Comprehensive Guide to Parametrics, Vectors, and Matrices

Parametric Functions

Functions that define curves based on a third variable called the parameter, usually denoted by t.

Coordinates for parametric functions

At a specific value t, the coordinates are (f(t), g(t)).

Parametric equations notation

A parametric function is written as a set of equations: x = f(t) and y = g(t).

Orientation of a curve

The direction in which the graph is traced as the parameter t increases.

Table for graphing parametric functions

A table with columns for t, x(t), and y(t) used to graph parametric equations.

How to eliminate the parameter

1. Solve one equation for t. 2. Substitute t into the other equation.

Example of converting parametric to Cartesian

x = 2t and y = t^2 + 1 leads to y = (x/2)^2 + 1.

Particle motion in parametric equations

Parametric equations often model the position of a particle at time t with x(t) and y(t).

Domain of motion

The range of t values, usually t >= 0.

Horizontal extremum

Maximum or minimum values of x(t) representing the rightmost and leftmost points.

Vertical extremum

Maximum or minimum values of y(t), representing the highest and lowest points.

Rate of change of x

Describes if the particle moves to the right (increasing) or left (decreasing).

Average Rate of Change (AROC) of x

Calculated as (x(b) - x(a)) / (b - a).

Parametrizing a circle

x(t) = r cos(b(t - h)) + xc; y(t) = r sin(b(t - h)) + yc.

Center of circle in parametric form

(xc, yc) represents the center of the circle in the parametric equations.

Standard circle period

Standard circle completes in 2π; period changes with argument to 2π/b.

Parametrizing line segments

Describes the motion from point A (x1,y1) to point B (x2,y2) using equations for x(t) and y(t).

Implicitly defined functions

Relations between x and y defined by a single equation without isolating one variable.

Verifying points on implicitly defined functions

Substitute x=a and y=b into the equation; true statement means the point is on the graph.

Graphing implicitly defined functions

Often involve non-functions (like circles or ellipses) requiring solving for y.

Conic sections as parametric equations

Easy representation of conic sections helping in graphing and analysis.

Parametric equation for ellipses

x(t) = a cos(t) + h; y(t) = b sin(t) + k.

Parametric equation for hyperbolas

x(t) = h + a sec(t); y(t) = k + b tan(t).

Definition of vector

A quantity defined by both magnitude (length) and direction, represented as an arrow.

Magnitude of a vector

||v|| = √(a^2 + b^2), representing the length of the vector.

Direction angle of a vector

θ = tan^{-1}(b/a), used to find vector direction based on quadrant.

Vector addition

Sum of two vectors u and v results in a new vector by adding corresponding components.

Scalar multiplication of a vector

Multiplying vector u by scalar k results in ku = ⟨ku1, ku2⟩.

Vector-valued function

Describes position of a particle at time t as a vector from the origin.

Position vector

p(t) = ⟨x(t), y(t)⟩, used to represent locations in vector form.

Velocity vector

The instantaneous speed and direction of the particle's motion.

Matrix definition

A rectangular array of numbers, with dimensions given as Rows x Columns.

Determinant of a 2x2 matrix

For A = [[a, b], [c, d]], det(A) = ad - bc.

Inverse matrix definition

A^-1 undoes matrix A; A · A^-1 = I, where I is the identity matrix.

Finding the inverse of a matrix

Inverse for A = [[a, b], [c, d]] is A^-1 = (1/(ad-bc)) [[d, -b], [-c, a]].

Linear transformations

Preserve vector addition and scalar multiplication, keeping grid lines parallel.

Transition matrix in Markov chains

Represents probabilities of states changing over time.

Next state equation

S{n+1} = T · Sn, where T is the transition matrix.

Common mistake in matrix multiplication

Incorrectly multiplying vector as Sn · T instead of T · Sn.

Calculator mode for parametric equations

Use Radian mode for trigonometric calculations unless specified otherwise.

Vector notation

Use chevrons ⟨a, b⟩ for vectors, not parentheses (a, b).