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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

Try It

8.1 The Ellipse

1.

x 2 + y 2 16 =1 x 2 + y 2 16 =1

2.

( x1 ) 2 16 + ( y3 ) 2 4 =1 ( x1 ) 2 16 + ( y3 ) 2 4 =1

3.

center: ( 0,0 ); ( 0,0 ); vertices: ( ±6,0 ); ( ±6,0 ); co-vertices: ( 0,±2 ); ( 0,±2 ); foci: ( ±4 2 ,0 ) ( ±4 2 ,0 )

4.

Standard form: x 2 16 + y 2 49 =1; x 2 16 + y 2 49 =1; center: ( 0,0 ); ( 0,0 ); vertices: ( 0,±7 ); ( 0,±7 ); co-vertices: ( ±4,0 ); ( ±4,0 ); foci: ( 0,± 33 ) ( 0,± 33 )

5.

Center: ( 4,2 ); ( 4,2 ); vertices: ( 2,2 ) ( 2,2 ) and ( 10,2 ); ( 10,2 ); co-vertices: ( 4,22 5 ) ( 4,22 5 ) and ( 4,2+2 5 ); ( 4,2+2 5 ); foci: ( 0,2 ) ( 0,2 ) and ( 8,2 ) ( 8,2 )

6.

(x3) 2 4 + ( y+1 ) 2 16 =1; (x3) 2 4 + ( y+1 ) 2 16 =1; center: ( 3,1 ); ( 3,1 ); vertices: ( 3,5 ) ( 3,5 ) and ( 3,3 ); ( 3,3 ); co-vertices: ( 1,1 ) ( 1,1 ) and ( 5,1 ); ( 5,1 ); foci: ( 3,12 3 ) ( 3,12 3 ) and ( 3,1+2 3 ) ( 3,1+2 3 )

7.
  1. x 2 57,600 + y 2 25,600 =1 x 2 57,600 + y 2 25,600 =1
  2. The people are standing 358 feet apart.

8.2 The Hyperbola

1.

Vertices: ( ±3,0 ); ( ±3,0 ); Foci: ( ± 34 ,0 ) ( ± 34 ,0 )

2.

y 2 4 x 2 16 =1 y 2 4 x 2 16 =1

3.

( y3 ) 2 25 + ( x1 ) 2 144 =1 ( y3 ) 2 25 + ( x1 ) 2 144 =1

4.

vertices: ( ±12,0 ); ( ±12,0 ); co-vertices: ( 0,±9 ); ( 0,±9 ); foci: ( ±15,0 ); ( ±15,0 ); asymptotes: y=± 3 4 x; y=± 3 4 x;

5.

center: ( 3,4 ); ( 3,4 ); vertices: ( 3,14 ) ( 3,14 ) and ( 3,6 ); ( 3,6 ); co-vertices: ( 5,4 ); ( 5,4 ); and ( 11,4 ); ( 11,4 ); foci: ( 3,42 41 ) ( 3,42 41 ) and ( 3,4+2 41 ); ( 3,4+2 41 ); asymptotes: y=± 5 4 ( x3 )4 y=± 5 4 ( x3 )4

6.

The sides of the tower can be modeled by the hyperbolic equation. x 2 400 y 2 3600 =1or  x 2 20 2 y 2 60 2 =1. x 2 400 y 2 3600 =1or  x 2 20 2 y 2 60 2 =1.

8.3 The Parabola

1.

Focus: ( 4,0 ); ( 4,0 ); Directrix: x=4; x=4; Endpoints of the latus rectum: ( 4,±8 ) ( 4,±8 )

2.

Focus: ( 0,2 ); ( 0,2 ); Directrix: y=−2; y=−2; Endpoints of the latus rectum: ( ±4,2 ). ( ±4,2 ).

3.

x 2 =14y. x 2 =14y.

4.

Vertex: ( 8,1 ); ( 8,1 ); Axis of symmetry: y=−1; y=−1; Focus: ( 9,1 ); ( 9,1 ); Directrix: x=7; x=7; Endpoints of the latus rectum: ( 9,3 ) ( 9,3 ) and ( 9,1 ). ( 9,1 ).

5.

