Package of problems

## Problem

### 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:

• 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?