Question 1

Rotation of Axes
1 Identify Conics Without Completing the Square
-$$5 x ^ { 2 } - 6 y ^ { 2 } + 2 x - 3 y - 5 = 0$$

Accepted Answer

A)  hyperbola 
B)  circle 
C)  ellipse 
D)  parabola 
A)  hyperbola 
B)  circle 
C)  ellipse 
D)  parabola

Question 2

Graph the Polar Equations of Conics
-$r = \frac { 2 } { 2 - 2 \cos \theta } \quad$ Identify the directrix and vertex.

Accepted Answer

A)  directrix: 1 unit(s) to the left of
the pole at $x = - 1$
vertex: $\left( \frac { 1 } { 2 } , \pi \right)$
  
B)  directrix: 1 unit(s) to the right of
the pole at $x = 1$
vertex: $\left( \frac { 1 } { 2 } , 0 \right)$
  
C)  directrix: 1 unit(s) above
the pole at $\mathrm { y } = 1$
vertex: $\left( \frac { 1 } { 2 } , \frac { \pi } { 2 } \right)$
  
D)  directrix: 1 unit(s) below
the pole at $\mathrm { y } = - 1$
vertex: $\left( \frac { 1 } { 2 } , \frac { 3 \pi } { 2 } \right)$
  
A)  directrix: 1 unit(s) to the left of
the pole at $x = - 1$
vertex: $\left( \frac { 1 } { 2 } , \pi \right)$
  
B)  directrix: 1 unit(s) to the right of
the pole at $x = 1$
vertex: $\left( \frac { 1 } { 2 } , 0 \right)$
  
C)  directrix: 1 unit(s) above
the pole at $\mathrm { y } = 1$
vertex: $\left( \frac { 1 } { 2 } , \frac { \pi } { 2 } \right)$
  
D)  directrix: 1 unit(s) below
the pole at $\mathrm { y } = - 1$
vertex: $\left( \frac { 1 } { 2 } , \frac { 3 \pi } { 2 } \right)$

Question 3

Graph Parabolas with Vertices Not at the Origin
-$( \mathrm { y } + 1 ) ^ { 2 } = - 12 ( \mathrm { x } + 2 )$

Accepted Answer

A)  vertex: $( - 2 , - 1 )$
focus: $( - 5 , - 1 )$
directrix: $x = 1$ 
B)  vertex: $( 2,1 )$
focus: $( - 1,1 )$
directrix: $x = 5$ 
C)  vertex: $( - 1 , - 2 )$
focus: $( - 4 , - 2 )$
directrix: $x = 2$ 
D)  vertex: $( - 2 , - 1 )$
focus: $( 1 , - 1 )$
directrix: $x = - 5$ 
A)  vertex: $( - 2 , - 1 )$
focus: $( - 5 , - 1 )$
directrix: $x = 1$ 
B)  vertex: $( 2,1 )$
focus: $( - 1,1 )$
directrix: $x = 5$ 
C)  vertex: $( - 1 , - 2 )$
focus: $( - 4 , - 2 )$
directrix: $x = 2$ 
D)  vertex: $( - 2 , - 1 )$
focus: $( 1 , - 1 )$
directrix: $x = - 5$

Question 4

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection.
-$$\begin{array} { l } 
x = ( y + 2 ) ^ { 2 } - 1 \
( x - 2 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 1
\end{array}$$

Accepted Answer

A)  $\{ ( 2 , - 2 ) \}$ 
B)  $\{ ( - 1 , - 2 ) \}$ 
C)  $\{ ( - 1 , - 2 ) , ( 2 , - 2 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 2 , - 2 ) \}$ 
B)  $\{ ( - 1 , - 2 ) \}$ 
C)  $\{ ( - 1 , - 2 ) , ( 2 , - 2 ) \}$ 
D)  $\varnothing$

Question 5

Eliminate the Parameter
-$$\mathrm { x } = \mathrm { t } ^ { 3 } + 1 , \mathrm { y } = \mathrm { t } ^ { 3 } - 10 ; - 2 \leq \mathrm { t } \leq 2$$

