Precalculus

Description

In this Precalculus course , you will learn the foundational level mathematics needed to study differential and integral calculus. Here is an outline of the course materials:

1. Functions and Graphs

    • Rectangular Coordinate System: Distance Formula, Midpoint Formula, Circle, Standard Equation of a Circle, Unit Circle

    • Functions: Relations and Functions, Set Builder and Interval Notation, Domain, Range, Vertical Line Test

    • Properties of Functions: Even and Odd Functions, Increasing, Decreasing and Constant Functions, Absolute and Local Extrema, Average Rate of Change, Difference Quotient

    • Library of Functions: Constant, Identity, Linear, Square, Cube, Square root, Cube Root, Reciprocal, Piece-wise Defined Functions, Greatest Integer Function, Absolute Value Function

    • Graph Transformations: Vertical and Horizontal Shifts, Vertical Stretching and Compressing, Horizontal Stretching and Compressing, Reflections about the x and y-axis

2. Linear and Quadratic Functions

    • Linear Functions: Definition of a Linear Function, Slope-Intercept Form, Point-Slope Form, Finding Intercepts, Parallel and Perpendicular Lines     

    • Linear Equations and Inequalities: Equations, Linear Equations in One Variable, Properties of Equality, Solving Linear Equations, Linear Inequalities in One Variable, Properties of Inequalities, Solving Linear Inequalities, Compound Inequalities, Absolute Value Inequalities     

    • Quadratic Functions: Definition of a Quadratic Function, Vertex Form, Completing the Square, Vertex Formula     

    • Quadratic Equations and Inequalities: Quadratic Equations, Square Root Property, Solution by Factoring, Solution by Completing the Square, Solution by Quadratic Formula, Discriminant, Graphical Solutions, Quadratic Inequalities, Solving Quadratic Inequalities

     

    • Complex Numbers: Imaginary Unit i, Square Root of a Negative Number, Definition of a Complex Number, Complex Conjugates, Complex Numbers and Radicals, Operations on Complex Numbers, Operations with Powers of i, Quadratic Equations with Complex Solutions

3. Polynomial and Rational Functions

    • Polynomial Functions: Definition of a Polynomial Function, Division Algorithm, Remainder Theorem, Division of Polynomials, Synthetic Division, x-Intercepts Behavior, Leading Coefficient Test

     

    • Real and Complex Zeros of Polynomial Functions: Factor Theorem, Fundamental Theorem of Algebra, Rational Zeros Theorem, n-Zeros Theorem, Conjugate Zeros Theorem

     

    • Rational Functions: Definition of a Rational Function, Vertical Asymptotes, Horizontal Asymptotes, Oblique Asymptotes, Graphing Rational Functions

     

    • Power Functions: Rational Exponents and Radical Notation, Power Functions, Root Functions, Solving Radical Equations

4. Inverse, Exponential, and Logarithmic Functions

    • Operations on Functions: Sum, Difference, Product, and Quotient of the functions, Composition of Functions

     

    • Inverse Functions:One-to-one Functions, Horizontal Line Test, Inverse of a Function, Properties of Inverse Functions

     

    • Exponential Functions: Laws of Exponents, Definition of an Exponential Function, Properties of Exponential Functions, Compound Interest Formula

     

    • Logarithmic Functions: Definition of a Logarithmic Function, Properties of Logarithmic Functions, Inverse Properties of Exponential and Logarithmic Functions, Continuous Compound Interest

     

    • Properties of Logarithms: Rules of Logarithms, Change of Base Formula

     

    • Exponential and Logarithmic Equations: One-to-One Property of Exponential Equality, One-to-One Property of Logarithmic Equality, Solve Exponential and Logarithmic Equations

5. Trigonometric Functions

    • Angles and Their Measure: Degree Measure, Minutes and Seconds, Radian Measure, Arc Length, Degree and Radian Conversion, Coterminal Angles, Linear Velocity, Angular Velocity

     

    • Trigonometric Functions - Unit Circle: Sine, Cosine, Cosecant, Secant, Tangent, Cotangent, Unit Circle, Fundamental Identities of Trigonometry: Reciprocal Identities, Quotient Identities, and Pythagorean Identities

     

    • Graphs of Sine and Cosine Functions: Periodic Functions, Even-Odd Properties, Graphing Sinusoidal Functions, Amplitude, Period, Frequency, Phase Shift, and Vertical Translation

