Area of a parallelogram
[A = b h]
Base times perpendicular height.
Where:
- \(A\) - area
- \(b\) - base length
- \(h\) - perpendicular height
When: Any parallelogram.
AA SLAA HLAI SLAI HL
Prior learning
Area of a triangle
[A = \tfrac{1}{2} b h]
Half of base times perpendicular height.
Where:
- \(A\) - area
- \(b\) - base length
- \(h\) - perpendicular height
When: When you know a base and its height.
AA SLAA HLAI SLAI HL
Area of a trapezoid
[A = \tfrac{1}{2}(a + b)h]
Average of the two parallel sides times the height between them.
Where:
- \(A\) - area
- \(a, b\) - the two parallel side lengths
- \(h\) - perpendicular distance between them
When: Four-sided shapes with one pair of parallel sides.
AA SLAA HLAI SLAI HL
Area of a circle
[A = \pi r^2]
π times the radius squared.
Where:
- \(A\) - area
- \(r\) - radius
- \(\pi \approx 3.142\)
When: Circles and disc-shaped regions.
AA SLAA HLAI SLAI HL
Circumference of a circle
[C = 2\pi r]
The distance around a circle.
Where:
- \(C\) - circumference
- \(r\) - radius
When: Perimeter of a circle.
AA SLAA HLAI SLAI HL
Volume of a cuboid
[V = l w h]
Length × width × height.
Where:
- \(V\) - volume
- \(l\) - length
- \(w\) - width
- \(h\) - height
When: Rectangular boxes.
AA SLAA HLAI SLAI HL
Volume of a cylinder
[V = \pi r^2 h]
Circle area times height.
Where:
- \(V\) - volume
- \(r\) - base radius
- \(h\) - height
When: Cylinders / tubes.
AA SLAA HLAI SLAI HL
Volume of a prism
[V = A h]
Cross-section area times length.
Where:
- \(V\) - volume
- \(A\) - cross-section area
- \(h\) - length of the prism
When: Any solid with a constant cross-section.
AA SLAA HLAI SLAI HL
Curved surface of a cylinder
[A = 2\pi r h]
Unroll the side into a rectangle: circumference × height.
Where:
- \(A\) - curved surface area
- \(r\) - radius
- \(h\) - height
When: Surface area of the curved part.
AA SLAA HLAI SLAI HL
Distance between two points
[d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}]
Pythagoras on the horizontal and vertical gaps.
Where:
- \(d\) - distance
- \(x_1\) - \(x\)-coordinate of the first point
- \(x_2\) - \(x\)-coordinate of the second point
- \(y_1\) - \(y\)-coordinate of the first point
- \(y_2\) - \(y\)-coordinate of the second point
When: Length of a segment from coordinates.
AA SLAA HLAI SLAI HL
Midpoint of a segment
[\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)]
Average the x-coordinates and the y-coordinates.
Where:
- \(x_1\) - \(x\)-coordinate of the first point
- \(x_2\) - \(x\)-coordinate of the second point
- \(y_1\) - \(y\)-coordinate of the first point
- \(y_2\) - \(y\)-coordinate of the second point
When: The point halfway between two points.
AA SLAA HLAI SLAI HL
Number & Algebra
Arithmetic sequence - nth term
[u_n = u_1 + (n-1)d]
Start at the first term and add the common difference once per step.
Where:
- \(u_n\) - the \(n\)th term
- \(u_1\) - first term
- \(n\) - term number
- \(d\) - common difference
When: Sequences that go up/down by a fixed amount.
AA SLAA HLAI SLAI HL
Arithmetic series - sum (first term & difference)
[S_n = \tfrac{n}{2}\big(2u_1 + (n-1)d\big)]
Use when you know the first term and the common difference.
Where:
- \(S_n\) - sum of the first \(n\) terms
- \(u_1\) - first term
- \(n\) - number of terms
- \(d\) - common difference
When: Totalling an arithmetic sequence from \(u_1\) and \(d\).
AA SLAA HLAI SLAI HL
Arithmetic series - sum (first & last term)
[S_n = \tfrac{n}{2}(u_1 + u_n)]
Use when you know the first and last term.
Where:
- \(S_n\) - sum of the first \(n\) terms
- \(u_1\) - first term
- \(u_n\) - last (\(n\)th) term
- \(n\) - number of terms
When: Totalling an arithmetic sequence from \(u_1\) and \(u_n\).
AA SLAA HLAI SLAI HL
Geometric sequence - nth term
[u_n = u_1 r^{\,n-1}]
Multiply the first term by the ratio once per step.
Where:
- \(u_n\) - the \(n\)th term
- \(u_1\) - first term
- \(r\) - common ratio
- \(n\) - term number
When: Growth/decay by a fixed factor.
AA SLAA HLAI SLAI HL
Geometric series - sum
[S_n = \dfrac{u_1(r^n - 1)}{r - 1}, \quad r \ne 1]
Shortcut for adding terms that keep multiplying by r.
Where:
- \(S_n\) - sum of the first \(n\) terms
- \(u_1\) - first term
- \(r\) - common ratio \((r \ne 1)\)
- \(n\) - number of terms
When: Repeated percentage growth (e.g. savings).
