Every formula in one page, organised by unit then by topic. Variables defined inline. No worked examples — pure reference for last-minute revision. Use the table of contents to jump to any topic.
1. Distribute parentheses
2. Combine like terms
3. Move x-terms to one side, constants to the other
4. Divide by the coefficient of x
Same algebra applies whether the equation has fractions, decimals, or negatives. Multiply through by the LCD to clear fractions.
y = mx + b
m = slope (steepness); b = y-intercept (where the line crosses the y-axis at x = 0).
Ax + By = C
Useful for finding intercepts. Set x = 0 for the y-intercept, y = 0 for the x-intercept.
y − y₁ = m(x − x₁)
Builds a line equation from a known slope m and a single point (x₁, y₁).
m = (y₂ − y₁) / (x₂ − x₁)
Rise over run. Sign matters — falling line has negative slope.
1 solution ⇔ different slopes
No solution ⇔ same slope, different y-intercepts (parallel)
Infinite ⇔ same slope AND same y-intercept (same line)
First convert each equation to slope-intercept form, then compare. The geometric question — do the lines cross, never meet, or coincide?
Solve one equation for one variable
→ Substitute into the other
→ Solve
Best when one equation already has a variable isolated.
Multiply equations so one variable has
opposite coefficients
→ Add to cancel
→ Solve, back-substitute
Best when neither variable is isolated and you can scale to get matching coefficients.
Multiply or divide by NEGATIVE
→ FLIP the inequality sign
This is the only rule that's different from solving equations. Adding or subtracting — even of negatives — does NOT flip the sign.
Open circle (○): strict < >
Closed circle (●): inclusive ≤ ≥
Arrow direction indicates the values that satisfy.
Dashed line: strict < >
Solid line: inclusive ≤ ≥
Shade the half-plane where the inequality is true
Quick test: plug (0, 0) into the inequality. If true, shade the side containing the origin; if false, shade the other side.
m = change in y per 1-unit change in x
Always read with units. e.g. dollars per hour, km per minute, cm per week.
b = value of y when x = 0
In real-world contexts: fixed fee, initial position, base salary, upfront cost.
C = (5/9)(F − 32)
→ F = (9/5)C + 32
Multiply both sides by 9/5, then add 32.
I = Prt
→ t = I / (Pr) r = I / (Pt) P = I / (rt)
I = interest, P = principal, r = rate, t = time.
y = mx + b
→ x = (y − b) / m
Subtract b, then divide by m.
A = (1/2)bh
→ h = 2A / b
Multiply by 2 to clear the half, then divide by b.
Total cost = (rate per item) × (number of items) + (fixed fee)
Fixed fee = y-intercept; rate per item = slope. Words like 'per', 'each', 'every' signal a rate; 'flat fee', 'sign-up', 'membership', 'initial' signal a fixed cost.
Distance = (speed) × (time) + (initial position)
If starting at zero, the simpler form is just d = vt.
Profit = (price per unit) × (units sold) − (fixed cost)
Break-even point: where profit = 0.
Pay = (commission per sale) × (sales) + (base pay)
Same shape as cost — base pay is the y-intercept.
Set the two cost equations equal
→ Solve for the variable
Below break-even one plan wins; above it, the other does. Comparing slopes tells you which is which.
x = (−b ± √(b² − 4ac)) / (2a)
Solves any quadratic ax² + bx + c = 0 with a ≠ 0. The ± gives the two roots.
Δ = b² − 4ac
Δ > 0 → two distinct real roots
Δ = 0 → one repeated (double) root
Δ < 0 → no real roots
Tells you the number of real solutions before you bother computing them. Geometrically — how the parabola sits relative to the x-axis.
Sum = −b / a
Product = c / a
Quick check after you've found the roots: their sum should equal −b/a and their product c/a.
ax² + bx + c = 0
→ (x + b/(2a))² = (b² − 4ac)/(4a²)
Useful when factoring fails and you want a clean square form. Take square root, solve for x.
y = a(x − h)² + k
Vertex at (h, k). Sign of a = direction; |a| = vertical stretch.
y = ax² + bx + c
c is the y-intercept (set x = 0). The leading coefficient a controls direction and stretch.
y = a(x − p)(x − q)
x-intercepts directly readable as p and q.
x = −b / (2a)
Vertical line through the vertex. The vertex's y-coordinate comes from plugging this x into the equation.
a > 0 → opens up (has minimum)
a < 0 → opens down (has maximum)
Sign of the leading coefficient is the only thing that matters for direction.
(x − a) is a factor of P(x) ⇔ P(a) = 0
Equivalently: a is a root iff (x − a) divides the polynomial.
a² − b² = (a − b)(a + b)
Spot any squared term minus another squared term — instant factoring.
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
If the middle term is twice the product of the square roots of the outer terms, it's a perfect square.