Accepted Answer

A)  $\mathrm { y } = \mathrm { x } - 11 ; - 7 \leq \mathrm { x } \leq 9$ 
B)  $y = - x - 11 ; - 7 \leq x \leq 9$ 
C)  $\mathrm { y } = - \mathrm { x } ^ { 2 } ; - 4 \leq \mathrm { x } \leq 4$ 
D)  $y = x ^ { 3 } ; - 3 \leq x \leq 1$ 
A)  $\mathrm { y } = \mathrm { x } - 11 ; - 7 \leq \mathrm { x } \leq 9$ 
B)  $y = - x - 11 ; - 7 \leq x \leq 9$ 
C)  $\mathrm { y } = - \mathrm { x } ^ { 2 } ; - 4 \leq \mathrm { x } \leq 4$ 
D)  $y = x ^ { 3 } ; - 3 \leq x \leq 1$

Question 6

Identify Conics Without Rotating Axes
-$3 x ^ { 2 } - 7 x y + 3 y ^ { 2 } - 4 x - 4 y + 5 = 0$

Accepted Answer

A)  hyperbola 
B)  ellipse 
C)  circle 
D)  parabola 
A)  hyperbola 
B)  ellipse 
C)  circle 
D)  parabola

Question 7

Graph the ellipse and locate the foci.
-$$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$

Accepted Answer

A)  foci at $( 2 \sqrt { 7 } , 0 )$ and $( - 2 \sqrt { 7 } , 0 )$
  
B)  foci at $( 0,2 \sqrt { 7 } )$ and $( 0 , - 2 \sqrt { 7 } )$
  
C)  foci at $( 3 \sqrt { 5 } , 0 )$ and $( - 3 \sqrt { 5 } , 0 )$
  
D)  foci at $( 0,3 \sqrt { 5 } )$ and $( 0 , - 3 \sqrt { 5 } )$
  
A)  foci at $( 2 \sqrt { 7 } , 0 )$ and $( - 2 \sqrt { 7 } , 0 )$
  
B)  foci at $( 0,2 \sqrt { 7 } )$ and $( 0 , - 2 \sqrt { 7 } )$
  
C)  foci at $( 3 \sqrt { 5 } , 0 )$ and $( - 3 \sqrt { 5 } , 0 )$
  
D)  foci at $( 0,3 \sqrt { 5 } )$ and $( 0 , - 3 \sqrt { 5 } )$

Question 8

Conic Sections in Polar Coordinates
1 Define Conics in Terms of a Focus and a Directrix
-$$r = \frac { 3 } { 6 - 3 \sin \theta }$$

Accepted Answer

A)  ellipse; The directrix is 1 unit(s) below the pole at $y = - 1$. 
B)  ellipse; The directrix is 1 unit(s) to the left of the pole at $x = - 1$. 
C)  ellipse; The directrix is 1 unit(s) to the right of the pole at $x = 1$. 
D)  ellipse; The directrix is 1 unit(s) above the pole at $\mathrm { y } = 1$. 
A)  ellipse; The directrix is 1 unit(s) below the pole at $y = - 1$. 
B)  ellipse; The directrix is 1 unit(s) to the left of the pole at $x = - 1$. 
C)  ellipse; The directrix is 1 unit(s) to the right of the pole at $x = 1$. 
D)  ellipse; The directrix is 1 unit(s) above the pole at $\mathrm { y } = 1$.

Question 9

Additional Concepts
-$$x = - ( y + 3 ) ^ { 2 } - 8$$

Accepted Answer

A)  Opens to left; $( - 8 , - 3 )$ 
B)  Opens to left; $( - 8,3 )$ 
C)  Opens to left; $( - 3 , - 8 )$ 
D)  Opens to right; $( - 3 , - 8 )$ Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. 
A)  Opens to left; $( - 8 , - 3 )$ 
B)  Opens to left; $( - 8,3 )$ 
C)  Opens to left; $( - 3 , - 8 )$ 
D)  Opens to right; $( - 3 , - 8 )$ Use the vertex and the direction in which the parabola opens to determine the relation's domain and range.

