Knowledge organisers

Functions: IB Maths AI SL 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

Linear model: y = mx + c: y changes by the same amount m for each step of 1 in x.
Exponential model: y = k aˣ + c: y is multiplied by the same factor a for each step of 1 in x.
Asymptote: A line the graph gets closer and closer to but does not reach, such as y = c for an exponential model.
Domain of a model: The input values that make sense in the context, such as t ≥ 0 for time.

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.

Gradient; forms of a lineIn the formula booklet	\(m=\frac{y_2-y_1}{x_2-x_1},\) \(y=mx+c,\) \(ax+by+d=0,\) \(y-y_1=m(x-x_1)\)
Parallel; perpendicular	\(m_1=m_2;\) \(m_1m_2=-1\)
Quadratic: axis of symmetryIn the formula booklet	\(x=-\frac{b}{2a}\)
Linear; quadratic models	\(f(x)=mx+c;\) \(f(x)=ax^2+bx+c\)
Exponential models	\(f(x)=ka^x+c,\) \(f(x)=ka^{-x}+c,\) \(f(x)=ke^{rx}+c\)
Direct / inverse variation	\(f(x)=ax^n,\) \(n\in\mathbb Z\)
Cubic model	\(f(x)=ax^3+bx^2+cx+d\)
Sinusoidal model (degrees)	\(f(x)=a\sin(bx)+d:\) \(\text{amplitude }\|a\|,\) \(\text{period }\tfrac{360^\circ}{b}\)
Inverse function: swap \(x\) and \(y\); graph reflects in \(y=x\); domain of \(f^{-1}\) = range of \(f\)
Horizontal asymptote of \(ka^x+c\)	\(y=c\)

Worked example

A population is modelled by P = 500 × 1.04ᵗ, where t is in years. Find P when t = 10, and the first whole year in which P is more than 1000.

P(10) = 500 × 1.04¹⁰ = 740.1…
500 × 1.04ᵗ = 1000 gives 1.04ᵗ = 2, so t = ln 2 / ln 1.04 = 17.7…

Answer: P(10) ≈ 740, and P first passes 1000 when t = 18

Common mistakes

Exponential and log models: e⁰ = 1, the equation vs the inequality, reading the model
Interpreting in context: parameters, derivatives and justifications
Mixing up the perpendicular gradient rule (m₁·m₂ = −1)
Reading the zeros of a(x − p)(x − q) as p and −q (or −p and −q) instead of p and q

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

You should be able to…

Use function notation, find the domain and range of a function from its equation or graph, and interpret them in a modelling context.
Model growth and decay with functions such as f(x) = ka^x + c, identify the asymptote and interpret the parameters in context.
Understand logarithms as the inverse of exponentials, evaluate log₁₀ and ln with technology and solve equations such as 5ˣ = 40.
Use technology to fit and analyse cubic models, finding maxima, minima and intersections in context.
Use exponential models to find half-life and doubling time and solve for time with technology.
Answer long-response functions questions that combine choosing, fitting and using models, linking each part's answer to the context.

The printable sheet

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

Revise it next

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