math

x

Standard independent variable

y

Standard dependent variable

z

Standard spatial variable (3D)

t

Standard parametric or time variable

θ

Standard symbol for an unknown angle

r

Radius or distance from pole in polar coordinates

≤

Less than or equal to

≥

Greater than or equal to

≠

Not equal to

≈

Approximately equal to

≡

Identically equal to / equivalence

∈

Element of; set membership

∉

Not an element of

⊂

Proper subset

⊆

Subset (proper or equal)

∀

"For all" universal quantifier

∃

"There exists" existential quantifier

∴

Therefore (logical shorthand)

∵

Because (logical shorthand)

∘

Function composition operator (f ∘ g)(x)

Σ

Summation operator; sum a sequence

Π

Product operator; multiply a sequence

|A|

Absolute value of a scalar or determinant of matrix A

‖v‖

Magnitude (norm) of vector v

Real Number System

Hierarchy: Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real (includes irrational)

Order of Operations

PEMDAS: Parentheses, Exponents, Multiplication/Division (L→R), Addition/Subtraction (L→R)

Product Rule for Exponents

a^m · a^n = a^(m+n)

Quotient Rule for Exponents

a^m / a^n = a^(m-n)

Power Rule for Exponents

(a^m)^n = a^(mn)

Zero Exponent Rule

a^0 = 1 (a ≠ 0)

Negative Exponent Rule

a^(-n) = 1/(a^n)

Power of a Product Rule

(ab)^n = a^n b^n

Power of a Quotient Rule

(a/b)^n = a^n / b^n

Interval Notation

[a,b] includes endpoints; (a,b) excludes; [a,b) half-open; unbounded uses ∞ or -∞

Inequality Notation

Use

Compound Inequalities

"And" (intersection) means both conditions must hold; "Or" (union) means at least one holds

Absolute Value Inequality (less than)

|x| < a ⇔ -a < x < a

Absolute Value Inequality (greater than)

|x| > a ⇔ x < -a or x > a

Difference of Squares

a² - b² = (a-b)(a+b)

Perfect Square Trinomial (sum)

a² + 2ab + b² = (a+b)²

Perfect Square Trinomial (difference)

a² - 2ab + b² = (a-b)²

Completing the Square

Rewrite ax²+bx+c as a(x-h)²+k by adding/subtracting (b/(2a))²

Quadratic Formula

x = (-b ± √(b²-4ac))/(2a)

Discriminant

D = b² - 4ac; D>0: 2 real roots; D=0: 1 real root; D<0: 2 complex conjugate roots

Polynomial Long Division

Divide polynomials similarly to numeric long division; yields quotient and remainder

Synthetic Division

Shortcut for dividing polynomial by x-c: bring down coefficients, multiply-add

Remainder Theorem

If polynomial f(x) is divided by x-c, remainder = f(c)

Factor Theorem

x-c is a factor of f(x) ⇔ f(c) = 0

Rational Root Theorem

Possible rational roots p/q where p divides constant term, q divides leading coefficient

Descartes' Rule of Signs

Number of positive real roots ≤ number of sign changes in f(x); negative roots from f(-x). Subtract multiples of 2

Multiplicity of Roots

If root r has multiplicity m: m odd → graph crosses x-axis; m even → graph touches and turns

Polynomial End Behavior

Leading coefficient test: sign and degree determine behavior as x → ±∞

Fundamental Theorem of Algebra

Every nonconstant polynomial with complex coefficients has at least one complex root; degree n polynomial has exactly n complex roots (counting multiplicity)

Domain Definition

The set of all possible input values (x) for which a function is defined

Range Definition

The set of all possible output values (y) a function can produce

Increasing/Decreasing Intervals

Function is increasing where slope > 0, decreasing where slope < 0 (over x-intervals)

Even Function Condition

f(-x) = f(x); symmetric about y-axis

Odd Function Condition

f(-x) = -f(x); symmetric about the origin

One-to-One Function

A function where each y-value corresponds to exactly one x-value; passes horizontal line test