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
SOAP — Same, Opposite, Always Positive — for the sign pattern.
A polynomial of degree n has at most n real zeros.
Tight constraint on what the SAT can ask.
x^a · x^b = x^(a+b)
Same base — add exponents.
x^a / x^b = x^(a−b)
Same base — subtract exponents.
(x^a)^b = x^(ab)
Multiply the exponents.
(xy)^n = x^n · y^n
Distribute the exponent across multiplied factors.
x^(−n) = 1 / x^n
Flip to the reciprocal.
x^(1/n) = ⁿ√x
x^(m/n) = (ⁿ√x)^m = ⁿ√(x^m)
Numerator = power; denominator = root. Either order works.
x⁰ = 1 (for any x ≠ 0)
Common edge case to remember.
√(a · b) = √a · √b
→ √50 = √(25 · 2) = 5√2
Pull out perfect-square factors.
y = a · b^x
a = starting value (y-intercept at x = 0); b = growth/decay factor per period.
b > 1 → exponential growth
0 < b < 1 → exponential decay
Read the percentage from b: b = 1.05 ⇒ +5% per period; b = 0.85 ⇒ −15% per period.
A = P(1 + r)^t
A = amount, P = principal, r = annual interest rate (decimal), t = years compounded annually.
A = P(1 + r/n)^(nt)
n = compounding periods per year (4 = quarterly, 12 = monthly, 365 = daily).
y = a · (1/2)^(t/T)
T = half-life (the time for the quantity to halve).
(f ∘ g)(x) = f(g(x))
Apply the inner function first, then feed the result into the outer.
y = f(x − h) → shifts RIGHT by h
y = f(x + h) → shifts LEFT by h
Counter-intuitive but true: the inside change goes in the OPPOSITE direction.
y = f(x) + k → shifts UP by k
y = f(x) − k → shifts DOWN by k
Outside change does what you expect.
y = a · f(x)
|a| > 1 stretches vertically; 0 < |a| < 1 compresses; a < 0 also reflects across the x-axis.
y = −f(x) → reflects across x-axis
y = f(−x) → reflects across y-axis
Negation outside flips the y-values; negation inside flips the x-values.
Even: f(−x) = f(x) (y-axis symmetry)
Odd: f(−x) = −f(x) (origin symmetry)
Examples: f(x) = x² is even; f(x) = x³ is odd.
Solve one equation for one variable
→ Plug into the other
→ Simplify (typically a quadratic)
→ Solve, then back-substitute
Standard approach when at least one equation isn't linear.
Discriminant of the resulting quadratic = 0
Exactly one intersection means the line just touches the parabola.
At most 2 points
The substitution always produces a quadratic — at most 2 real roots.
a / b = c / d → a · d = b · c
Standard tool for solving any proportion.
Direct: y = kx (both grow together)
Inverse: x · y = k (one shrinks as the other grows)
Workforce-time problems are usually inverse: more workers → less time, but workers × time stays constant.
v = d / t d = v · t t = d / v
All three forms come up. Match units carefully.
Multiply by 1 in disguise:
60 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 16.67 m/s
Build conversion fractions whose value is 1 and chain them so unwanted units cancel.
p% of N = (p / 100) · N
20% of 80 ⇒ 0.20 × 80 = 16.
% change = (new − old) / old × 100%
Always divide by the OLD value, not the new one.
× (1 + p/100)
e.g. up 8% ⇒ multiply by 1.08.
× (1 − p/100)
e.g. 25% off ⇒ multiply by 0.75.
Multiply the factors — DO NOT add the percentages.
+10% then −10% ⇒ 1.10 × 0.90 = 0.99 (net −1%)
Classic SAT trap: equal-and-opposite percent changes don't cancel out.
If new = factor × original → original = new / factor
After 25% off, sale = 0.75 × original → original = sale / 0.75
Divide by the multiplier, not by the percent.
mean = (sum of values) / count
Sensitive to outliers — pulled toward extreme values.
Middle value when sorted.
Even count: average of the two middle values.
Robust to outliers.
Most frequently occurring value(s).
Datasets can have one mode, more than one, or none at all.
range = max − min
Simple measure of spread.
Standard deviation = typical distance of values from the mean.
SD = 0 means every value is identical. The SAT mostly asks comparative questions.
Add k to every value:
mean → mean + k
SD → unchanged
Shifting moves the mean but not the spread.
Multiply every value by k:
mean → k · mean
SD → |k| · SD
Scaling changes both the centre and the spread proportionally.
P(A and B) = (cell count) / (grand total)
Both events occurring.
P(A) = (row or column total) / (grand total)
Total occurrence of one event, regardless of the other.
P(A | B) = (cell count) / (B's row or column total)
Read 'probability of A given B'. The 'given' restricts the denominator.
