Package of problems

Things you might have noticed

Other multiples

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.

What do you notice?

Here are mapping diagrams of our original $f(x)=3x$ together with the other functions given. We have decided to centre each of the input numberlines on zero and use a scale of one unit per tick mark; as we discovered in the Warm-up, this makes no difference to the mapping diagram, only to the labelling.

 $f(x)=3x$ $f(x)=2x+1$ $f(x)=\frac{1}{2}x-1$ $f(x)=x$ $f(x)=-x$ $f(x)=-2x+2$

Clearly, the multiplier $a$ in $f(x)=ax+b$ makes a significant difference to the mapping diagrams:

• $f(x)=2x$ is less “spread out” than $f(x)=3x$, and $f(x)=\frac{1}{2}x$ results in the arrows becoming closer together on the output numberline.
• $f(x)=x$ consists of parallel arrows.
• $f(x)=-x$ and $f(x)=-2x$ both lead to arrows which cross themselves. (Incidentally, this is one way to see that multiplying an inequality by a negative number reverses the inequality.)

As we saw in the Warm-up, though, the mapping diagram for $f(x)=ax+b$ remains the same if we change the “$+b$” part (as long as we stick to our conventions).

What would $f(x)=0x$ look like?

Extending the arrows

We could extend the arrows into straight lines.

• 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.

Here are the mapping diagrams for the same six functions as above, with extended lines.

 $f(x)=3x$ $f(x)=2x+1$ $f(x)=\frac{1}{2}x-1$ $f(x)=x$ $f(x)=-x$ $f(x)=-2x+2$

The extended lines all meet at a single point (except for the function $f(x)=x$), which we could think of as a centre of enlargement or a focal point. This means that we can describe a linear function as a scaling of the left-hand numberline into the right-hand one.

What is the scale factor for each of the functions considered above (such as $f(x)=3x$, $f(x)=2x+1$, $f(x)=\frac{1}{2}x-1$ and so on)? Can you suggest a general rule?

Some further questions:

• Can you prove that every linear function behaves in this way: that it is a scaling of the left-hand numberline into the right-hand one?

• Can you find a rule for the location of the focal point for $f(x)=ax+b$?

Taking it further: changing the conventions

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.

Here are the mapping diagrams for $f(x)=3x+1$ with the left one centred as we have been doing, and the right one with both numberlines centred at $0$. For each one, we have drawn five arrows starting at $-1$, $-\frac{1}{2}$, $0$, $\frac{1}{2}$ and $1$.

 $f(x)=3x+1$ $f(x)=3x+1$

Some observations we might make:

• The second diagram looks fairly similar to the first in many ways.

• The focal point is the same distance away from the left-hand numberline in both diagrams.

• The second diagram looks just like our earlier diagrams if we imagine the left-hand numberline being centred on $-\frac{1}{2}$.

• The second diagram looks like the first, just with the right-hand numberline moved up by $1$ unit and everything else pulled along with it. (The technical name for this transformation is a shear.)

Can you explain these observations?