Vertex: ( 2,3 ); ( 2,3 ); Axis of symmetry: x=−2; x=−2; Focus: ( 2,2 ); ( 2,2 ); Directrix: y=8; y=8; Endpoints of the latus rectum: ( 12,2 ) ( 12,2 ) and ( 8,2 ). ( 8,2 ).

6.
  1. y 2 =1280x y 2 =1280x
  2. The depth of the cooker is 500 mm

8.4 Rotation of Axes

1.
  1. hyperbola
  2. ellipse
2.

x 2 4 + y 2 1 =1 x 2 4 + y 2 1 =1

3.
  1. hyperbola
  2. ellipse

8.5 Conic Sections in Polar Coordinates

1.

ellipse; e= 1 3 ;x=2 e= 1 3 ;x=2

2.
3.

r= 1 1cosθ r= 1 1cosθ

4.

48x+3 x 2 y 2 =0 48x+3 x 2 y 2 =0

8.1 Section Exercises

1.

An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

3.

This special case would be a circle.

5.

It is symmetric about the x-axis, y-axis, and the origin.

7.

yes; x 2 3 2 + y 2 2 2 =1 x 2 3 2 + y 2 2 2 =1

9.

yes; x 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1 x 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1

11.

x 2 2 2 + y 2 7 2 =1; x 2 2 2 + y 2 7 2 =1; Endpoints of major axis ( 0,7 ) ( 0,7 ) and ( 0,7 ). ( 0,7 ). Endpoints of minor axis ( 2,0 ) ( 2,0 ) and ( 2,0 ). ( 2,0 ). Foci at ( 0,3 5 ),( 0,3 5 ). ( 0,3 5 ),( 0,3 5 ).

13.

x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1; x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1; Endpoints of major axis ( 1,0 ) ( 1,0 ) and ( 1,0 ). ( 1,0 ). Endpoints of minor axis ( 0, 1 3 ),( 0, 1 3 ). ( 0, 1 3 ),( 0, 1 3 ). Foci at ( 2 2 3 ,0 ),( 2 2 3 ,0 ). ( 2 2 3 ,0 ),( 2 2 3 ,0 ).

15.

( x2 ) 2 7 2 + ( y4 ) 2 5 2 =1; ( x2 ) 2 7 2 + ( y4 ) 2 5 2 =1; Endpoints of major axis ( 9,4 ),( 5,4 ). ( 9,4 ),( 5,4 ). Endpoints of minor axis ( 2,9 ),( 2,1 ). ( 2,9 ),( 2,1 ). Foci at ( 2+2 6 ,4 ),( 22 6 ,4 ). ( 2+2 6 ,4 ),( 22 6 ,4 ).

17.

( x+5 ) 2 2 2 + ( y7 ) 2 3 2 =1; ( x+5 ) 2 2 2 + ( y7 ) 2 3 2 =1; Endpoints of major axis ( 5,10 ),( 5,4 ). ( 5,10 ),( 5,4 ). Endpoints of minor axis ( 3,7 ),( 7,7 ). ( 3,7 ),( 7,7 ). Foci at ( 5,7+ 5 ),( 5,7 5 ). ( 5,7+ 5 ),( 5,7 5 ).

19.

( x1 ) 2 3 2 + ( y4 ) 2 2 2 =1; ( x1 ) 2 3 2 + ( y4 ) 2 2 2 =1; Endpoints of major axis ( 4,4 ),( 2,4 ). ( 4,4 ),( 2,4 ). Endpoints of minor axis ( 1,6 ),( 1,2 ). ( 1,6 ),( 1,2 ). Foci at ( 1+ 5 ,4 ),( 1 5 ,4 ). ( 1+ 5 ,4 ),( 1 5 ,4 ).

21.

( x3 ) 2 ( 3 2 ) 2 + ( y5 ) 2 ( 2 ) 2 =1; ( x3 ) 2 ( 3 2 ) 2 + ( y5 ) 2 ( 2 ) 2 =1; Endpoints of major axis ( 3+3 2 ,5 ),( 33 2 ,5 ). ( 3+3 2 ,5 ),( 33 2 ,5 ). Endpoints of minor axis ( 3,5+ 2 ),( 3,5 2 ). ( 3,5+ 2 ),( 3,5 2 ). Foci at ( 7,5 ),( 1,5 ). ( 7,5 ),( 1,5 ).

23.

