IB Mathematics: Analysis and Approaches HL practice

Independent practice matched to the topics of this course. It is not official course material, so check your own course outline for exactly what your teacher covers. This page includes all the IB Math AA SL skills plus the higher-level ones (marked HL).

Functions

• Functions: notation, domain, range and the vertical line test
• Inverse functions
• Composite functions f(g(x))
• Parent functions and their key features
• Transformations: y = a f(k(x − d)) + c
• Odd and even functions HL
• Multiply, divide, add and subtract rational expressions HL
• Absolute value equations and inequalities HL

Quadratic Functions

• Maximum or minimum of a quadratic function
• Roots, zeros and x-intercepts
• Quadratic inequalities
• Where a line meets a parabola
• Write a quadratic from its zeros, vertex or a family

Polynomial and Rational Functions

• Polynomial functions: degree and end behaviour HL
• Zeros and multiplicity HL
• The remainder theorem HL
• The factor theorem: factor cubics HL
• Sum and product of the roots HL
• Rational functions: asymptotes and holes
• Rational functions: zeros, intercepts and end behaviour

Exponential and Logarithmic Functions

• Rational exponents
• Simplify with the exponent laws
• Exponential functions: graphs and transformations
• Exponential growth and decay models
• Logarithmic functions and their graphs
• Log laws and change of base
• Solve exponential and logarithmic equations
• Exponential and logarithmic inequalities

Sequences and Series

• Recursive sequences
• Arithmetic sequences: general term
• Geometric sequences: general term
• Arithmetic series
• Geometric series
• Infinite geometric series
• Pascal's triangle and the binomial theorem

Financial Mathematics

• Compound interest
• Present value
• Annuities: future value and present value
• Rounding, significant figures and percentage error

Trigonometry

• Angles in standard position and the CAST rule
• Exact values for special angles up to 360°
• Find all angles from 0° to 360° with a given ratio
• Reciprocal ratios: csc, sec and cot HL
• Radians and the unit circle
• Trigonometric identities
• Double-angle and compound-angle formulas
• Solve trigonometric equations
• The sine law: the ambiguous case (SSA)
• Trigonometry in 3D and with bearings
• Inverse trigonometric functions HL

Periodic and Sinusoidal Functions

• Graphs of y = sin x and y = cos x
• Transformations: amplitude, period, phase shift and axis
• Write the equation of a sinusoidal function
• The tangent function and its graph
• Polar coordinates HL

Complex Numbers, Counting and Proof

• Complex numbers: add, subtract, multiply and divide HL
• Quadratics with complex roots HL
• Complex numbers in polar form and De Moivre's theorem HL
• Permutations: arrangements where order matters HL
• Combinations: selections where order doesn't matter HL
• Systems of equations in three unknowns HL
• Proof: deductive proof and induction
• Partial fractions HL

Vectors, Matrices and Conics

• Vectors: magnitude, addition and scalar multiples HL
• The dot product and the angle between vectors HL
• Vector and parametric equations of lines HL

3D Geometry, Voronoi Diagrams and Graphs

• Distance and midpoint in 3D
• 3D solids: volume, surface area and angles

Statistics

• Sampling methods, bias and study design
• Summary statistics, IQR and outliers
• Regression lines, residuals and r²

Probability

• Probability rules: complements, or, and independence
• Conditional probability and Bayes' theorem HL
• Discrete random variables and expected value
• The binomial distribution
• The normal distribution and z-scores

Introduction to Calculus

• Limits
• The derivative from first principles
• Derivatives with the power rule
• Tangent and normal lines
• Increasing, decreasing and stationary points
• Chain, product and quotient rules; eˣ, ln x, sin x, cos x
• Optimization problems
• Motion: displacement, velocity and acceleration
• Introduction to integration and area

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