A-level and IB · Mathematics

IB Maths: analysis and approaches

IB Maths AA flashcards on algebra, functions, trigonometry, vectors, probability, statistics and calculus, with worked examples and HL extension cards.

36 cards6 topicsAbout 15 minutes a roundIB Diploma Programme Mathematics: analysis and approaches (SL and HL) style content, comparable to Grade 12 calculus and first-year algebra

Independent study material written by StudyBrix, not produced or reviewed by any exam board or by the International Baccalaureate Organization. Check your own specification or subject guide.

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About this deck

This deck supports students taking IB Diploma Mathematics: analysis and approaches at SL or HL, and Canadian students comparing it with Grade 12 calculus. It follows the five topic areas of the current course: number and algebra, functions, geometry and trigonometry, statistics and probability, and calculus. Cards marked (HL) go beyond the SL content, such as proof by induction, complex numbers, Bayes' theorem and Maclaurin series.

The current course has papers without and with technology (plus a third problem-solving paper at HL) and an internally assessed mathematical exploration. The IB has announced a revised course with first teaching in August 2027 and first assessment in May 2029, which removes some content (for example financial applications and bivariate data), so check which guide applies to you. If you are comparing with A-level, specifications differ between exam boards (AQA, Edexcel, OCR), so readers should check theirs.

You should know MYP extended maths or Grade 10 algebra well: factoring, exponents, linear and quadratic functions and right-angle trigonometry. Work each example on paper, first without a calculator, then check with one, because some papers do not allow technology. Write down the formula from the booklet before substituting.

Common mistakes: leaving answers unsimplified when an exact value is asked for; using degree mode in calculus; dropping the second solution of a trig equation; and forgetting that a probability must lie between 0 and 1. In the exploration, aim for your own mathematics rather than a textbook summary.

Before you start: MYP extended mathematics or Grade 10 to 11 algebra and functions, including quadratics, exponents and basic trigonometry.

All 36 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Number and algebra

Compound interest: An amount grows by a fixed percentage each period, applied to the growing total: FV = PV(1 + r/(100k))^(kn) for k periods a year over n years.Example: 1000 dollars at 4% a year compounded monthly for 5 years: 1000(1 + 0.04/12)^60, about 1221.00 dollars.
Geometric sequence: A sequence where each term is the previous one times a common ratio r: u(n) = u1 r^(n - 1).Example: Population 1000 growing 3% a year: after 10 years 1000 * 1.03^10, about 1344.
Logarithm definition: log base a of b is the power to which a must be raised to give b, so a^x = b means x = log_a b.Example: Solve 1.05^n = 2: n = ln 2/ln 1.05, about 14.2, so money doubles in about 15 years at 5%.
Binomial theorem: (a + b)^n is the sum of nCr a^(n - r) b^r for r from 0 to n, where nCr counts combinations.Example: The x^3 coefficient in (2 + x)^5 is 5C3 * 2^2 = 10 * 4 = 40.
Proof by induction (HL): Show a statement holds for n = 1, assume it holds for n = k, prove it then holds for n = k + 1, and conclude it holds for all positive integers.Example: 1 + 2 + ... + n = n(n + 1)/2: if true for k, adding k + 1 gives (k + 1)(k + 2)/2, the formula for k + 1.
Complex number arithmetic (HL): Numbers a + bi with i^2 = -1; add parts separately and multiply by expanding.Example: (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 5 + i.
De Moivre's theorem (HL): For z = r cis(theta), z^n = r^n cis(n theta). It makes powers and roots of complex numbers quick.Example: 1 + i = sqrt(2) cis(pi/4), so (1 + i)^8 = 16 cis(2 pi) = 16.

Functions

Domain and range: The domain is the set of allowed inputs; the range is the set of outputs the function actually produces.Example: f(x) = sqrt(x - 2) has domain x >= 2 and range f(x) >= 0.
Composite and inverse functions: (f o g)(x) = f(g(x)); the inverse undoes f and exists only for one-to-one functions, its graph reflected in y = x.Example: f(x) = 2x + 3: the inverse is (x - 3)/2, and f(f^-1(5)) = 5.
Quadratic vertex form: f(x) = a(x - h)^2 + k has vertex (h, k) and axis of symmetry x = h.Example: f(x) = 2(x - 1)^2 - 8 has vertex (1, -8) and zeros at x = -1 and x = 3.
Rational function asymptotes: For f(x) = (ax + b)/(cx + d), the vertical asymptote is x = -d/c and the horizontal asymptote is y = a/c.Example: f(x) = (2x + 1)/(x - 3): vertical x = 3, horizontal y = 2.
Transformations of graphs: f(x) + b shifts up b, f(x - a) shifts right a, pf(x) stretches vertically by p, f(qx) stretches horizontally by 1/q, and -f(x) reflects in the x-axis.Example: y = -2(x + 1)^2 is y = x^2 shifted left 1, stretched by 2 and reflected, so it opens downward.
Exponential decay: A quantity that falls by a constant factor over equal time steps, modelled by N = N0 e^(-kt) or N0 a^t with 0 < a < 1.Example: A drug at 200 mg loses 20% each hour: 200 * 0.8^3 = 102.4 mg after 3 hours.

