Precalculus
Course Content
Complex numbers

Intro to the imaginary numbers

Powers of the imaginary unit

Simplifying roots of negative numbers

i as the principal root of 1

What are the imaginary numbers?

What are the complex numbers?

Intro to complex numbers

Classifying complex numbers

The complex plane

Plotting numbers on the complex plane

Adding and subtracting complex numbers

Adding complex numbers

Subtracting complex numbers

Distance and midpoint of complex numbers

Distance & midpoint of complex numbers

Multiplying complex numbers

Multiplying complex numbers

Complex conjugates and dividing complex numbers

Intro to complex number conjugates

Complex number conjugates

Dividing complex numbers

Identities with complex numbers

Complex numbers & sum of squares factorization

Factoring sum of squares

Absolute value and angle of complex numbers

Absolute value of complex numbers

Absolute value & angle of complex numbers

Polar form of complex numbers

Polar & rectangular forms of complex numbers

Multiplying and dividing complex numbers in polar form

Dividing complex numbers: polar & exponential form

Powers of complex numbers

Complex number equations: x³=1
Polynomials

Adding and subtracting polynomials: two variables

Adding polynomials: two variables (intro)

Subtracting polynomials: two variables (intro)

Subtracting polynomials: two variables

Finding an error in polynomial subtraction

Polynomials review

The binomial theorem

Intro to the Binomial Theorem

Pascal’s triangle and binomial expansion

Expanding binomials

Expanding binomials w/o Pascal’s triangle

Binomial expansion & combinatorics

Pascal’s triangle & combinatorics

The fundamental theorem of algebra

The Fundamental theorem of Algebra (Opens a modal)

Quadratics & the Fundamental Theorem of Algebra

Number of possible real roots of a polynomial

Dividing polynomials with remainders

Polynomial division

Divide polynomials by monomials (with remainders)

Solving equations by graphing

Interpreting equations graphically

Interpreting equations graphically (example 2)

Solving equations graphically (1 of 2)

Solving equations graphically (2 of 2)

Composing functions

Intro to composing functions

Evaluating composite functions

Evaluating composite functions: using tables

Evaluating composite functions: using graphs

Finding composite functions

Evaluating composite functions (advanced)
Composite functions

Modeling with composite functions

Modeling with composite functions

Modeling with composite functions (example 2)

Verifying inverse functions by composition

Verifying inverse functions by composition

Verifying inverse functions by composition: not inverse

Invertible functions

Determining if a function is invertible

Restricting domains of functions to make them invertible

Inverse trigonometric functions

Intro to arcsine

Intro to arctangent

Intro to arccosine

Restricting domains of functions to make them invertible

Domain & range of inverse tangent function

Using inverse trig functions with a calculator

Sinusoidal equations

Solving sinusoidal equations of the form sin(x)=d

Solving cos(θ)=1 and cos(θ)=1
Trigonometry

Sinusoidal models

Trig word problem: solving for temperature

Trigonometric identities

Sine & cosine identities: symmetry

Tangent identities: symmetry

Sine & cosine identities: periodicity

Tangent identities: periodicity

Angle addition identities

Review of trig angle addition identities

Using the cosine angle addition identity

Using the cosine doubleangle identity

Proof of the sine angle addition identity

Proof of the cosine angle addition identity

Using trigonometric identities

Finding trig values using angle addition identities

Using trig angle addition identities: finding side lengths

Using trig angle addition identities: manipulating expressions

Using trigonometric identities

Vectors introduction

Intro to vectors and scalars

Recognizing vectors

Equivalent vectors

Components of vectors

Components of vectors (example 2)

Vector magnitude from components

Vector magnitude from initial & terminal points

Magnitude of vectors

Vector magnitude from graph

Scalar multiplication

Scalar multiplication of vectors

Analyzing scalar multiplication
Vectors

Vector addition and subtraction

Adding & subtracting vectors

Graphically adding & subtracting vectors

Graphically subtracting vectors

Adding vectors algebraically & graphically

Combined vector operations

Combined vector operations

Unit vectors

Unit vectors intro

Worked example: finding unit vector with given direction

Worked example: Scaling unit vectors

Magnitude and direction form of vectors

Direction of vectors from components: 1st & 2nd quadrants

Direction of vectors from components: 3rd & 4th quadrants

Component form of vectors

Vector components from magnitude & direction

Adding vectors in magnitude and direction form

Adding vectors in magnitude & direction form (1 of 2)

Adding vectors in magnitude & direction form (2 of 2)

Vector addition & magnitude

Vectors word problems

Vectors word problem: pushing a box

Vectors word problem: tug of war

Vectors word problem: hiking

Introduction to matrices

Intro to matrices

Rowechelon form and Gaussian elimination

Solving linear systems with matrices

Adding and subtracting matrices

Adding & subtracting matrices

Multiplying matrices by scalars

Multiplying matrices by scalars
Matrices

Multiplying matrices by matrices

Intro to matrix multiplication

Multiplying matrices

Properties of matrix multiplication

Defined matrix operations

Intro to identity matrix

Dimensions of identity matrix

Is matrix multiplication commutative?

