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

4.1

Parametric Functions

Parametric Functions Modeling Planar Motion

Parametric Functions and Rates of Change

Parametrically Defined Circles and Lines

Implicitly Defined Functions

Conic Sections

Parametrization of Implicitly Defined Functions

Vectors

Vector-Valued Functions

Matrices

The Inverse and Determinant of a Matrix

Linear Transformations and Matrices

Matrices as Functions

Matrices Modeling Contexts

Lesson

Diagnostic

after diagnostic

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

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 14 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

Unit 4 is not assessed on the AP exam; this assessment is for schools that teach it.