Key Idea: Topic 2.4 is about describing the shape and behaviour of a graph: where it rises, where it falls, where it peaks or troughs, and how it behaves at its extremes. A local maximum is a peak — the function is higher there than at nearby points. A local minimum is a trough. An asymptote is a line the graph approaches but never reaches.
✅ Graph features and what they tell you
- Local maximum
- A point where the function value is higher than all nearby points. The graph goes up to it, then comes back down. Example: the peak of a hill shape.
- Local minimum
- A point where the function value is lower than all nearby points. The graph comes down to it, then rises again. Example: the bottom of a valley shape.
- Increasing interval
- A range of x where the function's output gets larger as x increases. Written as a < x < b or using interval notation (a, b).
- Decreasing interval
- A range of x where the function's output gets smaller as x increases. The graph slopes downward.
- Vertical asymptote
- A vertical line x = a that the graph approaches but never crosses. Typically occurs where the denominator is zero. Example: f(x) = 1/(x−2) has a vertical asymptote at x = 2.
- Horizontal asymptote
- A horizontal line y = k that the graph approaches as x → ±∞. Common in exponential decay: f(x) = 2⁻ˣ approaches y = 0 as x → ∞.
Example: Describe f(x) = x³ − 3x: GDC shows: local max at (−1, 2), local min at (1, −2). Increasing on: x < −1 and x > 1. Decreasing on: −1 < x < 1. Exponential: f(x) = 3⁻ˣ + 1: As x → ∞, 3⁻ˣ → 0, so f(x) → 1. Horizontal asymptote: y = 1 (the graph gets closer and closer but never reaches 1).
When describing increasing/decreasing intervals, use the x-values (not y-values) for the interval. A function can have more than one local maximum or minimum — use the GDC to find all of them within the domain shown.
Paper 2 (GDC allowed): Use the GDC 'maximum' and 'minimum' functions to find turning point coordinates. Then state the intervals of increase/decrease around those points. Paper 1: You may be given a sketch and asked to describe features in words. Use precise vocabulary: 'local maximum at x = 2, f(2) = 5', not just 'it goes up then down'.
IB-style question [6 marks]
The number of active users of an app, N thousand, t months after a redesign is modelled by N(t) = 400 · 0.7ᵗ + 60, for t ≥ 0. (a) Write down the number of active users at the moment of the redesign (t = 0). (b) State whether N is increasing or decreasing, and explain why. (c) Write down the equation of the horizontal asymptote and interpret it in context.
Step by step:
(a) Substitute t = 0. Since 0.7⁰ = 1, the model gives the starting value.
(b) The base 0.7 is between 0 and 1, so the term 400 · 0.7ᵗ shrinks as t grows. The output falls over time, so N is decreasing.
(c) As t → ∞, the term 400 · 0.7ᵗ → 0, leaving the constant.
Interpret it: in the long run the user base levels off at about 60 thousand active users — the model never drops below this floor.
(a) 460 thousand users. (b) Decreasing — the base 0.7 < 1 makes the curve fall. (c) N = 60; the user base settles at about 60 thousand in the long run.