AA SLAA HLAI SLAI HL
Sum to infinity
[S_\infty = \dfrac{u_1}{1 - r}, \quad |r| < 1]
If terms shrink, infinitely many add to a finite total.
Where:
- \(S_\infty\) - sum of all the terms
- \(u_1\) - first term
- \(r\) - common ratio \((|r| < 1)\)
When: Only when |r| < 1.
AA SLAA HLAI HL
Compound interest
[FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}]
PV = start; r% = yearly rate; k = compounds per year; n = years.
Where:
- \(FV\) - future value
- \(PV\) - present value (initial amount)
- \(r\) - annual interest rate (%)
- \(k\) - compounding periods per year
- \(n\) - number of years
When: Investments/loans compounding more than once a year.
AA SLAA HLAI SLAI HL
Percentage error
[\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100\%]
How far an approximate value is from the exact one, as a %.
Where:
- \(\varepsilon\) - percentage error
- \(v_A\) - approximate value
- \(v_E\) - exact value
When: Rounded/measured vs true value.
AI SLAI HL
Exponent ↔ logarithm
[a^x = b \iff x = \log_a b]
Logs undo exponentials; use a log when the unknown is a power.
Where:
- \(a\) - base \((a > 0,\ a \ne 1)\)
- \(x\) - exponent
- \(b\) - result \((b > 0)\)
When: Solving for an exponent.
AA SLAA HLAI SLAI HL
Laws of logarithms
[\begin{aligned}\log_a(xy)&=\log_a x+\log_a y\\\log_a\tfrac{x}{y}&=\log_a x-\log_a y\\\log_a x^m&=m\log_a x\end{aligned}]
Products → sums, quotients → differences, powers come out front.
Where:
- \(a\) - base of the logarithm
- \(x, y\) - positive numbers
- \(m\) - any power
When: Combining/splitting logs to solve equations.
AA SLAA HLAI HL
Change of base
[\log_a x = \dfrac{\log_b x}{\log_b a}]
Rewrite a log in a base your calculator has (10 or e).
Where:
- \(a\) - original base
- \(b\) - new base (e.g. 10 or \(e\))
- \(x\) - argument \((x > 0)\)
When: Evaluating e.g. log₅30.
AA SLAA HL
Binomial theorem
[(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \dots + b^n]
Expands a bracket to a power; each term chooses r b's via C(n,r).
Where:
- \(a, b\) - the two terms in the bracket
- \(n\) - power (positive integer)
- \(\binom{n}{r}\) - binomial coefficient
When: Expanding powers or finding one term.
AA SLAA HL
Binomial - general term
[\binom{n}{r} a^{\,n-r} b^{\,r}]
The (r+1)th term - set the power of x to find a specific term.
Where:
- \(n\) - power
- \(r\) - term index \((0 \le r \le n)\)
- \(a, b\) - the two terms
When: Finding a coefficient without full expansion.
AA HL
Combinations and permutations
[\begin{aligned}\binom{n}{r}&=\dfrac{n!}{r!(n-r)!}\\[6pt]{}^nP_r&=\dfrac{n!}{(n-r)!}\end{aligned}]
C counts selections (order doesn't matter); P counts arrangements (order matters).
Where:
- \(n\) - total number of items
- \(r\) - number chosen
- \(n!\) - factorial of \(n\)
When: Counting problems, binomial coefficients.
AA HL
Complex number - modulus & argument
[\begin{aligned}z&=a+bi\\|z|&=\sqrt{a^2+b^2}\\\arg z&=\arctan\tfrac{b}{a}\end{aligned}]
Modulus = distance from origin; argument = angle from positive real axis.
Where:
- \(z\) - complex number
- \(a\) - real part
- \(b\) - imaginary part
- \(|z|\) - modulus (distance from origin)
- \(\arg z\) - argument (angle)
When: Polar form, Argand diagrams.
AA HLAI HL
Complex - polar / Euler form
[z = r(\cos\theta + i\sin\theta) = r\,e^{i\theta}]
Two compact ways to write a complex number using its modulus and angle.
Where:
- \(z\) - complex number
- \(r\) - modulus
- \(\theta\) - argument (angle)
- \(i\) - imaginary unit \((i^2 = -1)\)
When: Multiplying, dividing and powering complex numbers.
AA HLAI HL
De Moivre's theorem
[(r\,\text{cis}\,\theta)^n = r^n\,\text{cis}(n\theta)]
Power the modulus, multiply the angle by n.
Where:
- \(r\) - modulus
- \(\theta\) - argument
- \(n\) - power
- \(\text{cis}\,\theta = \cos\theta + i\sin\theta\)
When: Powers and nth roots of complex numbers.
AA HL
Determinant & inverse of a 2×2 matrix
[\begin{aligned}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=ad-bc\\[8pt]M^{-1}&=\tfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\end{aligned}]
Cross-multiply diagonals and subtract; inverse swaps the diagonal and negates the off-diagonal.
Where:
- \(a, b, c, d\) - the four matrix entries
- \(\det\) - determinant
- \(M^{-1}\) - inverse matrix
When: Solving systems, transformations (AI HL).
AI HL
Power of a matrix (diagonalisation)
[M^n = P D^n P^{-1}]
Diagonalise a matrix to raise it to a power quickly.
