 4 y + 9 = 0$$

Accepted Answer

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

Question 5

Find parametric equations for the line described below.
-The line through the point $\mathrm { P } ( - 5 , - 4,4 )$ in the direction of the vector

Accepted Answer

A)  $x = - 8 t + 5 , y = 6 t + 4 , z = - 6 t - 4$ 
B)  $x = - 8 t - 5 , y = - 6 t - 4 , z = 6 t + 4$ 
C)  $x = 8 t - 5 , y = - 6 t - 4 , z = - 6 t + 4$ 
D)  $x = 8 t + 5 , y = - 6 t + 4 , z = - 6 t - 4$ 
A)  $x = - 8 t + 5 , y = 6 t + 4 , z = - 6 t - 4$ 
B)  $x = - 8 t - 5 , y = - 6 t - 4 , z = 6 t + 4$ 
C)  $x = 8 t - 5 , y = - 6 t - 4 , z = - 6 t + 4$ 
D)  $x = 8 t + 5 , y = - 6 t + 4 , z = - 6 t - 4$

Question 6

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.
-$$x ^ { 2 } + 2 x y + y ^ { 2 } + 4 \sqrt { 2 } x - 4 \sqrt { 2 } y = 0$$

Accepted Answer

A)  $\frac { 3 u ^ { 2 } } { 8 } + \frac { v ^ { 2 } } { 8 } = 1$ 
B)  $\mathrm { v } = \frac { \mathrm { u } 2 } { 4 }$ 
C)  $\frac { \mathrm { u } ^ { 2 } } { 4 } + \frac { 3 \mathrm { v } ^ { 2 } } { 4 } = 1$ 
D)  $4 u = v ^ { 2 }$ 
A)  $\frac { 3 u ^ { 2 } } { 8 } + \frac { v ^ { 2 } } { 8 } = 1$ 
B)  $\mathrm { v } = \frac { \mathrm { u } 2 } { 4 }$ 
C)  $\frac { \mathrm { u } ^ { 2 } } { 4 } + \frac { 3 \mathrm { v } ^ { 2 } } { 4 } = 1$ 
D)  $4 u = v ^ { 2 }$

Question 7

Find the midpoint of the segment PQ.
-$$\mathrm { P } ( 3 , - 5,4 ) , \mathrm { Q } ( 6,0,0 )$$

Accepted Answer

A)  $\left( \frac { 9 } { 2 } , - \frac { 5 } { 2 } , 2 \right)$ 
B)  $( 3,0,0 )$ 
C)  $\left( \frac { 9 } { 2 } , - 5,0 \right)$ 
D)  $\left( \frac { 3 } { 2 } , - \frac { 5 } { 2 } , 2 \right)$ 
A)  $\left( \frac { 9 } { 2 } , - \frac { 5 } { 2 } , 2 \right)$ 
B)  $( 3,0,0 )$ 
C)  $\left( \frac { 9 } { 2 } , - 5,0 \right)$ 
D)  $\left( \frac { 3 } { 2 } , - \frac { 5 } { 2 } , 2 \right)$

Question 8

Find the polar equation of the conic with focus at the pole, having the given properties. Identify the conic.
-Directrix $x = 5$, eccentricity 1

Accepted Answer

A)  $r = \frac { 5 } { 1 + \sin \theta }$, parabola 
B)  $r = \frac { 1 } { 1 + \cos 5 \theta }$, hyperbola 
C)  $r = \frac { 5 } { 1 + \cos \theta }$, parabola 
D)  $r = \frac { 5 } { 1 - \cos \theta }$, ellipse 
A)  $r = \frac { 5 } { 1 + \sin \theta }$, parabola 
B)  $r = \frac { 1 } { 1 + \cos 5 \theta }$, hyperbola 
C)  $r = \frac { 5 } { 1 + \cos \theta }$, parabola 
D)  $r = \frac { 5 } { 1 - \cos \theta }$, ellipse

Question 9

Find the midpoint of the segment PQ.
-$$\mathrm { P } ( 8 , - 7 , - 4 ) , \mathrm { Q } ( 9 , - 6 , - 5 )$$

