Complete study guide
SAT Math Reference Sheet.
Formulas, rules, definitions, shortcuts, worked examples, and SAT tips: everything in one scannable reference. Covers all four Digital SAT Math domains.
Reference Sheet vs. Formula Sheet: which one?
This Reference Sheet is the explain-everything version: each formula with a worked example, when to use it, and the mistake to avoid, plus conversion tables, function transformations, and probability logic, organized to read and skim. To quiz yourself on the formulas until they stick, use the flip-card Formula Sheet; it shows the concept, you recall the formula, then flip to check. When you're ready to test yourself, open the Question Bank or a timed practice test.
Algebra & Linear Equations
Forms of a Linear Equation
| Form | Equation | When to use |
|---|---|---|
| Slope-intercept | y = mx + b | Read slope and y-intercept directly |
| Point-slope | y − y₁ = m(x − x₁) | Given a point + slope, or two points |
| Standard | Ax + By = C | Find intercepts quickly; slope = −A/B |
Slope Rules
| Condition | Rule | Example |
|---|---|---|
| Slope formula | m = (y₂ − y₁) / (x₂ − x₁) | Always Δy ÷ Δx |
| Parallel lines | m₁ = m₂ | Same slope, different intercepts |
| Perpendicular lines | m₁ × m₂ = −1 | Negative reciprocal |
| Horizontal line | m = 0 | y = constant |
| Vertical line | Undefined slope | x = constant |
"Which line is perpendicular to y = 3x − 2?" → slope must be −⅓. The SAT also tests "for what value of k does the system have no solution?": match slopes, then check intercepts differ.
Systems of Equations
| Case | Condition | Graph |
|---|---|---|
| One solution | Different slopes | Lines intersect |
| No solution | Same slope, different intercepts | Parallel lines |
| Infinite solutions | Identical equations | Same line |
Percent & Ratio Shortcuts
| Concept | Formula |
|---|---|
| Percent change | (New − Old) / Old × 100 |
| Increase by r% | New = Original × (1 + r/100) |
| Decrease by r% | New = Original × (1 − r/100) |
| Direct variation | y = kx → y/x = k (constant) |
| Inverse variation | y = k/x → xy = k (constant) |
A price drops from $80 to $68. Percent change = (68 − 80) / 80 × 100 = −15%. The price decreased by 15%.
Absolute Value
Quadratics & Polynomials
Three Forms of a Quadratic
| Form | Equation | What you can read off |
|---|---|---|
| Standard | y = ax² + bx + c | y-intercept = c; vertex x = −b/2a |
| Vertex | y = a(x − h)² + k | Vertex (h, k); opens up if a > 0 |
| Factored | y = a(x − r₁)(x − r₂) | x-intercepts r₁ and r₂ |
In vertex form y = (x − 3)², the vertex is x = +3, not x = −3. The sign inside flips.
Key Quadratic Formulas
| Formula | Expression | Use |
|---|---|---|
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a | Solve any quadratic |
| Discriminant | D = b² − 4ac | D>0: 2 roots · D=0: 1 root · D<0: no real roots |
| Vertex x-coord | h = −b / (2a) | Axis of symmetry; plug back in for k |
| Sum of roots | r₁ + r₂ = −b/a | Find missing coefficient without solving |
| Product of roots | r₁ × r₂ = c/a | Find missing coefficient without solving |
Factoring Patterns
| Pattern | Identity |
|---|---|
| Difference of squares | a² − b² = (a + b)(a − b) |
| Perfect square (sum) | (a + b)² = a² + 2ab + b² |
| Perfect square (diff) | (a − b)² = a² − 2ab + b² |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) |
Polynomial Remainder Theorem
If f(x) is divided by (x − k), the remainder equals f(k).
Example: f(x) = x² + 3x − 4 divided by (x − 2) → remainder = f(2) = 4 + 6 − 4 = 6
Systems with a Quadratic
To find where a line meets a parabola, substitute the linear equation into the quadratic, then solve the resulting quadratic. The number of real solutions equals the number of intersection points: use the discriminant to check (2, 1, or 0).
Example: y = x² and y = x + 2 → x² = x + 2 → x² − x − 2 = 0 → x = 2 or x = −1 (two intersection points). Don't forget to find each matching y-value.
