Successive Transformations

Year 12 returned from their AS study leave a couple of weeks ago and we have started to teach some C3 topics; Successive Transformations, Functions and 'e & ln'. This is new territory for me as a teacher so I've been playing around with making a few new sketches in GeoGebra to tackle these.

The topic of graph transformations contains quite a few concepts for students to get their heads around and things start to get even more complex when combinations of transformations are applied in succession.

I made the following sketch to help.

You could use this sketch at the board or better get students using it. Use the input box to enter a function. Use one of the four options at the bottom of the right-hand screen and the sliders to select a new function based on the original (use a = -1 for a reflection). Now consider the two possible sequences of transformations that you could apply to reach the new function; use the check boxes to reveal the graphs of the successive transformations; do they both work?

Consider the order that transformations in succession are carried out in:

When is the order important, when is it not? 

In which cases do you need to translate then stretch and when should you stretch and then translate? 

Test out your ideas on some different functions.

The key points that came out in our discussions were;

1. If you consider transformations as belonging to one of the following two families:

The order must be carefully considered when transformations are combined from the same family "in-breeding can lead to complications!".

2. When transformations combined are from separate families the order isn't important as they affect the x and y values independently.

3. Just as transformations from the blue family are highly intuitive, so is the order that transformations should be carried out in i.e. $y = 2f(x) + 3$ is a stretch of a scale factor 2 parallel to the y axis followed by a translation of 3 units upwards. On the other-hand as transformations in the green  family do not execute as you may first expect, combinations from this family are also not so intuitive i.e.$y = f(2x+3)$ is a translation 3 units to the left followed by a stretch with a scale factor 0.5 parallel to the x-axis; so the 'expected' order is reversed.

The sketch allows students to examine why these point are so, algebraically as well as graphically.