Precalculus

A function's behavior read from its rule, symmetry, end behavior and asymptotes of every kind, and the difference quotient as the bridge to the derivative.

Choosing, fitting, linearizing and judging exponential, logarithmic, power and logistic models for real data.

Angles of any size in standard position, radian measure as arc length, and sine, cosine and tangent defined as coordinates on the unit circle.

Graphs of the six trig functions, amplitude, period, phase and midline, the inverse trig functions, and sinusoidal models of things that repeat.

The reciprocal, quotient, Pythagorean and cofunction identities, verifying identities, and solving trig equations and inequalities on an interval and in general.

The angle-sum formulas and everything derived from them (double angle, half angle, product to sum, one combined wave) and the equations they unlock.

Solving any triangle with the laws of sines and cosines, the ambiguous case, triangle area from trig and from Heron, and surveying and navigation problems.

Vectors in components and in magnitude and direction, their sums, scalings and dot product, vectors in space, and curves traced by a parameter.

Points, equations and graphs in polar coordinates, and complex numbers in polar form, where multiplying is rotating and De Moivre gives powers and roots.

Parabolas, ellipses and hyperbolas as slices of a cone and as loci with foci and directrices, and identifying a conic from its equation.

Matrix arithmetic, systems as matrix equations solved by row reduction, inverses and Cramer's rule, and the determinant as area, leading to the cross product.

Proof by induction on sums and sequences, and the limit of a function read from graphs and tables, the doorway to calculus.