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?

For this question we could consider any two of cards A, F and H. Let’s say we choose to focus on cards F and H.

Card F shows the lines $y=-(x-1)$ in blue and $y=2-x$ in pink
Card F
Card H shows the lines $y=2(x-1)$ in blue and $y=8x$ in pink
Card H

In One way to sort the two, apparently parallel, lines on card F are described as a translation of each other. If we were to label the blue line \(f(x)\), we might label the pink line \(f(x-3)\) or perhaps \(f(x)+3\).

How would you describe in words the two translations that these represent?

The blue line on card H can also be transformed into the pink line in more than one way. In One way to sort it is categorised as a combination of a translation and a stretch and if we were to label the blue line \(f(x)\) we might therefore label the pink line \(2(f(x)+2)\).

Are the brackets important here? Can you describe in words the precise transformation that this represents?

For both cards you might thought about using a reflection to transform the blue line into the pink line.

  • Could you draw the line of reflection on each card?
  • Why might we have chosen to describe alternative transformations in the text above?

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?

Let’s select card D.

Card D shows two sine curves with an amplitude of 3. The blue curve has a minimum at $x=2$, the pink curve has a maximum at $x=0$
Card D

When looking at this card it is particularly helpful to focus on the pair of integer coordinates, \((2,0)\) (blue curve) and \((0,0)\) (pink curve), and to think about how you would transform the blue curve so that the first point maps onto the second. In One way to sort this is discussed in more detail and if we were to label the blue curve \(f(x)\) we might choose to label the pink curve \(-f(x+2)\).

If instead we chose to label the pink curve \(f(x)\), how would the function notation for the blue curve need to be modified from the previous answer (\(-f(x+2)\))?

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 \(f(x)\)?
Card J shows an unusual trigonometric curve (blue)
Card J

Firstly we need to recognise that just because we can’t see a pink function on this card, it doesn’t mean there isn’t one! The pink function could exist in a region not shown in the existing image or it could be ‘hidden’ underneath the blue curve.

An example is illustrated below:

Blank card K with the blue curve from card J plus a translation of it vertically by 8 units
  • Can you describe in words a translation that would transform the blue curve into the pink curve?
  • Can you suggest another translation that might transform the blue curve into the pink curve?
  • If we label the blue curve \(f(x)\) how could you label the pink curve in terms of \(f(x)\) in each case?

If the original blue function is \(f(x)\), then \(f(x+2n\pi)+c\) describes some possible pink curves.

What transformations is this representing?

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 \(f(x)\)?
In the first question we discussed card H and described the transformation of the blue line as a translation and a stretch. The function notation used to represent this was \(2(f(x)+2)\), (if the blue curve is labelled \(f(x)\)).
  • How did we know that the stretch factor was 2?
  • What’s the difference between calling the pink line \(2(f(x)+2)\) and calling it \(f(2x)+2\)?

Card B also seems to show two curves related by a stretch.

Card B shows two quadratic curves, $y=x^2$ in blue and $y=3(x+3)^2$
Card B

Here, if the blue curve is labelled \(f(x)\), the pink curve could be labelled \(3f(x+3)\) suggesting a stretch of factor 3 parallel to the \(y\)-axis.

Can this example be rewritten to represent a stretch parallel to the \(x\)-axis? What would the stretch factor be in this case?

It might be helpful and informative to use graphing software such as Desmos to play about with the function notation for this question.