P(A and B) = P(A) · P(B)
Definition: outcome of one doesn't affect the other.
residual = observed y − predicted y
Positive: point above the best-fit line. Negative: below.
∑ (residual)² (sum of squared residuals)
Least-squares regression is the standard fit.
Range: −1 ≤ r ≤ +1
|r| near 1 → strong linear pattern
|r| near 0 → weak / no linear pattern
Correlation ≠ causation. Two variables varying together aren't necessarily linked by cause-and-effect.
Margin of error ∝ 1 / √n
Quadrupling the sample roughly halves the margin of error.
p̂ = (count of successes) / (sample size)
Estimate of the true population proportion.
estimate ± margin of error
e.g. 55% ± 3% means the true value is most likely between 52% and 58%.
Random SAMPLING → generalise to the population (correlation only)
Random ASSIGNMENT → cause-and-effect (in a controlled experiment)
Sampling answers 'about whom can I make claims?'; assignment answers 'can I claim cause?'
Supplementary: sum = 180° (straight line)
Complementary: sum = 90° (right angle)
C for 'corner' = 90°. S for 'straight' = 180°.
Vertical angles are CONGRUENT (equal in measure).
Formed by two intersecting lines, opposite each other.
Two adjacent angles whose non-shared sides form a straight line.
Always supplementary.
Sum to 180°.
Corresponding angles → congruent
Alternate interior angles → congruent
Same-side interior → supplementary
Memorise these three patterns — they unlock most parallel-line angle chases.
a + b + c = 180°
The three interior angles of any triangle always sum to 180°.
exterior angle = sum of the two REMOTE interior angles
(Equivalently: 180° − adjacent interior angle.)
Sides in ratio 1 : √3 : 2
(opposite 30° : 60° : 90°)
Hypotenuse = 2 × shorter leg. Longer leg = shorter leg × √3.
Sides in ratio 1 : 1 : √2
Isosceles right triangle. Hypotenuse = leg × √2.
Linear ratio: a : b
Area ratio: a² : b²
Volume ratio: a³ : b³
When two figures are similar, areas scale as the square and volumes as the cube of the linear ratio.
A = (1/2) · base · height
Height is the perpendicular distance from the chosen base to the opposite vertex.
a² + b² = c²
c = hypotenuse (the side opposite the right angle); a, b = the two legs.
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Pythagoras applied to coordinates: Δx and Δy are the legs of a right triangle whose hypotenuse is the distance.
3-4-5 5-12-13 8-15-17 7-24-25
…and any multiple (e.g. 6-8-10, 9-12-15)
Recognising these saves time — no need to compute square roots.
C = 2πr = πd
Where d = 2r is the diameter.
A = πr²
Standard area of a circle.
L = (θ / 360°) · 2πr (θ in degrees)
A central angle of θ sweeps out a fraction θ / 360 of the circumference.
A_sector = (θ / 360°) · πr²
Same fraction of the full area.
inscribed angle = (1/2) × central angle
(when both subtend the same arc)
An angle vertex on the circle is half of the corresponding angle vertex at the centre.
(x − h)² + (y − k)² = r²
Centre (h, k), radius r.
sin θ = opposite / hypotenuse
SOH — Sine = Opposite over Hypotenuse.
cos θ = adjacent / hypotenuse
CAH — Cosine = Adjacent over Hypotenuse.
tan θ = opposite / adjacent
TOA — Tangent = Opposite over Adjacent.
tan θ = sin θ / cos θ
Useful when you only know sin and cos.
sin²θ + cos²θ = 1
Holds for any angle — directly from the Pythagorean theorem on the unit circle.
sin θ = cos(90° − θ)
cos θ = sin(90° − θ)
The two acute angles in a right triangle are complementary, and they swap their opposite/adjacent roles.
V = lwh
SA = 2(lw + lh + wh)
Box: length × width × height.
V = s³
SA = 6s²
All edges equal length s. 6 identical square faces.
V = πr²h
Lateral SA = 2πrh
Total SA = 2πr² + 2πrh
Lateral (the curved side) plus the two circular caps.
V = (1/3)πr²h
Exactly one-third the volume of a cylinder with the same r and h.
V = (4/3)πr³
SA = 4πr²
Surface area is 4× the area of one of its great circles.
m = (y₂ − y₁) / (x₂ − x₁)
Rise over run. Sign matters.
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Average the x's and average the y's.
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Pythagoras on coordinates.
Equal slopes: m₁ = m₂
Different y-intercepts (otherwise they're the same line).
Slopes are negative reciprocals: m₁ · m₂ = −1
e.g. slope 4 ⇒ perpendicular slope −1/4
Exception: a horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).
Horizontal line (y = k): slope = 0
Vertical line (x = h): slope = undefined
Vertical lines have zero run, so the slope formula divides by zero.
(x − h)² + (y − k)² = r²
Same form as in the Circles topic — centre (h, k), radius r.