Question 1

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
-$$y = \pm \sqrt { x ^ { 2 } - 10 }$$

Accepted Answer

A)  vertices: $( - \sqrt { 10 } , 0 ) , ( \sqrt { 10 } , 0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$ 
B)  vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - \sqrt { 10 } , 0 ) , ( \sqrt { 10 } , 0 )$ 
C)  vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$ 
D)  vertices: $( 0 , - \sqrt { 10 } ) , ( 0 , \sqrt { 10 } )$

foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$ 
A)  vertices: $( - \sqrt { 10 } , 0 ) , ( \sqrt { 10 } , 0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$ 
B)  vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - \sqrt { 10 } , 0 ) , ( \sqrt { 10 } , 0 )$ 
C)  vertices: $( - 10,0 ) , ( 10,0 )$
foci: $( - 2 \sqrt { 5 } , 0 ) , ( 2 \sqrt { 5 } , 0 )$ 
D)  vertices: $( 0 , - \sqrt { 10 } ) , ( 0 , \sqrt { 10 } )$

foci: $( 0 , - 2 \sqrt { 5 } ) , ( 0,2 \sqrt { 5 } )$

Question 2

Find Parametric Equations for Functions
-Ellipse: Center: $( 5 , - 5 )$; Vertices: 6 units above and below the center; Endpoints of Minor Axis: 4 units left and right of the center.

Accepted Answer

A)  $x = 5 + 4 \cos t , y = - 5 + 6 \sin t$ 
B)  $x = 5 + 6 \cos t , y = - 5 + 4 \sin t$ 
C)  $x = 5 - 4 \cos t , y = - 5 - 6 \sin t$ 
D)  $x = 4 + 5 \cos t , y = 6 - 5 \sin t$ 
A)  $x = 5 + 4 \cos t , y = - 5 + 6 \sin t$ 
B)  $x = 5 + 6 \cos t , y = - 5 + 4 \sin t$ 
C)  $x = 5 - 4 \cos t , y = - 5 - 6 \sin t$ 
D)  $x = 4 + 5 \cos t , y = 6 - 5 \sin t$

Question 3

Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate.
-$$x ^ { 2 } + 2 x - 7 y + 36 = 0$$

Accepted Answer

A)  $( x + 1 ) ^ { 2 } = 7 ( y - 5 )$ 
B)  $( x - 1 ) ^ { 2 } = 7 ( y - 5 )$ 
C)  $( x - 1 ) ^ { 2 } = - 7 ( y - 5 )$ 
D)  $( x + 1 ) ^ { 2 } = 7 ( y + 5 )$ 
A)  $( x + 1 ) ^ { 2 } = 7 ( y - 5 )$ 
B)  $( x - 1 ) ^ { 2 } = 7 ( y - 5 )$ 
C)  $( x - 1 ) ^ { 2 } = - 7 ( y - 5 )$ 
D)  $( x + 1 ) ^ { 2 } = 7 ( y + 5 )$

Question 4

Find Parametric Equations for Functions
-Hyperbola: Vertices: $( 6,0 )$; Vertices: $( - 6,0 )$; Foci: $( 10,0 )$ and $( - 10,0 )$

Accepted Answer

A)  $x = 6 \sec t , y = 8 \tan t$ 
B)  $x = 6 \sec t , y = 10 \tan t$ 
C)  $x = 8 \sec t , y = 6 \tan t$ 
D)  $x = 36 \sec t , y = 64 \tan t$ 
A)  $x = 6 \sec t , y = 8 \tan t$ 
B)  $x = 6 \sec t , y = 10 \tan t$ 
C)  $x = 8 \sec t , y = 6 \tan t$ 
D)  $x = 36 \sec t , y = 64 \tan t$

Question 5

Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term.
-$$14 x ^ { 2 } - 24 x y + 7 y ^ { 2 } + 30 x - 40 y - 45 = 0$$

