WitCH 59: Stretching the Truth

This double-barrelled WitCH comes from Maths Quest Mathematical Methods 11 (top). and Cambridge Mathematical Methods 1 & 2 (bottom). It is the final in our series of Quest-bashing, at least for now.

20 Replies to “WitCH 59: Stretching the Truth”

  1. I’m pretty sure dilatations preserve similarity, where as the examples shown are more stretches or non-uniform scaling.

    1. Thanks, Potii. Very good point, and you’re correct on the standard usage. although this not my point. The “scaling in one direction” usage is standard in VCE. My issue with the above excerpts is practical, not legal.

      1. I think I know what you mean.

        In the NSW new syllabus they have included this concept (essentially they did a copy and past of a lot of the Australian Curriculum for stage 6) and so some textbooks try to talk about such dilations with respect to the x-axis and y-axis or by using the words like horizontal or vertical dialations; this gets very confusing as it can be taken in different ways.

        For example: is 2f(x) for the function f(x)=x^2 a horizontal or vertical dilation (to use their words)? Who knows what it should be? It gets closer to the y-axis (horizontal movement) but also moves away from the x-axis (vertical movement). And for some reason the textbooks try to use these different words to make distinctions between af(x) and f(ax). However, f(√2 x) gives the same results as 2f(x) for f(x)=x^2 and so these definitions are useless.

        When I teach I feel it is better to talk about what these transformations to do the input and/or output. Then take it from there with what the transformations look visually.

  2. Following on from the above comments: unless the domain is restricted, aren’t there two correct dilations that produce the same image?

    1. Hi, RF. You mean the choice of either an x-stretch or a y-stretch? Yes, for both the examples above that is correct. It’s not really a problem for how things are presented: they have correctly indicated the effect of the chosen dilations. Also, Cambridge almost immediately goes on to make your point.

  3. I’m assuming that MQ goes on to describe more general graphs than just the standard parabola… so that students have some hope of actually doing a variety of questions?

      1. Is your point the potential confusion students may have about translating “dilation by a factor of a” into a more precise notion? I can reasonably see students looking at these and thinking they replace y by ay (see first line of Cambridge text) or y by y/a (if they interpret it as “…from the x-axis”.

        1. Hi Glen. My point is that the excerpts are woeful.

          This one seems to be prompting a similarly puzzled reaction to the partial fractions WitCH that I posted recently. And, true, there’s nothing as awful here as in the recent differentiability WitCH, or in many of the other WitCHes. But the above are still examples of objectively horrible mathematical exposition. The excerpts make a very simple mathematical concept – going from y = f(x) to y = 2f(x) = TWO TIMES WHAT IT WAS, is just a stretch in the y direction – into something barely comprehensible.

          I think the ceaseless bombardment of shit writing and spooned-on pedantry have lowered people’s expectations. The above excerpts are considered acceptable; they are not even close.

          1. Marty, I take your point, but now I’m curious about how you might explain stretches from/towards the y-axis (in the x-direction) by a factor of 2 (i) in a simple and intelligible way, and (ii) that helps students to see why it’s not y=f(2x) in such a way that they are hardly tempted to make that mistake and (iii) is logically analogous to the explanation of why a stretch by a factor of 2 from the x-axis maps y=f(x) to y=2f(x). (I suppose one might reasonably reject the third requirement because of the difference between transformations on the pre-image versus the image, but in my experience, having a unified explanation for both x and y transformations helps students).

            So I am more sympathetic to the Cambridge excerpt because I think the authors are trying to ease students into the general method by showing how it applies to a simple case, and the general method is one that a lot of students prefer because they struggle with this topic intuitively (ie. without a systematic method, a lot of students will think that dilating from the y-axis by a factor of 2 maps y=f(x) to y=f(2x)).

            I really don’t like the MQ presentation though, because stretches from the x-axis are described in terms of making the graph wider/narrower, which is thoroughly confusing.

            1. Hi, SRK. Addressing your last and easiest point first, yes MQ neon-signing \boldsymbol{ax^2} as resulting in the graph being “wider or narrower” is batshit insane. It is pointing kids in exactly the wrong direction. But I am certain that it is not just MQ sinning on this; I hear this stuff all the fucking time, almost without fail. I keep a stun gun with me, just to use on tutees who refer to \boldsymbol{2x^2} or whatever as “narrower”.

              On your second point, one might give Cambridge the benefit of the doubt, and accept that they’re trying to introduce the general notion of transforming a space (with the particular graph also being transformed as a consequence.) Given Cambridge loves to cut and paste its own texts, I’m not sure I’m willing to concede the point, but for the sake of argument let’s concede it here. What i am definitely willing to concede is that VCAA makes a mess of transformations, with no clear instruction. However, even conceding all that, and accepting that the Cambridge excerpt is not as neon stupid as the MQ excerpt, Cambridge still screws up.

              A fundamental principle of teaching is make things as easy as possible, and no easier. In particular, one should never make easy things hard. That principle applies even if, for some good reason, one wishes to present an easy idea in a more sophisticated manner. First make the easy thing easy; then, and only then, one can look to make the easy thing harder. So, does Cambridge above ever make the easy thing easy? I think the answer is a clear no.

              Finally to your, first and very good point. I agree that transformations are intrinsically confusing, at least for me, making them difficult to teach or write about clearly. The inversy nature of f(2x) and f(x + 3) is difficult to internalise, together with occasional commutativity, and combos like f(2x + 3), and so on. It’s not an easy topic. But, Christ, that’s all the more reason to be super-sure to make the easy parts easy. And, a*f(x) is easy.

              As I wrote above, what I think is going on here is a general acceptance of very poor writing on this topic:

              “Let (x’,y’) be the image of the point with coordinates (x,y) on the curve.”

              Who can write that and think “Yep, that’ll do”? Who can read it without a double-take? Of course the reality is that no one reads it, not even the writer. And such sloppy, incomprehensible writing is endemic in this topic. So, although I think the excerpts above are well-deserving of a bashing, they are also proxies for the general presentation of this topic.

              To give one simple but telling example: the notation x’ y’ is appalling, demanding constant strain to determine what is original and what is image. Simply switching to using X and Y for the images would assist the teaching of this topic no end.

                1. No, this one is fine (except that I can never stand 3b1b1). The σ gives both a scale factor for y out the front, and a scale for x inside.

                  1. Oh right, I thought that using ‘skinner/narrower/f*tter/thinner etc’ at all to describe transformations was the problem. My mistake.

                    1. No worries. That language isn’t great, but yes, that’s not the main problem. The problem, for example, is to say “skinnier” when the natural interpretation of the functions is to go with “taller”.

  4. I think there is an issue with “the amount of stretching or compression”. Increase by given amount or in a given ratio…?
    Years ago one of MC section questions of MM examination used the wording “dilation by 2 units…”.
    Not very nice.

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