Advanced Functions

How degree, leading coefficient and factors control the shape of a polynomial graph — and how the factor theorem lets you factor, solve and sign-chart any polynomial you meet.

What happens when a variable sits in the denominator: reciprocals, vertical and horizontal asymptotes, holes, full sketches, and solving rational equations and inequalities.

Logarithms from scratch — what a logarithm actually means, the laws that come straight from the exponent laws, the graph of the logarithm function and its transformations, and solving exponential and logarithmic equations (always checking for extraneous roots).

Retire the degree: radian measure, the unit circle in radians, exact values, sinusoidal transformations, and the three reciprocal trig functions.

The identity toolkit — reciprocal, quotient, Pythagorean, compound angle and double angle formulas — and how to use it to prove identities and find every solution of a trigonometric equation on a given interval.

How fast a function is changing — averaged over an interval or captured at a single instant — and how to build new functions by adding, multiplying, dividing and composing the ones you already know.