( x+5 ) 2 ( 5 ) 2 + ( y2 ) 2 ( 2 ) 2 =1; ( x+5 ) 2 ( 5 ) 2 + ( y2 ) 2 ( 2 ) 2 =1; Endpoints of major axis ( 0,2 ),( 10,2 ). ( 0,2 ),( 10,2 ). Endpoints of minor axis ( 5,4 ),( 5,0 ). ( 5,4 ),( 5,0 ). Foci at ( 5+ 21 ,2 ),( 5 21 ,2 ). ( 5+ 21 ,2 ),( 5 21 ,2 ).

25.

( x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1; ( x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1; Endpoints of major axis ( 2,4 ),( 8,4 ). ( 2,4 ),( 8,4 ). Endpoints of minor axis ( 3,2 ),( 3,6 ). ( 3,2 ),( 3,6 ). Foci at ( 3+ 21 ,4 ),( 3 21 ,4 ). ( 3+ 21 ,4 ),( 3 21 ,4 ).

27.

Foci ( 3,1+ 11 ),( 3,1 11 ) ( 3,1+ 11 ),( 3,1 11 )

29.

Focus ( 0,0 ) ( 0,0 )

31.

Foci ( 10,30 ),( 10,30 ) ( 10,30 ),( 10,30 )

33.

Center ( 0,0 ), ( 0,0 ), Vertices ( 4,0 ),( 4,0 ),(0,3),(0,3), ( 4,0 ),( 4,0 ),(0,3),(0,3), Foci ( 7 ,0 ),( 7 ,0 ) ( 7 ,0 ),( 7 ,0 )

35.

Center ( 0,0 ), ( 0,0 ), Vertices ( 1 9 ,0 ),( 1 9 ,0 ),( 0, 1 7 ),( 0, 1 7 ), ( 1 9 ,0 ),( 1 9 ,0 ),( 0, 1 7 ),( 0, 1 7 ), Foci ( 0, 4 2 63 ),( 0, 4 2 63 ) ( 0, 4 2 63 ),( 0, 4 2 63 )

37.

Center ( 3,3 ), ( 3,3 ), Vertices ( 0,3 ),( 6,3 ),( 3,0 ),( 3,6 ), ( 0,3 ),( 6,3 ),( 3,0 ),( 3,6 ), Focus ( 3,3 ) ( 3,3 )

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

39.

Center ( 1,1 ), ( 1,1 ), Vertices ( 5,1 ),( 3,1 ),( 1,3 ),( 1,1 ), ( 5,1 ),( 3,1 ),( 1,3 ),( 1,1 ), Foci ( 1,1+4 3 ),( 1,14 3 ) ( 1,1+4 3 ),( 1,14 3 )

41.

Center ( 4,5 ), ( 4,5 ), Vertices ( 2,5 ),( 6,4 ),( 4,6 ),( 4,4 ), ( 2,5 ),( 6,4 ),( 4,6 ),( 4,4 ), Foci ( 4+ 3 ,5 ),( 4 3 ,5 ) ( 4+ 3 ,5 ),( 4 3 ,5 )

43.

Center ( 2,1 ), ( 2,1 ), Vertices ( 0,1 ),( 4,1 ),( 2,5 ),( 2,3 ), ( 0,1 ),( 4,1 ),( 2,5 ),( 2,3 ), Foci ( 2,1+2 3 ),( 2,12 3 ) ( 2,1+2 3 ),( 2,12 3 )

45.

Center ( 2,2 ), ( 2,2 ), Vertices ( 0,2 ),( 4,2 ),( 2,0 ),( 2,4 ), ( 0,2 ),( 4,2 ),( 2,0 ),( 2,4 ), Focus ( 2,2 ) ( 2,2 )

47.

x 2 25 + y 2 29 =1 x 2 25 + y 2 29 =1

49.

( x4 ) 2 25 + ( y2 ) 2 1 =1 ( x4 ) 2 25 + ( y2 ) 2 1 =1

51.

( x+3 ) 2 16 + ( y4 ) 2 4 =1 ( x+3 ) 2 16 + ( y4 ) 2 4 =1

53.

x 2 81 + y 2 9 =1 x 2 81 + y 2 9 =1

55.

