Transformations of functions at GCSE - enlargement vs stretch vs translation (ordering)

Question

1. What happens when you have a function f(x) and apply a transformation af(x/a)? Is this just equivalent to an enlargement of f(x) with scale factor a with centre of enlargement (0,0)? (Because we have stretch in each of x and y directions by the same scale factor - this is equivalent to an enlargement with centre the origin I'm assuming...)

I can't seem to find this described anywhere in books at GCSE or even online.

2. Also, to confirm, if we have another function f(x) and apply the transformation af(x)+b, is it first a stretch of scale factor a in the y direction and then a translation by vector (0 b)? And not vice versa? Similar to how we apply BIDMAS with multiplication first?

3. Finally, if we had the transformation af(bx), then it doesn't matter which order we say stretch in y -direction by scale factor a and stretch in x- direction by scale factor 1/b?

If there is a whole list of such transformations which summarises these, it would be great if you can point me to these:) As only the "standard" ones seem to exist from wherever I've been looking.

Thanks very much in advance!

Answer 1

Welcome to Math Stackexchange. It’s very good that you’re thinking about these generalisations.

1. You are correct.
2. Your first assumption is correct. To get the “other order”, you can try $a(f(x)+b)$. Think about it.
3. Your observation is again correct.

If operations can be done in either order (as in number 3), we say that the operations commute.

The observation at (3) is a special case of the fact that two “purely stretching” linear maps that have “the same axes of stretching” always commute. In technical language, one can say that two matrices that are diagonalised in the same basis will commute.

In general, the rules you have stated are all you need. You might want to throw in $f(x-a)$. You don’t need a “whole list” of such transformations, because you can always compose these transformations as you need, or decipher them when given one.