Where:
- \(M\) - the matrix
- \(n\) - the power
- \(P\) - matrix of eigenvectors
- \(D\) - diagonal matrix of eigenvalues
- \(P^{-1}\) - inverse of \(P\)
When: Markov chains, repeated transformations (AI HL).
AI HL
Sum & product of polynomial roots
[\text{sum} = -\dfrac{a_{n-1}}{a_n}, \quad \text{product} = (-1)^n\dfrac{a_0}{a_n}]
Read the sum and product of all roots straight from the coefficients.
Where:
- \(a_n\) - leading coefficient
- \(a_{n-1}\) - next coefficient
- \(a_0\) - constant term
- \(n\) - degree of the polynomial
When: Polynomials \(\sum a_r x^r = 0\) (AA HL).
AA HL
Functions
Axis of symmetry of a parabola
[x = -\dfrac{b}{2a}]
The vertical line through the vertex - halfway between the roots.
Where:
- \(x\) - equation of the vertical line
- \(a, b\) - coefficients of \(ax^2 + bx + c\)
When: Max/min of a quadratic.
AA SLAA HLAI SLAI HL
Quadratic formula
[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}]
Solves any quadratic; the ± gives the two roots.
Where:
- \(x\) - the solutions (roots)
- \(a, b, c\) - coefficients of \(ax^2 + bx + c = 0\)
When: When it won't factorise nicely.
AA SLAA HLAI HL
Discriminant
[\Delta = b^2 - 4ac]
Δ>0 → two roots; Δ=0 → one; Δ<0 → none.
Where:
- \(\Delta\) - discriminant
- \(a, b, c\) - coefficients of \(ax^2 + bx + c\)
When: How many real roots a quadratic has.
AA SLAA HLAI HL
Equation of a straight line
[\begin{aligned}y &= mx + c\\ax + by + d &= 0\\y - y_1 &= m(x - x_1)\end{aligned}]
Three equivalent forms: gradient–intercept, general, and point–gradient.
Where:
- \(m\) - gradient (slope)
- \(c\) - \(y\)-intercept
- \(a, b, d\) - constants in the general form
- \((x_1, y_1)\) - a known point on the line
When: Lines, gradients and intercepts.
AA SLAA HLAI SLAI HL
Gradient of a line
[m = \dfrac{y_2 - y_1}{x_2 - x_1}]
Rise over run between two points.
Where:
- \(m\) - gradient
- \(x_1\) - \(x\)-coordinate of the first point
- \(x_2\) - \(x\)-coordinate of the second point
- \(y_1\) - \(y\)-coordinate of the first point
- \(y_2\) - \(y\)-coordinate of the second point
When: Finding the slope from two points.
AA SLAA HLAI SLAI HL
Exponential & logarithmic functions
[a^x = e^{x\ln a}, \quad \log_a a^x = x = a^{\log_a x}]
Rewrite any exponential in base \(e\); logs and powers of the same base undo each other.
Where:
- \(a\) - base \((a > 0,\ a \ne 1)\)
- \(x\) - exponent / argument \((x > 0)\)
- \(e \approx 2.718\)
When: Switching bases (AA).
AA SLAA HL
Logistic function
[f(x) = \dfrac{L}{1 + Ce^{-kx}}]
An S-shaped growth curve that levels off at a carrying capacity.
Where:
- \(L\) - carrying capacity (upper limit)
- \(C\) - constant set by the initial value
- \(k\) - growth rate \((k > 0)\)
- \(x\) - input (often time)
When: Limited-growth models (AI HL).
AI HL
Geometry & Trigonometry
Volume of a sphere
[V = \tfrac{4}{3}\pi r^3]
Grows with the cube of the radius.
Where:
- \(V\) - volume
- \(r\) - radius
When: Balls, domes.
AA SLAA HLAI SLAI HL
Surface area of a sphere
[A = 4\pi r^2]
Exactly four circle-areas.
Where:
- \(A\) - surface area
- \(r\) - radius
When: Outer area of a sphere.
AA SLAA HLAI SLAI HL
Volume of a cone
[V = \tfrac{1}{3}\pi r^2 h]
A third of the cylinder with the same base and height.
Where:
- \(V\) - volume
- \(r\) - base radius
- \(h\) - vertical height
When: Cones.
AA SLAA HLAI SLAI HL
Curved surface of a cone
[A = \pi r l]
Uses the slant height l, not the vertical height.
Where:
- \(A\) - curved surface area
- \(r\) - base radius
- \(l\) - slant height
When: Surface of the sloping part.
AA SLAA HLAI SLAI HL
Volume of a pyramid
[V = \tfrac{1}{3} A h]
A third of base area times height.
Where:
- \(V\) - volume
- \(A\) - base area
- \(h\) - vertical height
When: Pyramids.
AA SLAA HLAI SLAI HL
Right-angled trig (SOH-CAH-TOA)
[\begin{aligned}\sin\theta&=\tfrac{\text{opp}}{\text{hyp}}\\\cos\theta&=\tfrac{\text{adj}}{\text{hyp}}\\\tan\theta&=\tfrac{\text{opp}}{\text{adj}}\end{aligned}]
The three ratios linking an angle to the sides of a right triangle.
