### Combining Functions

Many ways problem

## Problem

There are lots of questions that we could ask about the functions shown on each of the cards in this problem. Here we ask a few of them and invite you to explore others.

When thinking about any of these questions you might find it helpful to use graphing software such as Desmos to check your answers and to explore other possibilities.

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 $f(x)$?

The quadratic curves shown on card I are also featured in Name that graph.

You might like to think about how transformations can help you to tackle this and similar problems.