Introduction to Key Features of Graphs
When analysing a function, we rarely look at it in isolation , we want to extract meaningful information from its graph. The key features of a graph are the specific characteristics that tell us how a function behaves: where it reaches its highest or lowest points, where it crosses the axes, whether it has any symmetry, whether it approaches certain lines without ever touching them, and how its output values change across its domain.
In IB Maths AA SL, you are expected to identify and interpret these features both analytically (by hand) and using technology (graphing calculators or software such as GeoGebra or Desmos). Technology becomes especially powerful when dealing with functions that are difficult to analyse algebraically.
The key features you need to be confident with are:
- Maximum and minimum values (extrema / turning points)
- x-intercepts (roots/zeros) and y-intercepts
- Domain and range
- Increasing and decreasing intervals
- Symmetry
- Vertical and horizontal asymptotes
- Intersection points of two graphs
Maximum and Minimum Values
Local (Relative) Maximum/Minimum: A point on a graph where the function value is higher (or lower) than all nearby points. The function may reach higher (or lower) values elsewhere on its domain.
Global (Absolute) Maximum/Minimum: The single highest (or lowest) function value across the entire domain. Not every function has one , for example, a line extending to infinity has neither.
Turning Point: A point where the graph changes from increasing to decreasing (a maximum turning point) or from decreasing to increasing (a minimum turning point). The IB uses this term extensively in mark schemes , it is synonymous with a local extremum for smooth curves.
Maximum and minimum points are collectively called extrema or turning points. They are critical in optimisation problems, where you need to find the best (largest or smallest) value of some quantity.
On a graph, a maximum turning point appears as a peak and a minimum turning point appears as a trough.
Consider .
Using a graphing calculator:
- Enter the function and view the graph , you can see a downward-opening parabola.
- Use the maximum feature (often found under Calc or Analysis menus).
- The calculator identifies the maximum point at (2, 9).
This tells us the highest y-value the function reaches is 9, occurring at . Because this is a parabola opening downward, this local maximum is also the global maximum.
Always determine whether an extremum is local or global. A function like has two local extrema (a local max and a local min), but no global maximum or minimum because as , and as , , so there is no single highest or lowest value across the entire domain.