( x+2 ) 2 4 + ( y2 ) 2 9 =1 ( x+2 ) 2 4 + ( y2 ) 2 9 =1

57.

Area = 12πsquareunits Area = 12πsquareunits

59.

Area = 2 5 πsquareunits Area = 2 5 πsquareunits

61.

Area = 9πsquareunits Area = 9πsquareunits

63.

x 2 4 h 2 + y 2 1 4 h 2 =1 x 2 4 h 2 + y 2 1 4 h 2 =1

65.

x 2 400 + y 2 144 =1 x 2 400 + y 2 144 =1 . Distance = 17.32 feet

67.

Approximately 51.96 feet

8.2 Section Exercises

1.

A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

3.

The foci must lie on the transverse axis and be in the interior of the hyperbola.

5.

The center must be the midpoint of the line segment joining the foci.

7.

yes x 2 6 2 y 2 3 2 =1 x 2 6 2 y 2 3 2 =1

9.

yes x 2 4 2 y 2 5 2 =1 x 2 4 2 y 2 5 2 =1

11.

x 2 5 2 y 2 6 2 =1; x 2 5 2 y 2 6 2 =1; vertices: ( 5,0 ),( 5,0 ); ( 5,0 ),( 5,0 ); foci: ( 61 ,0 ),( 61 ,0 ); ( 61 ,0 ),( 61 ,0 ); asymptotes: y= 6 5 x,y= 6 5 x y= 6 5 x,y= 6 5 x

13.

y 2 2 2 x 2 9 2 =1; y 2 2 2 x 2 9 2 =1; vertices: ( 0,2 ),( 0,2 ); ( 0,2 ),( 0,2 ); foci: ( 0, 85 ),( 0, 85 ); ( 0, 85 ),( 0, 85 ); asymptotes: y= 2 9 x,y= 2 9 x y= 2 9 x,y= 2 9 x

15.

( x1 ) 2 3 2 ( y2 ) 2 4 2 =1; ( x1 ) 2 3 2 ( y2 ) 2 4 2 =1; vertices: ( 4,2 ),( 2,2 ); ( 4,2 ),( 2,2 ); foci: ( 6,2 ),( 4,2 ); ( 6,2 ),( 4,2 ); asymptotes: y= 4 3 ( x1 )+2,y= 4 3 ( x1 )+2 y= 4 3 ( x1 )+2,y= 4 3 ( x1 )+2

17.

( x2 ) 2 7 2 ( y+7 ) 2 7 2 =1; ( x2 ) 2 7 2 ( y+7 ) 2 7 2 =1; vertices: ( 9,7 ),( 5,7 ); ( 9,7 ),( 5,7 ); foci: ( 2+7 2 ,7 ),( 27 2 ,7 ); ( 2+7 2 ,7 ),( 27 2 ,7 ); asymptotes: y=x9,y=x5 y=x9,y=x5

19.

( x+3 ) 2 3 2 ( y3 ) 2 3 2 =1; ( x+3 ) 2 3 2 ( y3 ) 2 3 2 =1; vertices: ( 0,3 ),( 6,3 ); ( 0,3 ),( 6,3 ); foci: ( 3+3 2 ,1 ),( 33 2 ,1 ); ( 3+3 2 ,1 ),( 33 2 ,1 ); asymptotes: y=x+6,y=x y=x+6,y=x

21.

( y4 ) 2 2 2 ( x3 ) 2 4 2 =1; ( y4 ) 2 2 2 ( x3 ) 2 4 2 =1; vertices: ( 3,6 ),( 3,2 ); ( 3,6 ),( 3,2 ); foci: ( 3,4+2 5 ),( 3,42 5 ); ( 3,4+2 5 ),( 3,42 5 ); asymptotes: y= 1 2 ( x3 )+4,y= 1 2 ( x3 )+4 y= 1 2 ( x3 )+4,y= 1 2 ( x3 )+4

23.

( y+5 ) 2 7 2 ( x+1 ) 2 70 2 =1; ( y+5 ) 2 7 2 ( x+1 ) 2 70 2 =1; vertices: ( 1,2 ),( 1,12 ); ( 1,2 ),( 1,12 ); foci: ( 1,5+7 101 ),( 1,57 101 ); ( 1,5+7 101 ),( 1,57 101 ); asymptotes: y= 1 10 ( x+1 )5,y= 1 10 ( x+1 )5 y= 1 10 ( x+1 )5,y= 1 10 ( x+1 )5

25.

