Functions, Graphs, and Trigonometry Concepts

Function

A rule that assigns each input (x) exactly one output (y).

Vertical Line Test

A test that determines if a graph is a function.

Slope-Intercept Form

f(x) = mx + b

Positive Slope

The function is increasing.

Negative Slope

The function is decreasing.

Constant Function

A horizontal line with slope 0.

General Form of a Quadratic Function

f(x) = ax² + bx + c

Standard (Vertex) Form of a Quadratic

f(x) = a(x − h)² + k

Quadratic Formula

x = (-b ± √(b² − 4ac)) / 2a

Graph of a Quadratic Function

A parabola.

Parabola Opening Upward

When a > 0.

Polynomial Function

A continuous, smooth function with no cusps or breaks.

Domain of Polynomial Function

All real numbers.

Factoring Techniques

GCF, difference of squares, and grouping.

Steps of Polynomial Long Division

Divide, multiply, subtract, bring down, repeat.

Vertical Asymptote

A factor in the denominator that does not cancel with the numerator.

Removable Discontinuity

A factor that cancels in both the numerator and denominator.

Horizontal Asymptote at y = 0

When degree of numerator < degree of denominator.

Horizontal Asymptote at Ratio of Leading Coefficients

When degrees of numerator and denominator are equal.

Slant Asymptote

When degree of numerator > degree of denominator.

Finding a Slant Asymptote

Use long or synthetic division; ignore the remainder.

General Form of an Exponential Function

f(x) = ab^x

Base b in Exponential Function

b > 0, b ≠ 1

Range of a > 0 Exponential Function

(0, ∞)

Domain of an Exponential Function

(−∞, ∞)

Constant e

Approximately 2.72; the base of natural exponential functions.

Definition of e Using a Limit

lim (1 + 1/n)^n as n → ∞ = e

Inverse of an Exponential Function

A logarithmic function.

Domain of a Log Function

(0, ∞)

Range of a Log Function

(−∞, ∞)

Change of Base Formula

log_b(M) = log_n(M) / log_n(b)

Common Log

log(x) with no base specified, base 10.

Natural Log

ln(x), base e.

Angle in Standard Position

Vertex at the origin, initial side along the positive x-axis.

Relationship Between Degrees and Radians

180° = π radians

Coterminal Angle

Angles that share the same terminal side.

Arc Length Formula

s = rθ (θ in radians)

Area of a Sector Formula

A = ½ r²θ (θ in radians)

Unit Circle

Helps determine trig values for standard angles.

Coordinates on the Unit Circle at Angle θ

(cos(θ), sin(θ))

Pythagorean Identity

sin²θ + cos²θ = 1

Definition of Sine in a Right Triangle

Opposite / Hypotenuse

Definition of Cosine

Adjacent / Hypotenuse

Definition of Tangent

Opposite / Adjacent

SOH-CAH-TOA

Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj

Sinusoid

A function modeled by sine or cosine transformations.

General Form of a Sinusoid

y = A sin(Bx − C) + D or y = A cos(Bx − C) + D

Inverse Trig Functions

Used for finding angles given a trig value.

sin⁻¹(x) = y

Implies sin(y) = x

Cosine Function Property

Even: cos(−θ) = cos(θ)

Sine Function Property

Odd: sin(−θ) = −sin(θ)

Tangent Function Property

Odd: tan(−θ) = −tan(θ)

Trig Identities Involving Sec, Csc, Cot

sec(−θ) = sec(θ) (even), csc(−θ) = −csc(θ) (odd), cot(−θ) = −cot(θ) (odd)

Double-Angle Identity

An identity expressing sin(2θ), cos(2θ), etc., in terms of sin(θ), cos(θ).

Half-Angle Identity

An identity that rewrites a trig function of θ/2.