Quadratic Roots
Solutions for ax² + bx + c = 0 with a ≠ 0; Δ = b² − 4ac is the discriminant.
Roots
x = [-b ± √(b² − 4ac)] / (2a) Nature Test
Δ = b² − 4ac IB Diploma Programme 2026
📐 IBDP Maths AA Formula Sheet
All Analysis & Approaches SL formulas with highlighted HL-only additions for Paper 1 & Paper 2 success.
Algebra Calculus HL Extensions
Memorise Less, Understand More
Use this dual-level sheet to keep algebraic identities, derivative rules, and probability relationships organised. Each formula lists what the symbols represent so you can translate quickly during GDC or non-GDC assessments.
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Standard + Higher Level labelling
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Statistics notation reminders
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Vector direction checks
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Fast calculus refresh
Algebra & Functions (SL Core)
Core Paper 1 algebra identities plus transformation language you must reference when sketching or solving inequalities.
General Binomial Term
nC r selects combinations, a and b are terms, r counts from 0 to n.
T_{r+1} = C(n, r) · a^{n−r} · b^r Logarithm Laws
a, b > 0 and logarithms share the same base.
log_a(xy) = log_a x + log_a y log_a(x / y) = log_a x − log_a y a^{log_a x} = x Function Transformations
k scales vertically, h shifts horizontally, c shifts vertically.
Use b to stretch/compress horizontally (b > 1 stretches).
y = k · f((x − h)/b) + c Topic Focus
Quadratics & Sequences
- Switch between expanded, factorised, and completed-square forms to justify intercepts or turning points.
- Use discriminant sign to discuss number of solutions when reasoning about intersections.
Binomial & Series
- State combination limits (r from 0 to n) when extracting a specific power for proofs.
- Explain when to use nCr vs. factorial notation to link with calculator outputs.
Function Composition
- Apply horizontal (input) transformations before vertical (output) ones when sketching.
- Mention domain/range adjustments after reflections or translations for full method marks.
Calculus & Vectors (SL Core)
Derivative definitions, integral language, and vector magnitude/unit results that underpin Paper 2 modelling questions.
Derivative Definition
f'(x) measures instantaneous rate of change at x.
f'(x) = lim_{h→0} [f(x + h) − f(x)] / h Power & Exponential Rules
n is real, a > 0, e is Euler's constant.
d/dx (x^n) = n x^{n−1} d/dx (e^{kx}) = k e^{kx} Definite Integral Area
∫_a^b f(x) dx accumulates signed area between x = a and x = b.
F(x) is any antiderivative of f(x).
∫_a^b f(x) dx = F(b) − F(a) Vector Magnitude & Direction
For vector v = ⟨v_x, v_y, v_z⟩, |v| is length, ũ is unit vector.
Magnitude
|v| = √(v_x² + v_y² + v_z²) Unit vector
ũ = v / |v| Topic Focus
Differentiation Techniques
- Always state the rule used (power, chain, product) before writing the derivative in Paper 1 explanations.
- Link gradient interpretation back to rate-of-change context (speed, cost, population).
Integrals & Area
- Clarify limits and orientation when interpreting definite integrals as net area.
- Show antiderivative plus constant when performing indefinite integration steps.
Vectors & Geometry
- Normalize vectors before using them in direction or projection problems.
- Quote magnitude when discussing distance or speed along a vector path.
Probability & Statistics (SL Core)
Summary statistics and discrete probability laws that appear in Paper 2 modelling and technology-required prompts.
Expected Value
X takes values x_i with probabilities p_i where Σp_i = 1.
E(X) = Σ x_i p_i Variance & Standard Deviation
μ is the mean of X.
Variance
Var(X) = Σ (x_i − μ)² p_i Std. Dev.
σ = √Var(X) Binomial Probability
n trials, p success probability, r successes.
P(X = r) = C(n, r) p^r (1 − p)^{n−r} z-score
x is observed value, μ mean, σ standard deviation.
z = (x − μ) / σ Topic Focus
Choosing Distributions
- Check model assumptions (independence, constant probability, fixed interval) before declaring binomial or Poisson.
- State parameter values (n, p, λ) explicitly to earn communication marks.
Normal & z-scores
- Sketch the bell curve and shade the region you are calculating to communicate understanding.
- Always mention mean and standard deviation units when interpreting z-scores.
Expectation & Variance
- For discrete tables, rewrite formula as Σ(x·P(x)) so examiners follow the substitution.
- When comparing two datasets, quote both mean and spread to justify conclusions.
Higher Level Extensions
Use these when HL papers push beyond SL techniques.
Targeted Paper 3 methods covering expansion, advanced integration, vector products, and differential equations.
Maclaurin Series (f up to x³)
f⁽ⁿ⁾(0) is the nth derivative evaluated at 0.
f(x) ≈ f(0) + f'(0)x + [f''(0)/2!]x² + [f'''(0)/3!]x³ Integration by Parts
u and v functions of x chosen so that du/dx simplifies.
∫ u dv = u v − ∫ v du Vector Products
θ is angle between vectors a and b.
Dot
a · b = |a||b| cos θ Cross magnitude
|a × b| = |a||b| sin θ First-order Differential Equation
k is growth/decay constant; y₀ value when t = 0.
y = y₀ e^{kt} Topic Focus
Series & Approximations
- Quote the order of accuracy when truncating a Maclaurin expansion.
- Link remainder size to |x| < 1 style conditions when required.
Advanced Calculus
- State u and dv choices before applying integration by parts to score method marks.
- For differential equations, connect constants back to given initial conditions.
Vector Products
- Clarify whether you are proving perpendicularity (dot) or area (cross) when referencing products.
- Mention direction of a × b using right-hand rule if asked about orientation.
How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Label HL Steps
When a method (e.g., integration by parts) is HL-only, annotate HL beside your working so examiners see you applied the right technique.
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Check GDC Syntax
Store functions as y₁(x) on your calculator so you can evaluate derivatives and definite integrals quickly during Paper 2.
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Derive Once Weekly
Re-derive the power, product, and chain rules from first principles to keep conceptual understanding fresh.
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Group Identities
Keep trigonometric identities beside calculus rules so you can simplify integrands before integrating.
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Content aligns with the 2025–2027 IBDP Maths Analysis & Approaches guide, including HL extensions flagged separately.
Always round calculator answers to the number of significant figures requested in the question stem.