What is a modulus function?

The modulus refers to the magnitude of a number, regardless of its sign. It is denoted by two straight lines either side of a function or variable – |x| –  which represents the non-negative value of x.

A simple way to think of it is as representing the distance from 0 on the number line. As a result, it’s sometimes referred to as the absolute value of a number.

The modulus function is used to calculate the absolute value or magnitude of a number, and will give the same answer irrespective of whether the number is positive or negative.

Another important property to understand is the domain and range of the modulus function. Its domain is all real numbers, as any real value can be substituted into |x|. Its range is [0, ∞) because the output of a modulus function is always zero or a positive value, regardless of whether the original input is positive or negative.

What do modulus function graphs look like?

When answering exam questions, your students need to understand the relationship between the functions they’ve been given. It’s easy to become absorbed in the algebra and lose sight of what the graphs are showing, which can lead to incorrect answers. Sketching or graphing the functions makes it much easier to identify where they intersect and how many solutions there are.

Modulus functions often have a characteristic V-shaped graph. For linear functions, this consists of two straight lines that meet at a vertex. This shape is created because any part of the graph below the x-axis is reflected above it, ensuring the output is always zero or positive.

The V shape provides a simple way for students to recognise a modulus function, and many A-level questions require them to sketch or interpret these graphs. However, your students should also be prepared for non-linear modulus functions, such as |sin x|, where the same transformation applies to a curve rather than a straight line.

Step-by-step guide to graphing modulus functions

 
When sketching a modulus function graph by hand, your students will usually begin by creating a table of values before plotting the corresponding points. A graphing calculator can speed up this process, making it easier to visualise the graph and investigate its key features.
 
Graphing modulus functions on the fx-CG100
 
The fx-CG100 includes powerful graphing tools that allow students to plot and analyse modulus functions, making it particularly useful for exploring transformations and checking sketch graphs in A-level maths.
 
To plot a modulus function on the fx-CG100:
• Press the Home key and select Graph & Table.
• On the function entry screen (Y1), press Catalog.
• Select Numeric Calc, then choose Absolute Value.
• Enter the expression inside the modulus bars, for example x − 5, then press OK or EXE.
• Either press the right tab key or select Draw Graph to display the graph.
 
Once the graph has been plotted, press Tools and select Graph Solve to investigate features such as the vertex, roots and y-intercept. If the graph isn’t displayed clearly, open View Window from the Tools menu to adjust the minimum and maximum values for the x- and y-axes. The View Window pop-up also appears by default when you draw a graph.
 
For a more in-depth exploration, you can watch our full video guide to plotting and analysing modulus functions on the fx-CG100.

How modulus functions appear in A-level maths exams

Your students are likely to encounter the modulus function in a variety of A-level exam questions, from sketching graphs and identifying transformations to solving modulus equations and stating the domain and range of a function.

For example, one question in a past AQA A-level maths paper asked students to sketch the graph of y = |3x − 6| before finding its points of intersection with y = |2x|.

The accompanying examiner report noted that while many students recognised the characteristic V-shaped graph, some lost marks by placing the apex incorrectly. It also noted that while most students found at least one solution to the resulting equation, only around half went on to find both solutions – a useful reminder for students to check for additional solutions when solving modulus equations.

Encouraging your students to sketch graphs alongside their calculations, and check their work using a calculator where appropriate, can help reinforce these key exam skills.