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Precalculus

Key Terms

PrecalculusKey Terms
  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
arrow notation
a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value
axis of symmetry
a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by x= b 2a . x= b 2a .
coefficient
a nonzero real number multiplied by a variable raised to an exponent
complex conjugate
the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number
complex number
the sum of a real number and an imaginary number, written in the standard form a+bi, a+bi, where a a is the real part, and bi bi is the imaginary part
complex plane
a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number
constant of variation
the non-zero value k k that helps define the relationship between variables in direct or inverse variation
continuous function
a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph
degree
the highest power of the variable that occurs in a polynomial
Descartes’ Rule of Signs
a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of f(x) f(x)and f(x) f(x)
direct variation
the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other
Division Algorithm
given a polynomial dividend f(x) f(x) and a non-zero polynomial divisor d(x) d(x) where the degree of d(x) d(x) is less than or equal to the degree of f(x), f(x), there exist unique polynomials q(x) q(x) and r(x) r(x) such that f(x)=d(x)q(x)+r(x) f(x)=d(x)q(x)+r(x) where q(x) q(x) is the quotient and r(x) r(x) is the remainder. The remainder is either equal to zero or has degree strictly less than d(x). d(x).
end behavior
the behavior of the graph of a function as the input decreases without bound and increases without bound
Factor Theorem
k kis a zero of polynomial function f(x) f(x)if and only if (xk) (xk) is a factor of f(x) f(x)
Fundamental Theorem of Algebra
a polynomial function with degree greater than 0 has at least one complex zero
general form of a quadratic function
the function that describes a parabola, written in the form f(x)=a x 2 +bx+c, f(x)=a x 2 +bx+c, where a,b, a,b, and c c are real numbers and a0. a0.
global maximum
highest turning point on a graph; f(a) f(a) where f(a)f(x) f(a)f(x) for all x. x.
global minimum
lowest turning point on a graph; f(a) f(a) where f(a)f(x) f(a)f(x) for all x. x.
horizontal asymptote
a horizontal line y=b y=b where the graph approaches the line as the inputs increase or decrease without bound.
imaginary number
a number in the form bi bi where i= 1 i= 1
Intermediate Value Theorem
for two numbers a a and b b in the domain of f, f, if a<b a<b and f( a )f( b ), f( a )f( b ), then the function f f takes on every value between f( a ) f( a ) and f( b ); f( b ); specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x- x- axis
inverse variation
the relationship between two variables in which the product of the variables is a constant
inversely proportional
a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases
invertible function
any function that has an inverse function
joint variation
a relationship where a variable varies directly or inversely with multiple variables
leading coefficient
the coefficient of the leading term
leading term
the term containing the highest power of the variable
Linear Factorization Theorem
allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form (xc), (xc), where c cis a complex number
multiplicity
the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form (xh) p , (xh) p , x=h x=h is a zero of multiplicity p. p.
polynomial function
a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.
power function
a function that can be represented in the form f(x)=k x p f(x)=k x p where k k is a constant, the base is a variable, and the exponent, p, p, is a constant
rational function
a function that can be written as the ratio of two polynomials
Rational Zero Theorem
the possible rational zeros of a polynomial function have the form p q p q where p pis a factor of the constant term and q qis a factor of the leading coefficient.
Remainder Theorem
if a polynomial f(x) f(x)is divided by xk, xk,then the remainder is equal to the value f(k) f(k)
removable discontinuity
a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function
smooth curve
a graph with no sharp corners
standard form of a quadratic function
the function that describes a parabola, written in the form f(x)=a (xh) 2 +k, f(x)=a (xh) 2 +k, where ( h, k ) ( h, k ) is the vertex.
synthetic division
a shortcut method that can be used to divide a polynomial by a binomial of the form xk xk
term of a polynomial function
any a i x i a i x i of a polynomial function in the form f(x)= a n x n + a n - 1 x n - 1 + ...+ a 2 x 2 + a 1 x+ a 1 f(x)= a n x n + a n - 1 x n - 1 + ...+ a 2 x 2 + a 1 x+ a 1
turning point
the location at which the graph of a function changes direction
varies directly
a relationship where one quantity is a constant multiplied by the other quantity
varies inversely
a relationship where one quantity is a constant divided by the other quantity
vertex
the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function
vertex form of a quadratic function
another name for the standard form of a quadratic function
vertical asymptote
a vertical line x=a x=a where the graph tends toward positive or negative infinity as the inputs approach a a
zeros
in a given function, the values of x x at which y=0, y=0, also called roots
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