HL AI Formula Booklet — Applications & Interpretation HL

All 18 formulas you need for IB Mathematics Applications & Interpretation HL, grouped by syllabus topic and written in the same notation as the official IB formula booklet. Use it to revise which formulas are given in the exam, and which you still need to understand how to apply.

Open the print-ready one-page booklet → Practise HL AI questions
Everything in SL AI, plus… · Topic 3 (HL only) — Geometry & Trig extension · Topic 4 (HL only) — Statistics extension · Topic 5 (HL only) — Calculus extension

Everything in SL AI, plus…

Complex numbers — Cartesian
\(z = a + bi,\ |z| = \sqrt{a^2 + b^2},\ \arg z = \arctan\!\left(\tfrac{b}{a}\right)\)
Modulus of a complex number
\(|z_1 z_2| = |z_1| \cdot |z_2|,\ \left|\tfrac{z_1}{z_2}\right| = \tfrac{|z_1|}{|z_2|}\)

Topic 3 (HL only) — Geometry & Trig extension

Vector dot product
\(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3\)
Vector magnitude
\(|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\)
Angle between two vectors
\(\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\)
Vector equation of a line
\(\vec{r} = \vec{a} + t\,\vec{d}\)
Matrix multiplication (2×2)
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\!\begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}\)
Determinant (2×2)
\(\det\!\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc\)
Inverse of a 2×2 matrix
\(M^{-1} = \dfrac{1}{\det M}\!\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)
Eigenvalue equation
\(M \vec{v} = \lambda \vec{v} \implies \det(M - \lambda I) = 0\)

Topic 4 (HL only) — Statistics extension

Poisson distribution
\(P(X = r) = \dfrac{e^{-\lambda}\lambda^{\,r}}{r!}\)
Mean & variance of Poisson
\(E(X) = \lambda,\ \mathrm{Var}(X) = \lambda\)
Confidence interval for a mean (large n)
\(\bar{x} \pm z^* \cdot \dfrac{\sigma}{\sqrt{n}}\)
Chi-squared test statistic
\(\chi^2_{\text{calc}} = \sum \dfrac{(f_o - f_e)^2}{f_e}\)
Transition matrix (Markov chain)
\(\vec{s}_{n+1} = T\,\vec{s}_n\)

Topic 5 (HL only) — Calculus extension

Euler's method
\(y_{n+1} = y_n + h\, f(x_n, y_n),\ x_{n+1} = x_n + h\)
Separable differential equation
\(\dfrac{dy}{dx} = f(x) g(y) \implies \int \dfrac{1}{g(y)}\, dy = \int f(x)\, dx\)
Trapezoidal rule (refresher)
\(A \approx \tfrac{h}{2}\bigl(y_0 + y_n + 2\sum_{i=1}^{n-1} y_i\bigr)\)

Other courses: SL AI formulas · SL AA formulas · HL AA formulas · HL AI revision notes · HL AI questions by topic