( x+3 ) 2 5 2 ( y4 ) 2 2 2 =1; ( x+3 ) 2 5 2 ( y4 ) 2 2 2 =1; vertices: ( 2,4 ),( 8,4 ); ( 2,4 ),( 8,4 ); foci: ( 3+ 29 ,4 ),( 3 29 ,4 ); ( 3+ 29 ,4 ),( 3 29 ,4 ); asymptotes: y= 2 5 ( x+3 )+4,y= 2 5 ( x+3 )+4 y= 2 5 ( x+3 )+4,y= 2 5 ( x+3 )+4

27.

y= 2 5 ( x3 )4,y= 2 5 ( x3 )4 y= 2 5 ( x3 )4,y= 2 5 ( x3 )4

29.

y= 3 4 ( x1 )+1,y= 3 4 ( x1 )+1 y= 3 4 ( x1 )+1,y= 3 4 ( x1 )+1

31.
33.
35.
37.
39.
41.
43.
45.

x 2 9 y 2 16 =1 x 2 9 y 2 16 =1

47.

( x6 ) 2 25 ( y1 ) 2 11 =1 ( x6 ) 2 25 ( y1 ) 2 11 =1

49.

( x4 ) 2 25 ( y2 ) 2 1 =1 ( x4 ) 2 25 ( y2 ) 2 1 =1

51.

y 2 16 x 2 25 =1 y 2 16 x 2 25 =1

53.

y 2 9 ( x+1 ) 2 9 =1 y 2 9 ( x+1 ) 2 9 =1

55.

( x+3 ) 2 25 ( y+3 ) 2 25 =1 ( x+3 ) 2 25 ( y+3 ) 2 25 =1

57.

y( x )=3 x 2 +1 ,y( x )=3 x 2 +1 y( x )=3 x 2 +1 ,y( x )=3 x 2 +1

59.

y( x )=1+2 x 2 +4x+5 ,y( x )=12 x 2 +4x+5 y( x )=1+2 x 2 +4x+5 ,y( x )=12 x 2 +4x+5

61.

x 2 25 y 2 25 =1 x 2 25 y 2 25 =1

63.

x 2 100 y 2 25 =1 x 2 100 y 2 25 =1

65.

x 2 400 y 2 225 =1 x 2 400 y 2 225 =1

67.

4(x-1)2-y22=16 4(x-1)2-y22=16

69.

( xh ) 2 a2 =4 - (y-k)2 b2 =(x-3)2-9y2=4 ( xh ) 2 a2 =4 -(y-k)2 b2=(x-3)2-9y2=4

8.3 Section Exercises

1.

A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

3.

The graph will open down.

5.

The distance between the focus and directrix will increase.

7.

yes y=4(1) x 2 y=4(1) x 2

9.

yes ( y3 ) 2 =4(2)( x2 ) ( y3 ) 2 =4(2)( x2 )

11.

y 2 = 1 8 x,V:(0,0);F:( 1 32 ,0 );d:x= 1 32 y 2 = 1 8 x,V:(0,0);F:( 1 32 ,0 );d:x= 1 32

13.

x 2 = 1 4 y,V:( 0,0 );F:( 0, 1 16 );d:y= 1 16 x 2 = 1 4 y,V:( 0,0 );F:( 0, 1 16 );d:y= 1 16

15.

y 2 = 1 36 x,V:( 0,0 );F:( 1 144 ,0 );d:x= 1 144 y 2 = 1 36 x,V:( 0,0 );F:( 1 144 ,0 );d:x= 1 144

17.

( x1 ) 2 =4( y1 ),V:( 1,1 );F:( 1,2 );d:y=0 ( x1 ) 2 =4( y1 ),V:( 1,1 );F:( 1,2 );d:y=0

19.

( y4 ) 2 =2( x+3 ),V:( 3,4 );F:( 5 2 ,4 );d:x= 7 2 ( y4 ) 2 =2( x+3 ),V:( 3,4 );F:( 5 2 ,4 );d:x= 7 2

21.

( x+4 ) 2 =24( y+1 ),V:( 4,1 );F:( 4,5 );d:y=−7 ( x+4 ) 2 =24( y+1 ),V:( 4,1 );F:( 4,5 );d:y=−7

23.

