Pre-Calculus (Advanced · AP Pre-Calculus)

Analyze functions deeply — determine domain/range from all representations, compose functions (f∘g), perform function arithmetic, and graph with precision.

Find inverse functions algebraically and graphically, verify inverses using composition (f(f⁻¹(x)) = x), understand the reflection relationship over y = x.

Analyze end behavior, find real and complex zeros, apply the Rational Root Theorem and Descartes' Rule of Signs, and graph polynomials completely.

Find vertical/horizontal/slant asymptotes, identify holes from common factors, graph rational functions, and solve rational equations and inequalities.

Work with e and natural log, solve exponential/logarithmic equations, model real-world phenomena (population, radioactive decay, pH), and analyze graphs.

Master the unit circle, graph all six trig functions with full transformations (amplitude, period, phase shift, vertical shift), and model periodic phenomena.

Verify and apply Pythagorean identities, sum/difference formulas, double-angle and half-angle formulas, and use identities to simplify expressions.

Solve trig equations on specific intervals and find general solutions. Apply inverse trig functions and use identities as solving strategies.

Write equations of conics in standard form, identify key features (vertices, foci, directrix, asymptotes), graph all four types, and classify from general form.

Represent vectors in component form, perform vector addition/subtraction/scalar multiplication, calculate dot products, and apply to physics problems.

Convert between polar and rectangular coordinates, graph polar curves (roses, limaçons, cardioids), and work with parametric equations.

Determine convergence/divergence of series, evaluate sums using sigma notation, and develop intuition for limits as a preview of calculus.

Perform matrix arithmetic, calculate determinants, find inverse matrices, and use matrices to solve systems of equations (Cramer's Rule, row reduction).