$f^{-1}$Precalculus

Hunting Zeros

Track down every zero of a polynomial with the rational zero test, Descartes' rule of signs and the intermediate value theorem. Then sketch rational functions, including the ones with slanted asymptotes.

Growth, Decay and Logarithms

Continuous growth with e, change of base on any calculator, carbon dating a fossil, cooling a cup of coffee, and solving equations where the unknown hides in an exponent.

Circular Functions

Radians, all six trig functions from one unit circle, reference angles in a flash, and sinusoids that model the temperature through a day.

Identities in Action

Test identities by overlaying graphs before proving them, solve equations with multiple angles, and turn products into sums — the trick behind beat frequencies and old slide-rule tables.

Navigating with Triangles

The laws of sines and cosines solve any triangle; Heron finds its area from the sides alone; and vectors steer planes through crosswinds and measure the work a force does.

Rotations and Roots

Complex numbers in polar form turn multiplication into rotation. Compute huge powers with DeMoivre, find every root of a number as the corners of a polygon, and draw spirals and figure-eights in polar coordinates.

Matrices at Work

Gaussian elimination as a recipe, matrix inverses that undo transformations, determinants as area scale factors, and partial fractions that split hard fractions into easy ones.

Curves in Motion

Eccentricity measures how squashed a conic is, and parametric equations describe where something is at each moment. Trace planets, thrown balls and a point on a rolling wheel.

Patterns and Proofs

Recursive sequences, sigma notation, infinite geometric series, proof by induction (falling dominoes), and the binomial theorem hiding inside Pascal's triangle.

Getting Infinitely Close

A limit asks where a function is heading, not where it is. Approach a hole, watch one-sided limits disagree, run off to infinity, and turn secant lines into the tangent line: the doorway to calculus.