( y3 ) 2 =−12( x+1 ),V:( 1,3 );F:( 4,3 );d:x=2 ( y3 ) 2 =−12( x+1 ),V:( 1,3 );F:( 4,3 );d:x=2

25.

( x5 ) 2 = 4 5 ( y+3 ),V:( 5,3 );F:( 5, 14 5 );d:y= 16 5 ( x5 ) 2 = 4 5 ( y+3 ),V:( 5,3 );F:( 5, 14 5 );d:y= 16 5

27.

( x2 ) 2 =−2( y5 ),V:( 2,5 );F:( 2, 9 2 );d:y= 11 2 ( x2 ) 2 =−2( y5 ),V:( 2,5 );F:( 2, 9 2 );d:y= 11 2

29.

( y1 ) 2 = 4 3 ( x5 ),V:( 5,1 );F:( 16 3 ,1 );d:x= 14 3 ( y1 ) 2 = 4 3 ( x5 ),V:( 5,1 );F:( 16 3 ,1 );d:x= 14 3

31.
33.
35.
37.
39.
41.
43.
45.

x 2 =−16y x 2 =−16y

47.

( y2 ) 2 =4 2 ( x2 ) ( y2 ) 2 =4 2 ( x2 )

49.

( y+ 3 ) 2 =−4 2 ( x 2 ) ( y+ 3 ) 2 =−4 2 ( x 2 )

51.

x 2 =y x 2 =y

53.

( y2 ) 2 = 1 4 ( x+2 ) ( y2 ) 2 = 1 4 ( x+2 )

55.

( y 3 ) 2 =4 5 ( x+ 2 ) ( y 3 ) 2 =4 5 ( x+ 2 )

57.

y 2 =−8x y 2 =−8x

59.

( y+1 ) 2 =12( x+3 ) ( y+1 ) 2 =12( x+3 )

61.

( 0,1 ) ( 0,1 )

63.

At the point 2.25 feet above the vertex.

65.

0.5625 feet

67.

x 2 =−125( y20 ), x 2 =−125( y20 ), height is 7.2 feet

69.

2304 feet

8.4 Section Exercises

1.

The xy xy term causes a rotation of the graph to occur.

3.

The conic section is a hyperbola.

5.

It gives the angle of rotation of the axes in order to eliminate the xy xy term.

7.

AB=0, AB=0, parabola

9.

AB=4<0, AB=4<0, hyperbola

11.

AB=6>0, AB=6>0, ellipse

13.

B 2 4AC=0, B 2 4AC=0, parabola

15.

B 2 4AC=0, B 2 4AC=0, parabola

17.

B 2 4AC=96<0, B 2 4AC=96<0, ellipse

19.

7 x 2 +9 y 2 4=0 7 x 2 +9 y 2 4=0

21.

3 x 2 +2 x y 5 y 2 +1=0 3 x 2 +2 x y 5 y 2 +1=0

23.

θ= 60 ,11 x 2 y 2 + 3 x + y 4=0 θ= 60 ,11 x 2 y 2 + 3 x + y 4=0

25.

θ= 150 ,21 x 2 +9 y 2 +4 x 4 3 y 6=0 θ= 150 ,21 x 2 +9 y 2 +4 x 4 3 y 6=0

27.

θ 36.9 ,125 x 2 +6 x 42 y +10=0 θ 36.9 ,125 x 2 +6 x 42 y +10=0

29.

θ= 45 ,3 x 2 y 2 2 x + 2 y +1=0 θ= 45 ,3 x 2 y 2 2 x + 2 y +1=0

31.

2 2 ( x + y )= 1 2 ( x y ) 2 2 2 ( x + y )= 1 2 ( x y ) 2

33.

( x y ) 2 8 + ( x + y ) 2 2 =1 ( x y ) 2 8 + ( x + y ) 2 2 =1

35.

( x + y ) 2 2 ( x y ) 2 2 =1 ( x + y ) 2 2 ( x y ) 2 2 =1

37.

3 2 x 1 2 y = ( 1 2 x + 3 2 y 1 ) 2 3 2 x 1 2 y = ( 1 2 x + 3 2 y 1 ) 2

39.
41.
43.
45.
47.
49.
51.

