In this resource, we will explore linear functions and how they can be represented. This is interesting in its own right, and also gives us a tool to think about an important idea in calculus. We will take this further in Mapping a derivative.

Here are mapping diagrams representing functions of the form \(f(x)=ax+b\) for \(x\in\mathbb{R}\).

(In these diagrams, the left-hand numberline is the input and the right-hand one is the output, for example in Diagram 1, \(f(1)=3\).)

What do you notice about these diagrams and the functions they represent? (You might find it helpful to work out some or all of the functions first.)

What features do the diagrams and functions have in common? What are the differences between them?