Associative property of matrix multiplication

Zero matrix & matrix multiplication

Using properties of matrix operations

Using identity & zero matrices

Matrices as transformations

Transforming vectors using matrices

Transforming polygons using matrices

Visual representation of transformation from matrix

The determinant of a 2×2 matrix

Determinant of a 2×2 matrix

Introduction to matrix inverses

Intro to matrix inverses

Determining invertible matrices

Finding the inverse of a matrix using its determinant

Finding inverse of a 2×2 matrix using determinant & adjugate

Solving equations with inverse matrices

Representing linear systems with matrix equations

Solving linear systems with matrix equations

Matrix word problem: vector combination

Model realworld situations with matrices

Matrix word problem: prices

Arithmetic sequences

Sequences intro

Intro to arithmetic sequences

Sequences word problem: growth pattern

Explicit formulas for arithmetic sequences

Geometric sequences

Intro to geometric sequences

Intro to geometric sequences (advanced)

Explicit & recursive formulas for geometric sequences

Converting recursive & explicit forms of geometric sequences

Geometric series introduction

Geometric series intro

Finite geometric series formula

Worked examples: finite geometric series

Geometric series word problems: swing

Geometric series word problems: hike

Geometric series

Geometric series (with summation notation)

Summation notation

Geometric series with sigma notation

Worked example: finite geometric series (sigma notation)

Finite geometric series word problem: social media

Finite geometric series word problem: mortgage

Arithmetic series

Arithmetic series intro

Arithmetic series formula

Worked example: arithmetic series (sigma notation)

Worked example: arithmetic series (sum expression)

Worked example: arithmetic series (recursive formula)

Proof of finite arithmetic series formula
Series

Introduction to conic sections

Intro to conic sections

The features of a circle

Graphing circles from features

Features of a circle from its graph

Standard equation of a circle

Features of a circle from its standard equation

Graphing a circle from its standard equation

Writing standard equation of a circle

Expanded equation of a circle

Features of a circle from its expanded equation

Center and radii of an ellipse

Intro to ellipses

Ellipse standard equation from graph

Ellipse graph from standard equation

Foci of an ellipse

Foci of an ellipse from equation
Conic sections

Focus and directrix of a parabola

Intro to focus & directrix

Equation of a parabola from focus & directrix

Focus & directrix of a parabola from equation

Introduction to hyperbolas

Intro to hyperbolas

Vertices & direction of a hyperbola

Vertices & direction of a hyperbola (example 2)

Graphing hyperbolas (old example)

Foci of a hyperbola

Foci of a hyperbola from equation

Proof of the hyperbola foci formula

Hyperbolas not centered at the origin

Equation of a hyperbola not centered at the origin

Identifying conic sections from their equations

Conic section from expanded equation: circle & parabola

Conic section from expanded equation: ellipse

Conic section from expanded equation: hyperbola

Challenging conic section problems (IIT JEE)

Representing a line tangent to a hyperbola

Common tangent of circle & hyperbola (1 of 5)

Common tangent of circle & hyperbola (2 of 5)

Common tangent of circle & hyperbola (3 of 5)

Common tangent of circle & hyperbola (4 of 5)

Common tangent of circle & hyperbola (5 of 5)

Intersection of circle & hyperbola

Basic probability

Intro to theoretical probability

Simple probability: yellow marble

Simple probability: nonblue marble

Venn diagrams and the addition rule

Venn diagrams and the addition rule

Addition rule for probability

Compound probability of independent events using diagrams

Die rolling probability

Probability with counting outcomes

Compound events example with tree diagram

Compound probability of independent events using the multiplication rule

Compound probability of independent events

Probability without equally likely events

Independent events example: test taking

Die rolling probability with independent events

Coin flipping probability

Freethrow probability

Threepointer vs freethrow probability

Dependent events

Dependent probability introduction

Dependent probability: coins

Dependent probability example

Independent & dependent probability

The Monty Hall problem

Permutations

Factorial and counting seat arrangements

Permutation formula

Possible three letter words

Zero factorial or 0!

Ways to arrange colors

Ways to pick officers

Combinations

Intro to combinations

Combination formula

Handshaking combinations

Combination example: 9 card hands
Probability and combinatorics

Probability using combinatorics

Probability using combinations

Probability & combinations (2 of 2)

Probability with counting outcomes

Getting exactly two heads (combinatorics)

Exactly three heads in five flips

Generalizing with binomial coefficients (bit advanced)

Example: Different ways to pick officers

Example: Combinatorics and probability

Example: Lottery probability

Mega millions jackpot probability

Birthday probability problem
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