More in-depth exploration of graph transformation

I have already went through in class/lecture on the individual effects of the 3 basic transformations (translation, scaling and reflection) as well as the procedures for drawing y = \f(x)\, y = f(\x\), y = 1/f(x) and y^2 = f(x). So I won’t repeat them here.

Likewise, the AMMA rule has been covered with several examples, thus I won’t elaborate further. But remember the rule is coined by PJC and hence, not an standardised rule to quote during tests and exams (especially the A-levels). Use the rule as a guide to prioritise which transformation to do first. Write down these successive transformations instead!

Instead, let’s explore further what happens if we combine the various forms of transformations together. In other words, we don’t restrict ourselves to just combining the 3 simple transformation types.

There’s no quick and fast rule, but generally, going by the x changes before y changes is pretty useful, while working inside out as shown by the examples below:

y = 5f(\x\)
Draw the y = f(\x\) graph first, then scale it parallel to y-axis by factor 5.

y = \2f(x)\ – 5
Scale parallel to y-axis by factor 2 first, then apply the \f( )\ procedure to the intermediate graph (i.e. reflect the bottom parts about x-axis). Finally, move the graph down by 5 units.

y^2 = 3f(4x – 1)
To draw the y^2 graph, first you must have the y graph. Hence, first apply AMMA rule on 3f(4x – 1), then apply the y^2 procedure on the intermediate graph.

y = 1/f(x – 3)
To draw the reciprocal graph, first you need the f(x – 3) graph. Hence, first translate y = f(x) to the right by 3 units. Then, apply the reciprocal curve procedure to this intermediate graph.

y = –2f(\x + 1\)
This is a tricky one. First, apply the f(\ \) procedure. Then, translate this intermediate graph left by 1 unit. Then, scale it parallel to y-axis by factor 2. Finally, reflect about x-axis.

Hopefully, the above examples illustrated enough about the generic sequence to follow when your combined transformations involve more than just translation, scaling and reflection.