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.

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