Accepted Answer

A)  $x = \frac { 3 } { 5 } x ^ { \prime } - \frac { 4 } { 5 } y ^ { \prime } ; y = \frac { 4 } { 5 } x ^ { \prime } + \frac { 3 } { 5 } y ^ { \prime }$ 
B)  $x = \frac { 7 } { 25 } x ^ { \prime } - \frac { 24 } { 25 } y ^ { \prime } ; y = \frac { 24 } { 25 } x ^ { \prime } + \frac { 7 } { 25 } y ^ { \prime }$ 
C)  $x = \frac { 89 } { 100 } x ^ { \prime } - \frac { 9 } { 20 } y ^ { \prime } ; y = \frac { 9 } { 20 } x ^ { \prime } + \frac { 89 } { 100 } y ^ { \prime }$ 
D)  $x = \frac { 1 } { 2 } x ^ { \prime } - \frac { \sqrt { 3 } } { 2 } y ^ { \prime } ; y = \frac { \sqrt { 3 } } { 2 } x ^ { \prime } + \frac { 1 } { 2 } y ^ { \prime }$ 
A)  $x = \frac { 3 } { 5 } x ^ { \prime } - \frac { 4 } { 5 } y ^ { \prime } ; y = \frac { 4 } { 5 } x ^ { \prime } + \frac { 3 } { 5 } y ^ { \prime }$ 
B)  $x = \frac { 7 } { 25 } x ^ { \prime } - \frac { 24 } { 25 } y ^ { \prime } ; y = \frac { 24 } { 25 } x ^ { \prime } + \frac { 7 } { 25 } y ^ { \prime }$ 
C)  $x = \frac { 89 } { 100 } x ^ { \prime } - \frac { 9 } { 20 } y ^ { \prime } ; y = \frac { 9 } { 20 } x ^ { \prime } + \frac { 89 } { 100 } y ^ { \prime }$ 
D)  $x = \frac { 1 } { 2 } x ^ { \prime } - \frac { \sqrt { 3 } } { 2 } y ^ { \prime } ; y = \frac { \sqrt { 3 } } { 2 } x ^ { \prime } + \frac { 1 } { 2 } y ^ { \prime }$

Question 6

Graph Parabolas with Vertices Not at the Origin
-$$( y + 4 ) ^ { 2 } = 8 x$$

Accepted Answer

A)  vertex: $( 0 , - 4 )$
focus: $( 2 , - 4 )$
directrix: $x = - 2$ 
B)  vertex: $( 0,4 )$
focus: $( 3,4 )$
directrix: $x = - 1$ 
C)  vertex: $( - 4,0 )$
focus: $( 2 , - 4 )$
directrix: $x = - 6$ 
D)  vertex: $( 0 , - 4 )$
focus: $( - 3 , - 4 )$
directrix: $x = - 2$ 
A)  vertex: $( 0 , - 4 )$
focus: $( 2 , - 4 )$
directrix: $x = - 2$ 
B)  vertex: $( 0,4 )$
focus: $( 3,4 )$
directrix: $x = - 1$ 
C)  vertex: $( - 4,0 )$
focus: $( 2 , - 4 )$
directrix: $x = - 6$ 
D)  vertex: $( 0 , - 4 )$
focus: $( - 3 , - 4 )$
directrix: $x = - 2$

Question 7

Use point plotting to graph the plane curve described by the given parametric equations.
-

Accepted Answer

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

Question 8

Write Equations of Rotated Conics in Standard Form
-$$17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$$

Accepted Answer

A)  $\frac { \left( x ^ { \prime } - \frac { 2 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } + \frac { 4 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 4 } = 1$ 
B)  $x ^ { \prime 2 } = - 16 y ^ { \prime }$ 
C)  $\frac { x ^ { \prime 2 } } { 16 } - \frac { y ^ { \prime 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 16 } = 1$ ) 
A)  $\frac { \left( x ^ { \prime } - \frac { 2 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } + \frac { 4 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 4 } = 1$ 
B)  $x ^ { \prime 2 } = - 16 y ^ { \prime }$ 
C)  $\frac { x ^ { \prime 2 } } { 16 } - \frac { y ^ { \prime 2 } } { 4 } = 1$ 
D)  $\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 16 } = 1$ )

