Precalculus

Learn the Fundamentals of Functions, Graphs, Trigonometry, and Analytic Geometry

What you'll learn

Functions and Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Trigonometric Identities

Applications of Trigonometric Functions

Systems of Equations and Inequalities

Conic Sections

Sequences, Series, and Probability

Requirements

Math equivalent to Intermediate Algebra

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-axis2. 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 Solutions3. 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 Equations4. 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 Equations5. 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 Functions6. 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 Equations7. 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 Function9. 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 Matrix10. 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 Induction11. 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

Overview

Section 1: Functions and Graphs

Lecture 1 Rectangular Coordinate System

Lecture 2 Functions

Lecture 3 Properties of Functions

Lecture 4 Library of Functions

Lecture 5 Graph Transformations

Section 2: Linear and Quadratic Functions

Lecture 6 Linear Functions

Lecture 7 Linear Equations and Inequalities

Lecture 8 Quadratic Functions

Lecture 9 Quadratic Equations and Inequalities

Lecture 10 Complex Numbers

Section 3: Polynomial and Rational Functions

Lecture 11 Polynomial Functions

Lecture 12 Real and Complex Zeros of Polynomial Functions

Lecture 13 Rational Functions

Lecture 14 Power Functions

Section 4: Inverse, Exponential, and Logarithmic Functions

Lecture 15 Operations on Functions

Lecture 16 Inverse Functions

Lecture 17 Exponential Functions

Lecture 18 Logarithmic Functions

Lecture 19 Properties of Logarithms

Lecture 20 Exponential and Logarithmic Equations

Section 5: Trigonometric Functions

Lecture 21 Angles and Their Measure

Lecture 22 Trigonometric Functions: Unit Circle

Lecture 23 Graphs of Sine and Cosine Functions

Lecture 24 Graphs of Tangent, Cotangent, Secant, and Cosecant Functions

Lecture 25 Inverse Trigonometric Functions

Section 6: Analytic Trigonometry

Lecture 26 Trigonometric Identities

Lecture 27 Sum and Difference Formulas

Lecture 28 Double-Angle and Half-Angle Formulas

Lecture 29 Product-to-Sum and Sum-to-Product Formulas

Lecture 30 Trigonometric Equations

Section 7: Applications of Trigonometry

Lecture 31 Right Triangle Trigonometry

Lecture 32 Law of Sines

Lecture 33 Law of Cosines

Lecture 34 Vectors

Lecture 35 Trigonometric Form of Complex Numbers

Lecture 36 Polar Coordinates

Section 8: Systems of Equations and Inequalities

Lecture 37 Two Variable Linear Systems of Equations

Lecture 38 Nonlinear Systems of Equations

Lecture 39 Partial Fractions

Lecture 40 Two Variable Systems of Inequalities

Lecture 41 Linear Programming

Section 9: Matrices and Determinants

Lecture 42 Linear Systems and Matrices

Lecture 43 Operations with Matrices

Lecture 44 Inverse of a Matrix

Lecture 45 Determinants

Section 10: Sequences, Series, and Probability

Lecture 46 Sequences

Lecture 47 Series

Lecture 48 Counting

Lecture 49 The Binomial Theorem

Lecture 50 Probability

Lecture 51 Mathematical Induction

Section 11: Analytic Geometry

Lecture 52 Conic Sections: Parabola

Lecture 53 Conic Sections: Ellipse

Lecture 54 Conic Sections: Hyperbola

Lecture 55 Conic Sections: Rotation of Axes

Lecture 56 Conic Sections: Polar Equations

Section 12: Final Examination

This course is for those interested in learning about precalculus/trigomometry and as prerequisite math for calculus