Where:
- \(\theta\) - the angle
- opp - side opposite the angle
- adj - side next to the angle
- hyp - hypotenuse (longest side)
When: Right-angled triangles.
AA SLAA HLAI SLAI HL
Sine rule
[\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}]
Pairs each side with its opposite angle.
Where:
- \(a, b, c\) - side lengths
- \(A, B, C\) - the angles opposite those sides
When: A side + its opposite angle are known.
AA SLAA HLAI SLAI HL
Cosine rule
[c^2 = a^2 + b^2 - 2ab\cos C, \quad \cos C = \dfrac{a^2 + b^2 - c^2}{2ab}]
Pythagoras plus an angle-correction term.
Where:
- \(a, b, c\) - side lengths
- \(C\) - angle opposite side \(c\)
When: Two sides + included angle, or all three sides.
AA SLAA HLAI SLAI HL
Area of a triangle (two sides & angle)
[A = \tfrac{1}{2}ab\sin C]
Half the product of two sides times the sine of the angle between.
Where:
- \(A\) - area
- \(a, b\) - two side lengths
- \(C\) - angle between them
When: No perpendicular height available.
AA SLAA HLAI SLAI HL
Arc length (degrees)
[l = \dfrac{\theta}{360}\times 2\pi r]
The fraction θ/360 of the full circumference.
Where:
- \(l\) - arc length
- \(\theta\) - angle in degrees
- \(r\) - radius
When: AI - angles in degrees.
AI SLAI HL
Sector area (degrees)
[A = \dfrac{\theta}{360}\times \pi r^2]
The fraction θ/360 of the full circle area.
Where:
- \(A\) - sector area
- \(\theta\) - angle in degrees
- \(r\) - radius
When: AI - angles in degrees.
AI SLAI HL
Arc length (radians)
[l = r\theta]
Radius times the angle (θ in radians).
Where:
- \(l\) - arc length
- \(r\) - radius
- \(\theta\) - angle in radians
When: AA - angles in radians.
AA SLAA HLAI HL
Sector area (radians)
[A = \tfrac{1}{2}r^2\theta]
The 'pizza slice' area (θ in radians).
Where:
- \(A\) - sector area
- \(r\) - radius
- \(\theta\) - angle in radians
When: AA - angles in radians.
AA SLAA HLAI HL
Pythagorean identity
[\sin^2\theta + \cos^2\theta = 1]
Lets you swap between sin and cos.
Where:
- \(\theta\) - any angle
When: Given one of sinθ/cosθ, find the other.
AA SLAA HLAI HL
Tangent identity
[\tan\theta = \dfrac{\sin\theta}{\cos\theta}]
Definition of tan in terms of sin and cos.
Where:
- \(\theta\) - any angle \((\cos\theta \ne 0)\)
When: Simplifying / solving trig equations.
AA SLAA HLAI HL
Double angle identities
[\begin{aligned}\sin 2\theta&=2\sin\theta\cos\theta\\\cos 2\theta&=\cos^2\theta-\sin^2\theta\end{aligned}]
Rewrite trig of 2θ in terms of θ.
Where:
- \(\theta\) - any angle
When: Simplifying, integrating, exact values.
AA SLAA HL
Compound angle identities
[\begin{aligned}\sin(A\pm B)&=\sin A\cos B\pm\cos A\sin B\\\cos(A\pm B)&=\cos A\cos B\mp\sin A\sin B\end{aligned}]
Break the sin/cos of a sum into parts.
Where:
- \(A, B\) - any two angles
When: Proofs, exact values (AA HL).
AA HL
Vector magnitude & scalar product
[\begin{aligned}|\mathbf{v}|&=\sqrt{v_1^2+v_2^2+v_3^2}\\\mathbf{a}\cdot\mathbf{b}&=|\mathbf{a}||\mathbf{b}|\cos\theta\end{aligned}]
Magnitude = length; the dot product gives the angle between vectors.
Where:
- \(\mathbf{v}\) - vector with components \(v_1, v_2, v_3\)
- \(|\mathbf{v}|\) - magnitude (length)
- \(\mathbf{a}, \mathbf{b}\) - two vectors
- \(\theta\) - angle between \(\mathbf{a}\) and \(\mathbf{b}\)
When: Angles, projections, perpendicularity.
AA HLAI HL
Vector (cross) product
[|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta]
Gives a vector perpendicular to both; its magnitude is the area of the parallelogram they span.
Where:
- \(\mathbf{a}, \mathbf{b}\) - two vectors
- \(\theta\) - angle between them
- \(|\mathbf{a} \times \mathbf{b}|\) - magnitude (= parallelogram area)
When: Normals, areas, planes (AA HL).
AA HLAI HL
Distance between two points (3D)
[d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}]
Pythagoras in three dimensions.
Where:
- \(d\) - distance
- \(x_1\) - \(x\)-coordinate of the first point
- \(x_2\) - \(x\)-coordinate of the second point
- \(y_1\) - \(y\)-coordinate of the first point
- \(y_2\) - \(y\)-coordinate of the second point
- \(z_1\) - \(z\)-coordinate of the first point
- \(z_2\) - \(z\)-coordinate of the second point
When: Length of a segment in 3-D space.