Question 9

Use Rotation of Axes Formulas
-$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0 ; \theta = 45 ^ { \circ }$

Accepted Answer

A)  $\frac { x ^ { \prime 2 } } { 4 } + y ^ { \prime 2 } = 1$ 
B)  $\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { \prime 2 } } { 4 } + 2 y ^ { \prime 2 } = 1$ 
D)  $x ^ { \prime 2 } + 4 y ^ { \prime 2 } = 1$ 
A)  $\frac { x ^ { \prime 2 } } { 4 } + y ^ { \prime 2 } = 1$ 
B)  $\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { \prime 2 } } { 4 } + 2 y ^ { \prime 2 } = 1$ 
D)  $x ^ { \prime 2 } + 4 y ^ { \prime 2 } = 1$

Question 10

Write Equations of Parabolas in Standard Form
-Focus: $( 13,0 )$; Directrix: $x = - 13$

Accepted Answer

A)  $y ^ { 2 } = 52 x$ 
B)  $y ^ { 2 } = - 52 x$ 
C)  $y ^ { 2 } = 13 x$ 
D)  $x ^ { 2 } = 52 y$ 
A)  $y ^ { 2 } = 52 x$ 
B)  $y ^ { 2 } = - 52 x$ 
C)  $y ^ { 2 } = 13 x$ 
D)  $x ^ { 2 } = 52 y$

Question 11

Eliminate the Parameter
-$$x = 2 t , y = t + 1 ; - 2 \leq t \leq 3$$

Accepted Answer

A)  $y = \frac { 1 } { 2 } x + 1 ; - 4 \leq x \leq 6$ 
B)  $y = - 2 x + 1 ; - \infty < x < \infty$ 
C)  $y = \frac { 1 } { 2 } x - 1 ; - \infty < x < \infty$ 
D)  $\mathrm { y } = \mathrm { x } ^ { 2 } + 1 ; - 2 \leq \mathrm { x } \leq 2$ 
A)  $y = \frac { 1 } { 2 } x + 1 ; - 4 \leq x \leq 6$ 
B)  $y = - 2 x + 1 ; - \infty < x < \infty$ 
C)  $y = \frac { 1 } { 2 } x - 1 ; - \infty < x < \infty$ 
D)  $\mathrm { y } = \mathrm { x } ^ { 2 } + 1 ; - 2 \leq \mathrm { x } \leq 2$

Question 12

Write Equations of Ellipses in Standard Form
-

Accepted Answer

A)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0 , - 2 \sqrt { 7 } )$ and $( 0,2 \sqrt { 7 } )$ 
B)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$
foci at $( 0 , - 2 \sqrt { 7 } )$ and $( 0,2 \sqrt { 7 } )$ 
C)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0 , - 8 )$ and $( 0,8 )$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0,8 )$ and $( 6,0 )$ 
A)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0 , - 2 \sqrt { 7 } )$ and $( 0,2 \sqrt { 7 } )$ 
B)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$
foci at $( 0 , - 2 \sqrt { 7 } )$ and $( 0,2 \sqrt { 7 } )$ 
C)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0 , - 8 )$ and $( 0,8 )$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 64 } = 1$
foci at $( 0,8 )$ and $( 6,0 )$

Question 13

Write Equations of Hyperbolas in Standard Form
-Foci: $( 0 , - 9 ) , ( 0,9 )$; vertices: $( 0 , - 4 ) , ( 0,4 )$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 65 } = 1$ 
B)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 65 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 81 } = 1$ 
D)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 81 } = 1$ 
A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 65 } = 1$ 
B)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 65 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 81 } = 1$ 
D)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 81 } = 1$

Question 14

Graph Hyperbolas Not Centered at the Origin
-$$( \mathrm { y } + 2 ) ^ { 2 } - 4 ( \mathrm { x } + 1 ) ^ { 2 } = 4$$