Accepted Answer

A)  $\left( 4 , - \frac { 7 } { 2 } , - 2 \right)$ 
B)  $\left( \frac { 17 } { 2 } , - 7 , - 5 \right)$ 
C)  $\left( \frac { 9 } { 2 } , - 3 , - \frac { 5 } { 2 } \right)$ 
D)  $\left( \frac { 17 } { 2 } , - \frac { 13 } { 2 } , - \frac { 9 } { 2 } \right)$ 
A)  $\left( 4 , - \frac { 7 } { 2 } , - 2 \right)$ 
B)  $\left( \frac { 17 } { 2 } , - 7 , - 5 \right)$ 
C)  $\left( \frac { 9 } { 2 } , - 3 , - \frac { 5 } { 2 } \right)$ 
D)  $\left( \frac { 17 } { 2 } , - \frac { 13 } { 2 } , - \frac { 9 } { 2 } \right)$

Question 10

Solve the problem.
-The roof of a building is in the shape of the hyperbola $y ^ { 2 } - 6 x ^ { 2 } = 9$, where $x$ and $y$ are in meters. Refer to the figure and determine the height, $h$, of the outside walls.

$\mathrm { a } = \mathrm { b } = 9 \mathrm {~m}$

Accepted Answer

A)  $9.3 \mathrm {~m}$ 
B)  $22.2 \mathrm {~m}$ 
C)  $495 \mathrm {~m}$ 
D)  $87 \mathrm {~m}$ 
A)  $9.3 \mathrm {~m}$ 
B)  $22.2 \mathrm {~m}$ 
C)  $495 \mathrm {~m}$ 
D)  $87 \mathrm {~m}$

Question 11

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$x ^ { 2 } - 8 x y + 6 y ^ { 2 } - 13 = 0$$

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Evaluate the expression.
-$$r = \langle 9 , - 9 , - 4 \rangle , \mathbf { v } = \langle 6 , - 9 , - 6 \rangle , \mathbf { w } = \langle 6 , - 1 , - 3 \rangle$$
$( \mathbf { r } + \mathbf { v } ) \cdot \mathbf { w }$

Accepted Answer

A)  138 
B)  $\langle - 90 , - 18 , - 30 \rangle$ 
C)  $\langle - 54 , - 81 , - 24 \rangle$ 
D)  $- 138$ 
A)  138 
B)  $\langle - 90 , - 18 , - 30 \rangle$ 
C)  $\langle - 54 , - 81 , - 24 \rangle$ 
D)  $- 138$

Question 14

Match the polar equation with its graph.
-$$r = \frac { 12 } { 4 - 4 \cos \theta }$$

Accepted Answer

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

Question 15

Find the center, vertices, and foci of the ellipse with the given equation.
-$$9 x ^ { 2 } + 3 y ^ { 2 } = 27$$

Accepted Answer

A)  Center: $( 0,0 )$; Vertices: $( - 9,0 ) , ( 9,0 ) ;$ Foci: $( - 6 \sqrt { 2 } , 0 ) , ( 6 \sqrt { 2 } , 0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( 0 , - 3 ) , ( 0,3 ) ;$ Foci: $( 0 , - \sqrt { 6 } ) , ( 0 , \sqrt { 6 } )$ 
C)  Center: $( 0,0 )$; Vertices: $( - 3,0 ) , ( - 3,0 ) ;$ Foci: $( - \sqrt { 6 } , 0 ) , ( \sqrt { 6 } , 0 )$ 
D)  Center: $( 0,0 )$; Vertices: $( 0 , - 9 ) , ( 0,9 )$; Foci: $( 0 , - 6 \sqrt { 2 } ) , ( 0,6 \sqrt { 2 }$, 
A)  Center: $( 0,0 )$; Vertices: $( - 9,0 ) , ( 9,0 ) ;$ Foci: $( - 6 \sqrt { 2 } , 0 ) , ( 6 \sqrt { 2 } , 0 )$ 
B)  Center: $( 0,0 )$; Vertices: $( 0 , - 3 ) , ( 0,3 ) ;$ Foci: $( 0 , - \sqrt { 6 } ) , ( 0 , \sqrt { 6 } )$ 
C)  Center: $( 0,0 )$; Vertices: $( - 3,0 ) , ( - 3,0 ) ;$ Foci: $( - \sqrt { 6 } , 0 ) , ( \sqrt { 6 } , 0 )$ 
D)  Center: $( 0,0 )$; Vertices: $( 0 , - 9 ) , ( 0,9 )$; Foci: $( 0 , - 6 \sqrt { 2 } ) , ( 0,6 \sqrt { 2 }$,

