Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 2: Functions | Topic 2.4
📋 Learning Intentions
- Identify local maxima, local minima, and global extrema of a function from its graph or equation using the GDC.
- Find the coordinates of x-intercepts (zeros) and y-intercepts of a function algebraically and using the GDC.
- Describe the behaviour of a function near vertical and horizontal asymptotes, using correct limit notation where required.
- Determine the intersection points of two functions using the GDC and interpret these points in context.
- Sketch a graph showing all key features — intercepts, turning points, and asymptotes — labelled with correct coordinates.
- Interpret key features of a graph in a real-world modelling context, explaining what each feature represents.
Real-World Introduction
A pharmaceutical company models the concentration of a drug in the bloodstream over time using a function. The concentration rises to a peak — a maximum — before declining as the body metabolises the drug. Knowing exactly when and at what level this peak occurs determines the safe dosing interval. This peak is a local maximum of the concentration–time graph.
Similarly, an engineer designing a suspension bridge needs to know the lowest point of each cable — a minimum — to ensure clearance above the road deck. In economics, a firm identifies the output level that minimises average cost, which corresponds to the minimum of a cost function. In every case, the critical information is encoded in the key features of a graph.
🌍 Why This Matters Every mathematical model — whether a population growth curve, a profit function, or a physics trajectory — communicates its most important information through its key features. Reading these features accurately from a graph or GDC output is one of the most frequently tested skills across both papers of the IB Math AI exam.
Core Concepts & Definitions
When a function increases and then decreases, it passes through a turning point — a point where the gradient changes from positive to negative. This turning point is a local maximum. Conversely, when a function decreases and then increases, it passes through a local minimum. These turning points are collectively called local extrema.
📖 Key Definition — Local Maximum and Local Minimum
- A local maximum is a point \((a, f(a))\) where \(f(a) \geq f(x)\) for all \(x\) in some open interval around \(a\). The function value is greater than all nearby values.
- A local minimum is a point \((a, f(a))\) where \(f(a) \leq f(x)\) for all \(x\) in some open interval around \(a\). The function value is less than all nearby values.
- A global (absolute) maximum is the largest function value over the entire domain. A global minimum is the smallest.
- Local extrema are also called turning points . At a turning point, the gradient of the curve equals zero: \(f'(x) = 0\).
On a smooth curve, a local