AA SLAA HLAI SLAI HL
Midpoint of a segment (3D)
[\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2},\ \dfrac{z_1+z_2}{2}\right)]
Average each coordinate of the two endpoints.
Where:
- \(x_1\) - \(x\)-coordinate of the first point
- \(x_2\) - \(x\)-coordinate of the second point
- \(y_1\) - \(y\)-coordinate of the first point
- \(y_2\) - \(y\)-coordinate of the second point
- \(z_1\) - \(z\)-coordinate of the first point
- \(z_2\) - \(z\)-coordinate of the second point
When: Midpoint in 3-D space.
AA SLAA HLAI SLAI HL
Reciprocal trig identities
[\begin{aligned}\sec\theta = \tfrac{1}{\cos\theta},\ \ &\csc\theta = \tfrac{1}{\sin\theta},\ \ \cot\theta = \tfrac{1}{\tan\theta}\\1 + \tan^2\theta &= \sec^2\theta\\1 + \cot^2\theta &= \csc^2\theta\end{aligned}]
The three reciprocal ratios and the two Pythagorean identities they give.
Where:
- \(\theta\) - any angle
- \(\sec, \csc, \cot\) - reciprocals of \(\cos, \sin, \tan\)
When: Trig proofs and integrals (AA HL).
AA HL
Transformation matrices
[\begin{aligned}\begin{pmatrix}k&0\\0&1\end{pmatrix} &\ \text{stretch } \parallel x\text{-axis, factor } k\\[2pt]\begin{pmatrix}1&0\\0&k\end{pmatrix} &\ \text{stretch } \parallel y\text{-axis, factor } k\\[2pt]\begin{pmatrix}k&0\\0&k\end{pmatrix} &\ \text{enlargement, factor } k,\ \text{centre } O\\[2pt]\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix} &\ \text{rotation } \theta \text{ about } O\end{aligned}]
Matrices that stretch, enlarge, rotate or reflect points about the origin.
Where:
- \(k\) - scale factor
- \(\theta\) - angle (anticlockwise for \(\theta > 0\))
- \(O\) - origin \((0,0)\)
When: Geometric transformations (AI HL).
AI HL
Vector equation of a line
[\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}; \quad x = x_0 + \lambda l,\ y = y_0 + \lambda m,\ z = z_0 + \lambda n]
A known point plus multiples of a direction vector trace out the line.
Where:
- \(\mathbf{r}\) - position vector of a general point
- \(\mathbf{a}\) - position vector of a known point
- \(\mathbf{b}\) - direction vector \((l, m, n)\)
- \(\lambda\) - parameter (varies along the line)
When: Lines in 2-D / 3-D (HL).
AA HLAI HL
Vector equation of a plane
[\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}; \quad \mathbf{r}\cdot\mathbf{n} = \mathbf{a}\cdot\mathbf{n}; \quad ax + by + cz = d]
A plane as a base point plus two directions, or via its normal vector.
Where:
- \(\mathbf{r}\) - general point on the plane
- \(\mathbf{a}\) - a known point
- \(\mathbf{b}, \mathbf{c}\) - two direction vectors
- \(\mathbf{n}\) - normal vector \((a, b, c)\)
- \(\lambda, \mu\) - parameters
When: Planes in 3-D (AA HL).
AA HL
Statistics & Probability
Mean from a frequency table
[\bar x = \dfrac{\sum f x}{\sum f}]
Each value weighted by how often it occurs.
Where:
- \(\bar x\) - mean
- \(x\) - a data value
- \(f\) - frequency of that value
- \(\sum\) - sum over all values
When: Grouped or repeated data.
AA SLAA HLAI SLAI HL
Probability of an event
[P(A) = \dfrac{n(A)}{n(U)}]
Favourable outcomes over total outcomes.
Where:
- \(P(A)\) - probability of event \(A\)
- \(n(A)\) - number of outcomes in \(A\)
- \(n(U)\) - total outcomes in the sample space
When: Equally-likely outcomes.
AA SLAA HLAI SLAI HL
Combined events (addition rule)
[P(A\cup B) = P(A) + P(B) - P(A\cap B)]
Add the chances, subtract the overlap.
Where:
- \(P(A), P(B)\) - probabilities of \(A\) and \(B\)
- \(P(A \cup B)\) - probability of \(A\) or \(B\)
- \(P(A \cap B)\) - probability of \(A\) and \(B\)
When: P(A or B).
AA SLAA HLAI SLAI HL
Conditional probability
[P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}]
Chance of A once B is known - sample space shrinks to B.
Where:
- \(P(A \mid B)\) - probability of \(A\) given \(B\)
- \(P(A \cap B)\) - probability of both
- \(P(B)\) - probability of \(B\) \((>0)\)
When: 'Given that…' problems.
AA SLAA HLAI SLAI HL
Independent events
[P(A\cap B) = P(A)\,P(B)]
Multiply when one event doesn't affect the other.
Where:
- \(P(A), P(B)\) - individual probabilities
- \(P(A \cap B)\) - probability both occur
When: Both happen; also the test for independence.
AA SLAA HLAI SLAI HL
Expected value
[E(X) = \sum x\,P(X=x)]
Long-run average: outcomes weighted by probability.