Question 16

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

Accepted Answer

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

Question 17

A quadratic equation has been transformed using the rotation equations x = 0.8u - 0.6v and y = 0.6u + 0.8v. 
Convert the given point from (u, v) coordinates back to (x, y) coordinates.
-parabola focus (-1, 9)

Accepted Answer

A)  (-8, 4.8) 
B)  (6.6, -6.2) 
C)  (4.6, -7.8) 
D)  (-6.2, 6.6) 
A)  (-8, 4.8) 
B)  (6.6, -6.2) 
C)  (4.6, -7.8) 
D)  (-6.2, 6.6)

Question 18

Graph the parabola.
-$y=-6(x-9)^{2}+3$

Accepted Answer

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

Question 19

Graph the parabola.
-$x=-5(y-4)^{2}-4$

Accepted Answer

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

Question 20

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

Accepted Answer

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

Graph the parabola. - $12 y=x^{2}$ $Graph the parabola. - 12 y=x^{2}$

Find the vertex, focus, directrix, and focal width of the parabola. - $- \frac { 1 } { 40 } x ^ { 2 } = y$

Determine the eccentricity, the type of conic, and the directrix. - $r = \frac { 5 } { 2 + 8 \sin \theta }$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $5 x ^ { 2 } + 5 x y - 6 y ^ { 2 } + 3 x - 4 y + 9 = 0$

Find parametric equations for the line described below. -The line through the point $\mathrm { P } ( - 5 , - 4,4 )$ in the direction of the vector $< 8 , - 6 , - 6 >$

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. - $x ^ { 2 } + 2 x y + y ^ { 2 } + 4 \sqrt { 2 } x - 4 \sqrt { 2 } y = 0$

Find the midpoint of the segment PQ. - $\mathrm { P } ( 3 , - 5,4 ) , \mathrm { Q } ( 6,0,0 )$

Find the polar equation of the conic with focus at the pole, having the given properties. Identify the conic. -Directrix $x = 5$ , eccentricity 1

Find the midpoint of the segment PQ. - $\mathrm { P } ( 8 , - 7 , - 4 ) , \mathrm { Q } ( 9 , - 6 , - 5 )$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $x ^ { 2 } - 8 x y + 6 y ^ { 2 } - 13 = 0$

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

Evaluate the expression. - $r = \langle 9 , - 9 , - 4 \rangle , \mathbf { v } = \langle 6 , - 9 , - 6 \rangle , \mathbf { w } = \langle 6 , - 1 , - 3 \rangle$ $( \mathbf { r } + \mathbf { v } ) \cdot \mathbf { w }$

Match the polar equation with its graph. - $r = \frac { 12 } { 4 - 4 \cos \theta }$

Find the center, vertices, and foci of the ellipse with the given equation. - $9 x ^ { 2 } + 3 y ^ { 2 } = 27$

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

A quadratic equation has been transformed using the rotation equations x = 0.8u - 0.6v and y = 0.6u + 0.8v. Convert the given point from (u, v) coordinates back to (x, y) coordinates. -parabola focus (-1, 9)

Graph the parabola. - $y=-6(x-9)^{2}+3$ $Graph the parabola. - y=-6(x-9)^{2}+3$

Graph the parabola. - $x=-5(y-4)^{2}-4$ $Graph the parabola. - x=-5(y-4)^{2}-4$

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

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

Graph the parabola. - 12y=x212 y=x^{2}12y=x2

Find the vertex, focus, directrix, and focal width of the parabola. - −140x2=y- \frac { 1 } { 40 } x ^ { 2 } = y−401​x2=y

Determine the eccentricity, the type of conic, and the directrix. - r=52+8sin⁡θr = \frac { 5 } { 2 + 8 \sin \theta }r=2+8sinθ5​

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - 5x2+5xy−6y2+3x−4y+9=05 x ^ { 2 } + 5 x y - 6 y ^ { 2 } + 3 x - 4 y + 9 = 05x2+5xy−6y2+3x−4y+9=0

Find parametric equations for the line described below. -The line through the point P(−5,−4,4)\mathrm { P } ( - 5 , - 4,4 )P(−5,−4,4) in the direction of the vector <8,−6,−6>< 8 , - 6 , - 6 ><8,−6,−6>

