### Combining Functions

Many ways problem

## Things you might have noticed

Select any two of the cards showing, what appear to be, straight line graphs:

• What is the same and what is different about the transformations that take the blue line to the pink line?

Select any card and consider the following questions:

• Describe in words how you can transform the blue function into the pink function.
• How would you need to modify your description if instead you transformed the pink function into the blue function?

• Label one of the functions $f(x)$.
• Can you now label the other function in terms of $f(x)$?
• Does it matter which function you choose to be $f(x)$ in the first place?

Consider card J:

• Where is the second (pink) function on this card?
• Could you sketch another possible second (pink) function on one of the blank cards, K or L?
• If you label the function on card J as $f(x)$, can you label your suggested second function in terms of $f(x)$?

Select a card showing a transformation that involves a stretch:

• Is the stretch parallel to the $y$-axis or parallel to the $x$-axis? How do you know?

For most of the examples shown in this problem we might choose to represent them in function notation as stretches parallel to the $y$-axis.

• What difference does it make to this notation if we choose to think instead of the stretch being parallel to the $x$-axis?
• Can you sketch an example of a pair of functions related by a stretch for which it is sensible to think of it as being parallel to the $x$-axis?
• If you label one of your functions $f(x)$, can you label the other in terms of $f(x)$?