Question 10

Use the rotated system to graph the equation.
-$x^{2}-8 x y+y^{2}+25=0$

Accepted Answer

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

Question 11

Eliminate the Parameter
-$$x = 2 + \sec t , y = 5 + 2 \tan t ; 0 < t < \frac { \pi } { 2 }$$

Accepted Answer

A) $( x - 2 ) ^ { 2 } - \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1 ; x > 3$ B) $( x - 2 ) ^ { 2 } + \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1 ; 1 \leq x \leq 3$ C) $( \mathrm { y } - 2 ) ^ { 2 } - \frac { ( \mathrm { x } - 5 ) ^ { 2 } } { 4 } = 1 ; - \infty < \mathrm { x } < \infty$ D) $( x - 2 ) ^ { 2 } - ( y - 5 ) ^ { 2 } = 4 ; - \infty < x < \infty$ A) $( x - 2 ) ^ { 2 } - \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1 ; x > 3$ B) $( x - 2 ) ^ { 2 } + \frac { ( y - 5 ) ^ { 2 } } { 4 } = 1 ; 1 \leq x \leq 3$ C) $( \mathrm { y } - 2 ) ^ { 2 } - \frac { ( \mathrm { x } - 5 ) ^ { 2 } } { 4 } = 1 ; - \infty < \mathrm { x } < \infty$ D) $( x - 2 ) ^ { 2 } - ( y - 5 ) ^ { 2 } = 4 ; - \infty < x < \infty$

Question 12

Additional Concepts
-$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$$

Accepted Answer

A)  Domain: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
B)  Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 )$ or $( 3 , \infty )$ 
C)  Domain: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
D)  Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$ 
A)  Domain: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
B)  Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 )$ or $( 3 , \infty )$ 
C)  Domain: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
D)  Domain: $( - \infty , \infty )$
Range: $( - \infty , \infty )$

Question 13

Solve Apps: Conic Sections in Polar Coordinates
-Halley's comet has an elliptical orbit with the sun at one focus. Its orbit shown below is given approximately by $\mathrm { r } = \frac { 10.82 } { 1 + 0.891 \sin \theta }$. In the formula, $\mathrm { r }$ is measured in astronomical units. (One astronomical unit is the average distance from Earth to the sun, approximately 93 million miles.)
Find the distance from Halley's comet to the sun at its greatest distance from the sun. Round to the nearest hundredth of an astronomical unit and the nearest million miles.

Accepted Answer

A)  $99.27$ astronomical units; 9232 million miles 
B)  $5.72$ astronomical units; 532 million miles 
C)  $12.14$ astronomical units; 1129 million miles 
D)  $6.07$ astronomical units; 565 million miles 
A)  $99.27$ astronomical units; 9232 million miles 
B)  $5.72$ astronomical units; 532 million miles 
C)  $12.14$ astronomical units; 1129 million miles 
D)  $6.07$ astronomical units; 565 million miles

Question 14

Additional Concepts
-$$\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$$

Accepted Answer

A)  Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 ]$ or $[ 3 , \infty )$ 
B)  Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 ]$ and $[ 3 , \infty )$ 
C)  Domain: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
D)  Domain: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
A)  Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 ]$ or $[ 3 , \infty )$ 
B)  Domain: $( - \infty , \infty )$
Range: $( - \infty , - 3 ]$ and $[ 3 , \infty )$ 
C)  Domain: $( - \infty , - 3 ]$ or $[ 3 , \infty )$
Range: $( - \infty , \infty )$ 
D)  Domain: $( - \infty , - 3 ]$ and $[ 3 , \infty )$
Range: $( - \infty , \infty )$

Question 15

Tech: Rotation of Axes
-

Accepted Answer

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

Question 16

Parametric Equations
1 Use Point Plotting to Graph Plane Curves Described by Parametric Equations
-$x = 4 + 5 \cos t , y = 6 + 4 \sin t ; t = \frac { \pi } { 2 }$

Accepted Answer

A)  $( 4,10 )$ 
B)  $( 9,6 )$ 
C)  $\left( 4 + \frac { 5 \sqrt { 2 } } { 2 } , 6 + 2 \sqrt { 2 } \right)$ 
D)  $( 4,8 )$ 
A)  $( 4,10 )$ 
B)  $( 9,6 )$ 
C)  $\left( 4 + \frac { 5 \sqrt { 2 } } { 2 } , 6 + 2 \sqrt { 2 } \right)$ 
D)  $( 4,8 )$

Question 17

Graph the Polar Equations of Conics
-$r = \frac { 2 } { 2 - \cos \theta } \quad$ Identify the directrix and vertices.

