Knowledge organisers

Functions: IB Maths AI HL knowledge organiser

Everything to know about functions on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Logistic model: Growth that starts nearly exponential and levels off at a carrying capacity L.
Sinusoidal model: y = a sin(b(x − c)) + d for repeating patterns: amplitude a, period 2π/b, principal axis y = d.
Linearising data: Taking logs to turn a power or exponential relationship into a straight line.
Composite transformation: Several transformations applied one after another; the order can change the result.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

LinesIn the formula booklet	\(m=\frac{y_2-y_1}{x_2-x_1},\) \(y=mx+c,\) \(ax+by+d=0,\) \(y-y_1=m(x-x_1)\)
Perpendicular	\(m_1m_2=-1\)
Quadratic: axis of symmetryIn the formula booklet	\(x=-\frac b{2a}\)
Models	\(f(x)=ka^x+c,\) \(ke^{rx}+c;\) \(ax^n;\) \(a\sin(bx)+d\ (\text{period }\tfrac{360^\circ}b)\)
Logistic model	\(f(x)=\frac{L}{1+Ce^{-kx}}\) \((L,C,k>0)\)
Linearising	\(y=ax^b\Rightarrow\ln y=\ln a+b\ln x;\) \(y=ka^x\Rightarrow\ln y=\ln k+x\ln a\)
Composite; inverse	\((f\circ g)(x)=f(g(x));\) \(f\big(f^{-1}(x)\big)=x\)
Transformations: \(f(x)+b\) up; \(f(x-a)\) right; \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\)

Worked example

Data fit y = a xᵇ. A graph of ln y against ln x is the straight line ln y = 0.7 + 1.5 ln x. Find a and b.

ln y = ln a + b ln x
So b = 1.5 and ln a = 0.7

Answer: a = e^0.7 = 2.01 (3 s.f.) and b = 1.5

Common mistakes

HL algebra: product rule, brackets, logs and exact solutions
Sketches from the GDC: domain, endpoints, turning points, angle mode
Mixing up the perpendicular gradient rule (m₁·m₂ = −1)
Misreading the y-intercept from an equation not in y = mx + c form

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find gradients and intercepts and write equations of lines in the forms y = mx + c, ax + by + d = 0 and y − y₁ = m(x − x₁).
Model situations with cubic functions and power models y = axⁿ, including direct and inverse variation, finding parameters from data.
Fit quadratic models with technology and use the vertex and zeros to answer projectile and optimisation questions.
Model growth and decay with exponential functions, identify the asymptote and interpret the parameters in context.
Interpret the parameters of a logistic model, including the carrying capacity, and sketch logistic curves for given values.
Fit a logistic model with technology and interpret L, k and x₀ in context.

The printable sheet

Functions knowledge organiser for IB Maths AI HL: one A4 page of key definitions, formulas, a worked example and common mistakes — Functions knowledge organiser (IB Maths AI HL), A4. Download the PDF.

Revise it next

Other IB Maths AI HL topics: Number and algebra · Geometry and trigonometry · Statistics and probability · Calculus · All IB Maths AI HL organisers