Exponents & Radicals
Laws of Exponents
| Rule | Expression | Example |
|---|---|---|
| Product rule | aᵐ · aⁿ = aᵐ⁺ⁿ | x³ · x⁴ = x⁷ |
| Quotient rule | aᵐ / aⁿ = aᵐ⁻ⁿ | x⁶ / x² = x⁴ |
| Power rule | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | x⁻² = 1/x² |
| Fractional exponent | aᵐ/ⁿ = ⁿ√(aᵐ) | 8^(2/3) = (∛8)² = 4 |
| Product to power | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
x² · x³ ≠ x⁶. When multiplying same base, add the exponents: x² · x³ = x⁵. The bases never multiply together.
Radical Rules
| Rule | Expression |
|---|---|
| Product | √(ab) = √a · √b |
| Quotient | √(a/b) = √a / √b |
| Simplify | √(x²) = |x| |
| Rationalize | 1/√a = √a / a |
| Conjugate | 1/(a+√b) · (a−√b)/(a−√b) → removes radical |
Exponential Growth & Decay
| Model | Formula | Notes |
|---|---|---|
| Growth (per period) | f(t) = a(1 + r)ᵗ | r as decimal; b = 1 + r > 1 |
| Decay (per period) | f(t) = a(1 − r)ᵗ | b = 1 − r, where 0 < b < 1 |
| Compound interest | A = P(1 + r/n)^(nt) | n = compounds per year |
| Simple interest | A = P(1 + rt) | Linear; constant dollar growth |
Functions & Transformations
Transformation Rules
| Transformation | Notation | Effect on graph |
|---|---|---|
| Shift up k | f(x) + k | Every point moves up k units |
| Shift down k | f(x) − k | Every point moves down k units |
| Shift right h | f(x − h) | Every point moves right h (sign flips!) |
| Shift left h | f(x + h) | Every point moves left h (sign flips!) |
| Reflect over x-axis | −f(x) | Negate all y-values |
| Reflect over y-axis | f(−x) | Negate all x-values |
| Vertical stretch | a · f(x), |a| > 1 | Stretches away from x-axis |
| Vertical compression | a · f(x), 0 < |a| < 1 | Compresses toward x-axis |
Changes inside the function (to x) are horizontal and counter-intuitive in direction. Changes outside the function (to the whole expression) are vertical and intuitive.
Composition & Inverse
Average Rate of Change
ARC = [f(b) − f(a)] / (b − a): the slope of the secant line between two points on the curve. It's the slope formula applied to a function over an interval [a, b], and it works for any function, not just lines.
Example: f(x) = x² on [1, 3] → (f(3) − f(1)) / (3 − 1) = (9 − 1) / 2 = 4. Read the interval the question specifies; the rate changes across the curve.
Linear vs. Exponential vs. Quadratic Growth
| Type | Pattern in table | Formula shape |
|---|---|---|
| Linear | Constant differences (+2, +2, +2…) | f(x) = mx + b |
| Exponential | Constant ratios (×3, ×3, ×3…) | f(x) = a · bˣ |
| Quadratic | Constant second differences | f(x) = ax² + bx + c |
Domain & Range Quick Rules
| Function type | Domain restriction | Range |
|---|---|---|
| Even root (√x) | Radicand ≥ 0 | y ≥ 0 |
| Rational (1/x) | Denominator ≠ 0 | Depends on function |
| Logarithm (log x) | Argument > 0 | All reals |
| Polynomial | All real numbers | Depends on degree/leading coeff |
Geometry & Measurement
Area & Perimeter Formulas
| Shape | Area | Perimeter / Circumference |
|---|---|---|
| Rectangle | A = lw | P = 2(l + w) |
| Square | A = s² | P = 4s |
| Triangle | A = ½bh | Sum of all sides |
| Trapezoid | A = ½(b₁ + b₂)h | Sum of all sides |
| Circle | A = πr² | C = 2πr = πd |
| Parallelogram | A = bh | Sum of all sides |
Height must always be perpendicular to the base; never use the slant side as h for triangles or trapezoids.