Accepted Answer

A)  directrix: 2 unit(s) to the left of
the pole at $x = - 2$
vertices: $\left( \frac { 2 } { 3 } , \pi \right) , ( 2,0 )$
  
B)  directrix: 2 unit(s) to the right of
the pole at $x = 2$
vertices: $\left( - \frac { 2 } { 3 } , \pi \right) , ( 2 , \pi )$
  
C)  directrix: 2 unit(s) below
the pole at $\mathrm { y } = - 2$
vertices: $\left( - \frac { 2 } { 3 } , \frac { \pi } { 2 } \right) , \left( 2 , \frac { \pi } { 2 } \right)$
  
D)  directrix: 2 unit(s) above
the pole at $\mathrm { y } = 2$
vertices: $\left( \frac { 2 } { 3 } , \frac { \pi } { 2 } \right) , \left( - 2 , \frac { \pi } { 2 } \right)$
  
A)  directrix: 2 unit(s) to the left of
the pole at $x = - 2$
vertices: $\left( \frac { 2 } { 3 } , \pi \right) , ( 2,0 )$
  
B)  directrix: 2 unit(s) to the right of
the pole at $x = 2$
vertices: $\left( - \frac { 2 } { 3 } , \pi \right) , ( 2 , \pi )$
  
C)  directrix: 2 unit(s) below
the pole at $\mathrm { y } = - 2$
vertices: $\left( - \frac { 2 } { 3 } , \frac { \pi } { 2 } \right) , \left( 2 , \frac { \pi } { 2 } \right)$
  
D)  directrix: 2 unit(s) above
the pole at $\mathrm { y } = 2$
vertices: $\left( \frac { 2 } { 3 } , \frac { \pi } { 2 } \right) , \left( - 2 , \frac { \pi } { 2 } \right)$

Question 18

Write Equations of Rotated Conics in Standard Form
-$x^{2}+x y+y^{2}-3 y-6=0$

Accepted Answer

A) $\frac{\left(x^{\prime}-\frac{\sqrt{2}}{2}\right)^{2}}{6}+\frac{\left(y^{\prime}-\frac{3 \sqrt{2}}{2}\right)^{2}}{18}=1$ 
B)  $y ^ { \prime 2 } = - 18 x ^ { \prime }$ 
C)  $\frac { x ^ { \prime 2 } } { 6 } - \frac { y ^ { \prime 2 } } { 8 } = 1$ 
D)  $\frac { x ^ { \prime 2 } } { 3 } + \frac { y ^ { \prime 2 } } { 4 } = 1$ ) 
A) $\frac{\left(x^{\prime}-\frac{\sqrt{2}}{2}\right)^{2}}{6}+\frac{\left(y^{\prime}-\frac{3 \sqrt{2}}{2}\right)^{2}}{18}=1$ 
B)  $y ^ { \prime 2 } = - 18 x ^ { \prime }$ 
C)  $\frac { x ^ { \prime 2 } } { 6 } - \frac { y ^ { \prime 2 } } { 8 } = 1$ 
D)  $\frac { x ^ { \prime 2 } } { 3 } + \frac { y ^ { \prime 2 } } { 4 } = 1$ )

Question 19

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection.
-$$\begin{array} { l } 
x = y ^ { 2 } - 7 \
x = y ^ { 2 } - 7 y
\end{array}$$

Accepted Answer

A)  $\{ ( - 6,1 ) \}$ 
B)  $\{ ( 1 , - 6 ) \}$ 
C)  $\{ ( 6,1 ) \}$ 
D)  $\{ ( - 6 , - 1 ) \}$ 
A)  $\{ ( - 6,1 ) \}$ 
B)  $\{ ( 1 , - 6 ) \}$ 
C)  $\{ ( 6,1 ) \}$ 
D)  $\{ ( - 6 , - 1 ) \}$

Question 20

Graph the parabola with the given equation.
-$$( x - 2 ) ^ { 2 } = - 8 ( y + 2 )$$

Accepted Answer

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

Rotation of Axes 1 Identify Conics Without Completing the Square - $5 x ^ { 2 } - 6 y ^ { 2 } + 2 x - 3 y - 5 = 0$

