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Precalculus

Key Concepts

PrecalculusKey Concepts
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  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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. 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 Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. 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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. 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

3.1 Complex Numbers

  • The square root of any negative number can be written as a multiple of i. i. See Example 1.
  • To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See Example 2.
  • Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See Example 3.
  • Complex numbers can be multiplied and divided.
  • To multiply complex numbers, distribute just as with polynomials. See Example 4, Example 5, and Example 8.
  • To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See Example 6, Example 7, and Example 9.
  • The powers of i i are cyclic, repeating every fourth one. See Example 10.

3.2 Quadratic Functions

  • A polynomial function of degree two is called a quadratic function.
  • The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
  • The axis of symmetry is the vertical line passing through the vertex. The zeros, or x- x- intercepts, are the points at which the parabola crosses the x- x- axis. The y- y- intercept is the point at which the parabola crosses the y- y- axis. See Example 1, Example 7, and Example 8.
  • Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See Example 2.
  • The vertex can be found from an equation representing a quadratic function. See Example 3.
  • The domain of a quadratic function is all real numbers. The range varies with the function. See Example 4.
  • A quadratic function’s minimum or maximum value is given by the y- y- value of the vertex.
  • The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See Example 5 and Example 6.
  • Some quadratic equations must be solved by using the quadratic formula. See Example 9.
  • The vertex and the intercepts can be identified and interpreted to solve real-world problems. See Example 10.

3.3 Power Functions and Polynomial Functions

  • A power function is a variable base raised to a number power. See Example 1.
  • The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
  • The end behavior depends on whether the power is even or odd. See Example 2 and Example 3.
  • A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See Example 4.
  • The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See Example 5.
  • The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See Example 6 and Example 7.
  • A polynomial of degree n n will have at most n n x-intercepts and at most n1 n1 turning points. See Example 8, Example 9, Example 10, Example 11, and Example 12.

3.4 Graphs of Polynomial Functions

  • Polynomial functions of degree 2 or more are smooth, continuous functions. See Example 1.
  • To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See Example 2, Example 3, and Example 4.
  • Another way to find the x- x- intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x- x- axis. See Example 5.
  • The multiplicity of a zero determines how the graph behaves at the x- x- intercepts. See Example 6.
  • The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
  • The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
  • The end behavior of a polynomial function depends on the leading term.
  • The graph of a polynomial function changes direction at its turning points.
  • A polynomial function of degree n n has at most n1 n1 turning points. See Example 7.
  • To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n1 n1 turning points. See Example 8 and Example 10.
  • Graphing a polynomial function helps to estimate local and global extremas. See Example 11.
  • The Intermediate Value Theorem tells us that if f(a) and f(b) f(a) and f(b) have opposite signs, then there exists at least one value c c between a a and b b for which f( c )=0. f( c )=0. See Example 9.

3.5 Dividing Polynomials

  • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See Example 1 and Example 2.
  • The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
  • Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form xk. xk. See Example 3, Example 4, and Example 5.
  • Polynomial division can be used to solve application problems, including area and volume. See Example 6.

3.6 Zeros of Polynomial Functions

  • To find f(k), f(k),determine the remainder of the polynomial f(x) f(x)when it is divided by xk. xk.See Example 1.
  • k kis a zero of f(x) f(x) if and only if (xk) (xk) is a factor of f(x). f(x).See Example 2.
  • Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See Example 3 and Example 4.
  • When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
  • Synthetic division can be used to find the zeros of a polynomial function. See Example 5.
  • According to the Fundamental Theorem, every polynomial function has at least one complex zero. See Example 6.
  • Every polynomial function with degree greater than 0 has at least one complex zero.
  • Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form (xc), (xc),where c c is a complex number. See Example 7.
  • The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
  • The number of negative real zeros of a polynomial function is either the number of sign changes of f(x) f(x) or less than the number of sign changes by an even integer. See Example 8.
  • Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See Example 9.

3.7 Rational Functions

  • We can use arrow notation to describe local behavior and end behavior of the toolkit functions f(x)= 1 x f(x)= 1 x and f(x)= 1 x 2 . f(x)= 1 x 2 . See Example 1.
  • A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See Example 2.
  • Application problems involving rates and concentrations often involve rational functions. See Example 3.
  • The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See Example 4.
  • The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See Example 5.
  • A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See Example 6.
  • A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See Example 7, Example 8, Example 9, and Example 10.
  • Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See Example 11.
  • If a rational function has x-intercepts at x= x 1 , x 2 ,, x n , x= x 1 , x 2 ,, x n , vertical asymptotes at x= v 1 , v 2 ,, v m , x= v 1 , v 2 ,, v m , and no x i =any  v j , x i =any  v j , then the function can be written in the form
    f(x)=a (x x 1 ) p 1 (x x 2 ) p 2 (x x n ) p n (x v 1 ) q 1 (x v 2 ) q 2 (x v m ) q n f(x)=a (x x 1 ) p 1 (x x 2 ) p 2 (x x n ) p n (x v 1 ) q 1 (x v 2 ) q 2 (x v m ) q n

    See Example 12.

3.8 Inverses and Radical Functions

  • The inverse of a quadratic function is a square root function.
  • If f 1 f 1 is the inverse of a function f, f, then f f is the inverse of the function f 1 . f 1 . See Example 1.
  • While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See Example 2.
  • To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See Example 3 and Example 4.
  • When finding the inverse of a radical function, we need a restriction on the domain of the answer. See Example 5 and Example 7.
  • Inverse and radical and functions can be used to solve application problems. See Example 6 and Example 8.

3.9 Modeling Using Variation

  • A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See Example 1.
  • Two variables that are directly proportional to one another will have a constant ratio.
  • A relationship where one quantity is a constant divided by another quantity is called inverse variation. See Example 2.
  • Two variables that are inversely proportional to one another will have a constant multiple. See Example 3.
  • In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See Example 4.
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