AP Precalculus Unit 4 Notes

Parametric functions

Parametric functions

Functions that are expressed in terms of a parameter, often denoted as t, and encapsulate two functions, x(t) and y(t), dependent on the parameter.

Graphing and Tabulating Parametric Functions

The process of creating a graph by plotting points from a table of values for parametric functions, aiding in the analysis and interpretation of their properties.

Parametric Function Overview

Parametric functions involve two dependent variables, x and y, that are influenced by a single independent variable, t.

Horizontal and Vertical Extrema

The furthest points in the horizontal and vertical directions of a parametric function, determined by the maximum and minimum values of x(t) and y(t), respectively.

Direction and Rate of Change

The analysis of the direction and rate of change in parametric planar motion functions by evaluating the functions x(t) and y(t) as the parameter t increases.

Average Rate of Change

The average rate of change in parametric functions provides insights into the horizontal and vertical changes in a particle's position over time.

Function and Its Inverse

The parametrization of a function f of x involves expressing both x and y as functions of a parameter t, while the parametrization of its inverse involves expressing x and y as functions of t.

Parametric Formulas for Conic Sections

Specific formulas for parametric representation of conic sections, such as parabolas, ellipses, and hyperbolas, using trigonometric functions.

Characteristics of Vectors

Vectors are defined as directed line segments with both direction and magnitude, and can be decomposed into components in the two-dimensional plane.

Representation in the Plane

Placing a vector in the plane involves defining its tail and head, representing the beginning and end of the line segment, respectively.

Vector Components

Vectors can be represented as (a,b) or ⟨a,b⟩, with a and b being the components calculated along each dimension.

Magnitude and Direction

The magnitude of a vector is determined by the Pythagorean theorem, while the direction aligns with the line segment from the origin to the point represented by the vector.

Scalar Multiplication

Scaling a vector by multiplying it with a scalar results in a vector parallel to the original vector.

Vector Addition

The sum of vectors involves combining their corresponding components.

Dot Product

The dot product of two vectors is the sum of the products of their corresponding components, and is related to magnitudes and angles.

Unit Vectors

Unit vectors have a magnitude of 1 and represent direction, often obtained by dividing a vector by its magnitude.

Angle Between Vectors

The angle between two vectors can be computed using the arccosine function.

Law of Cosines

The Law of Cosines relates the cosine of an angle between vectors to their magnitudes and dot product.

Law of Sines

The Law of Sines relates the sine of angles and the lengths of sides in vector-formed triangles.

Position Vectors

Vectors that describe a particle's position using a vector originating from the origin and ending at the particle's location.

Position Vector

A vector-valued function that represents the position of a particle in a plane.

Velocity Vector

A vector-valued function that represents the velocity of a particle in a plane.

Magnitude of Velocity

The speed of a particle at a specific time, calculated by taking the square root of the sum of squared x and y components of the velocity vector.

Matrix Dimensions

The number of rows and columns in a matrix, denoted as n×m.

Square Matrix

A matrix with an equal number of rows and columns, denoted as n×n.

Row Matrix

A matrix with only one row, denoted as 1×n.

Column Matrix

A matrix with only one column, denoted as n×1.

Matrix Elements

The individual values in a matrix, denoted as a_mn, representing the element in the m-th row and n-th column.

Matrix Equality

Two matrices are equal if they have the same order and each corresponding element is equal.

Matrix Multiplication

The multiplication of two matrices is possible if the number of columns in the first matrix equals the number of rows in the second matrix. The resulting matrix has dimensions m×n, where m is the number of rows of the first matrix and n is the number of columns of the second matrix.

Identity Matrix

A special square matrix with 1's on the main diagonal and 0's elsewhere.

Determinant

A scalar value that has applications in invertibility and vector areas. For a 2×2 matrix A=[ab​cd​], the determinant is calculated as det(A)=ad−bc.

Area of Parallelogram

The absolute value of the determinant represents the area of the parallelogram formed by vectors.

Inverse of a Matrix

A square matrix A has an inverse (A−1) if and only if the determinant of A is not equal to 0. The inverse of a 2×2 matrix [ab​cd​] is given by A−1=det(A)1​[d−b​−ca​].

Linear Transformation

A function that maps input vectors to output vectors while preserving vector addition and scalar multiplication operations.

Transformation Matrix

For a linear transformation L from 2R2 to 2R2, there exists a unique 2×2 matrix A such that L(v)=Av for all vectors in 2R2.

Composition of Linear Transformations

Linear transformations can be composed, where one transformation is applied followed by another. The composition of two linear transformations is itself a linear transformation.

Inverse of a Linear Transformation

The inverse of a linear transformation undoes the effect of the original one. It is represented by the inverse of the matrix for the original transformation.

Matrix Models

Matrices can be used as dynamic models to represent transitions between different states.

Transition Matrix

A square matrix, often 2×2, that encodes the rates of transition between different states.

Predictive Matrix Multiplication

The systematic multiplication of the transition matrix with the current state vector to predict future states.

Steady State

A dynamic equilibrium undisturbed by subsequent transitions, predicted through iterative multiplication of the transition matrix.

Retrospective Projection

Unraveling the past by projecting past states through multiplication with the current state vector using the inverse of the transition matrix.