Graph the Polar Equations of Conics - $r = \frac { 2 } { 2 - 2 \cos \theta } \quad$ Identify the directrix and vertex. $Graph the Polar Equations of Conics - r = \frac { 2 } { 2 - 2 \cos \theta } \quad Identify the directrix and vertex.$

Graph Parabolas with Vertices Not at the Origin - $( \mathrm { y } + 1 ) ^ { 2 } = - 12 ( \mathrm { x } + 2 )$

Eliminate the Parameter - $\mathrm { x } = \mathrm { t } ^ { 3 } + 1 , \mathrm { y } = \mathrm { t } ^ { 3 } - 10 ; - 2 \leq \mathrm { t } \leq 2$

Identify Conics Without Rotating Axes - $3 x ^ { 2 } - 7 x y + 3 y ^ { 2 } - 4 x - 4 y + 5 = 0$

Graph the ellipse and locate the foci. - $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$ $Graph the ellipse and locate the foci. - \frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$

Conic Sections in Polar Coordinates 1 Define Conics in Terms of a Focus and a Directrix - $r = \frac { 3 } { 6 - 3 \sin \theta }$

Additional Concepts - $x = - ( y + 3 ) ^ { 2 } - 8$

Use the rotated system to graph the equation. - $x^{2}-8 x y+y^{2}+25=0$ $Use the rotated system to graph the equation. - x^{2}-8 x y+y^{2}+25=0$

Eliminate the Parameter - $x = 2 + \sec t , y = 5 + 2 \tan t ; 0 < t < \frac { \pi } { 2 }$

Additional Concepts - $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ $Additional Concepts - \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$

Additional Concepts - $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$ $Additional Concepts - \frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$

Tech: Rotation of Axes -

Parametric Equations 1 Use Point Plotting to Graph Plane Curves Described by Parametric Equations - $x = 4 + 5 \cos t , y = 6 + 4 \sin t ; t = \frac { \pi } { 2 }$

Graph the Polar Equations of Conics - $r = \frac { 2 } { 2 - \cos \theta } \quad$ Identify the directrix and vertices. $Graph the Polar Equations of Conics - r = \frac { 2 } { 2 - \cos \theta } \quad Identify the directrix and vertices.$

Write Equations of Rotated Conics in Standard Form - $x^{2}+x y+y^{2}-3 y-6=0$

Graph the parabola with the given equation. - $( x - 2 ) ^ { 2 } = - 8 ( y + 2 )$ $Graph the parabola with the given equation. - ( x - 2 ) ^ { 2 } = - 8 ( y + 2 )$

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

Rotation of Axes 1 Identify Conics Without Completing the Square - 5x2−6y2+2x−3y−5=05 x ^ { 2 } - 6 y ^ { 2 } + 2 x - 3 y - 5 = 05x2−6y2+2x−3y−5=0

Graph the Polar Equations of Conics - r=22−2cos⁡θr = \frac { 2 } { 2 - 2 \cos \theta } \quadr=2−2cosθ2​ Identify the directrix and vertex.

Graph Parabolas with Vertices Not at the Origin - (y+1)2=−12(x+2)( \mathrm { y } + 1 ) ^ { 2 } = - 12 ( \mathrm { x } + 2 )(y+1)2=−12(x+2)

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. - x=(y+2-1 (x-2+(y+2=1

Eliminate the Parameter - x=t3+1,y=t3−10;−2≤t≤2\mathrm { x } = \mathrm { t } ^ { 3 } + 1 , \mathrm { y } = \mathrm { t } ^ { 3 } - 10 ; - 2 \leq \mathrm { t } \leq 2x=t3+1,y=t3−10;−2≤t≤2

Identify Conics Without Rotating Axes - 3x2−7xy+3y2−4x−4y+5=03 x ^ { 2 } - 7 x y + 3 y ^ { 2 } - 4 x - 4 y + 5 = 03x2−7xy+3y2−4x−4y+5=0

Graph the ellipse and locate the foci. - x264+y236=1\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 164x2​+36y2​=1

Conic Sections in Polar Coordinates 1 Define Conics in Terms of a Focus and a Directrix - r=36−3sin⁡θr = \frac { 3 } { 6 - 3 \sin \theta }r=6−3sinθ3​