Geometry and trigonometry

Cosine rule: In any triangle, c^2 = a^2 + b^2 - 2ab cos C. Used for a side from two sides and the included angle, or an angle from three sides.Example: a = 5, b = 8, C = 60 degrees: c^2 = 25 + 64 - 40 = 49, so c = 7.
Sine rule ambiguous case: a/sin A = b/sin B. When the given angle is opposite the shorter given side, two different triangles may fit.Example: sin B = 0.8 gives B = 53.1 or 126.9 degrees; check each against the angle sum.
Area of a triangle with sine: Area = (1/2)ab sin C, using two sides and the angle between them.Example: a = 5, b = 8, C = 60 degrees: area = 20 sin 60, about 17.3.
Unit circle: A circle of radius 1 where a point at angle theta has coordinates (cos theta, sin theta), which extends trig to any angle.Example: cos(2pi/3) = -1/2 and sin(2pi/3) = sqrt(3)/2.
Scalar product (HL): a . b = |a||b| cos theta = a1b1 + a2b2 + a3b3; a zero result means the vectors are perpendicular.Example: a = (1, 2, 2), b = (2, -1, 2): a . b = 4, so cos theta = 4/9 and theta is about 63.6 degrees.
Volume and surface of solids: Cone volume (1/3) pi r^2 h, sphere volume (4/3) pi r^3, and curved cone surface pi r l with slant height l.Example: Cone r = 3 cm, h = 4 cm: volume = 12 pi, about 37.7 cm^3; slant height 5 cm, curved area 15 pi.

Statistics and probability

Conditional probability: P(A | B) = P(A and B)/P(B), the chance of A once B is known to have occurred.Example: From a two-way table: 18 of 60 students who walk are late, so P(late | walks) = 0.3.
Bayes' theorem (HL): Reverses a conditional probability: P(A | B) = P(B | A)P(A)/P(B), with P(B) found from all routes in a tree diagram.Example: 1% have a disease, the test detects 95% and gives 5% false positives: P(disease | positive) = 0.0095/0.059, about 0.161.
Expected value: The long-run mean of a discrete random variable: the sum of each value times its probability.Example: Win 10 with p = 0.2, win 2 with p = 0.3, lose 3 with p = 0.5: E(X) = 2 + 0.6 - 1.5 = 1.1.
Binomial model: Counts successes in n independent trials with constant probability p; mean np and variance np(1 - p).Example: n = 20, p = 0.25: mean 5, variance 3.75.
Normal distribution inverse: Given a probability, find the value with that area to its left, using the inverse normal function on a GDC.Example: Heights N(170, 8^2): the top 10% are above 170 + 1.2816 * 8, about 180.3 cm.
Regression line of y on x: The least-squares line used to predict y from x; reliable only within the data range and when r is strong.Example: y = 1.8x + 12 predicts y = 30 at x = 10.

Calculus

Derivative as gradient: f'(x) gives the gradient of the tangent at each point and the instantaneous rate of change.Example: f(x) = x^3 - 2x: f'(2) = 12 - 2 = 10, the tangent slope at (2, 4).
Equation of a tangent: Use the point (a, f(a)) and gradient f'(a) in y - f(a) = f'(a)(x - a); the normal has gradient -1/f'(a).Example: y = x^2 at x = 3: tangent y - 9 = 6(x - 3), so y = 6x - 9.
Kinematics with calculus: Velocity is the derivative of displacement and acceleration the derivative of velocity; distance travelled is the integral of |v|.Example: v = 3t^2 - 12: the particle is at rest at t = 2 and changes direction there.
Optimization: Find a maximum or minimum by setting the derivative to zero, then confirm with the second derivative or endpoints.Example: A rectangle with perimeter 20 has area x(10 - x); the derivative 10 - 2x = 0 gives x = 5, area 25.
Area between two curves: The integral from a to b of (upper function - lower function), with limits at the intersection points.Example: y = x and y = x^2 meet at 0 and 1: area = integral of (x - x^2) = 1/2 - 1/3 = 1/6.
Volume of revolution (HL): Rotating y = f(x) about the x-axis from a to b gives volume pi times the integral of y^2.Example: y = sqrt(x) from 0 to 4: pi times integral of x dx = 8 pi, about 25.1.
Maclaurin series (HL): A power series for f(x) built from derivatives at 0: f(0) + f'(0)x + f''(0)x^2/2! + ...Example: e^x is about 1 + x + x^2/2 + x^3/6; at x = 0.1 this gives 1.10517, matching e^0.1.
Chain rule: To differentiate a composite function, multiply the derivative of the outer function (evaluated at the inner one) by the derivative of the inner function.Example: y = sin(3x^2): dy/dx = cos(3x^2) * 6x.
Separable differential equation (HL): An equation that can be written g(y) dy = f(x) dx and integrated on both sides.Example: dy/dx = 0.5y with y(0) = 4 gives y = 4e^(0.5x).

Assessment and technology

Mathematical exploration: The internally assessed piece of writing in which a student investigates an area of mathematics of their own choice and shows personal engagement.Example: Modelling the cooling of a cup of tea with an exponential function fitted to measured data.
Technology-free paper skills: Some papers do not allow a calculator, so exact values, surds, fractions and standard trig values must be handled by hand.Example: Write cos(pi/6) = sqrt(3)/2 rather than 0.866 when an exact answer is required.