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Precalculus 2e

Key Concepts

Precalculus 2eKey Concepts
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Table of contents
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 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
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Functions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 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
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 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
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. 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
  16. Index

Key Concepts

6.1 Graphs of the Sine and Cosine Functions

  • Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of 2π. 2π.
  • The function sinx sinx is odd, so its graph is symmetric about the origin. The function cosx cosx is even, so its graph is symmetric about the y-axis.
  • The graph of a sinusoidal function has the same general shape as a sine or cosine function.
  • In the general formula for a sinusoidal function, the period is P= 2π | B | . P= 2π | B | . See Example 1.
  • In the general formula for a sinusoidal function, | A | | A | represents amplitude. If | A |>1, | A |>1, the function is stretched, whereas if | A |<1, | A |<1, the function is compressed. See Example 2.
  • The value C B C B in the general formula for a sinusoidal function indicates the phase shift. See Example 3.
  • The value D D in the general formula for a sinusoidal function indicates the vertical shift from the midline. See Example 4.
  • Combinations of variations of sinusoidal functions can be detected from an equation. See Example 5.
  • The equation for a sinusoidal function can be determined from a graph. See Example 6 and Example 7.
  • A function can be graphed by identifying its amplitude and period. See Example 8 and Example 9.
  • A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See Example 10.
  • Sinusoidal functions can be used to solve real-world problems. See Example 11, Example 12, and Example 13.

6.2 Graphs of the Other Trigonometric Functions

  • The tangent function has period π. π.
  • f( x )=Atan( Bx−C )+D f( x )=Atan( Bx−C )+D is a tangent with vertical and/or horizontal stretch/compression and shift. See Example 1, Example 2, and Example 3.
  • The secant and cosecant are both periodic functions with a period of 2π. 2π. f( x )=Asec( Bx−C )+D f( x )=Asec( Bx−C )+D gives a shifted, compressed, and/or stretched secant function graph. See Example 4 and Example 5.
  • f( x )=Acsc( Bx−C )+D f( x )=Acsc( Bx−C )+D gives a shifted, compressed, and/or stretched cosecant function graph. See Example 6 and Example 7.
  • The cotangent function has period π π and vertical asymptotes at 0,±π,±2π,... 0,±π,±2π,...
  • The range of cotangent is ( −∞,∞ ), ( −∞,∞ ), and the function is decreasing at each point in its range.
  • The cotangent is zero at ± π 2 ,± 3π 2 ,... ± π 2 ,± 3π 2 ,...
  • f( x )=Acot( Bx−C )+D f( x )=Acot( Bx−C )+D is a cotangent with vertical and/or horizontal stretch/compression and shift. See Example 8 and Example 9.
  • Real-world scenarios can be solved using graphs of trigonometric functions. See Example 10.

6.3 Inverse Trigonometric Functions

  • An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
  • Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
  • For any trigonometric function f(x), f(x), if x= f −1 (y), x= f −1 (y), then f(x)=y. f(x)=y. However, f(x)=y f(x)=y only implies x= f −1 (y) x= f −1 (y) if x x is in the restricted domain of f. f. See Example 1.
  • Special angles are the outputs of inverse trigonometric functions for special input values; for example, π 4 = tan −1 (1)and π 6 = sin −1 ( 1 2 ). π 4 = tan −1 (1)and π 6 = sin −1 ( 1 2 ). See Example 2.
  • A calculator will return an angle within the restricted domain of the original trigonometric function. See Example 3.
  • Inverse functions allow us to find an angle when given two sides of a right triangle. See Example 4.
  • In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, sin( cos −1 ( x ) )= 1− x 2 . sin( cos −1 ( x ) )= 1− x 2 . See Example 5.
  • If the inside function is a trigonometric function, then the only possible combinations are sin −1 ( cosx )= π 2 −x sin −1 ( cosx )= π 2 −x if 0≤x≤π 0≤x≤π and cos −1 ( sinx )= π 2 −x cos −1 ( sinx )= π 2 −x if − π 2 ≤x≤ π 2 . − π 2 ≤x≤ π 2 . See Example 6 and Example 7.
  • When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See Example 8.
  • When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See Example 9.
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