Where:
- \(E(X)\) - expected (mean) value
- \(x\) - an outcome value
- \(P(X=x)\) - probability of that outcome
When: Fair games, average payoff.
AA SLAA HLAI SLAI HL
Binomial distribution
[\begin{aligned}X&\sim B(n,p)\\E(X)&=np\\\operatorname{Var}(X)&=np(1-p)\end{aligned}]
n independent trials, success chance p; on average np successes.
Where:
- \(X\) - number of successes
- \(n\) - number of trials
- \(p\) - probability of success
- \(E(X)\) - mean
- \(\operatorname{Var}(X)\) - variance
When: Fixed number of yes/no trials.
AA SLAA HLAI SLAI HL
Standardising (z-score)
[z = \dfrac{x - \mu}{\sigma}]
How many standard deviations a value is from the mean.
Where:
- \(z\) - standardised value
- \(x\) - data value
- \(\mu\) - mean
- \(\sigma\) - standard deviation
When: Normal-distribution problems by hand.
AA SLAA HL
Pearson's correlation & regression
[\begin{aligned}y&=ax+b\;\text{(least squares)}\\-1&\le r\le 1\end{aligned}]
Line of best fit; r measures strength/direction of linear association.
Where:
- \(y\) - predicted value
- \(x\) - explanatory variable
- \(a\) - gradient
- \(b\) - \(y\)-intercept
- \(r\) - correlation coefficient \((-1 \le r \le 1)\)
When: Bivariate data, prediction.
AA SLAA HLAI SLAI HL
Poisson distribution
[P(X=r) = \dfrac{e^{-\lambda}\lambda^{\,r}}{r!}]
Counts of random events at average rate λ.
Where:
- \(X\) - number of events
- \(m\) - mean number of events
- \(r\) - a specific count
- \(e \approx 2.718\)
When: Events per interval (calls/hour etc.).
AI HL
Chi-squared test statistic
[\chi^2 = \sum \dfrac{(f_o - f_e)^2}{f_e}]
Compares observed with expected frequencies.
Where:
- \(\chi^2\) - test statistic
- \(f_o\) - observed frequency
- \(f_e\) - expected frequency
When: Tests of independence / goodness-of-fit (AI).
AI SLAI HL
Bayes' theorem
[P(A\mid B) = \dfrac{P(A)P(B\mid A)}{P(B)}]
Reverses a conditional probability using the overall rate.
Where:
- \(P(A \mid B)\) - probability of \(A\) given \(B\)
- \(P(B \mid A)\) - probability of \(B\) given \(A\)
- \(P(A), P(B)\) - individual probabilities
When: Test-accuracy / diagnostic problems (HL).
AA HL
Interquartile range
[\text{IQR} = Q_3 - Q_1]
The spread of the middle 50% of the data.
Where:
- \(Q_1\) - lower quartile
- \(Q_3\) - upper quartile
When: Spread; outlier boundaries.
AA SLAA HLAI SLAI HL
Complementary events
[P(A) + P(A') = 1]
An event and its non-occurrence have probabilities that sum to 1.
Where:
- \(P(A)\) - probability of \(A\)
- \(P(A')\) - probability \(A\) does not occur
When: Finding \(P(\text{not } A)\).
AA SLAA HLAI SLAI HL
Mutually exclusive events
[P(A\cup B) = P(A) + P(B)]
When two events cannot both happen, just add their probabilities.
Where:
- \(P(A), P(B)\) - individual probabilities
- \(P(A\cup B)\) - probability of \(A\) or \(B\)
When: Events with no overlap.
AA SLAA HLAI SLAI HL
Variance & standard deviation
[\sigma^2 = \dfrac{\sum f_i(x_i - \mu)^2}{n} = \dfrac{\sum f_i x_i^2}{n} - \mu^2, \quad \sigma = \sqrt{\sigma^2}]
Average squared distance from the mean; its square root is the standard deviation.
Where:
- \(\sigma^2\) - variance
- \(\sigma\) - standard deviation
- \(x_i\) - a data value
- \(f_i\) - its frequency
- \(\mu\) - mean
- \(n\) - total frequency
When: Spread of a data set (AA HL).
AA HL
Linear transformation of a random variable
[\begin{aligned}E(aX + b) &= aE(X) + b\\\operatorname{Var}(aX + b) &= a^2\operatorname{Var}(X)\end{aligned}]
Scaling and shifting: the mean scales and shifts; the variance scales by \(a^2\).
Where:
- \(X\) - random variable
- \(a, b\) - constants
- \(E\) - expected value
- \(\operatorname{Var}\) - variance
When: Transforming a random variable (HL).
AA HLAI HL
Linear combinations of random variables
[\begin{aligned}E\!\left(\textstyle\sum a_i X_i\right) &= \textstyle\sum a_i E(X_i)\\\operatorname{Var}\!\left(\textstyle\sum a_i X_i\right) &= \textstyle\sum a_i^2 \operatorname{Var}(X_i)\end{aligned}]
For independent variables: means add; variances add with squared weights.
Where:
- \(X_i\) - independent random variables
- \(a_i\) - constant weights
- \(E\) - expected value
- \(\operatorname{Var}\) - variance
When: Sums of independent variables (AI HL).
