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Precalculus

Functions in depth, complex numbers, sequences, conics and vectors.

Lessons

Core Complex numbers i² = −1, arithmetic in the complex plane, modulus and conjugates. (1 + 2i)*(3 - i) Core Polynomial functions Degree, end behaviour, zeros, turning points, and what the graph must look like. y = x^3 - 3x Core Rational functions Asymptotes, holes and domain: where a fraction of polynomials blows up. y = 1/(x^2 - 1) Core Sequences and series Arithmetic and geometric sequences, partial sums, and the geometric series formula. sum of 2^k for k = 0 to 10 Core The binomial theorem Expanding (a + b)ⁿ with Pascal's triangle and binomial coefficients. expand (a + b)^5 Advanced Conic sections Circles, ellipses, parabolas and hyperbolas as equations and as surfaces sliced. x^2 + y^2 - 25 Core Vectors Magnitude, direction, components, and adding vectors head to tail. [[3],[4]] Core Exponential and logarithmic functions Growth and decay, the number e, log rules, and change of base. y = e^x Core Polynomial division and the remainder theorem Long division, synthetic division, and why f(a) is the remainder on dividing by x − a. simplify (x^3 - 1)/(x - 1) Core Parametric equations and polar coordinates Curves traced by a parameter, and the (r, θ) description of the plane. polar form of 1 + i

The bridge between algebra and calculus: functions studied for their own sake (polynomial, rational, exponential, logarithmic) plus the complex numbers, sequences and series, conic sections and vectors that calculus will assume you know.

More lessons

Linear Functions

Introduction to Linear Functions Linear Functions Graphs of Linear Functions Modeling with Linear Functions Fitting Linear Models to Data

Polynomial and Rational Functions

Introduction to Polynomial and Rational Functions Quadratic Functions Power Functions and Polynomial Functions Graphs of Polynomial Functions Dividing Polynomials Zeros of Polynomial Functions Inverses and Radical Functions Modeling Using Variation

Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic Functions Exponential Functions Graphs of Exponential Functions Logarithmic Functions Graphs of Logarithmic Functions Logarithmic Properties Exponential and Logarithmic Equations Exponential and Logarithmic Models Fitting Exponential Models to Data Finding Composite and Inverse Functions Evaluate and Graph Exponential Functions Evaluate and Graph Logarithmic Functions Use the Properties of Logarithms Solve Exponential and Logarithmic Equations

Analytic Geometry

Introduction to Analytic Geometry The Ellipse The Hyperbola The Parabola Rotation of Axes Conic Sections in Polar Coordinates

Sequences, Probability and Counting Theory

Introduction to Sequences, Probability and Counting Theory Sequences and Their Notations Arithmetic Sequences Geometric Sequences Series and Their Notations Counting Principles Probability

Conics

Distance and Midpoint Formulas; Circles Parabolas Ellipses Hyperbolas Solve Systems of Nonlinear Equations

Sequences, Series and Binomial Theorem

Sequences Geometric Sequences and Series

Some of these lessons are adapted from open texts: OpenStax Intermediate Algebra 2e, OpenStax Precalculus 2e, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

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