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. - x2+2xy+y2+42x−42y=0x ^ { 2 } + 2 x y + y ^ { 2 } + 4 \sqrt { 2 } x - 4 \sqrt { 2 } y = 0x2+2xy+y2+42​x−42​y=0

Find the midpoint of the segment PQ. - P(3,−5,4),Q(6,0,0)\mathrm { P } ( 3 , - 5,4 ) , \mathrm { Q } ( 6,0,0 )P(3,−5,4),Q(6,0,0)

Find the polar equation of the conic with focus at the pole, having the given properties. Identify the conic. -Directrix x=5x = 5x=5 , eccentricity 1

Find the midpoint of the segment PQ. - P(8,−7,−4),Q(9,−6,−5)\mathrm { P } ( 8 , - 7 , - 4 ) , \mathrm { Q } ( 9 , - 6 , - 5 )P(8,−7,−4),Q(9,−6,−5)

Solve the problem. -The roof of a building is in the shape of the hyperbola y2−6x2=9y ^ { 2 } - 6 x ^ { 2 } = 9y2−6x2=9 , where xxx and yyy are in meters. Refer to the figure and determine the height, hhh , of the outside walls. a=b=9 m\mathrm { a } = \mathrm { b } = 9 \mathrm {~m}a=b=9 m

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - x2−8xy+6y2−13=0x ^ { 2 } - 8 x y + 6 y ^ { 2 } - 13 = 0x2−8xy+6y2−13=0

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

Match the polar equation with its graph. - r=124−4cos⁡θr = \frac { 12 } { 4 - 4 \cos \theta }r=4−4cosθ12​

Find the center, vertices, and foci of the ellipse with the given equation. - 9x2+3y2=279 x ^ { 2 } + 3 y ^ { 2 } = 279x2+3y2=27

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

A quadratic equation has been transformed using the rotation equations x = 0.8u - 0.6v and y = 0.6u + 0.8v. Convert the given point from (u, v) coordinates back to (x, y) coordinates. -parabola focus (-1, 9)

Graph the parabola. - y=−6(x−9)2+3y=-6(x-9)^{2}+3y=−6(x−9)2+3

Graph the parabola. - x=−5(y−4)2−4x=-5(y-4)^{2}-4x=−5(y−4)2−4

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

Graph the parabola. - $12 y=x^{2}$ $Graph the parabola. - 12 y=x^{2}$

Find the vertex, focus, directrix, and focal width of the parabola. - $- \frac { 1 } { 40 } x ^ { 2 } = y$

Determine the eccentricity, the type of conic, and the directrix. - $r = \frac { 5 } { 2 + 8 \sin \theta }$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $5 x ^ { 2 } + 5 x y - 6 y ^ { 2 } + 3 x - 4 y + 9 = 0$

Find parametric equations for the line described below. -The line through the point $\mathrm { P } ( - 5 , - 4,4 )$ in the direction of the vector $< 8 , - 6 , - 6 >$

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. - $x ^ { 2 } + 2 x y + y ^ { 2 } + 4 \sqrt { 2 } x - 4 \sqrt { 2 } y = 0$

Find the midpoint of the segment PQ. - $\mathrm { P } ( 3 , - 5,4 ) , \mathrm { Q } ( 6,0,0 )$

Find the polar equation of the conic with focus at the pole, having the given properties. Identify the conic. -Directrix $x = 5$ , eccentricity 1

Find the midpoint of the segment PQ. - $\mathrm { P } ( 8 , - 7 , - 4 ) , \mathrm { Q } ( 9 , - 6 , - 5 )$

Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola. - $x ^ { 2 } - 8 x y + 6 y ^ { 2 } - 13 = 0$

Match the polar equation with its graph. - $r = \frac { 12 } { 4 - 4 \cos \theta }$

Find the center, vertices, and foci of the ellipse with the given equation. - $9 x ^ { 2 } + 3 y ^ { 2 } = 27$

Graph the parabola. - $y=-6(x-9)^{2}+3$ $Graph the parabola. - y=-6(x-9)^{2}+3$

Graph the parabola. - $x=-5(y-4)^{2}-4$ $Graph the parabola. - x=-5(y-4)^{2}-4$

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