Combining Functions

Many ways problem

One way to sort

Can you sort the cards into categories by thinking about transformations?

• Which categories could you choose?

Stage 1

I started by selecting just one card to think about, card F.

• The two straight line graphs appear to be parallel to each other.
• If I am thinking about transformations then this tells me that I am looking at a translation of some kind.

In order to identify the nature of the translation I can select a point on the blue line and consider the translation needed to take it to an equivalent position on the pink line.

So it seems that a translation of $3$ units parallel to the $x$-axis or $3$ units parallel to the $y$-axis will transform the blue line into the pink line.

Of course, there are many other, equivalent, translations that would also transform the blue line into the pink line. For example the translation represented by the vector $\big( \begin{smallmatrix}1\\2\end{smallmatrix}\big)$ is illustrated below.

How might this be represented using function notation?

What difference does it make if I start instead with the pink line and consider how to transform it into the blue line?

What other transformations could transform the blue line into the pink line?

Stage 2

I now look at the remaining cards and select those that have something in common with card F. I select cards C, D and G as these looks like translations.

When I look at card D I notice that the size of the translation is hard to identify precisely. This card could still fit into the same category as those above, but I cannot easily describe the translation from blue to pink as I will have to estimate the horizontal element.

I do however notice that if I reflect the blue function in the $x$-axis then a horizontal translation of exactly $2$ units would transform it into the pink function.

Can you visualise this?

How would you use function notation to show a reflection like this? What about the combined transformation from the blue function to the pink function?

Stage 3

Now I select cards that also seem to show a reflection. Cards A, E and I perhaps.

It is worth bearing in mind that there is more than one possibility for the line of reflection. For example on Card A, reflecting the blue function in the vertical line that passes through the point at which the two lines cross would also work.

In this case I chose to reflect the blue function in the line $y=0$ because this is easier to describe, both in words and if I wanted to use function notation.

Card I seems to show a reflection and a translation of some sort but there is something else going on as well. If I simply reflect the blue function and translate it so that the maximum point of the blue function lines up with that of the pink function, the rest of the curve doesn’t match.

It seems as though the blue function must also be ‘stretched’, parallel to the $x$-axis, or indeed ‘squashed’ parallel to the $y$-axis, in order for it to match the pink function.

It is not immediately clear to me how much the blue function needs to be ‘stretched’ or ‘squashed’ by. In this first instance I will continue to sort the cards and will return to the detail of this transformation if and when I need to be more precise.

Stage 4

The two remaining cards are B and H. Both of these seem to show a ‘stretch’ of some kind in addition to a translation.

Can you describe in detail the transformation shown on card B? Can you use function notation to do this?

What makes the transformation on this card easier to describe than that on card I?

Summary

I sorted the cards in the following way:

• Cards F, C and G show two functions where one is a translation of the other.
• Cards A, D, and E show two functions where the blue function needs to be reflected and translated to become the pink function.
• Card I shows two functions where the blue function needs to be reflected, translated and ‘stretched’ in order to become the pink function.
• Cards B and H show two functions where the blue function needs to be translated and ‘stretched’ to become the pink function.

There are three cards that have not yet been considered: cards J, K and L. Card J shows just one function - why might I have chosen to place it in the category above?

Cards K and L are blank. It would seem sensible to use these to show transformations in categories that don’t currently have any examples. For instance, two functions that illustrate a reflection only or a stretch only or perhaps a combination of a reflection and a stretch.

Did you sort the cards differently?

What choices did you make and how do these differ from mine?