AP Precalculus Unit 4 Essentials: Key Terms & Core Concepts

Key Concept 1

Parametric and implicit functions describe curves that y = f(x) cannot

A parametric function gives x and y each as a function of a parameter t, so the point (x(t), y(t)) traces a curve — with a built-in direction and timing — even when that curve loops or fails the vertical line test. You can eliminate the parameter to recover a rectangular relationship, though doing so can hide the orientation. Implicitly defined functions and the conic sections (circles, ellipses, parabolas, hyperbolas) extend this further, describing curves through relations in x and y that are not solved for y.

Parametric Functions Implicit Functions Conic Sections

Key Concept 2

Vectors carry magnitude and direction and combine componentwise

A vector ⟨a, b⟩ has a magnitude √(a² + b²) and a direction. Vectors add and subtract component by component, and scalar multiplication scales the magnitude (reversing direction for a negative scalar). A vector-valued function returns a vector for each input, which makes it a natural tool for modeling an object's position or velocity as time changes — connecting directly to the parametric view of motion.

Vectors Vector Operations Vector-Valued Functions

Key Concept 3

Matrices are functions that transform vectors

A matrix can be read as a function: it takes an input vector and returns an output vector, and a 2×2 matrix transforms the whole plane through rotation, scaling, reflection, or shear. Composing two such functions corresponds to multiplying their matrices (order matters — AB ≠ BA). The determinant ad − bc reveals whether the transformation can be undone: an inverse exists exactly when the determinant is nonzero, and repeatedly applying a matrix models a system's change over discrete steps.

Matrices Determinant & Inverse Linear Transformations

Parameter

An independent variable (often t) that x and y each depend on in a parametric function.

Parametric

Parametric function

A pair x = x(t), y = y(t) whose points (x(t), y(t)) trace a curve as t varies.

Parametric

Orientation

The direction in which a parametric curve is traced as the parameter increases.

Parametric

Eliminating the parameter

Combining x(t) and y(t) into a single rectangular equation in x and y by removing t.

Parametric

Planar motion

Movement in a plane described by position functions x(t) and y(t).

Parametric

Implicitly defined function

A relation between x and y, such as x² + y² = 25, in which y is not isolated.

Implicit & Conics

Conic section

A curve — circle, ellipse, parabola, or hyperbola — defined by a second-degree equation in x and y.

Implicit & Conics

Ellipse

A conic of the form (x − h)²/a² + (y − k)²/b² = 1; a stretched circle.

Implicit & Conics

Hyperbola

A conic defined by a difference of squared terms set equal to 1, opening along an axis.

Implicit & Conics

Vector

A quantity with magnitude and direction, written in component form ⟨a, b⟩.

Vectors

Magnitude

The length of a vector ⟨a, b⟩, equal to √(a² + b²).

Vectors

Component form

Writing a vector by its horizontal and vertical components, ⟨a, b⟩.

Vectors

Scalar multiplication

Multiplying a vector by a number, which scales its magnitude and may reverse its direction.

Vectors

Vector-valued function

A function that outputs a vector for each input, modeling position or velocity over time.

Vectors

Matrix

A rectangular array of numbers arranged in rows and columns.

Matrices

Matrix multiplication

Combining rows of one matrix with columns of another; it is not commutative.

Matrices

Identity matrix

The square matrix with 1s on the main diagonal and 0s elsewhere; it leaves vectors unchanged.

Matrices

Determinant

For a 2×2 matrix with rows (a, b) and (c, d), the value ad − bc; it scales area and signals invertibility.

Matrices

Inverse matrix

A matrix A⁻¹ with A·A⁻¹ = I; it exists only when the determinant is nonzero.

Matrices

Linear transformation

A mapping of vectors to vectors represented by a matrix — rotation, scaling, reflection, or shear.

Transformations

Matrix as a function

Viewing a matrix as a rule that sends each input vector to an output vector.

Transformations

Matrix power

Repeatedly applying a matrix, used to model a system's change over successive steps.

Transformations