Additional Concepts - x=−(y+3)2−8x = - ( y + 3 ) ^ { 2 } - 8x=−(y+3)2−8

Use the rotated system to graph the equation. - x2−8xy+y2+25=0x^{2}-8 x y+y^{2}+25=0x2−8xy+y2+25=0

Eliminate the Parameter - x=2+sec⁡t,y=5+2tan⁡t;0<t<π2x = 2 + \sec t , y = 5 + 2 \tan t ; 0 < t < \frac { \pi } { 2 }x=2+sect,y=5+2tant;0<t<2π​

Additional Concepts - x29−y225=1\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 19x2​−25y2​=1

Additional Concepts - y29−x236=1\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 19y2​−36x2​=1

Tech: Rotation of Axes -

Parametric Equations 1 Use Point Plotting to Graph Plane Curves Described by Parametric Equations - x=4+5cos⁡t,y=6+4sin⁡t;t=π2x = 4 + 5 \cos t , y = 6 + 4 \sin t ; t = \frac { \pi } { 2 }x=4+5cost,y=6+4sint;t=2π​

Graph the Polar Equations of Conics - r=22−cos⁡θr = \frac { 2 } { 2 - \cos \theta } \quadr=2−cosθ2​ Identify the directrix and vertices.

Write Equations of Rotated Conics in Standard Form - x2+xy+y2−3y−6=0x^{2}+x y+y^{2}-3 y-6=0x2+xy+y2−3y−6=0

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. - x=-7 x=-7y

Graph the parabola with the given equation. - (x−2)2=−8(y+2)( x - 2 ) ^ { 2 } = - 8 ( y + 2 )(x−2)2=−8(y+2)

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

Rotation of Axes 1 Identify Conics Without Completing the Square - $5 x ^ { 2 } - 6 y ^ { 2 } + 2 x - 3 y - 5 = 0$

Graph the Polar Equations of Conics - $r = \frac { 2 } { 2 - 2 \cos \theta } \quad$ Identify the directrix and vertex. $Graph the Polar Equations of Conics - r = \frac { 2 } { 2 - 2 \cos \theta } \quad Identify the directrix and vertex.$

Graph Parabolas with Vertices Not at the Origin - $( \mathrm { y } + 1 ) ^ { 2 } = - 12 ( \mathrm { x } + 2 )$

Eliminate the Parameter - $\mathrm { x } = \mathrm { t } ^ { 3 } + 1 , \mathrm { y } = \mathrm { t } ^ { 3 } - 10 ; - 2 \leq \mathrm { t } \leq 2$

Identify Conics Without Rotating Axes - $3 x ^ { 2 } - 7 x y + 3 y ^ { 2 } - 4 x - 4 y + 5 = 0$

Graph the ellipse and locate the foci. - $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$ $Graph the ellipse and locate the foci. - \frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$

Conic Sections in Polar Coordinates 1 Define Conics in Terms of a Focus and a Directrix - $r = \frac { 3 } { 6 - 3 \sin \theta }$

Additional Concepts - $x = - ( y + 3 ) ^ { 2 } - 8$

Use the rotated system to graph the equation. - $x^{2}-8 x y+y^{2}+25=0$ $Use the rotated system to graph the equation. - x^{2}-8 x y+y^{2}+25=0$

Eliminate the Parameter - $x = 2 + \sec t , y = 5 + 2 \tan t ; 0 < t < \frac { \pi } { 2 }$

Additional Concepts - $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ $Additional Concepts - \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$

Additional Concepts - $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$ $Additional Concepts - \frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$

Parametric Equations 1 Use Point Plotting to Graph Plane Curves Described by Parametric Equations - $x = 4 + 5 \cos t , y = 6 + 4 \sin t ; t = \frac { \pi } { 2 }$

Graph the Polar Equations of Conics - $r = \frac { 2 } { 2 - \cos \theta } \quad$ Identify the directrix and vertices. $Graph the Polar Equations of Conics - r = \frac { 2 } { 2 - \cos \theta } \quad Identify the directrix and vertices.$

Write Equations of Rotated Conics in Standard Form - $x^{2}+x y+y^{2}-3 y-6=0$

Graph the parabola with the given equation. - $( x - 2 ) ^ { 2 } = - 8 ( y + 2 )$ $Graph the parabola with the given equation. - ( x - 2 ) ^ { 2 } = - 8 ( y + 2 )$