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IB Mathematics Analysis and Approaches SL Formula Reference cheat sheet - grade 11-12

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Math Grade 11-12

IB Mathematics Analysis and Approaches SL Formula Reference Cheat Sheet

A printable reference covering functions, trigonometry, calculus, probability, statistics, vectors, and sequences for grades 11-12.

TopicsFunctionsTrigonometryDerivativesIntegralsProbabilityVectorsData AnalysisStudy Guide
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This IB Mathematics Analysis and Approaches SL formula reference brings together the main results students use across algebra, functions, trigonometry, calculus, statistics, probability, vectors, and sequences. It is designed for quick review before practice sets, topic tests, and final exam revision. Students need this cheat sheet to connect formulas with when they are used, rather than searching through notes during problem solving.

Key Facts

  • The quadratic formula solves ax2+bx+c=0ax^2 + bx + c = 0 using x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a≠0a \neq 0.
  • For an arithmetic sequence, un=u1+(n−1)du_n = u_1 + (n - 1)d and Sn=n2(2u1+(n−1)d)S_n = \frac{n}{2}(2u_1 + (n - 1)d).
  • For a geometric sequence, un=u1rn−1u_n = u_1 r^{n - 1} and Sn=u1(1−rn)1−rS_n = \frac{u_1(1 - r^n)}{1 - r} when r≠1r \neq 1.
  • The sine rule is asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, and the cosine rule is c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C.
  • The derivative rule for powers is ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n - 1}, and the tangent gradient at x=ax = a is f′(a)f'(a).
  • The definite integral ∫abf(x) dx\int_a^b f(x)\,dx gives signed area between y=f(x)y = f(x) and the xx-axis from x=ax = a to x=bx = b.
  • For a binomial distribution, X∼B(n,p)X \sim B(n,p) has mean E(X)=npE(X) = np and variance Var⁡(X)=np(1−p)\operatorname{Var}(X) = np(1-p).
  • The scalar product of two vectors is a⋅b=∣a∣∣b∣cos⁡θ\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos \theta, so perpendicular vectors satisfy a⋅b=0\mathbf{a}\cdot\mathbf{b} = 0.

Vocabulary

Function
A function is a rule that assigns each input xx in its domain exactly one output f(x)f(x).
Derivative
A derivative f′(x)f'(x) measures the instantaneous rate of change or gradient of the curve y=f(x)y = f(x).
Integral
An integral accumulates quantities and can represent signed area under a curve, written as ∫f(x) dx\int f(x)\,dx or ∫abf(x) dx\int_a^b f(x)\,dx.
Correlation coefficient
The correlation coefficient rr measures the strength and direction of a linear relationship, with −1≤r≤1-1 \leq r \leq 1.
Vector
A vector is a quantity with magnitude and direction, often written in component form such as (xy)\begin{pmatrix} x \\ y \end{pmatrix}.
Binomial distribution
A binomial distribution models the number of successes in nn independent trials when each trial has success probability pp.

Common Mistakes to Avoid

  • Using the quadratic formula with the wrong signs is wrong because x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} requires the opposite of bb and the full denominator 2a2a.
  • Forgetting degree and radian mode is wrong because trigonometric answers depend on angle units, especially in calculus where radians are required for standard derivative rules.
  • Treating ∫abf(x) dx\int_a^b f(x)\,dx as always positive area is wrong because a definite integral gives signed area, so regions below the xx-axis contribute negative values.
  • Using Sn=u1(1−rn)1−rS_n = \frac{u_1(1-r^n)}{1-r} when r=1r = 1 is wrong because the formula divides by zero, and the sum is instead Sn=nu1S_n = nu_1.
  • Assuming correlation proves causation is wrong because a strong value of rr only shows linear association, not that one variable directly causes the other.

Practice Questions

  1. 1 Solve 2x2−5x−3=02x^2 - 5x - 3 = 0 using x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  2. 2 Find the first 1010 terms sum of an arithmetic sequence with u1=7u_1 = 7 and d=4d = 4.
  3. 3 A random variable has X∼B(20,0.35)X \sim B(20,0.35). Find E(X)E(X) and Var⁡(X)\operatorname{Var}(X).
  4. 4 Explain why a turning point of f(x)f(x) occurs where f′(x)=0f'(x) = 0, and why this condition alone does not always guarantee a maximum or minimum.

Understanding IB Mathematics Analysis and Approaches SL Formula Reference

A formula reference is most useful when it helps you make decisions. Before choosing a formula, identify the mathematical object in the question. A table of values may suggest a sequence, a graph may show a function, and a diagram may contain a triangle or vectors.

Then identify what is known and what must be found. This prevents a common mistake, which is using a familiar formula because it contains the numbers given.

Units, labels, domain restrictions, and the wording of the problem often tell you more than the raw numbers do. In exam work, write down the values you substitute and keep enough working for someone else to follow your reasoning.

Functions link several representations of the same relationship. You should be able to move between an equation, a graph, a table, and a real situation. For example, a model for cooling, population growth, or loan interest may be useful only over a stated interval.

Outside that interval, the model can give unrealistic results. Pay close attention to transformations. A change inside a function affects horizontal position or scale, while a change outside affects vertical position or scale.

In calculus, derivatives describe instantaneous change. This can represent speed from position or marginal cost from a cost model. Integrals accumulate small changes.

Their signed nature matters because parts below the horizontal axis count negatively. For total area, split the interval where the graph crosses the axis and use positive areas.

Trigonometry and vectors are both tools for describing direction and shape. In a non-right-angled triangle, first match each side with its opposite angle. This matching is essential when using relationships involving sines.

The cosine relationship is especially useful when two sides and their included angle are known, or when all three sides are known. A calculated angle can sometimes have two possible values, so check whether the diagram and angle sum permit both. Vectors appear in displacement, forces, navigation, and computer graphics.

Their components describe movement in separate directions. The scalar product is useful because it tests perpendicularity and helps find angles. A result close to zero may reflect rounding, so retain calculator accuracy until the final answer.

Probability and statistics require interpretation, not only calculation. A binomial model applies when there is a fixed number of trials, each trial has two outcomes, the probability stays constant, and trials are independent. Real examples such as product checks or survey responses may fail one of these conditions.

State assumptions clearly when a model is imperfect. Mean describes the long-run average, while variance describes spread around that average. In data analysis, a correlation does not prove that one variable causes the other.

Sequences provide models for repeated change. Arithmetic patterns add a constant amount, while geometric patterns multiply by a constant factor.

This distinction matters in savings plans, depreciation, bacteria growth, and repeated discounts. Always test a formula with the first few terms, since an incorrect starting index can change every later result.