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Precalculus

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.

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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

Chapters from OpenStax Precalculus 2e

Every section of the book, condensed into a lesson with its own practice problems.

1. Functions

Introduction to FunctionsFunctions and Function NotationDomain and RangeRates of Change and Behavior of GraphsComposition of FunctionsTransformation of FunctionsAbsolute Value FunctionsInverse Functions

2. Linear Functions

Introduction to Linear FunctionsLinear FunctionsGraphs of Linear FunctionsModeling with Linear FunctionsFitting Linear Models to Data

3. Polynomial and Rational Functions

Introduction to Polynomial and Rational FunctionsQuadratic FunctionsPower Functions and Polynomial FunctionsGraphs of Polynomial FunctionsDividing PolynomialsZeros of Polynomial FunctionsInverses and Radical FunctionsModeling Using Variation

4. Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic FunctionsExponential FunctionsGraphs of Exponential FunctionsLogarithmic FunctionsGraphs of Logarithmic FunctionsLogarithmic PropertiesExponential and Logarithmic EquationsExponential and Logarithmic ModelsFitting Exponential Models to Data

6. Periodic Functions

Introduction to Periodic FunctionsGraphs of the Sine and Cosine FunctionsGraphs of the Other Trigonometric FunctionsInverse Trigonometric Functions

10. Analytic Geometry

Introduction to Analytic GeometryThe EllipseThe HyperbolaThe ParabolaRotation of AxesConic Sections in Polar Coordinates

11. Sequences, Probability and Counting Theory

Introduction to Sequences, Probability and Counting TheorySequences and Their NotationsArithmetic SequencesGeometric SequencesSeries and Their NotationsCounting PrinciplesProbability

Symbols used here

|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
\theta
theta
The usual name for an angle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.

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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