Question 1

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

Accepted Answer

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

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 $\mathrm { y } = - 20$

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

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)  $x = 8 t + 5 , y = - 6 t + 4 , z = - 6 t - 4$

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)  $4 u = v ^ { 2 }$

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)  $\left( \frac { 3 } { 2 } , - \frac { 5 } { 2 } , 2 \right)$

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

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)  $\left( \frac { 17 } { 2 } , - \frac { 13 } { 2 } , - \frac { 9 } { 2 } \right)$

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)  $87 \mathrm {~m}$

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

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)  $y = \frac { 1 } { 3 } x ^ { 2 }$

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)  $- 138$

Question 14

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

Accepted Answer

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: $( 0 , - 6 \sqrt { 2 } ) , ( 0,6 \sqrt { 2 }$,

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)  $y = - x ^ { 2 }$

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)

Question 18

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

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 19

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

Accepted Answer

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

Exam 8: Analytic Geometry in Two and Three Dimensions

Graph the parabola. - $12 y=x^{2}$ $Graph the parabola. - 12 y=x^{2} A) B) C) D)$

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. - $Find an equation that matches the parabola's graph. - A) x = \frac { 1 } { 3 } y ^ { 2 } B) x = - \frac { 1 } { 3 } y ^ { 2 } C) y = - \frac { 1 } { 3 } x ^ { 2 } D) y = \frac { 1 } { 3 } x ^ { 2 }$

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. - $Find an equation that matches the parabola's graph. - A) x = y ^ { 2 } B) x = - y ^ { 2 } C) y = x ^ { 2 } D) y = - x ^ { 2 }$

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

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

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

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

Graph the parabola. - $12 y=x^{2}$ $Graph the parabola. - 12 y=x^{2} A) B) C) D)$

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. - $Find an equation that matches the parabola's graph. - A) x = \frac { 1 } { 3 } y ^ { 2 } B) x = - \frac { 1 } { 3 } y ^ { 2 } C) y = - \frac { 1 } { 3 } x ^ { 2 } D) y = \frac { 1 } { 3 } x ^ { 2 }$

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. - $Find an equation that matches the parabola's graph. - A) x = y ^ { 2 } B) x = - y ^ { 2 } C) y = x ^ { 2 } D) y = - x ^ { 2 }$

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

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

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