Functions Involving Parameters, Vectors, and Matrices: every key term you need (+ practice quiz)
40 flashcard terms for AP Precalculus Unit 4, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 22-question quiz — free, no account needed.
A pair of functions x = f(t), y = g(t) that give the coordinates of a point in terms of a third variable, the parameter t (often time).
Parameter
The independent variable t in a parametric function; each t-value produces one point (x(t), y(t)) in the plane.
Direction (Orientation) of a Parametric Curve
The way the point moves along the curve as t increases; usually marked with arrows and determined by evaluating a few t-values in order.
Eliminating the Parameter
Solving one component for t and substituting into the other (or using an identity like cos^2 t + sin^2 t = 1) to obtain a rectangular equation for the path.
Domain Restriction After Eliminating t
The rectangular equation may describe more than the actual path; the range of x(t) and y(t) over the given t-interval limits which part of the curve is traced.
Horizontal and Vertical Motion
x(t) tells the horizontal position and y(t) the vertical position; the point moves right when x is increasing and up when y is increasing.
Parametric Rate of Change
The average rate of change of y with respect to x over an interval is Δy/Δx = (Δy/Δt)/(Δx/Δt), the ratio of the component rates.
Vertical Tangent Point on a Parametric Curve
Occurs where x(t) changes direction (a max or min of x) while y keeps changing; the curve is momentarily moving straight up or down.
Parametrizing a Circle
x = h + r cos t, y = k + r sin t for 0 ≤ t ≤ 2π traces the circle of radius r centered at (h, k) counterclockwise starting at (h + r, k).
Parametrizing a Line
x = x_0 + at, y = y_0 + bt passes through (x_0, y_0) with direction ⟨a, b⟩; different parametrizations can trace the same line at different speeds.
Parametrizing a Function Graph
Any y = f(x) can be written x = t, y = f(t); the inverse relation is x = f(t), y = t, swapping the roles of the coordinates.
Projectile Motion Parametrization
x(t) = (v cos θ)t and y(t) = −(g/2)t^2 + (v sin θ)t + h_0 model an object launched with speed v at angle θ from height h_0.
Implicitly Defined Function
A relation between x and y such as x^2 + y^2 = 25 or xy = 4 in which y is not isolated; the graph may fail the vertical line test.
Conic Sections
Circles, ellipses, parabolas, and hyperbolas; each is an implicitly defined relation of degree 2 that can also be parametrized.
Vector
A quantity with both magnitude and direction, written ⟨a, b⟩ in component form or as ai + bj; represented by an arrow whose position does not matter.
Vector Components
For a vector from P(x_1, y_1) to Q(x_2, y_2), the components are ⟨x_2 − x_1, y_2 − y_1⟩ (terminal minus initial).
Magnitude of a Vector
|⟨a, b⟩| = √(a^2 + b^2), the length of the arrow, found with the Pythagorean theorem.
Direction Angle of a Vector
The angle θ the vector makes with the positive x-axis, with tan θ = b/a; adjust for the correct quadrant based on the signs of a and b.
A vector of magnitude 1; the unit vector in the direction of v is v/|v|. i = ⟨1, 0⟩ and j = ⟨0, 1⟩ are the standard unit vectors.
Vector Addition
Add componentwise: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩; geometrically the tip-to-tail or parallelogram rule.
Scalar Multiplication
k⟨a, b⟩ = ⟨ka, kb⟩ stretches the vector by |k| and reverses direction if k < 0; the result is parallel to the original.
Dot Product
⟨a, b⟩ · ⟨c, d⟩ = ac + bd, a scalar; also equals |u||v|cos θ where θ is the angle between the vectors.
Angle Between Vectors
cos θ = (u · v)/(|u||v|); vectors are perpendicular exactly when their dot product is zero.
Position and Velocity Vectors
The position of a moving particle at time t is ⟨x(t), y(t)⟩; the velocity vector points in the direction of motion and its magnitude is speed.
Vector Form of a Parametric Line
A particle moving from position vector p in constant velocity v: r(t) = p + tv, which is the vector version of x = x_0 + at, y = y_0 + bt.
Matrix
A rectangular array of numbers with m rows and n columns (an m × n matrix); entries are indexed by row then column.
Matrix Addition and Scalar Multiplication
Matrices of the same dimensions add entrywise; a scalar multiplies every entry.
Matrix Multiplication
The (i, j) entry of AB is the dot product of row i of A with column j of B; defined only when the number of columns of A equals the number of rows of B.
Non-Commutativity of Matrix Multiplication
In general AB ≠ BA; the order of multiplication matters, which mirrors the fact that the order of transformations matters.
Identity Matrix
The square matrix I with 1s on the main diagonal and 0s elsewhere; AI = IA = A, so it is the transformation that leaves every vector unchanged.
Determinant of a 2×2 Matrix
For [[a, b],[c, d]], det = ad − bc; it is nonzero exactly when the matrix has an inverse.
Inverse Matrix
A^−1 satisfies AA^−1 = I; for a 2×2 matrix, A^−1 = (1/det)·[[d, −b],[−c, a]], which exists only when det ≠ 0.
Linear Transformation
A function that maps a vector ⟨x, y⟩ to A⟨x, y⟩ using a matrix A; it maps the origin to itself and sends lines to lines (or to a point).
Matrix Columns as Images of Unit Vectors
The first column of A is where the transformation sends i = ⟨1, 0⟩ and the second column is where it sends j = ⟨0, 1⟩.
Rotation Matrix
[[cos θ, −sin θ],[sin θ, cos θ]] rotates every vector counterclockwise by θ about the origin without changing its length.
Determinant as Area Scale Factor
|det A| is the factor by which the transformation multiplies areas; a negative determinant means orientation is reversed (a reflection is involved).