AI HL
Unbiased estimate of population variance
[s_{n-1}^2 = \dfrac{n}{n-1}\,s_n^2]
Corrects the sample variance so it doesn't underestimate the population.
Where:
- \(s_{n-1}^2\) - unbiased estimate of the variance
- \(s_n^2\) - sample variance
- \(n\) - sample size
When: Estimating population spread from a sample (AI HL).
AI HL
Continuous random variable - mean & variance
[\begin{aligned}E(X) &= \int_{-\infty}^{\infty} x\,f(x)\,dx\\\operatorname{Var}(X) &= E(X^2) - [E(X)]^2\end{aligned}]
Expected value and variance from the probability density function.
Where:
- \(X\) - continuous random variable
- \(f(x)\) - probability density function
- \(E(X)\) - mean
- \(\operatorname{Var}(X)\) - variance
When: Continuous distributions (AA HL).
AA HL
Transition matrices
[T^n\,\mathbf{s}_0 = \mathbf{s}_n]
Apply a transition matrix \(n\) times to find the state after \(n\) steps.
Where:
- \(T\) - transition matrix
- \(\mathbf{s}_0\) - initial state vector
- \(\mathbf{s}_n\) - state after \(n\) steps
- \(n\) - number of steps
When: Markov chains (AI HL).
AI HL
Calculus
Derivative - power rule
[\dfrac{d}{dx}(x^n) = n x^{\,n-1}]
Bring the power down, reduce it by one.
Where:
- \(x\) - the variable
- \(n\) - the power
When: Differentiating powers of x.
AA SLAA HLAI SLAI HL
Standard derivatives
[\begin{aligned}\tfrac{d}{dx}\sin x&=\cos x,\ \tfrac{d}{dx}\cos x=-\sin x\\\tfrac{d}{dx}\tan x&=\tfrac{1}{\cos^2 x}\\\tfrac{d}{dx}e^x&=e^x,\ \tfrac{d}{dx}\ln x=\tfrac{1}{x}\end{aligned}]
The common functions and their derivatives.
Where:
- \(x\) - the variable
- \(e \approx 2.718\)
- \(\ln\) - natural logarithm
When: Differentiating trig/exponential/log (AA).
AA SLAA HLAI HL
Chain rule
[\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}]
Differentiate the outside, times the derivative of the inside.
Where:
- \(y\) - the outer variable
- \(u\) - the inner function
- \(x\) - the variable
When: Function inside a function.
AA SLAA HLAI HL
Product & quotient rules
[\begin{aligned}(uv)'&=u'v+uv'\\[4pt]\left(\tfrac{u}{v}\right)'&=\dfrac{u'v-uv'}{v^2}\end{aligned}]
For products and fractions of two functions.
Where:
- \(u, v\) - two functions of \(x\)
- \(u', v'\) - their derivatives
When: Differentiating e.g. x²eˣ or x/(x+1).
AA SLAA HLAI HL
Integration - power rule
[\int x^n\,dx = \dfrac{x^{\,n+1}}{n+1} + C, \quad n\ne -1]
Reverse of differentiating: raise the power, divide by it.
Where:
- \(x\) - the variable
- \(n\) - the power \((n \ne -1)\)
- \(C\) - constant of integration
When: Integrating powers of x (don't forget +C).
AA SLAA HLAI SLAI HL
Standard integrals
[\begin{aligned}\int e^x\,dx&=e^x+C\\\int\tfrac{1}{x}\,dx&=\ln|x|+C\\\int\cos x\,dx&=\sin x+C\end{aligned}]
The common antiderivatives.
Where:
- \(x\) - the variable
- \(C\) - constant of integration
When: Integrating exponential/log/trig (AA).
AA SLAA HLAI HL
Area under a curve
[A = \int_a^b y\,dx]
A definite integral sums thin strips into a signed area.
Where:
- \(A\) - area
- \(y\) - the curve function
- \(a, b\) - lower and upper limits (\(x\) from \(a\) to \(b\))
When: Area, and (with velocity) distance.
AA SLAA HLAI SLAI HL
Volume of revolution
[V = \pi\int_a^b y^2\,dx]
Rotate a region about the x-axis and sum disc volumes.
Where:
- \(V\) - volume
- \(y\) - the curve function
- \(a, b\) - limits (region rotated about the \(x\)-axis)
When: Solids of revolution (AA HL).
AA HLAI HL
Kinematics
[\begin{aligned}v&=\dfrac{ds}{dt}\\a&=\dfrac{dv}{dt}\\s&=\int v\,dt\end{aligned}]
Differentiate to go displacement→velocity→acceleration; integrate to reverse.
Where:
- \(s\) - displacement
- \(v\) - velocity
- \(a\) - acceleration
- \(t\) - time
When: Motion problems.
AA SLAA HLAI HL
Trapezoidal rule
[\int_a^b y\,dx \approx \tfrac{h}{2}\big(y_0 + y_n + 2(y_1+\dots+y_{n-1})\big)]
Estimate area with trapezia of width h.
Where:
- \(h\) - strip width
- \(y_0, \dots, y_n\) - the ordinate (\(y\)) values
- \(n\) - number of strips
When: Approximating an integral (AI).
