scaling and translating x axis

Question

say $y_1(x) = y(2x)$ or $y_2(x) = y(x-2)$, we can get the results with simple transformations. However, when we have something like $y_3(x) = y(2x -2)$, do we shift first and then scale or is it the other way round. More importantly why?

Answer 1

There are a couple ways to view what happens in he case of $y_3(x) = y(2x - 2)$, and they all arrive at the same consistent answer at the end (as it should be with math).

The first way to look at it as as a translation of $x$ by 2 units to the right, which we do by replacing $x \to x - 2$. This results in $y(x) \to y(x - 2)$. Afterwards, we scale $x$ by $\frac{1}{2}$, which is the same as replacing $x \to 2x$. The final result is $y(2x - 2)$.

If we first want to look at it as a "scale then shift", we first scale by $\frac{1}{2}$ so that we have $x \to 2x$ and $y(x) \to y(2x)$. Notice, however, that if now we shift by 2 units to the right, we have $x \to 2x$ so that $y(2x) \to y(2(x - 2)) = y(2x - 4)$. Because we started off with scaling by $\frac{1}{2}$, the correct approach now is to scale by $\frac{1}{2} \cdot 2 = 1$ units to the right. Indeed, if we do $x \to x - 1$, then $y(2x) \to y(2(x - 1)) = y(2x - 2)$.

In summary, you can view it either way. As a shift then scale, or as a scale then shift, but the amount you shift/scale by could change depending on the order that you do them in.