     

    • Graphs of Tangent, Cotangent, Secant, and Cosecant Functions

     

    • Inverse Trigonometric Functions: Inverse Sine, Inverse Cosine, Inverse Tangent, Inverse Cotangent, Inverse Secant, Inverse Cosecant, Composition of Inverse Trigonometric Functions

6. Analytic Trigonometry

    • Trigonometric Identities: Reciprocal Identities, Quotient Identities, Pythagorean Identities, Even-Odd Identities, Cofunction Identities, Using Identities

     

    • Sum and Difference Formulas: Using Sum and Difference Formulas

     

    • Double-Angle and Half-Angle Formulas: Using Double-Angle, Half-Angle, and Power Reducing Formulas

     

    • Product-to-Sum and Sum-to-Product Formulas: Using Product-to-Sum and Sum-to-Product Formulas

     

    • Trigonometric Equations: Solving Trigonometric Equations

7. Applications of Trigonometry

    • Right Triangle Trigonometry: Trigonometric Functions of Right Triangles, Solving Right Triangles, Complementary Angle Theorem

     

    • Law of Sines: Use Law of Sines to Solve Oblique Triangles

     

    • Law of Cosines: Use Law of Cosines to Solve Oblique Triangles

     

    • Vectors: Basic Operations with Vectors, Unit Vectors, Dot Product, Angle Between Two Vectors

     

    • Trigonometric Form of Complex Number: Complex Plane, Absolute Value of a Complex Number, Trigonometric Form of a Complex Number, Product and Quotient of Complex Numbers, De Moivre's Theorem, Finding nth Roots of a Complex Number

     

    • Polar Coordinates: Polar Coordinates, Polar-Rectangular Coordinate Conversion

8. Systems of Equations and Inequalities

    • Two Variable Linear Systems of Equations: Graphical Solutions, Method of Substitution, and Method of Elimination

     

    • Nonlinear Systems of Equations: Solve Nonlinear Systems of Equations

     

    • Partial Fractions: Partial Fraction Decomposition

     

    • Two Variable Systems of Inequalities: Graphical Solutions for Two Variable Systems of Inequalities

     

    • Linear Programming: Apply Linear Programming to Optimize an Objective Function

9. Matrices and Determinants

    • Linear Systems and Matrices: Solve Systems of Equations with Matrices, Gaussian Elimination, Equivalent System Row Operations, Row-Echelon Form of a Matrix, Reduced Row-Echelon Form of a Matrix, and Gauss-Jordan Elimination

     

    • Operations with Matrices: Matrix Addition and Subtraction, Matrix Scalar Multiplication, and Matrix Multiplication

     

    • Inverse of a Matrix: Identity Matrix, Inverse of a Matrix, Find the Inverse of a Matrix, Inverse of a 2 x 2 Matrix, and Matrix System of Equations Solutions

     

    • Determinants: Determinant of a Square Matrix, Minors and Cofactors of a Square Matrix

10. Sequences, Series, and Probability

    • Sequences: Finite & Infinite Sequences, Factorials, Arithmetic & Geometric Sequences

    • Series: Finite & Infinite Series, Summation Notation, Arithmetic & Geometric Series

    • Counting: Fundamental Counting Principle, Permutations, and Combinatorics

    • The Binomial Theorem: Binomial Formula, Pascal's Triangle, and Binomial Coefficients

    • Probability: Probability of an Event, Probability of a Complementary Event, Probability of the Union of Two Events, Probability of Independent Events

    • Mathematical Induction: Generalized Principle of Mathematical Induction

11. Analytic Geometry

    • Conic Sections – Parabola: Equation of a Parabola, Vertex, Focus, and Directrix

     

    • Conic Sections – Ellipse: Equation of an Ellipse, Major and Minor Axis, Vertices, Foci, and Eccentricity

     

    • Conic Sections – Hyperbola: Equation of a Hyperbola, Transverse and Conjugate Axis, Vertices, Foci, Asymptotes, and Fundamental Rectangle

     

    • Conic Sections - Rotation of Axes: General Form of an Equation of a Conic, Rotation of Axes to Eliminate xy Term, Rotation Formulas, Identification of Conics with the Discriminant, Rotation of Axes: Parabola, Ellipse, and Hyperbola

     

    • Conic Sections - Polar Equations: Polar Definition of a Conic and Polar Equations of Conics with Focus at the Pole

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