AP Pre-Calc Unit 4 Cheatsheet: Functions Involving Parameters, Vectors, and Matrices

Definition: A parametric function in ℝ² is f(t) = (x(t), y(t)), where x and y depend on a single parameter t.

• x and y are dependent variables; t is the independent variable
• Each value of t produces one point (x_i, y_i) on the curve

To graph:

1. Build a table: evaluate x(t) and y(t) at several t-values
2. Plot points in order of increasing t
3. Connect points to show the curve and direction of motion

Domain matters: Restricted domain creates start and end points on the graph.

Vector from P₁ = (x₁, y₁) to P₂ = (x₂, y₂): ⟨a, b⟩ where a = x₂ − x₁, b = y₂ − y₁

Magnitude: |⟨a, b⟩| = √(a² + b²)

Unit vectors: i = ⟨1, 0⟩, j = ⟨0, 1⟩; any vector ⟨a, b⟩ = ai + bj

Unit vector in direction of v: v/|v|

Scalar multiplication: c⟨a, b⟩ = ⟨ca, cb⟩ (parallel to original)

Vector addition: ⟨a₁, b₁⟩ + ⟨a₂, b₂⟩ = ⟨a₁ + a₂, b₁ + b₂⟩

Dot product: ⟨a₁, b₁⟩ · ⟨a₂, b₂⟩ = a₁a₂ + b₁b₂ = |u| |v| cos θ

Perpendicularity: u · v = 0 ⟹ u ⊥ v

Definition: A linear transformation L: ℝ² → ℝ² maps input vector to output vector such that each output component is a sum of constant multiples of input components.

Matrix representation: For any linear transformation L, there exists a unique 2 × 2 matrix A such that L(v) = Av.

Finding the matrix: Apply L to unit vectors i and j. The outputs become the columns of A.

Rotation by angle θ (counterclockwise): A = [cos θ −sin θ; sin θ cos θ]

Dilation: |det(A)| scales areas by this factor.

Composition: (L₂ ∘ L₁)(v) = (A₂A₁)v. Matrix of composition = product of matrices.

Inverse transformation: If L(v) = Av, then L⁻¹(v) = A⁻¹v (when det(A) ≠ 0).