Functions Involving Parameters, Vectors, and Matrices
Fourteen topics on functions beyond y = f(x). Parametric functions and planar motion, implicitly defined curves and conic sections, vectors and vector-valued functions, and matrices as linear transformations and models. This unit is not assessed on the AP exam; schools teach it based on state or local requirements, and its drills stay out of exam-weighted mixes.
Topics
Key forms For every problem in this unit
Parametric curve
(x(t), y(t)) over a stated t interval, traced in order of t
Circle, radius r, counterclockwise
x = r·cos t, y = r·sin t
Vector magnitude
|⟨a, b⟩| = √(a² + b²)
Dot product
⟨a, b⟩·⟨c, d⟩ = ac + bd (a scalar)
Determinant (2×2)
det[[a, b], [c, d]] = ad − bc; invertible iff det ≠ 0
Inverse (2×2)
(1/det)·[[d, −b], [−c, a]]
Linear transformation
columns are the images of (1, 0) and (0, 1)
Area scale factor
|det| (orientation flips when det < 0)
Order matters
AB ≠ BA in general
Unit 4 tools