Does the amount of translation for depends on whether it goes before or after dilation?

Question

One question in my practice book, asks me to describe $y=f(-ax+b)$ based on y=f(x).

According to the book, the order of transformation must be listed in this order:

1. Reflection
2. Dilation (scaling)
3. Translation

So the first step in describing is to factor out the scale factor and get $-a(x-\frac{b}{a})$.

My question is, is the order my book gives used in the mathematical world and not just on the test I am taking. There are two ways of looking at what the translation applied to the equation is. Before dilation happen, translation is b units to the right. But after it, translation is only b/a units to the right.

Answer 1

The order depends on the goal of the author. If the goal is to teach plotting functions $f(ax-b)$, then doing dilation before translation is preferable because the typical elementary functions like $f(x)=x^2$ or $f(x)=\sin x$ are easier to dilate before they are translated. For example, compare two ways of dealing with $(2x-3)^2$

Dilation-translation order

1. Start with parabola $x^2$
2. Then $(2x)^2$ is a skinny parabola with the same vertex
3. And $(2(x-3/2))^2$ is a skinny parabola moved $3/2$ units to the right.

Translation-dilation order

1. Start with parabola $x^2$
2. Then $(x-3)^2$ is a parabola moved 3 units to the right, so the vertex is now at 3.
3. And $(2x-3)^2$ is a skinny parabola whose vertex is moved again, so it's now at $3/2$.

In the translation-dilation order the last step is more complicated because there is more going on.

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On the other hand, applying translation before dilation is useful when amount by which a function is translated somehow depends on how "spread out" it is. This is typical for wavelets in harmonic analysis. Consider the Haar wavelet (source of picture: Wikipedia)

This function $\psi$ should be translated by $-2, -1, 0, 1, 2, 3,\dots $ units to tile the line neatly. But we also need its scaled versions, such as $\psi (4x)$, which is supported on the interval $[0,1/4]$. This one needs to be translated by multiples of $1/4$, etc.

It is easier to describe the process as (1) we translate $\psi$ by integer amounts to tile the line; (2) then scale all of those translated functions by powers of $2$.

This way, we end up with the formula $\psi(2^nx-k)$, for $n,k\in\mathbb{Z}$.