Other multiples

In the Warm-up, we looked at mapping diagrams for functions of the form \(f(x)=3x+b\).

We also used certain conventions when we drew those mapping diagrams, both to allow for fair comparison and to restrict the number of things we varied between the diagrams:

  • the numberlines are a fixed distance apart
  • the scales on the two numberlines are equal (so \(\quantity{1}{cm}\) represents the same difference on each numberline)
  • the centre arrow is horizontal
  • the arrow start points are a fixed distance apart

What would the mapping diagrams of other functions of the form \(f(x)=ax+b\) look like?

Using the same conventions, draw mapping diagrams for these functions. You can choose what your centre input value will be and what scale you will use.

  • \(f(x)=2x+1\)
  • \(f(x)=\frac{1}{2}x-1\)
  • \(f(x)=x\)
  • \(f(x)=-x\)
  • \(f(x)=-2x+2\)
  • another function of your choosing

What do you notice?

You can download blank mapping diagrams or small blank mapping diagrams for this.

Extending the arrows

We could extend the arrows into straight lines, as shown here for \(f(x)=3x\), with just one arrow extended so far:

two parallel numberlines, left one centred on zero, right one centred on zero, both going up in steps of one, five arrows from left line to right line, starting at minus 1, minus a half, zero, a half and one, with the arrow from a half to 1.5 extended in both directions beyond the numberlines
  • What patterns do you notice when you extend the arrows for different linear functions?

  • What does this tell you about the nature of linear functions?

You can explore this further using the first of the interactivities.

A chance to reflect

How does this mapping diagram representation of a linear function \(f(x)=ax+b\) relate to the representation as the graph of \(y=ax+b\)?

  • Which features of the function are clearer in one representation rather than the other?

  • How are the features in one representation related to features in the other?

We have used several conventions in drawing our mapping diagrams. Many of them may seem sensible, but it is interesting to ask what would happen if we changed them. Possibly one of the most surprising conventions that we have used is to always make our centre arrow horizontal, by lining up the right-hand numberline appropriately.

How would things change if we did not require this? For example, what would \(f(x)=3x+1\) look like if we centred both numberlines at \(0\)? You can also explore this using the second of the interactivities.

You may also like to ponder what would happen if we changed our other conventions.