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Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises
  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

Quadratic Functions

For the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts. Finally, graph the function.

1.

f(x)= x 2 4x5 f(x)= x 2 4x5

2.

f(x)=2 x 2 4x f(x)=2 x 2 4x

For the following exercises, find the equation of the quadratic function using the given information.

3.

The vertex is (2,3) (2,3) and a point on the graph is (3,6). (3,6).

4.

The vertex is (3,6.5) (3,6.5) and a point on the graph is (2,6). (2,6).

For the following exercises, complete the task.

5.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

6.

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height, h, h, in terms of horizontal distance traveled, x, x, given by h(x)= 32 (120) 2 x 2 +x. h(x)= 32 (120) 2 x 2 +x.Find the maximum height the object attains.

Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

7.

f(x)=4 x 5 3 x 3 +2x1 f(x)=4 x 5 3 x 3 +2x1

8.

f(x)= 5 x+1 x 2 f(x)= 5 x+1 x 2

9.

f(x)= x 2 ( 36x+ x 2 ) f(x)= x 2 ( 36x+ x 2 )

For the following exercises, determine end behavior of the polynomial function.

10.

f(x)=2 x 4 +3 x 3 5 x 2 +7 f(x)=2 x 4 +3 x 3 5 x 2 +7

11.

f(x)=4 x 3 6 x 2 +2 f(x)=4 x 3 6 x 2 +2

12.

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

Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

13.

f(x)= (x+3) 2 (2x1) (x+1) 3 f(x)= (x+3) 2 (2x1) (x+1) 3

14.

f(x)= x 5 +4 x 4 +4 x 3 f(x)= x 5 +4 x 4 +4 x 3

15.

f(x)= x 3 4 x 2 +x4 f(x)= x 3 4 x 2 +x4

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

16.
Graph of an odd-degree polynomial with two turning points.
17.
Graph of an even-degree polynomial with two turning points.
18.

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function f(x)= x 3 5x+1 f(x)= x 3 5x+1

Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

19.

x 3 2 x 2 +4x+4 x2 x 3 2 x 2 +4x+4 x2

20.

3 x 4 4 x 2 +4x+8 x+1 3 x 4 4 x 2 +4x+8 x+1

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

21.

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

22.

x 3 +4x+10 x3 x 3 +4x+10 x3

23.

2 x 3 +6 x 2 11x12 x+4 2 x 3 +6 x 2 11x12 x+4

24.

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

Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

25.

2 x 3 3 x 2 18x8=0 2 x 3 3 x 2 18x8=0

26.

3 x 3 +11 x 2 +8x4=0 3 x 3 +11 x 2 +8x4=0

27.

2 x 4 17 x 3 +46 x 2 43x+12=0 2 x 4 17 x 3 +46 x 2 43x+12=0

28.

4 x 4 +8 x 3 +19 x 2 +32x+12=0 4 x 4 +8 x 3 +19 x 2 +32x+12=0

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

29.

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

30.

2 x 4 x 3 +4 x 2 5x+1=0 2 x 4 x 3 +4 x 2 5x+1=0

Rational Functions

For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.

31.

f(x)= x+2 x5 f(x)= x+2 x5

32.

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

33.

f(x)= 3 x 2 27 x 2 +x2 f(x)= 3 x 2 27 x 2 +x2

34.

f(x)= x+2 x 2 9 f(x)= x+2 x 2 9

For the following exercises, find the slant asymptote.

35.

f(x)= x 2 1 x+2 f(x)= x 2 1 x+2

36.

f(x)= 2 x 3 x 2 +4 x 2 +1 f(x)= 2 x 3 x 2 +4 x 2 +1

Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

37.

f(x)= (x2) 2 ,x2 f(x)= (x2) 2 ,x2

38.

f(x)= (x+4) 2 3,x4 f(x)= (x+4) 2 3,x4

39.

f(x)= x 2 +6x2,x3 f(x)= x 2 +6x2,x3

40.

f(x)=2 x 3 3 f(x)=2 x 3 3

41.

f(x)= 4x+5 3 f(x)= 4x+5 3

42.

f(x)= x3 2x+1 f(x)= x3 2x+1

Modeling Using Variation

For the following exercises, find the unknown value.

43.

y y varies directly as the square of x. x. If when x=3, y=36, x=3, y=36,find y yif x=4. x=4.

44.

y y varies inversely as the square root of x x If when x=25, y=2, x=25, y=2,find y yif x=4. x=4.

45.

y yvaries jointly as the cube of x xand as z. z. If when x=1 x=1and z=2, z=2, y=6, y=6,find y yif x=2 x=2and z=3. z=3.

46.

y yvaries jointly as x xand the square of z zand inversely as the cube of w. w. If when x=3, x=3, z=4, z=4,and w=2, w=2, y=48, y=48,find y yif x=4, x=4, z=5, z=5,and w=3. w=3.

For the following exercises, solve the application problem.

47.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

48.

The volume V Vof an ideal gas varies directly with the temperature T Tand inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

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