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Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises
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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 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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

The Rectangular Coordinate Systems and Graphs

For the following exercises, find the x-intercept and the y-intercept without graphing.

1.

4x3y=12 4x3y=12

2.

2y4=3x 2y4=3x

For the following exercises, solve for y in terms of x, putting the equation in slope–intercept form.

3.

5x=3y12 5x=3y12

4.

2x5y=7 2x5y=7

For the following exercises, find the distance between the two points.

5.

( −2,5 )( 4,−1 ) ( −2,5 )( 4,−1 )

6.

( −12,−3 )( −1,5 ) ( −12,−3 )( −1,5 )

7.

Find the distance between the two points (−71,432) (−71,432) and (511,218) (511,218) using your calculator, and round your answer to the nearest thousandth.

For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

8.

( −1,5 ) and ( 4,6 ) ( −1,5 ) and ( 4,6 )

9.

( −13,5 ) and ( 17,18 ) ( −13,5 ) and ( 17,18 )

For the following exercises, construct a table and graph the equation by plotting at least three points.

10.

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

11.

4x3y=6 4x3y=6

Linear Equations in One Variable

For the following exercises, solve for x. x.

12.

5x+2=7x8 5x+2=7x8

13.

3(x+2)10=x+4 3(x+2)10=x+4

14.

7x3=5 7x3=5

15.

125(x+1)=2x5 125(x+1)=2x5

16.

2x 3 3 4 = x 6 + 21 4 2x 3 3 4 = x 6 + 21 4

For the following exercises, solve for x. x. State all x-values that are excluded from the solution set.

17.

x x 2 9 + 4 x+3 = 3 x 2 9 x x 2 9 + 4 x+3 = 3 x 2 9 x3,−3 x3,−3

18.

1 2 + 2 x = 3 4 1 2 + 2 x = 3 4

For the following exercises, find the equation of the line using the point-slope formula.

19.

Passes through these two points: ( −2,1 ),( 4,2 ). ( −2,1 ),( 4,2 ).

20.

Passes through the point ( 3,4 ) ( 3,4 ) and has a slope of 1 3 . 1 3 .

21.

Passes through the point ( 3,4 ) ( 3,4 ) and is parallel to the graph y= 2 3 x+5. y= 2 3 x+5.

22.

Passes through these two points: ( 5,1 ),( 5,7 ). ( 5,1 ),( 5,7 ).

Models and Applications

For the following exercises, write and solve an equation to answer each question.

23.

The number of males in the classroom is five more than three times the number of females. If the total number of students is 73, how many of each gender are in the class?

24.

A man has 72 ft. of fencing to put around a rectangular garden. If the length is 3 times the width, find the dimensions of his garden.

25.

A truck rental is $25 plus $.30/mi. Find out how many miles Ken traveled if his bill was $50.20.

Complex Numbers

For the following exercises, use the quadratic equation to solve.

26.

x 2 5x+9=0 x 2 5x+9=0

27.

2 x 2 +3x+7=0 2 x 2 +3x+7=0

For the following exercises, name the horizontal component and the vertical component.

28.

43i 43i

29.

−2i −2i

For the following exercises, perform the operations indicated.

30.

( 9i )( 47i ) ( 9i )( 47i )

31.

( 2+3i )( 58i ) ( 2+3i )( 58i )

32.

2 75 +3 25 2 75 +3 25

33.

16 +4 9 16 +4 9

34.

6i(i5) 6i(i5)

35.

(35i) 2 (35i) 2

36.

4 · 12 4 · 12

37.

2 ( 8 5 ) 2 ( 8 5 )

38.

2 53i 2 53i

39.

3+7i i 3+7i i

Quadratic Equations

For the following exercises, solve the quadratic equation by factoring.

40.

2 x 2 7x4=0 2 x 2 7x4=0

41.

3 x 2 +18x+15=0 3 x 2 +18x+15=0

42.

25 x 2 9=0 25 x 2 9=0

43.

7 x 2 9x=0 7 x 2 9x=0

For the following exercises, solve the quadratic equation by using the square-root property.

44.

x 2 =49 x 2 =49

45.

( x4 ) 2 =36 ( x4 ) 2 =36

For the following exercises, solve the quadratic equation by completing the square.

46.

x 2 +8x5=0 x 2 +8x5=0

47.

4 x 2 +2x1=0 4 x 2 +2x1=0

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution.

48.

2 x 2 5x+1=0 2 x 2 5x+1=0

49.

15 x 2 x2=0 15 x 2 x2=0

For the following exercises, solve the quadratic equation by the method of your choice.

50.

(x2) 2 =16 (x2) 2 =16

51.

x 2 =10x+3 x 2 =10x+3

Other Types of Equations

For the following exercises, solve the equations.

52.

x 3 2 =27 x 3 2 =27

53.

x 1 2 4 x 1 4 =0 x 1 2 4 x 1 4 =0

54.

4 x 3 +8 x 2 9x18=0 4 x 3 +8 x 2 9x18=0

55.

3 x 5 6 x 3 =0 3 x 5 6 x 3 =0

56.

x+9 =x3 x+9 =x3

57.

3x+7 + x+2 =1 3x+7 + x+2 =1

58.

| 3x7 |=5 | 3x7 |=5

59.

| 2x+3 |5=9 | 2x+3 |5=9

Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.

60.

5x812 5x812

61.

2x+5>x7 2x+5>x7

62.

x1 3 + x+2 5 3 5 x1 3 + x+2 5 3 5

63.

| 3x+2 |+19 | 3x+2 |+19

64.

| 5x1 |>14 | 5x1 |>14

65.

| x3 |<−4 | x3 |<−4

For the following exercises, solve the compound inequality. Write your answer in interval notation.

66.

−4<3x+218 −4<3x+218

67.

3y<12y<5+y 3y<12y<5+y

For the following exercises, graph as described.

68.

Graph the absolute value function and graph the constant function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.

| x+3 |5 | x+3 |5

69.

Graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines. See the interval where the inequality is true.

x+3<3x4 x+3<3x4

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