θ= 45 θ= 45

53.

θ= 60 θ= 60

55.

θ 36.9 θ 36.9

57.

4 6 <k<4 6 4 6 <k<4 6

59.

k=2 k=2

8.5 Section Exercises

1.

If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

3.

The directrix will be parallel to the polar axis.

5.

One of the foci will be located at the origin.

7.

Parabola with e=1 e=1 and directrix 3 4 3 4 units below the pole.

9.

Hyperbola with e=2 e=2 and directrix 5 2 5 2 units above the pole.

11.

Parabola with e=1 e=1 and directrix 3 10 3 10 units to the right of the pole.

13.

Ellipse with e= 2 7 e= 2 7 and directrix 2 2 units to the right of the pole.

15.

Hyperbola with e= 5 3 e= 5 3 and directrix 11 5 11 5 units above the pole.

17.

Hyperbola with e= 8 7 e= 8 7 and directrix 7 8 7 8 units to the right of the pole.

19.

25 x 2 +16 y 2 12y4=0 25 x 2 +16 y 2 12y4=0

21.

21 x 2 4 y 2 30x+9=0 21 x 2 4 y 2 30x+9=0

23.

64 y 2 =48x+9 64 y 2 =48x+9

25.

96 y 2 25 x 2 +110y+25=0 96 y 2 25 x 2 +110y+25=0

27.

3 x 2 +4 y 2 2x1=0 3 x 2 +4 y 2 2x1=0

29.

5 x 2 +9 y 2 24x36=0 5 x 2 +9 y 2 24x36=0

31.
33.
35.
37.
39.
41.
43.

r= 4 5+cosθ r= 4 5+cosθ

45.

r= 4 1+2sinθ r= 4 1+2sinθ

47.

r= 1 1+cosθ r= 1 1+cosθ

49.

r= 7 828cosθ r= 7 828cosθ

51.

r= 12 2+3sinθ r= 12 2+3sinθ

53.

r= 15 43cosθ r= 15 43cosθ

55.

r= 3 33cosθ r= 3 33cosθ

57.

r=± 2 1+sinθcosθ r=± 2 1+sinθcosθ

59.

r=± 2 4cosθ+3sinθ r=± 2 4cosθ+3sinθ

Review Exercises

1.

x 2 5 2 + y 2 8 2 =1; x 2 5 2 + y 2 8 2 =1; center: ( 0,0 ); ( 0,0 ); vertices: ( 5,0 ),( −5,0 ),( 0,8 ),( 0,8 ); ( 5,0 ),( −5,0 ),( 0,8 ),( 0,8 ); foci: ( 0, 39 ),( 0, 39 ) ( 0, 39 ),( 0, 39 )

3.

(x+3) 2 1 2 + (y2) 2 3 2 =1(3,2);(2,2),(4,2),(3,5),(3,1);( 3,2+2 2 ),( 3,22 2 ) (x+3) 2 1 2 + (y2) 2 3 2 =1(3,2);(2,2),(4,2),(3,5),(3,1);( 3,2+2 2 ),( 3,22 2 )

5.

center: ( 0,0 ); ( 0,0 ); vertices: ( 6,0 ),( −6,0 ),( 0,3 ),( 0,−3 ); ( 6,0 ),( −6,0 ),( 0,3 ),( 0,−3 ); foci: ( 3 3 ,0 ),( 3 3 ,0 ) ( 3 3 ,0 ),( 3 3 ,0 )

7.

center: ( −2,−2 ); ( −2,−2 ); vertices: ( 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 ); ( 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 ); foci: ( −2,−2+4 3 , ),( −2,−2−4 3 ) ( −2,−2+4 3 , ),( −2,−2−4 3 )

9.

x 2 25 + y 2 16 =1 x 2 25 + y 2 16 =1

11.

Approximately 35.71 feet

13.

( y+1 ) 2 4 2 ( x4 ) 2 6 2 =1; ( y+1 ) 2 4 2 ( x4 ) 2 6 2 =1; center: ( 4,−1 ); ( 4,−1 ); vertices: ( 4,3 ),( 4,−5 ); ( 4,3 ),( 4,−5 ); foci: ( 4,−1+2 13 ),( 4,−12 13 ) ( 4,−1+2 13 ),( 4,−12 13 )

15.