Accepted Answer

A)  Center: $( - 1 , - 2 )$; Vertices: $( - 1 , - 4 )$ and $( - 1,0 )$; Foci: $( - 1 , - 2 - \sqrt { 5 } )$ and $( - 1 , - 2 + \sqrt { 5 } )$ 
B)  Center: $( 1,2 )$; Vertices: $( 1,0 )$ and $( 1,4 )$; Foci: $( 1,2 - \sqrt { 5 } )$ and $( 1,2 + \sqrt { 5 } )$ 
C)  Center: $( - 1 , - 2 )$; Vertices: $( 1 , - 2 )$ and $( - 1,2 )$; Foci: $( - 1 , - \sqrt { 5 } )$ and $( - 1 , \sqrt { 5 } )$ 
D)  Center: $( - 1 , - 2 )$; Vertices: $( 0 , - 3 )$ and $( 0,1 )$; Foci: $( 0 , - 1 - \sqrt { 5 } )$ and $( 0 , - 1 + \sqrt { 5 } )$ 
A)  Center: $( - 1 , - 2 )$; Vertices: $( - 1 , - 4 )$ and $( - 1,0 )$; Foci: $( - 1 , - 2 - \sqrt { 5 } )$ and $( - 1 , - 2 + \sqrt { 5 } )$ 
B)  Center: $( 1,2 )$; Vertices: $( 1,0 )$ and $( 1,4 )$; Foci: $( 1,2 - \sqrt { 5 } )$ and $( 1,2 + \sqrt { 5 } )$ 
C)  Center: $( - 1 , - 2 )$; Vertices: $( 1 , - 2 )$ and $( - 1,2 )$; Foci: $( - 1 , - \sqrt { 5 } )$ and $( - 1 , \sqrt { 5 } )$ 
D)  Center: $( - 1 , - 2 )$; Vertices: $( 0 , - 3 )$ and $( 0,1 )$; Foci: $( 0 , - 1 - \sqrt { 5 } )$ and $( 0 , - 1 + \sqrt { 5 } )$

Question 15

Conic Sections in Polar Coordinates
1 Define Conics in Terms of a Focus and a Directrix
-$$r = \frac { 4 } { 2 + 2 \sin \theta }$$

Accepted Answer

A)  parabola; The directrix is 2 unit(s) above the pole at $y = 2$. 
B)  parabola; The directrix is 2 unit(s) to the right of the pole at $x = 2$. 
C)  hyperbola; The directrix is 2 unit(s) above the pole at $y = 2$. 
D)  hyperbola; The directrix is 2 unit(s) to the right of the pole at $x = 2$. 
A)  parabola; The directrix is 2 unit(s) above the pole at $y = 2$. 
B)  parabola; The directrix is 2 unit(s) to the right of the pole at $x = 2$. 
C)  hyperbola; The directrix is 2 unit(s) above the pole at $y = 2$. 
D)  hyperbola; The directrix is 2 unit(s) to the right of the pole at $x = 2$.

Question 16

Graph Hyperbolas Centered at the Origin
-$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$$

Accepted Answer

A)  Asymptotes: $y = \pm \frac { 5 } { 3 } x$
  
B)  Asymptotes: $\mathrm { y } = \pm \frac { 3 } { 5 } \mathrm { x }$
  
C)  Asymptotes: $y = \pm \frac { 5 } { 3 } x$
  
D)  Asymptotes: $\mathrm { y } = \pm \frac { 3 } { 5 } \mathrm { x }$
  
A)  Asymptotes: $y = \pm \frac { 5 } { 3 } x$
  
B)  Asymptotes: $\mathrm { y } = \pm \frac { 3 } { 5 } \mathrm { x }$
  
C)  Asymptotes: $y = \pm \frac { 5 } { 3 } x$
  
D)  Asymptotes: $\mathrm { y } = \pm \frac { 3 } { 5 } \mathrm { x }$

Question 17

The Hyperbola
1 Locate a Hyperbola's Vertices and Foci
-$\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 81 } = 1$

