Math Formula Cheat Sheet

Slope

m = (y₂ − y₁) / (x₂ − x₁)

Rise over run. Positive slope goes up left-to-right; negative goes down.

Slope-Intercept Form

y = mx + b

m = slope, b = y-intercept (where the line crosses the y-axis).

Point-Slope Form

y − y₁ = m(x − x₁)

Use when you know a point and the slope but not the y-intercept.

Standard Form

Ax + By = C

A, B, and C are integers. Slope = −A/B; y-intercept = C/B.

Average Rate of Change

(f(b) − f(a)) / (b − a)

The slope between two points on a curve — a very common 'interpret the model' question.

Direct & Inverse Variation

Direct: y = kx\nInverse: y = k/x

k is the constant of proportionality. 'Varies directly' = multiply; 'inversely' = divide.

Percent Change

% Change = (New − Old) / Old × 100

Positive = increase; negative = decrease. Always divide by the original.

Percent of

Part = Percent × Whole

Convert percent to decimal first: 35% = 0.35.

Distance = Rate × Time

d = r × t

Rearrange as needed: r = d/t, t = d/r.

Simple Interest

I = P × r × t

P = principal, r = annual rate (decimal), t = time in years.

Standard Form

y = ax² + bx + c

Vertex x-coordinate = −b / (2a). Opens up if a > 0, down if a < 0.

Vertex Form

y = a(x − h)² + k

Vertex is at (h, k). a determines width and direction.

Factored Form

y = a(x − r₁)(x − r₂)

r₁ and r₂ are the x-intercepts (roots/zeros).

Quadratic Formula

x = [−b ± √(b² − 4ac)] / 2a

Use when factoring is difficult. Set the equation to 0 first.

Discriminant

b² − 4ac

> 0: two real roots · = 0: one real root · < 0: no real roots.

Sum & Product of Roots

Sum = −b/a\nProduct = c/a

A shortcut when a question asks about roots without solving the full equation.

Completing the Square

ax² + bx + c → a(x + b/2a)² + (c − b²/4a)

Convert to vertex form. Add (b/2)² to both sides when a = 1.

Difference of Squares

a² − b² = (a + b)(a − b)

One of the most-tested factoring patterns — recognize it on sight.

Exponential Growth

y = a(1 + r)ᵗ

a = starting amount, r = growth rate (decimal), t = time. Each period multiplies by (1 + r).

Exponential Decay

y = a(1 − r)ᵗ

Same form, but the quantity shrinks. Used for depreciation and half-life style problems.

General Exponential Model

y = a · bˣ

b > 1 means growth; 0 < b < 1 means decay. b is the per-step multiplier.

Compound Interest

A = P(1 + r/n)ⁿᵗ

n = times compounded per year. For annual compounding, n = 1 → A = P(1 + r)ᵗ.

Area of Triangle

A = ½ × base × height

Area of Rectangle

A = length × width

Area of Circle

A = πr²

Circumference of Circle

C = 2πr = πd

Area of Trapezoid

A = ½(b₁ + b₂) × h

b₁ and b₂ are the parallel bases.

Volume of Rectangular Prism

V = l × w × h

Volume of Cylinder

V = πr²h

Volume of Cone

V = ⅓πr²h

Volume of Sphere

V = (4/3)πr³

Pythagorean Theorem

a² + b² = c²

c is the hypotenuse (longest side, opposite the right angle).

Special Right Triangle 30-60-90

Sides: x, x√3, 2x

Short leg : long leg : hypotenuse.

Special Right Triangle 45-45-90

Sides: x, x, x√2

Both legs equal; hypotenuse = leg × √2.

Sum of Interior Angles

(n − 2) × 180°

n = number of sides. Each angle of a regular polygon = (n − 2)·180° / n.

Arc Length

(θ / 360) × 2πr

θ is the central angle in degrees.

Sector Area

(θ / 360) × πr²

The 'pizza slice' area for central angle θ.

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Distance Formula

d = √[(x₂−x₁)² + (y₂−y₁)²]

Derived from the Pythagorean theorem.

Equation of a Circle

(x − h)² + (y − k)² = r²

Center (h, k), radius r. Complete the square to get here.

Parallel Lines

Same slope (m₁ = m₂)\nDifferent y-intercepts

Parallel lines never intersect.

Perpendicular Lines

m₁ × m₂ = −1

Slopes are negative reciprocals — e.g., 2 and −½.

Mean (Average)

Mean = Sum of values / Number of values

Also: Sum = Mean × Count. Use this when a value is missing.

Median

Middle value of ordered data set

For an even count, average the two middle values. Resistant to outliers.

Mode

Most frequently occurring value

A set can have multiple modes or none.

Range

Range = Max − Min

Weighted Average

Σ(value × weight) / Σ(weights)

Use when groups have different sizes — don't just average the averages.

Standard Deviation

Spread of data around the mean

You won't compute it, but know: larger SD = more spread out data.

Line of Best Fit

ŷ = mx + b

Slope = predicted rate of change. Be careful predicting far outside the data range.

Margin of Error

Estimate ± margin

A larger random sample narrows the interval; only random samples justify generalizing.

Basic Probability

P(event) = Favorable / Total

Complement Rule

P(not A) = 1 − P(A)

P(A and B) — Independent

P(A ∩ B) = P(A) × P(B)

Only when events are independent (one doesn't affect the other).

P(A or B)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Conditional Probability

P(A | B) = P(A ∩ B) / P(B)

Read as 'probability of A given B.' Divide by the relevant row/column total.

Expected Value

E = Σ [x × P(x)]

Sum of each outcome multiplied by its probability.

Product Rule

aᵐ × aⁿ = aᵐ⁺ⁿ

Quotient Rule

aᵐ / aⁿ = aᵐ⁻ⁿ

Power Rule

(aᵐ)ⁿ = aᵐⁿ

Zero Exponent

a⁰ = 1 (a ≠ 0)

Negative Exponent

a⁻ⁿ = 1/aⁿ

Fractional Exponent

a^(m/n) = ⁿ√(aᵐ)

E.g., 8^(2/3) = (∛8)² = 4.

Square Root Rules

√(ab) = √a × √b\n√(a/b) = √a / √b

Scientific Notation

a × 10ⁿ (1 ≤ |a| < 10)

SOH-CAH-TOA

sin θ = opposite / hypotenuse\ncos θ = adjacent / hypotenuse\ntan θ = opposite / adjacent

Pythagorean Identity

sin²θ + cos²θ = 1

Reciprocal Identities

csc θ = 1/sin θ\nsec θ = 1/cos θ\ncot θ = 1/tan θ

Complementary Angles

sin θ = cos(90° − θ)\ncos θ = sin(90° − θ)

Co-function identity: sin and cos of complementary angles are equal.

Unit Circle Key Values

sin 30° = ½, cos 30° = √3/2\nsin 45° = √2/2, cos 45° = √2/2\nsin 60° = √3/2, cos 60° = ½

Radian-Degree Conversion

Radians = Degrees × π/180\nDegrees = Radians × 180/π