Circle Formulas
| Concept | Formula | Notes |
|---|---|---|
| Arc length | L = (θ/360) × 2πr | θ in degrees; fraction of full circumference |
| Sector area | A = (θ/360) × πr² | Same fraction, applied to area |
| Equation | (x − h)² + (y − k)² = r² | Center (h, k); watch the sign |
Volume Formulas
| Solid | Formula |
|---|---|
| Rectangular prism | V = lwh |
| Cube | V = s³ |
| Cylinder | V = πr²h |
| Cone | V = (1/3)πr²h |
| Sphere | V = (4/3)πr³ |
| Pyramid | V = (1/3)Bh (B = base area) |
Coordinate Geometry
| Concept | Formula |
|---|---|
| Distance | d = √[(x₂−x₁)² + (y₂−y₁)²] |
| Midpoint | M = ((x₁+x₂)/2, (y₁+y₂)/2) |
Pythagorean Theorem
a² + b² = c², where c is the hypotenuse (the side opposite the right angle, always the longest). Also the basis of the distance formula.
Example: legs 3 and 4 → c² = 9 + 16 = 25 → c = 5. Never use a leg as c.
3-4-5 · 5-12-13 · 8-15-17 · 7-24-25
And all multiples: 6-8-10, 9-12-15, 10-24-26, etc. Spotting these saves calculation time.
Trigonometry
SOH-CAH-TOA
| Ratio | Definition | Memory aid |
|---|---|---|
| sin θ | opposite / hypotenuse | Sine = Opposite / Hypotenuse |
| cos θ | adjacent / hypotenuse | Cosine = Adjacent / Hypotenuse |
| tan θ | opposite / adjacent | Tangent = Opposite / Adjacent |
Special Right Triangles
| Triangle | Sides | Key ratio |
|---|---|---|
| 30-60-90 | x : x√3 : 2x | Short leg × √3 = long leg; × 2 = hyp |
| 45-45-90 | x : x : x√2 | Leg × √2 = hypotenuse |
In 30-60-90, x is the shortest side (opposite 30°). Don't use the hypotenuse as x.
Special Angle Values
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Complementary Angle Identity
sin θ = cos(90° − θ) and cos θ = sin(90° − θ)
SAT test: "sin(x°) = cos(y°)" always means x + y = 90. Use this to set up an equation and solve.
Statistics & Data Analysis
Measures of Center & Spread
| Measure | Formula / Definition | Outlier effect |
|---|---|---|
| Mean | x̄ = Σx / n | Pulled toward outliers |
| Median | Middle value of sorted data | Resistant to outliers |
| Mode | Most frequent value | Resistant to outliers |
| Range | Max − Min | Very sensitive to outliers |
| IQR | Q3 − Q1 | Resistant to outliers |
| Standard deviation | Average distance from mean | Sensitive to outliers |
To find a missing value given the mean: Mean × n = Sum. Work backwards from the total sum. Median requires sorting; don't forget this step.
Weighted Average
x̄ = Σ(value × weight) / Σweights
20 students avg 80, 30 students avg 90: (20×80 + 30×90) / 50 = 86. Never just average the averages.
Scatter Plots & Line of Best Fit
| Term | Meaning on the SAT |
|---|---|
| Slope of best fit line | Rate of change of y per unit x (in context) |
| y-intercept | Predicted y-value when x = 0 (in context) |
| Positive association | As x increases, y generally increases |
| Negative association | As x increases, y generally decreases |
| Extrapolation | Predicting outside data range: unreliable |
| Correlation ≠ causation | Association doesn't prove cause and effect |
Surveys & Sampling
Probability
Core Probability Rules
| Rule | Formula | Notes |
|---|---|---|
| Basic probability | P(A) = favorable / total | 0 ≤ P(A) ≤ 1 |
| Complement | P(not A) = 1 − P(A) | "At least one" uses this |
| AND (independent) | P(A and B) = P(A) × P(B) | Events don't affect each other |
| AND (dependent) | P(A and B) = P(A) × P(B|A) | Drawing without replacement |
| OR (mutually exclusive) | P(A or B) = P(A) + P(B) | Events can't both happen |
| OR (general) | P(A or B) = P(A) + P(B) − P(A and B) | Subtract overlap once |
| Conditional | P(A|B) = P(A and B) / P(B) | "Given that B happened" |
P(at least one head in 3 flips) = 1 − P(all tails) = 1 − (0.5)³ = 1 − 0.125 = 0.875
Direct counting is tedious; always use the complement for "at least one" problems.
