Functions Involving Parameters, Vectors, and Matrices

Parametric function ● (core concept) — A pair of parametric equations in which two dependent variables x and y depend on a single independent variable t: f(t) = (x(t), y(t)). Tables come from evaluating x(t) and y(t) at values of t in the domain; the graph is sketched by connecting the resulting points in order of increasing t.

Parameter ● (core concept) — The single independent variable (t) on which both dependent variables of a parametric function depend. The domain is often restricted, giving start and end points on the graph.

Parametric planar motion ● (core concept) — f(t) = (x(t), y(t)) can model a particle's position in the plane at time t.

Intercepts and extrema in planar motion ● (core concept) — Zeros of x(t) correspond to y-intercepts of the motion's graph and zeros of y(t) to x-intercepts; horizontal and vertical extrema come from the maximum and minimum values of x(t) and y(t).

Direction of parametric motion ● (core concept) — As t increases, increasing x(t) means motion to the right and decreasing x(t) means motion to the left; increasing y(t) means motion up and decreasing y(t) means motion down. The direction at a given point in the plane can differ for different values of t.

Slope of a parametric curve ● (core concept) — Over [t₁, t₂], the ratio of the average rate of change of y to the average rate of change of x gives the slope of the graph between the corresponding points, so long as the average rate of change of x is nonzero.

Equivalent parametrizations ● (core concept) — The same curve can be parametrized in different ways, and can be traversed in different directions by different parametric functions.

Parametrization of the unit circle ● (core concept) — (x(t), y(t)) = (cos t, sin t) with 0 ≤ t ≤ 2π models one complete counterclockwise revolution around the unit circle, starting and ending at (1, 0); transformations of it model any circular path.

Parametrization of a line segment ● (core concept) — A path along the segment from (x₁, y₁) to (x₂, y₂) can be parametrized many ways, such as from the initial position with constant rates of change of x and y with respect to t.

Implicitly defined function ● (core concept) — An equation in two variables can implicitly describe one or more functions; it is graphed by finding solutions (ordered pairs) of the equation.

Extracting a function from an implicit equation ● (core concept) — Solving for one variable can define a function whose graph is part or all of the equation's graph. For nearby points on the graph: a positive ratio of the changes in the two variables means both increase or both decrease; a negative ratio means one increases while the other decreases; a zero rate of change of x with respect to y indicates vertical intervals, and a zero rate of change of y with respect to x indicates horizontal intervals.

Parabola ● (core concept) — With vertex (h, k) and a ≠ 0: x − h = a(y − k)² opens left or right; y − k = a(x − h)² opens up or down.

Ellipse ● (core concept) — Centered at (h, k) with horizontal radius a and vertical radius b: (x−h)²/a² + (y−k)²/b² = 1. The special case a = b is a circle.

Hyperbola ● (core concept) — Centered at (h, k) with horizontal and vertical lines of symmetry: (x−h)²/a² − (y−k)²/b² = 1 opens left and right; (y−k)²/b² − (x−h)²/a² = 1 opens up and down.

Asymptotes of a hyperbola ● (core concept) — y − k = ±(b/a)(x − h).

Parametrization of an implicit equation ● (core concept) — A parametrization (x(t), y(t)) is valid when substituting x(t), y(t) for x, y satisfies the equation for every t in the domain. The graph y = f(x) parametrizes as (t, f(t)); an equation solved for x instead parametrizes as (f(t), t); and if f is invertible, its inverse parametrizes as (f(t), t).

Parametrization of an ellipse ● (core concept) — x(t) = h + a cos t, y(t) = k + b sin t, for 0 ≤ t ≤ 2π.

Parametrization of a hyperbola ● (core concept) — Left-right opening: x(t) = h + a sec t, y(t) = k + b tan t; up-down opening: x(t) = h + a tan t, y(t) = k + b sec t, for 0 ≤ t ≤ 2π.

Vector ● (core concept) — A directed line segment with a tail (starting point) and a head (ending point); its length is the magnitude. A vector from P₁ = (x₁, y₁) to P₂ = (x₂, y₂) has components a = x₂ − x₁, b = y₂ − y₁, written ⟨a, b⟩; its direction is parallel to the line from the origin to (a, b), and components can be found with trigonometry.

Magnitude of a vector ● (core concept) — The length of the directed segment: for ⟨a, b⟩, √(a² + b²).

Zero vector ● (core concept) — ⟨0, 0⟩, the vector when P₁ = P₂.

