Precalculus | 1 Big School (By Blue Brain)

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 double-angle 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
Row-echelon 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 real-world 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: non-blue 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
Free-throw probability
Three-pointer vs free-throw 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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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