Two-Way Tables
Unit Conversions
Time
| 1 minute | = 60 seconds |
| 1 hour | = 60 minutes = 3,600 seconds |
| 1 day | = 24 hours |
| 1 week | = 7 days |
| 1 year | = 52 weeks ≈ 365 days |
Length
| 1 foot | = 12 inches |
| 1 yard | = 3 feet = 36 inches |
| 1 mile | = 5,280 feet = 1,760 yards |
| 1 meter | = 100 centimeters |
| 1 kilometer | = 1,000 meters |
Weight & Volume
| 1 pound | = 16 ounces |
| 1 ton | = 2,000 pounds |
| 1 gallon | = 4 quarts = 8 pints |
| 1 liter | = 1,000 milliliters |
| 1 kg | = 1,000 grams |
Fractions → Decimals → Percents
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 3/4 | 0.75 | 75% |
| 2/3 | 0.666… | 66.7% |
Unit conversion problems: set up fractions so the unwanted units cancel. Write out the conversion chain before calculating. The SAT provides some conversions; check the reference info first.
Common Mistakes
| Mistake | Incorrect | Correct |
|---|---|---|
| Forgetting the negative case in absolute value | |x| = 5 → x = 5 | x = 5 or x = −5 |
| Sign in vertex form | y = (x−3)² → vertex x = −3 | vertex x = +3 |
| Direction of horizontal shift | f(x+2) shifts right 2 | shifts LEFT 2 |
| Slope in standard form | Ax + By = C → slope = A/B | slope = −A/B |
| Exponent product rule | x² · x³ = x⁶ | x² · x³ = x⁵ |
| Perfect square expansion | (a + b)² = a² + b² | (a + b)² = a² + 2ab + b² |
| Percent change denominator | (New − Old) / New | (New − Old) / Old |
| Inverse function notation | f⁻¹(x) means 1/f(x) | f⁻¹(x) is the inverse function (swap x and y) |
| Slope formula order | m = (x₂−x₁) / (y₂−y₁) | m = (y₂−y₁) / (x₂−x₁) |
| Height in area formulas | Using slant side as height | Height is always perpendicular to the base |
| Radius vs diameter in circles | A = π(2r)² = 4πr² | A = πr² (use radius, not diameter) |
| Exponential rate setup | f(t) = a · r^t (r = 5%) | f(t) = a · (1.05)^t (b = 1 + rate) |
| Averaging two averages | Average of 80 and 90 = 85 (always) | Use weighted average when group sizes differ |
| Sum of roots sign | Sum = +b/a | Sum = −b/a |
SAT Strategy Tips
- Use the reference sheet. The Digital SAT provides formulas for area, volume, and special triangles. Check it before deriving from scratch.
- Plug in numbers. When a question asks "which expression is equivalent to…," plug in x = 2 into both the question and each answer. Eliminates algebra errors.
- Work backwards from answer choices. For multiple-choice, substitute the answer options into the problem. Start with the middle value for "find x" questions.
- Identify what's being asked before solving. Many errors come from solving for x when the question asks for 2x + 1, or finding the radius when the question asks for area.
- For system of equations questions about solutions: match coefficients to determine no solution (parallel) vs. infinite solutions (identical lines) without solving.
- Translate word problems step by step. Underline key quantities. Define variables before writing equations. Don't try to hold the whole setup in your head.
- Memorize Pythagorean triples (3-4-5, 5-12-13, 8-15-17). Spotting them in a diagram saves you from computing square roots.
- When in doubt about an exponential model, write out two data points and verify the ratio is constant. This also confirms whether it's growth (ratio > 1) or decay (ratio < 1).
- sin(x°) = cos(y°) means x + y = 90. Set up the equation and solve; this appears frequently in the trig section.
- For percent problems: always divide by the original (old) value, not the new one.
- For probability in two-way tables: identify the denominator carefully. If the question says "given that…," the denominator is that row or column total, not the grand total.
- Don't skip units. Unit conversion questions are free points if you cancel units systematically. Write out every conversion factor as a fraction.
- Harder questions appear later in each module. If a late-module problem seems straightforward, double-check: there's likely a trap you're missing.
- Check the discriminant first when a problem asks about x-intercepts or the number of real solutions. D > 0, D = 0, or D < 0 answers the question without solving.
- For average rate of change, use the slope formula: [f(b) − f(a)] / (b − a). It works for any function, not just lines.