Accepted Answer

A)  vertices: $( 0 , - 10 ) , ( 0,10 )$
 foci: $( 0 , - \sqrt { 181 } ) , ( 0 , \sqrt { 181 } )$ 
B)  vertices: $( - 9,0 ) , ( 9,0 )$
 foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$ 
C)  vertices: $( 0 , - 10 ) , ( 0,10 )$
foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$ 
D)  vertices: $( - 10,0 ) , ( 10,0 )$
 foci: $( - 9,0 ) , ( 9,0 )$ 
A)  vertices: $( 0 , - 10 ) , ( 0,10 )$
 foci: $( 0 , - \sqrt { 181 } ) , ( 0 , \sqrt { 181 } )$ 
B)  vertices: $( - 9,0 ) , ( 9,0 )$
 foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$ 
C)  vertices: $( 0 , - 10 ) , ( 0,10 )$
foci: $( - \sqrt { 181 } , 0 ) , ( \sqrt { 181 } , 0 )$ 
D)  vertices: $( - 10,0 ) , ( 10,0 )$
 foci: $( - 9,0 ) , ( 9,0 )$

Question 18

Find the standard form of the equation of the ellipse satisfying the given conditions.
-Foci: $( 0 , - 3 ) , ( 0,3 )$; vertices: $( 0 , - 5 ) , ( 0,5 )$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
A)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$

Question 19

Conic Sections in Polar Coordinates
1 Define Conics in Terms of a Focus and a Directrix
-$$r = \frac { 9 } { 3 - 3 \cos \theta }$$

Accepted Answer

A)  parabola; The directrix is 3 unit(s) to the left of the pole at $x = - 3$. 
B)  parabola; The directrix is 3 unit(s) to the right of the pole at $x = 3$. 
C)  parabola; The directrix is 3 unit(s) above the pole at $y = 3$. 
D)  parabola; The directrix is 3 units below the pole at $\mathrm { y } = - 3$. 
A)  parabola; The directrix is 3 unit(s) to the left of the pole at $x = - 3$. 
B)  parabola; The directrix is 3 unit(s) to the right of the pole at $x = 3$. 
C)  parabola; The directrix is 3 unit(s) above the pole at $y = 3$. 
D)  parabola; The directrix is 3 units below the pole at $\mathrm { y } = - 3$.

Question 20

Graph Ellipses Not Centered at the Origin
-$$36 ( x + 3 ) ^ { 2 } + 16 ( y - 2 ) ^ { 2 } = 576$$

Accepted Answer

A)  foci at $( - 3,2 - 2 \sqrt { 5 } )$ and $( - 3,2 + 2 \sqrt { 5 } )$ 
B)  foci at $( 2 , - 3 - 2 \sqrt { 5 } )$ and $( 2 , - 3 + 2 \sqrt { 5 } )$ 
C)  foci at $( 3,2 - 2 \sqrt { 5 } )$ and $( 3,2 + 2 \sqrt { 5 } )$ 
D)  foci at $( - 2,2 - 2 \sqrt { 5 } )$ and $( - 2,2 + 2 \sqrt { 5 } )$ 
A)  foci at $( - 3,2 - 2 \sqrt { 5 } )$ and $( - 3,2 + 2 \sqrt { 5 } )$ 
B)  foci at $( 2 , - 3 - 2 \sqrt { 5 } )$ and $( 2 , - 3 + 2 \sqrt { 5 } )$ 
C)  foci at $( 3,2 - 2 \sqrt { 5 } )$ and $( 3,2 + 2 \sqrt { 5 } )$ 
D)  foci at $( - 2,2 - 2 \sqrt { 5 } )$ and $( - 2,2 + 2 \sqrt { 5 } )$

The Hyperbola 1 Locate a Hyperbola's Vertices and Foci - $y = \pm \sqrt { x ^ { 2 } - 10 }$

Find Parametric Equations for Functions -Ellipse: Center: $( 5 , - 5 )$ ; Vertices: 6 units above and below the center; Endpoints of Minor Axis: 4 units left and right of the center.

Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate. - $x ^ { 2 } + 2 x - 7 y + 36 = 0$

Find Parametric Equations for Functions -Hyperbola: Vertices: $( 6,0 )$ ; Vertices: $( - 6,0 )$ ; Foci: $( 10,0 )$ and $( - 10,0 )$

Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term. - $14 x ^ { 2 } - 24 x y + 7 y ^ { 2 } + 30 x - 40 y - 45 = 0$

Graph Parabolas with Vertices Not at the Origin - $( y + 4 ) ^ { 2 } = 8 x$

Use point plotting to graph the plane curve described by the given parametric equations. -

Write Equations of Rotated Conics in Standard Form - $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$

Use Rotation of Axes Formulas - $5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0 ; \theta = 45 ^ { \circ }$

Write Equations of Parabolas in Standard Form -Focus: $( 13,0 )$ ; Directrix: $x = - 13$

Eliminate the Parameter - $x = 2 t , y = t + 1 ; - 2 \leq t \leq 3$

Write Equations of Ellipses in Standard Form -

Write Equations of Hyperbolas in Standard Form -Foci: $( 0 , - 9 ) , ( 0,9 )$ ; vertices: $( 0 , - 4 ) , ( 0,4 )$

Graph Hyperbolas Not Centered at the Origin - $( \mathrm { y } + 2 ) ^ { 2 } - 4 ( \mathrm { x } + 1 ) ^ { 2 } = 4$

Conic Sections in Polar Coordinates 1 Define Conics in Terms of a Focus and a Directrix - $r = \frac { 4 } { 2 + 2 \sin \theta }$

Graph Hyperbolas Centered at the Origin - $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ $Graph Hyperbolas Centered at the Origin - \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$

The Hyperbola 1 Locate a Hyperbola's Vertices and Foci - $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 81 } = 1$

Find the standard form of the equation of the ellipse satisfying the given conditions. -Foci: $( 0 , - 3 ) , ( 0,3 )$ ; vertices: $( 0 , - 5 ) , ( 0,5 )$

Conic Sections in Polar Coordinates 1 Define Conics in Terms of a Focus and a Directrix - $r = \frac { 9 } { 3 - 3 \cos \theta }$

Graph Ellipses Not Centered at the Origin - $36 ( x + 3 ) ^ { 2 } + 16 ( y - 2 ) ^ { 2 } = 576$

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Systems of Equations and Inequalities

Matrices and Determinants

Sequences, Induction, and Probability

Prerequisites: Fundamental Concepts of Algebra

Filters

Exam 10: Conic Sections and Analytic Geometry

The Hyperbola 1 Locate a Hyperbola's Vertices and Foci - y=±x2−10y = \pm \sqrt { x ^ { 2 } - 10 }y=±x2−10​

Find Parametric Equations for Functions -Ellipse: Center: (5,−5)( 5 , - 5 )(5,−5) ; Vertices: 6 units above and below the center; Endpoints of Minor Axis: 4 units left and right of the center.

Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate. - x2+2x−7y+36=0x ^ { 2 } + 2 x - 7 y + 36 = 0x2+2x−7y+36=0

Find Parametric Equations for Functions -Hyperbola: Vertices: (6,0)( 6,0 )(6,0) ; Vertices: (−6,0)( - 6,0 )(−6,0) ; Foci: (10,0)( 10,0 )(10,0) and (−10,0)( - 10,0 )(−10,0)

Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term. - 14x2−24xy+7y2+30x−40y−45=014 x ^ { 2 } - 24 x y + 7 y ^ { 2 } + 30 x - 40 y - 45 = 014x2−24xy+7y2+30x−40y−45=0

Graph Parabolas with Vertices Not at the Origin - (y+4)2=8x( y + 4 ) ^ { 2 } = 8 x(y+4)2=8x