AI SLAI HL
Integration by parts
[\int u\,dv = uv - \int v\,du]
Swaps a hard integral for an easier one; choose u to simplify when differentiated.
Where:
- \(u, v\) - two functions of \(x\)
- \(du, dv\) - their differentials
When: Products like x·eˣ (AA HL).
AA HL
Maclaurin series
[f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dots]
Approximates a function near 0 as an infinite polynomial.
Where:
- \(f(x)\) - the function
- \(f'(0), f''(0)\) - derivatives evaluated at 0
- \(n!\) - factorial
When: Series approximations (AA HL).
AA HL
Derivative from first principles
[f'(x) = \lim_{h\to0}\dfrac{f(x+h) - f(x)}{h}]
The limit definition of the derivative.
Where:
- \(f'(x)\) - derivative
- \(f(x)\) - the function
- \(h\) - a small change in \(x\)
When: Differentiating from the definition (AA HL).
AA HL
Standard derivatives (HL)
[\begin{aligned}\tfrac{d}{dx}\tan x &= \sec^2 x,\ \ \tfrac{d}{dx}\sec x = \sec x\tan x\\\tfrac{d}{dx}a^x &= a^x\ln a,\ \ \tfrac{d}{dx}\log_a x = \tfrac{1}{x\ln a}\\\tfrac{d}{dx}\arcsin x &= \tfrac{1}{\sqrt{1-x^2}},\ \ \tfrac{d}{dx}\arctan x = \tfrac{1}{1+x^2}\end{aligned}]
Derivatives of the further functions: tan/sec, general exponentials and logs, inverse trig.
Where:
- \(x\) - the variable
- \(a\) - base \((a > 0)\)
- \(\arcsin, \arctan\) - inverse trig functions
When: Differentiating these functions (AA HL).
AA HL
Standard integrals (HL)
[\begin{aligned}\int a^x\,dx &= \dfrac{a^x}{\ln a} + C\\\int\dfrac{dx}{a^2 + x^2} &= \dfrac{1}{a}\arctan\!\dfrac{x}{a} + C\\\int\dfrac{dx}{\sqrt{a^2 - x^2}} &= \arcsin\!\dfrac{x}{a} + C\end{aligned}]
Antiderivatives that produce logs, arctan and arcsin.
Where:
- \(x\) - the variable
- \(a\) - a constant
- \(C\) - constant of integration
When: Integrals giving inverse-trig results (AA HL).
AA HL
Area between a curve and the y-axis
[A = \int_a^b x\,dy]
Integrate \(x\) with respect to \(y\) for a region against the \(y\)-axis.
Where:
- \(A\) - area
- \(x\) - the curve as a function of \(y\)
- \(a, b\) - lower and upper limits on \(y\)
When: Regions measured along \(y\) (HL).
AA HLAI HL
Euler's method
[y_{n+1} = y_n + h\,f(x_n, y_n), \quad x_{n+1} = x_n + h]
Step forward along a differential equation using the gradient at each point.
Where:
- \(y_n\) - current \(y\)-value
- \(x_n\) - current \(x\)-value
- \(h\) - step length
- \(f(x_n, y_n)\) - gradient at the current point
When: Numerical solution of \(\tfrac{dy}{dx} = f(x,y)\) (HL).
AA HLAI HL
Euler's method for coupled systems
[\begin{aligned}x_{n+1} &= x_n + h\,f_1(x_n, y_n, t_n)\\y_{n+1} &= y_n + h\,f_2(x_n, y_n, t_n)\\t_{n+1} &= t_n + h\end{aligned}]
Euler's method applied to two equations at once.
Where:
- \(x_n, y_n\) - current values
- \(t_n\) - current time
- \(h\) - step length
- \(f_1, f_2\) - the two rate functions
When: Coupled differential equations (AI HL).
AI HL
Exact solution for coupled linear DEs
[\mathbf{x} = Ae^{\lambda_1 t}\mathbf{p}_1 + Be^{\lambda_2 t}\mathbf{p}_2]
Built from the eigenvalues and eigenvectors of the system.
Where:
- \(\lambda_1, \lambda_2\) - eigenvalues
- \(\mathbf{p}_1, \mathbf{p}_2\) - eigenvectors
- \(A, B\) - constants from initial conditions
- \(t\) - time
When: Solving coupled linear systems (AI HL).
AI HL
Integrating factor
[\text{IF} = e^{\int P(x)\,dx}]
Multiply through by this to solve a linear first-order ODE.
Where:
- \(P(x)\) - coefficient of \(y\)
- \(Q(x)\) - the right-hand side
- \(e \approx 2.718\)
When: Equations \(y' + P(x)y = Q(x)\) (AA HL).
AA HL
Maclaurin series for special functions
[\begin{aligned}e^x &= 1 + x + \tfrac{x^2}{2!} + \dots\\\ln(1+x) &= x - \tfrac{x^2}{2} + \tfrac{x^3}{3} - \dots\\\sin x &= x - \tfrac{x^3}{3!} + \dots,\ \ \cos x = 1 - \tfrac{x^2}{2!} + \dots\end{aligned}]
The standard series expansions to quote directly.
Where:
- \(x\) - the variable
- \(n!\) - factorial
- valid near \(x = 0\)
When: Series approximations near 0 (AA HL).
AA HL