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Algebra and Trigonometry 2e

Review Exercises

Algebra and Trigonometry 2eReview Exercises
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    6. Exercises
      1. Review Exercises
      2. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. 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. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. 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
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Review Exercises

The Ellipse

For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

1.

x 2 25 + y 2 64 =1 x 2 25 + y 2 64 =1

2.

(x−2) 2 100 + ( y+3 ) 2 36 =1 (x−2) 2 100 + ( y+3 ) 2 36 =1

3.

9 x 2 + y 2 +54x−4y+76=0 9 x 2 + y 2 +54x−4y+76=0

4.

9 x 2 +36 y 2 −36x+72y+36=0 9 x 2 +36 y 2 −36x+72y+36=0

For the following exercises, graph the ellipse, noting center, vertices, and foci.

5.

x 2 36 + y 2 9 =1 x 2 36 + y 2 9 =1

6.

(x−4) 2 25 + ( y+3 ) 2 49 =1 (x−4) 2 25 + ( y+3 ) 2 49 =1

7.

4 x 2 + y 2 +16x+4y−44=0 4 x 2 + y 2 +16x+4y−44=0

8.

2 x 2 +3 y 2 −20x+12y+38=0 2 x 2 +3 y 2 −20x+12y+38=0

For the following exercises, use the given information to find the equation for the ellipse.

9.

Center at ( 0,0 ), ( 0,0 ), focus at ( 3,0 ), ( 3,0 ), vertex at ( −5,0 ) ( −5,0 )

10.

Center at ( 2,−2 ), ( 2,−2 ), vertex at ( 7,−2 ), ( 7,−2 ), focus at ( 4,−2 ) ( 4,−2 )

11.

A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

The Hyperbola

For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.

12.

x 2 81 − y 2 9 =1 x 2 81 − y 2 9 =1

13.

( y+1 ) 2 16 − ( x−4 ) 2 36 =1 ( y+1 ) 2 16 − ( x−4 ) 2 36 =1

14.

9 y 2 −4 x 2 +54y−16x+29=0 9 y 2 −4 x 2 +54y−16x+29=0

15.

3 x 2 − y 2 −12x−6y−9=0 3 x 2 − y 2 −12x−6y−9=0

For the following exercises, graph the hyperbola, labeling vertices and foci.

16.

x 2 9 − y 2 16 =1 x 2 9 − y 2 16 =1

17.

( y−1 ) 2 49 − ( x+1 ) 2 4 =1 ( y−1 ) 2 49 − ( x+1 ) 2 4 =1

18.

x 2 −4 y 2 +6x+32y−91=0 x 2 −4 y 2 +6x+32y−91=0

19.

2 y 2 − x 2 −12y−6=0 2 y 2 − x 2 −12y−6=0

For the following exercises, find the equation of the hyperbola.

20.

Center at ( 0,0 ), ( 0,0 ), vertex at ( 0,4 ), ( 0,4 ), focus at ( 0,−6 ) ( 0,−6 )

21.

Foci at ( 3,7 ) ( 3,7 ) and ( 7,7 ), ( 7,7 ), vertex at ( 6,7 ) ( 6,7 )

The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

22.

y 2 =12x y 2 =12x

23.

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

24.

y 2 −6y−6x−3=0 y 2 −6y−6x−3=0

25.

x 2 +10x−y+23=0 x 2 +10x−y+23=0

For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

26.

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

27.

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

28.

x 2 −8x−10y+46=0 x 2 −8x−10y+46=0

29.

2 y 2 +12y+6x+15=0 2 y 2 +12y+6x+15=0

For the following exercises, write the equation of the parabola using the given information.

30.

Focus at ( −4,0 ); ( −4,0 ); directrix is x=4 x=4

31.

Focus at ( 2, 9 8 ); ( 2, 9 8 ); directrix is y= 7 8 y= 7 8

32.

A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

33.

16 x 2 +24xy+9 y 2 +24x−60y−60=0 16 x 2 +24xy+9 y 2 +24x−60y−60=0

34.

4 x 2 +14xy+5 y 2 +18x−6y+30=0 4 x 2 +14xy+5 y 2 +18x−6y+30=0

35.

4 x 2 +xy+2 y 2 +8x−26y+9=0 4 x 2 +xy+2 y 2 +8x−26y+9=0

For the following exercises, determine the angle θ θ that will eliminate the xy xy term, and write the corresponding equation without the xy xy term.

36.

x 2 +4xy−2 y 2 −6=0 x 2 +4xy−2 y 2 −6=0

37.

x 2 −xy+ y 2 −6=0 x 2 −xy+ y 2 −6=0

For the following exercises, graph the equation relative to the x ′ y ′ x ′ y ′ system in which the equation has no x ′ y ′ x ′ y ′ term.

38.

9 x 2 −24xy+16 y 2 −80x−60y+100=0 9 x 2 −24xy+16 y 2 −80x−60y+100=0

39.

x 2 −xy+ y 2 −2=0 x 2 −xy+ y 2 −2=0

40.

6 x 2 +24xy− y 2 −12x+26y+11=0 6 x 2 +24xy− y 2 −12x+26y+11=0

Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

41.

r= 10 1−5cosθ r= 10 1−5cosθ

42.

r= 6 3+2cosθ r= 6 3+2cosθ

43.

r= 1 4+3sinθ r= 1 4+3sinθ

44.

r= 3 5−5sinθ r= 3 5−5sinθ

For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.

45.

r= 3 1−sinθ r= 3 1−sinθ

46.

r= 8 4+3sinθ r= 8 4+3sinθ

47.

r= 10 4+5cosθ r= 10 4+5cosθ

48.

r= 9 3−6cosθ r= 9 3−6cosθ

For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

49.

Directrix is x=3 x=3 and eccentricity e=1 e=1

50.

Directrix is y=−2 y=−2 and eccentricity e=4 e=4

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