( x2 ) 2 2 2 ( y+3 ) 2 ( 2 3 ) 2 =1; ( x2 ) 2 2 2 ( y+3 ) 2 ( 2 3 ) 2 =1; center: ( 2,−3 ); ( 2,−3 ); vertices: ( 4,−3 ),( 0,−3 ); ( 4,−3 ),( 0,−3 ); foci: ( 6,−3 ),( −2,−3 ) ( 6,−3 ),( −2,−3 )

17.


19.


21.

( x5 ) 2 1 ( y7 ) 2 3 =1 ( x5 ) 2 1 ( y7 ) 2 3 =1

23.

( x+2 ) 2 = 1 2 ( y1 ); ( x+2 ) 2 = 1 2 ( y1 ); vertex: ( −2,1 ); ( −2,1 ); focus: ( −2, 9 8 ); ( −2, 9 8 ); directrix: y= 7 8 y= 7 8

25.

( x+5 ) 2 =( y+2 ); ( x+5 ) 2 =( y+2 ); vertex: ( 5,2 ); ( 5,2 ); focus: ( 5, 7 4 ); ( 5, 7 4 ); directrix: y= 9 4 y= 9 4

27.


29.


31.

( x2 ) 2 =( 1 2 )( y1 ) ( x2 ) 2 =( 1 2 )( y1 )

33.

B 2 4AC=0, B 2 4AC=0, parabola

35.

B 2 4AC=31<0, B 2 4AC=31<0, ellipse

37.

θ= 45 , x 2 +3 y 2 12=0 θ= 45 , x 2 +3 y 2 12=0

39.

θ= 45 θ= 45

41.

Hyperbola with e=5 e=5 and directrix 2 2 units to the left of the pole.

43.

Ellipse with e= 3 4 e= 3 4 and directrix 1 3 1 3 unit above the pole.

45.


47.


49.

r= 3 1+cos  θ r= 3 1+cos  θ

Practice Test

1.

x 2 3 2 + y 2 2 2 =1; x 2 3 2 + y 2 2 2 =1; center: ( 0,0 ); ( 0,0 ); vertices: ( 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 ); ( 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 ); foci: ( 5 ,0 ),( 5 ,0 ) ( 5 ,0 ),( 5 ,0 )

3.

center: ( 3,2 ); ( 3,2 ); vertices: ( 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 ); ( 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 ); foci: ( 3+2 7 ,2 ),( 32 7 ,2 ) ( 3+2 7 ,2 ),( 32 7 ,2 )

5.

( x1 ) 2 36 + ( y2 ) 2 27 =1 ( x1 ) 2 36 + ( y2 ) 2 27 =1

7.

x 2 7 2 y 2 9 2 =1; x 2 7 2 y 2 9 2 =1; center: ( 0,0 ); ( 0,0 ); vertices ( 7,0 ),( −7,0 ); ( 7,0 ),( −7,0 ); foci: ( 130 ,0 ),( 130 ,0 ); ( 130 ,0 ),( 130 ,0 ); asymptotes: y=± 9 7 x y=± 9 7 x

9.

center: ( 3,−3 ); ( 3,−3 ); vertices: ( 8,−3 ),( −2,−3 ); ( 8,−3 ),( −2,−3 ); foci: ( 3+ 26 ,−3 ),( 3 26 ,−3 ); ( 3+ 26 ,−3 ),( 3 26 ,−3 ); asymptotes: y=± 1 5 (x3)3 y=± 1 5 (x3)3

11.

( y3 ) 2 1 ( x1 ) 2 8 =1 ( y3 ) 2 1 ( x1 ) 2 8 =1

13.

( x2 ) 2 = 1 3 ( y+1 ); ( x2 ) 2 = 1 3 ( y+1 ); vertex: ( 2,−1 ); ( 2,−1 ); focus: ( 2, 11 12 ); ( 2, 11 12 ); directrix: y= 13 12 y= 13 12

15.


17.

Approximately 8.49 8.49 feet

19.

parabola; θ 63.4 θ 63.4

21.

x 2 4 x +3 y =0 x 2 4 x +3 y =0

23.

Hyperbola with e= 3 2 , e= 3 2 , and directrix 5 6 5 6 units to the right of the pole.

25.
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