# Confusing verbal descriptions of function transformations

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?

• Personally, I avoid the words "stretch" and "squeeze" as much as possible. Instead, I typically say that multiplication by $a$ scales the graph of $f$ vertically by a factor of $a$. – Xander Henderson Feb 23 '20 at 3:40
• @XanderHenderson: That sounds a better choice, thanks. Could you offer your comment as an answer? – reflectionalist Feb 23 '20 at 9:36
• Building on Xander's comment: you could say it scales the graph by a factor of a, and that it is a stretch when |a|>1 but a shrink when |a|<1. – Brendan W. Sullivan Feb 23 '20 at 20:47

I've found terminology differs as well, but this is how I think of the phrase "by a factor of" for $$af(x)$$

• If $$0<|a|<1$$ then it is a vertical "squeeze" to get a factor of $$a$$ between the new and old $$y$$ values
• If $$|a|>1$$ then it is a vertical "stretch" to get a factor of $$a$$...

And then a point $$(x,y)$$ becomes $$(x,ay)$$. So $$a$$ is the new scale or ratio, vertically-speaking.

Whenever I teach functions, I honestly just explain that the terminology differs and can be confusing, and that students should use the terms that fit the book we're using or the test they're preparing for.

• That is smart, similar to @XanderHenderson's. – reflectionalist Feb 23 '20 at 9:39

I tend to be very strict on how I define things in this area - I like to keep things formal in preparation for University study. I'll explain everything I use and why I believe it to be effective - and use your examples.

Let $$y = f(x)$$.

• $$y = 2f(x)$$ is said to be a vertical stretch of scale factor $$2$$ parallel to the y-axis.
• $$y = \frac{1}{2} f(x)$$ is said to be a vertical stretch of scale factor $$\frac{1}{2}$$ parallel to the y-axis.

I always clarify the type of stretch as when listing stretch as opposed to later - I find it helps clarify for examiners when marking. I always use scale factor instead of factor as I find factor can be ambiguous, a factor of $$2$$ could imply multiply or divide by this. I clarify that it is parallel to the y-axis, not "in" the y-axis. Finally, I define both these ideas of "squeezes" and "compression" as stretches and make it explicitly clear to students that if the scale factor of the stretch $$a$$ is such that $$0< |a| < 1$$ then it will have this "shrinking" affect.

When it comes to translations I often repeat this process of emphasising the type of translation when stating it.

• $$y = f(x) + 2$$ is said to be a vertical translation of 2 units upwards.
• $$y = f(x) + (-2)$$ is said to be a vertical translation of 2 units downwards.

I often find mixing the idea of translation with "shift" is confusing, and stick to one terminology and use it strictly.

These definitions are now what I use teaching the Canadian PEI curriculum as in the UK we don't say upwards/downwards/left/right when describing translations we would describe the following as:

• $$y = f(x+4) + 3$$ as being a translation being described by the vector $$\begin{pmatrix} -4 \\ 3 \end{pmatrix}$$.

This gets rid of confusing terminology.

I use these because they provide a similar structure for all transformations.

For example, looking at reflections, my students can look at these as "a type of" stretch and it helps when it comes to combinations of transformations too.

Hope this helps.

• Nice perspective. But it seems to me that "a translation being described by the vector $\binom {-4}3$" is not getting rid of confusing terminology compared to "a translation of four points to the left and three points up". – Matthew Daly Feb 25 '20 at 11:56
• @MatthewDaly Yes, this is true for this answer - I don't know how you would really eliminate which is the most essential terminology for this topic. I know in UK however, students often lose marks for not clarifying "units" and assume mm/cm/m and lose marks. – Mathew Duxbury Feb 26 '20 at 7:15