precalc

What is a function?

A rule describing the dependence of one quantity, A, on another, B.

Domain

The set of all inputs for a function.

Range

The set of all outputs for a function.

Interval Notation - Hard

Includes end values, denoted by [ ].

Interval Notation - Soft

Excludes end values, denoted by ( ).

Set Builder Notation Example

{ x | x > 0 } means 'The set of all x such that x is greater than 0'.

Denominator Restriction

The restriction that the denominator of a function cannot equal zero.

Radicand Restriction

The restriction that the radicand must be greater than or equal to zero.

Average Rate of Change Formula

ARC = (f(b) - f(a)) / (b - a).

Even Function Definition

A function symmetric over the y-axis, satisfying f(-x) = f(x).

Odd Function Definition

A function symmetric about the origin, satisfying f(-x) = -f(x).

Greatest Integer Function

Denoted as f(x) = [[x]], it is the greatest integer less than or equal to x.

Transformation Formula

y = a f(b(x - c)) + d describes the transformations of a function.

One-to-One Function

A function where only one x-value corresponds to every y-value.

Cancellation Property

If f-1(x) is the inverse of f(x), then f(f-1(x)) = x and f-1(f(x)) = x.

Involution

A function that is equal to its own inverse.

Law of Sines

In any triangle, the lengths of the sides are proportional to the sines of the corresponding angles.

Law of Cosines

Used for SAS and SSS cases, given by the formula a² = b² + c² - 2bc * cos(A).

Pythagorean Identity

An identity involving sine and cosine that holds true for any angle.

Area of a Non-Right Triangle

A = 1/2ab sin(C), where A is the area and a, b are sides forming angle C.

Angle Addition Formula

Double Angle Formula

Half-Angle Formula

Reciprocal Identities

Basic Trigonometric Values

Sine Wave Equation

y = a sin(bx) + d, where d represents mean position or midline.

Cosine Wave Period

The period of a cosine function is determined by 2/b.

Amplitude

In a trigonometric function, the amplitude is |a|, representing the maximum displacement.

Periodic Functions

Functions like sine and cosine that repeat values after a certain interval.

Damped Harmonic Motion

A phenomenon where the amplitude of oscillation decreases over time.

Trigonometric Inverses

Functions that 'undo' trigonometric functions, with limited domain and range.

Modulus of a Complex Number

The absolute value of a complex number z = a + bi, calculated as |z| = √(a² + b²).

Polar Coordinates to Rectangular Coordinates

x = r cos(θ), y = r sin(θ), transforming polar coordinates to rectangular form.

De Moivre's Theorem

Used to raise complex numbers in polar form to a power or to find roots.

Complex Number Multiplication

(z1 * z2) = r1 r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)].

Symmetry Tests for Polar Equations

Methods to determine symmetry such as using r = -f(θ) for origin symmetry.

Unit Circle Relationships

The relationships of sine and cosine for various angles on the unit circle.

Composition of Functions

Combining two functions to create a new function with its own domain and range.

Even-Odd Properties of Trigonometric Functions

Sine and cosecant are odd; cosine and secant are even; others depend on sine.

Greatest Integer Function Notation

Often written as [[x]], indicating the greatest integer ≤ x.

Transforming Rectangular to Polar Coordinates

Using r² = x² + y² and tan(θ) = y/x.

Phase Shift

Horizontal shift of a function graph, represented by c in the transformation.

Frequency in Simple Harmonic Motion

Number of cycles per unit time, calculated as f = 1/T.

Graphing Polar Equations

Finding the graph of polar equations by converting to rectangular form.

Cardioids and Limacons

Types of polar curves that have specific shapes and properties.

Trigonometric Identities

Equations involving trigonometric functions that are true for all angle values.

Unit Circle

A circle of radius one centered at the origin of the coordinate plane.

Characteristics of Sine and Cosine Graphs

Periodic, continuous functions defining oscillatory motion.