Unit 2 · Topic 2: Functions

What IB Mathematics: Applications and Interpretation Unit 2 covers

Use this map to connect each assessed skill to the relationship or representation that makes it visible.

Functions: choose, calculate, interpret

Connect the published share to the unit model

This topic develops representation, model choice, parameters, domain, and interpretation. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Functions: choose, calculate, interpret” page.

Vocabulary in use: domain, range, intercept, gradient, quadratic model, asymptote, piecewise function, residual. Each term is applied on these pages and collected again in the glossary with a checked page pointer on the “Functions: choose, calculate, interpret” page.

A strong functions solution defines input and output, selects a defensible family, records parameters, and checks the useful domain before interpreting a prediction. The sensor, fountain, cooling, and travel situations in this topic are independently written illustrations.

The decision that organizes this unit

Define the system and choose the route before calculating

Define the input and output with units, then mark the useful domain before selecting a model.

Mechanism route and repair branches

Relationships to preserve

• A linear model has a constant gradient linking output change to input change.
• A quadratic turning point can identify a modeled maximum or minimum inside the stated domain.
• A residual is observed output minus predicted output, so its sign shows which value is larger.

Representations to read

• an equation with parameters and a stated domain
• a graph with intercepts, turning points, or asymptotes
• a table of observed values, predicted values, and residuals

Branches to reject

• extrapolating beyond the measured domain without warning
• reporting a turning point without checking that it lies in the stated interval
• reversing observed minus predicted when calculating a residual

How IB Mathematics: Applications and Interpretation assesses Topic 2: Functions

What a complete response must make visible

Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.

Choose a linear model from two measurements

Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.

Practice 1: Build a linear calibration model

Practice 2: Interpret a residual

Key terms for Unit 2: Topic 2: Functions

Models, uses, and boundaries

Interpret a Gradient

Gradient divides output change by input change; retain the measurement units in that quotient. Use this relation when the named quantities and conditions in the question match the calculation described here.

Build a Linear Model

Compute 0.05 V per °C, then use 1.2 = 0.05(10) + b to find b = 0.70; keep the estimate at 28°C inside the measured domain from 10°C to 40°C. Use this relation when the named quantities and conditions in the question match the calculation described here.

Locate a Quadratic Turning Point

Find t = −12 ÷ (−10) = 1.2 and substitute to obtain h(1.2) = 8.0; check that 1.2 seconds lies inside the stated time domain and gives the maximum. Use this relation when the named quantities and conditions in the question match the calculation described here.

Solve an Exponential Half-Life Model

Solve 36(0.82)^t = 18 to obtain t = ln(0.5) ÷ ln(0.82) = 3.49; verify that substituting 3.49 hours makes the initial 36°C difference approximately 18°C. Use this relation when the named quantities and conditions in the question match the calculation described here.

Calculate a Residual

Subtract predicted from observed to calculate 45.1 − 42.6 = +2.5 minutes; use the positive sign to check that the observed journey was longer than predicted. Use this relation when the named quantities and conditions in the question match the calculation described here.

IB Mathematics: Applications and Interpretation Unit 2 FAQ

Evidence workshop

Continue from the free model into complete practice

The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.

• Choose a linear model from two measurements
• Find a quadratic turning point
• Interpret an exponential half-life model
• Check and answer: Choose a linear model from two measurements · Interpret an exponential half-life model
• Check a model with a residual
• Check and answer: Find a quadratic turning point · Check a model with a residual

Related IB Mathematics: Applications and Interpretation unit guides

How to study IB Mathematics: Applications and Interpretation Unit 2

Start with the organizing decision

Before solving, restate the decision in operational terms: Choose a function family and representation that fit the measurements, domain, and behavior the question describes.. Your first written move should be to Define the input and output with units, then mark the useful domain before selecting a model..

Practice the same idea in several representations

Rotate through an equation with parameters and a stated domain, a graph with intercepts, turning points, or asymptotes, a table of observed values, predicted values, and residuals. Use each representation to practice Domain, Range, Inverse, Graphs and Their Key Features, Straight Lines and Linear Models, Quadratic and Cubic Models, Exponential Growth and Decay, Direct and Inverse Variation, Sinusoidal Models, Composite and Inverse Functions and Transformations of Graphs (HL), Further Models: Half-Life, Natural Logarithm, Phase-Shifted Sinusoidal, Logistic, and Piecewise (HL), Logarithmic Scales, Linearizing Data, Log-Log and Semi-Log Graphs (HL), and explain what stays invariant when the surface form changes.

Turn each error into a repair check

After every attempt, audit the response for extrapolating beyond the measured domain without warning; reporting a turning point without checking that it lies in the stated interval; reversing observed minus predicted when calculating a residual. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.

Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.

Where to go next

Return to the IB Mathematics: Applications and Interpretation to see every unit and plan your revision.