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.
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 |
SAT tip
"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) |
Worked example
A price drops from $80 to $68. Percent change = (68 − 80) / 80 × 100 = −15%. The price decreased by 15%.
Absolute Value
|ax + b| = c → Split: ax + b = c or ax + b = −c (only valid when c ≥ 0)
|ax + b| < c → −c < ax + b < c (compound inequality)
|ax + b| > c → ax + b < −c or ax + b > c
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 |
Common mistake
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
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 |
Common mistake
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 |
Key rule
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
(f ∘ g)(x) = f(g(x)) → Apply g first, then apply f to that result. Work inside-out.
f⁻¹(x) → Swap x and y in y = f(x), then solve for y. NOT the same as 1/f(x).
f(f⁻¹(x)) = x → A function and its inverse cancel out. Graph: reflect over y = x.
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 |
Watch out
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 triples to memorize
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 |
Common mistake
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 |
SAT tip
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
Random sample → Results can be generalized to the whole population
Non-random / biased sample → Results cannot be reliably generalized
Larger sample size → Smaller margin of error; more reliable estimates
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" |
Worked example — "at least one"
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
Joint probability → Cell value / grand total
Marginal probability → Row or column total / grand total
Conditional probability → Cell value / row or column total (the "given" group)
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% |
SAT tip
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.