Scalar multiplication ● (core concept) — Multiplying a vector by a constant multiplies each component by that constant; the result is parallel to the original vector.

Vector addition ● (core concept) — The sum's components are the sums of the corresponding components. Graphically (tip-to-tail): place the second vector's tail at the first vector's head; the sum runs from the first tail to the second head.

Dot product ● (core concept) — ⟨a₁, b₁⟩ · ⟨a₂, b₂⟩ = a₁a₂ + b₁b₂: the sum of the products of corresponding components. Geometrically it equals the product of the magnitudes times the cosine of the angle between the vectors.

Unit vector ● (core concept) — A vector of magnitude 1. A unit vector in the direction of a nonzero vector v is (1/|v|)·v. In standard form ⟨a, b⟩ = a·i + b·j, where i = ⟨1, 0⟩ and j = ⟨0, 1⟩ are the unit vectors in the x- and y-directions.

Perpendicular vectors ● (core concept) — If the dot product of two nonzero vectors is zero, the vectors are perpendicular.

Law of Sines and Law of Cosines ○ — Prerequisite geometry tools for finding side lengths and angle measures of triangles formed by vector addition.

Vector-valued function ● (core concept) — The position of a particle with parametric function f(t) = (x(t), y(t)) can be written as p(t) = x(t)i + y(t)j, or p(t) = ⟨x(t), y(t)⟩; the magnitude of the position vector is the particle's distance from the origin.

Velocity vector ● (core concept) — v(t) = ⟨x(t), y(t)⟩ expresses a particle's velocity at time t: the sign of x(t) tells whether it moves right or left, and the sign of y(t) whether it moves up or down.

Speed ● (core concept) — The magnitude of the velocity vector at time t.

Matrix ● (core concept) — An n × m array with n rows and m columns.

Matrix multiplication ● (core concept) — Two matrices can be multiplied when the number of columns in the first equals the number of rows in the second; the entry in row i, column j of the product is the dot product of row i of the first matrix and column j of the second.

Identity matrix ● (core concept) — The square matrix I with 1s on the main diagonal (top-left to bottom-right) and 0s elsewhere; multiplying a square matrix by its identity leaves it unchanged.

Inverse of a matrix ● (core concept) — The product of a square matrix and its inverse (when it exists) is the identity matrix of the same size. A 2 × 2 inverse can be found with or without technology; a square matrix has an inverse if and only if its determinant is nonzero.

Determinant of a 2 × 2 matrix ● (core concept) — det(A) = ad − bc for A = [[a, b], [c, d]]. The nonzero absolute value of the determinant of a 2 × 2 matrix of column (or row) vectors is the area of the parallelogram they span; a zero determinant means the vectors are parallel.

Linear transformation ● (core concept) — A function mapping an input vector to an output vector so that each output component is a sum of constant multiples of the input components. A linear transformation always maps the zero vector to the zero vector.

Transformation matrix ● (core concept) — For every linear transformation L from ℝ² to ℝ² there is a unique 2 × 2 matrix A with L(v) = Av, and conversely every 2 × 2 matrix defines such a transformation. A single vector is a 2 × 1 matrix; n vectors form a 2 × n matrix; multiplying A by a 2 × n matrix of input vectors gives the 2 × n matrix of output vectors.

Matrix associated with a linear transformation ● (core concept) — The map (x, y) ↦ (a₁₁x + a₁₂y, a₂₁x + a₂₂y) is associated with the matrix [[a₁₁, a₁₂], [a₂₁, a₂₂]]; the images of the unit vectors i and j reveal the matrix's entries.

Rotation matrix ● (core concept) — [[cos θ, −sin θ], [sin θ, cos θ]] rotates every vector counterclockwise about the origin by angle θ.

Composition of linear transformations ● (core concept) — The composition of two linear transformations is a linear transformation, whose matrix is the product of the two matrices.

Inverse linear transformation ● (core concept) — Two linear transformations are inverses when their composition maps every vector to itself; if L(v) = Av then L⁻¹(v) = A⁻¹v. The absolute value of a transformation matrix's determinant gives the magnitude by which the transformation dilates regions of the plane.

Transition matrix ● (core concept) — A matrix built from the rates (percent changes) of transitions between states, modeling how states change over discrete intervals.

State vector ● (core concept) — The distribution between states at a step; multiplying the transition matrix by a state vector predicts future states, and repeated multiplication converges to the steady state. Multiplying the inverse of the transition matrix by a state vector predicts past states.

Steady state ● (core concept) — A distribution between states that does not change from one step to the next.