While teaching Function Transformations, I found the verbal descriptions of stretch and squeeze really confusing.
So for $y = f(x)$,
- $y = 2f(x)$ is said to stretch $f(x)$ vertically by a factor of $2$;
- $y = \frac{1}{2}f(x)$ is said to squeeze $f(x)$ vertically by a factor of ?
I found in textbooks and online both $2$ and $\frac{1}{2}$ and can not decide which one is right. The former makes more sense to me since "squeeze" has already indicated the opposite direction of stretch marked by the exponent $-1$ of $2^{-1} = \frac{1}{2}$. For me, this is also compatible with translation where
- $y = f(x) + 2$ is said to shift $f(x)$ up by $2$ units;
- $y = f(x) + (-2)$ is said to shift $f(x)$ down by $2$ units.
Note how the word "down" has indicated the opposite direction of "up" marked by the negative sign.
The only way I can make sense of the use of $\frac{1}{2}$ is to think the phrase "a factor of ?" refers to the factor in front of $f(x)$, hence "by a factor of $2$" for stretch and "by a factor of $\frac{1}{2}$" for squeeze. But then we should avoid words like "stretch" and "squeeze" that already indicate directions (away from or toward the axis), shouldn't we? I found the word "dilate" is used in some textbooks and online articles. But Mathwords says it is wrong and English has no words to cover both stretch and squeeze.
Or should I replace "by a factor of ?" with "by a multiple of ?" or "? times"?
So what description do you use and why?