Inverse Function Condition

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; domain and range swap

Vertical Asymptote

A vertical line x = a where f(x) → ±∞; occurs where denominator = 0 (non-removable)

Horizontal Asymptote

A horizontal line y = L where f(x) → L as x → ±∞

x and y Intercepts

x-intercept(s): set y=0, solve for x; y-intercept: set x=0, evaluate f(0)

Linear Graph Identification

Graph of y = mx + b is a straight line

Quadratic Graph Identification

Graph of y = ax²+bx+c is a parabola opening up if a>0, down if a<0

Cubic Graph Identification

Graph of y = ax³+bx²+cx+d is an S-shaped curve with one or two turns

Polynomial Graphs Domain & Range

Domain all real numbers; range: all reals if odd degree, bounded below/above if even degree

Square Root Graph Identification

Graph of y = √x starts at origin and increases gradually in first quadrant

Square Root Graph Domain & Range

Domain [0,∞), Range [0,∞)

Absolute Value Graph Identification

Graph of y = |x| is V-shaped with vertex at origin

Absolute Value Graph Domain & Range

Domain all reals, Range [0,∞) for standard vertex; general range [k,∞) for y = a|x-h|+k with a>0

Absolute Value Graph Symmetry

Standard y = |x| is an even function, symmetric about y-axis

Reciprocal Graph Identification

Graph of y = 1/x is a hyperbola with two branches in QI and QIII

Reciprocal Graph Asymptotes

Vertical asymptote x=0, Horizontal asymptote y=0

Reciprocal Squared Graph Identification

Graph of y = 1/x² has two branches both above x-axis, symmetric

Reciprocal Squared Graph Symmetry

y = 1/x² is an even function, symmetric about y-axis

Exponential Graph Identification

y = b^x (b>1) grows rapidly to right, approaches 0 to left; 0<b<1 decays

Exponential Graph Domain & Range

Domain all reals, Range (0,∞)

Exponential Graph Asymptotes

Horizontal asymptote y = 0

Exponential Graph End Behavior

As x→∞, y→∞ (b>1) or y→0 (0

Logarithmic Graph Identification

y = log_b x is reflection of exponential across y=x, passing through (1,0)

Logarithmic Graph Domain & Range

Domain (0,∞), Range all reals

Logarithmic Graph Asymptotes

Vertical asymptote x = 0

Logistic Graph Identification

S-shaped curve with two horizontal asymptotes, often y = 0 and y = L (carrying capacity)

Logistic Graph Asymptotes

Two horizontal asymptotes (typically y=0 and y=L)

Step Graphs Identification

Graph of floor function (⌊x⌋) steps down; ceiling (⌈x⌉) steps up

Piecewise Functions

Functions defined by different expressions over different intervals; can have jumps

Vertical Shift

f(x) + k shifts graph up (k>0) or down (k<0)

Horizontal Shift

f(x - h) shifts graph right (h>0) or left (h<0)

Vertical Stretch/Compression

a f(x) stretches vertically if |a|>1, compresses if 0<|a|<1; negative a reflects across x-axis

Horizontal Stretch/Compression

f(b x) compresses horizontally if |b|>1, stretches if 0<|b|<1; negative b reflects across y-axis

Reflection across x-axis

-f(x) reflects graph over x-axis

Reflection across y-axis

f(-x) reflects graph over y-axis

Logarithm Definition

y = log_b x ⇔ b^y = x (b>0, b≠1, x>0)

Product Rule for Logs

log_b(MN) = log_b M + log_b N

Quotient Rule for Logs

log_b(M/N) = log_b M - log_b N

Power Rule for Logs

log_b(M^p) = p log_b M

Change of Base Formula

log_b a = (log_c a) / (log_c b)

e (Euler's Number)

e ≈ 2.71828; the natural exponential base, limit of (1+1/n)^n

Continuous Compounding

A = P e^(rt) (P principal, r annual rate, t years)