How to distinguish between vertical and horizontal stretch/shrink when ambiguous? Please bear with me. I am trying to help my daughter with her Algebra 1 homework. We are asked to describe the transformation of function f to function g as follows:
$$f(x) = x$$
$$g(x) = 2x+3$$
The provided answer states that $g(x)=2x+3$ can be re-written as $$g(x)=2f(x)+3$$ and is therefore a vertical stretch by a factor of 2 (plus a vertical translation up by 3 units). Well and good. However, $g(x)=2x+3$ can also be re-written as $$g(x)=f(2x)+3$$ and be described as a horizontal shrink by a factor of 1/2. But even though this horizontal shrink gives exactly the same graph as the vertical stretch, it is not mentioned as a possible correct answer. I understand that the order of transformations is important and can give completely different graphs if you mess up the order, but this is not the case here. There is at least one more question in the study material that likewise lists the vertical stretch, but not the identical horizontal shrink, as the correct answer. Is it because g is originally expressed as $g(x)=2x+3$? Does this necessitate that we think of the transformation only in the vertical axis? Something to do with $y=mx+b$ where $m=2$? Many thanks.
 A: For a linear function like $f(x)=x$, you cannot distinguish between a horizontal scaling and vertical scaling.  It's equally valid to interpret it in both ways.
Even some nonlinear functions permit two interpretations too (say $g(x) = 4x^2+3=(2x)^2+3$ )
The vertical scaling is probably just the most apparent explanation, and I don't think it's a big deal that the other interpretation was omitted.
A: You're right that for a straight line, the graph is identical regardless of which way you consider the scaling.
But for every other type of curve (in general; there are always specific cases where some transformations are equivalent or can be obtained using a combination of others) they will not have the same result. Try playing with vertical scaling and horizontal shifting of $y=2^x$ to see another version of the issue you encountered.
So, why treat it as vertical scaling only? To some extent, they're really the same thing. Going up twice as fast as the same as going along at half the speed. When working with straight lines, the idea of relative rate of change is often what we are most concerned with, the vertical change per unit horizontal change. Horizontal scaling would mess with the "per unit" aspect. Vertical scaling corresponds directly to changing the rate. So, vertical scaling is a better choice on this case.
