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Algebra and Trigonometry 2e

Practice Test

Algebra and Trigonometry 2ePractice Test
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

Practice Test

For the following exercises, determine whether each of the following relations is a function.

1.

y=2x+8 y=2x+8

2.

{ (2,1),(3,2),(−1,1),(0,−2) } { (2,1),(3,2),(−1,1),(0,−2) }

For the following exercises, evaluate the function f(x)=−3 x 2 +2x f(x)=−3 x 2 +2x at the given input.

3.

f(−2) f(−2)

4.

f(a) f(a)

5.

Show that the function f(x)=−2 (x−1) 2 +3 f(x)=−2 (x−1) 2 +3 is not one-to-one.

6.

Write the domain of the function f(x)= 3−x f(x)= 3−x in interval notation.

7.

Given f(x)=2 x 2 −5x, f(x)=2 x 2 −5x, find f(a+1)−f(1) f(a+1)−f(1) in simplest form.

8.

Graph the function f(x)={ x+1  if −2<x<3   −x   if   x≥3 f(x)={ x+1  if −2<x<3   −x   if   x≥3

9.

Find the average rate of change of the function f(x)=3−2 x 2 +x f(x)=3−2 x 2 +x by finding f(b)−f(a) b−a f(b)−f(a) b−a in simplest form.

For the following exercises, use the functions f(x)=3−2 x 2 +xand g(x)= x f(x)=3−2 x 2 +xand g(x)= x to find the composite functions.

10.

( g∘f )(x) ( g∘f )(x)

11.

( g∘f )(1) ( g∘f )(1)

12.

Express H(x)= 5 x 2 −3x 3 H(x)= 5 x 2 −3x 3 as a composition of two functions, f f and g, g, where ( f∘g )(x)=H(x). ( f∘g )(x)=H(x).

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

13.

f(x)= x+6 −1 f(x)= x+6 −1

14.

f(x)= 1 x+2 −1 f(x)= 1 x+2 −1

For the following exercises, determine whether the functions are even, odd, or neither.

15.

f(x)=− 5 x 2 +9 x 6 f(x)=− 5 x 2 +9 x 6

16.

f(x)=− 5 x 3 +9 x 5 f(x)=− 5 x 3 +9 x 5

17.

f(x)= 1 x f(x)= 1 x

18.

Graph the absolute value function f(x)=−2| x−1 |+3. f(x)=−2| x−1 |+3.

For the following exercises, find the inverse of the function.

19.

f(x)=3x−5 f(x)=3x−5

20.

f(x)= 4 x+7 f(x)= 4 x+7

For the following exercises, use the graph of g g shown in Figure 1.

Graph of a cubic function.
Figure 1
21.

On what intervals is the function increasing?

22.

On what intervals is the function decreasing?

23.

Approximate the local minimum of the function. Express the answer as an ordered pair.

24.

Approximate the local maximum of the function. Express the answer as an ordered pair.

For the following exercises, use the graph of the piecewise function shown in Figure 2.

Graph of absolute function and step function.
Figure 2
25.

Find f(2). f(2).

26.

Find f(−2). f(−2).

27.

Write an equation for the piecewise function.

For the following exercises, use the values listed in Table 1.

x x 0 1 2 3 4 5 6 7 8
F(x) F(x) 1 3 5 7 9 11 13 15 17
Table 1
28.

Find F(6). F(6).

29.

Solve the equation F(x)=5. F(x)=5.

30.

Is the graph increasing or decreasing on its domain?

31.

Is the function represented by the graph one-to-one?

32.

Find F −1 (15). F −1 (15).

33.

Given f(x)=−2x+11, f(x)=−2x+11, find f −1 (x). f −1 (x).

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