Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 2: Functions
📋 Learning Intentions
- Sketch graphs of functions by identifying key features: intercepts, turning points, asymptotes, and end behaviour.
- Use GDC technology to graph functions, find intercepts, locate maxima and minima, and determine asymptotes.
- Construct graphs of sums of functions \( y = f(x) + g(x) \) by adding ordinates point by point.
- Interpret graphical features in context — including domain restrictions, range, and behaviour near asymptotes.
- Identify the effect of transformations on the graph of a function and link graphical behaviour to the algebraic form.
- Apply graphical methods to solve equations and inequalities using intersection points and sign analysis.
Real-World Introduction
A marine biologist tracking a dolphin population over time might model growth with a logistic function, while simultaneously modelling seasonal food availability with a sinusoidal function. To understand the net effect on the population — whether it thrives or crashes — the biologist needs to graph the sum of these two functions and read off critical values. No single formula captures the full picture; the graph does.
Engineers designing suspension bridges use the graph of a catenary (a hyperbolic cosine function) to check that the cable shape meets safety tolerances. Economists graph revenue and cost functions on the same axes to locate the profit-maximising output visually before confirming algebraically. In every case, the graph is the primary tool for understanding behaviour.
🌍 Why This Matters In IB Mathematics AI, graphing functions is not a preliminary step — it is the method. The GDC replaces algebraic manipulation as the primary tool, and reading graphs accurately is tested directly in every paper. Mastery of graphical analysis underpins optimisation, modelling, and calculus throughout the course.
Core Concepts & Definitions
Before sketching any function, a clear understanding of what a "graph" represents is essential. Every point \((x, y)\) on a graph satisfies the equation \(y = f(x)\). The graph is the complete visual record of how the output changes as the input varies across the domain.
📖 Key Definition — Graph of a Function The graph of a function \( f \) is the set of all ordered pairs \( (x, f(x)) \) where \( x \) belongs to the domain of \( f \). Graphically, this is the curve (or collection of curves) traced in the coordinate plane as \( x \) varies over its domain.
The graph encodes all information about a function: where it is positive or negative, where it increases or decreases, its extreme values, and its long-run behaviour. Identifying these features systematically is the skill tested in IB exams.
Concept A — Key Graphical Features to Identify
Every sketch or GDC graph should be annotated with the following features where they exist. Examiners award marks for correctly labelled features, not just a curve shape.
- x-intercepts (roots/zeros): Points where \( f(x) = 0 \). Found by solving \( f(x) = 0 \) or using GDC zero-finder.
- y-intercept: The point \( (0, f(0)) \). Substitute \( x = 0 \) directly.
- Turning points (local maxima/minima): Points where the gradient changes sign. On GDC: use maximum/minimum finder.
- Asymptotes: Lines the graph approaches but does not reach.
- Vertical asymptote: \( x = a \) where \( f(x) \to