AP Precalculus ›  Unit 4  ·  Optional · Not on the AP Exam

Unit 4 Parameters, Vectors, and Matrices

Meet functions that do more than one thing at a time. Split motion into horizontal and vertical parts with parameters, describe it with vectors, and use matrices to rotate, stretch, and transform whole sets of points at once.

📝 0% of AP Exam🕐 ~30 Class Periods📺 14 Episodes⭐ Optional Unit
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Episode Guide — Season 4 (Bonus Season)

EP.4.1

Parametric Functions

One input, two outputs. A parameter t drives both x(t) and y(t), tracing a curve that can double back on itself and fail the vertical line test while still being a function.

x(t), y(t)Parameter tNot a vertical-line-test world
🕐 2 class daysOpen lesson →
EP.4.2

Parametric Functions Modeling Planar Motion

Treat horizontal and vertical motion separately, then put them together. Use x(t) and y(t) to describe where a particle is, which way it is heading, and when it passes a point.

Position at time tDirection of motionHorizontal vs. vertical
🕐 2 class daysOpen lesson →
EP.4.3

Parametric Functions and Rates of Change

Find how fast x and y are changing over an interval of t. Average rates of change of each component describe the motion and support conclusions about speed and direction.

Rate of change of x(t)Rate of change of y(t)Justify with data
🕐 2 class daysOpen lesson →
EP.4.4

Parametrically Defined Circles and Lines

Write a line or a circle in parametric form, and choose the parameter restrictions that give you exactly the segment, arc, or orientation you want.

Lines: x = x₀ + atCircles: a cos t, a sin tRestricting t
🕐 2 class daysOpen lesson →
EP.4.5

Implicitly Defined Functions

Some relations are not written as y in terms of x at all. Graph and analyze equations that define y implicitly, and read their characteristics from graphs and tables.

F(x, y) = 0Graphs of relationsMultiple representations
🕐 2 class daysOpen lesson →
EP.4.6

Conic Sections

Circles, ellipses, parabolas, and hyperbolas all come from slicing a cone. Rewrite their equations into forms that reveal the center, vertices, and axes.

Circle & ellipseParabolaHyperbola
🕐 3 class daysOpen lesson →
EP.4.7

Parametrization of Implicitly Defined Functions

Take an implicit equation and give it a parameter. Parametrize circles, ellipses, and hyperbolas so you can trace them point by point.

x = a cos t, y = b sin tConic parametrizationsEquivalent forms
🕐 2 class daysOpen lesson →
EP.4.8

Vectors

A vector carries both magnitude and direction. Add, subtract, and scale vectors in component form, and use them to describe displacement and velocity.

Magnitude & directionComponent form ⟨a, b⟩Vector addition
🕐 3 class daysOpen lesson →
EP.4.9

Vector-Valued Functions

A vector-valued function outputs a vector. Describe planar motion as one function r(t) = ⟨x(t), y(t)⟩ and use it to support conclusions about a moving object.

r(t) = ⟨x(t), y(t)⟩Position vectorPaths in the plane
🕐 1 class daysOpen lesson →
EP.4.10

Matrices

A matrix is a rectangular array that organizes data and transformations. Build matrices, add them, scale them, and multiply them, with and without a graphing calculator.

Matrix dimensionsMatrix multiplicationCalculator practice
🕐 2 class daysOpen lesson →
EP.4.11

The Inverse and Determinant of a Matrix

The determinant tells you whether a matrix can be undone. Find determinants and inverses of 2 × 2 matrices and use them to solve systems of equations.

det(A) = ad − bcA⁻¹ when it existsSolving Ax = b
🕐 2 class daysOpen lesson →
EP.4.12

Linear Transformations and Matrices

A matrix acts on a vector and moves it somewhere new. Connect a linear transformation of the plane to the matrix that represents it.

L(v) = AvUnit vectors e₁, e₂Plane transformations
🕐 1 class daysOpen lesson →
EP.4.13

Matrices as Functions

Rotations and dilations are matrix functions. Compose transformations by multiplying their matrices, and reverse one with an inverse matrix.

Rotation by θ|det A| scales areaCompositions & inverses
🕐 3 class daysOpen lesson →
EP.4.14

Matrices Modeling Contexts

Populations, markets, and ecosystems shift between states from one step to the next. Model transitions with a matrix and predict future and past states after n steps.

Transition matricesState vectorsn-step predictions
🕐 3 class daysOpen lesson →
✅ Progress Checks in AP Classroom
Part 1 covers Topics 4.1–4.7 (24 multiple-choice, 2 free-response). Part 2 covers Topics 4.8–4.14 (21 multiple-choice, 2 free-response). These Progress Checks are optional. Assign each one after you finish its topics, then review the rationales for anything you missed.