DWG NO. 00 — Introduction
A course in functions, trigonometry, and the mathematics of change
If you've made it to pre-calculus, you've probably already asked the question every student in this room eventually asks: Why do I need this? Will I ever use this?
It's a fair question. Here's the honest answer: this course is where algebra and trigonometry stop being separate boxes of tricks and start becoming one connected language — the language you'll need to describe anything that changes. A population growing, a signal oscillating, a rocket's trajectory, a speaker's sound wave, a market's swing. Calculus, which comes next, is the mathematics of change itself. Pre-calculus is where you build the vocabulary and the fluency you'll need to read it.
That's really what a function is: a precise way of saying "this depends on that." Once you can read, graph, transform, and combine functions fluently — polynomial, exponential, logarithmic, trigonometric — you can describe almost anything that depends on something else. And once you understand the limit, the idea this course ends on, you have the one piece of machinery that turns "how something is changing right now" from a vague notion into a number you can actually compute.
None of this is new, and none of it is entirely yours to discover from scratch. The unit circle, the logarithm, the identities you'll prove — each was worked out by someone solving a real problem, then checked, refined, and passed down by everyone who came after.
"Mathematics is the language in which God has written the universe."
— often attributed to Galileo Galilei
You don't have to take that on faith to feel the truth in it. Every time you graph a curve that matches a real signal, or use an exponential to predict a real population, you're using the same language. This course is where you learn to speak it.
Table of Contents
Functions and Their Graphs
The core vocabulary — domain, range, notation, transformations, composition, inverses — that every function you meet from here on will use.
→Polynomial and Rational Functions
Graphing, dividing, and finding the zeros of polynomials, then what happens when you divide one polynomial by another.
→Exponential and Logarithmic Functions
Growth, decay, and the inverse relationship between exponentials and logarithms — the mathematics behind compound interest and half-life.
→Trigonometric Functions and the Unit Circle
Angles, radians, and the unit circle, then graphing sine, cosine, and their relatives as functions in their own right.
→Trigonometric Identities and Equations
The algebra of trigonometry: proving identities are always true, and solving equations where they aren't given, but needed.
→Additional Topics in Trigonometry
Solving any triangle with the Law of Sines and Cosines, an introduction to vectors, and polar coordinates.
→Systems of Equations and Matrices
Solving several equations at once, and matrices as a tool for organizing and automating that process.
→Conic Sections
Circles, ellipses, parabolas, and hyperbolas — the curves you get from slicing a cone, and how to recognize each from its equation.
→Sequences, Series, and Combinatorics
Patterns of numbers that follow a rule, what happens when you add them up, and how to count possibilities systematically.
→Limits and an Introduction to Calculus
The idea that unlocks calculus: what a function approaches, even at a value it never quite reaches — and the first step toward the derivative.
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