A reminder, WitCH 2, WitCH 3 and WitCH 7 are also open for business. Our new WitCH comes courtesy of John the Merciless. Once again, it is from Cambridge’s text Specialist Mathematics VCE Units 3 & 4 (2019). The text provides a general definition and some instruction, followed by a number of examples, one of which we have included below. Have fun.

**Update**

With John the Impatient’s permission, I’ve removed John’s comments for now, to create a clean slate. It’s up for other readers to do the work here, and (the royal) we are prepared to wait (as is the continuing case for WitCh 2 and Witch 3).

This WitCH is probably difficult for a Specialist teacher (and much more so for other teachers). But it is also important: the instruction and the example, and the subsequent exercises, are deeply flawed. (If anybody can confirm that exercise 6G 17(f) exists in a current electronic or hard copy version, please indicate so in the comments.)

Part (f) is not present in the current online version of the text. It is also not present in the current student book PDF available from the same website.

I will however confirm that the Specialist Mathematics Units 3&4 students I spoke with last year (and also at least one teacher of the subject) were unable to tell me whether or not the non-linear asymptotes required labelling on graphs.

My follow-up question to said teacher was whether or not students would be penalised for leaving these “construction lines” in their graphs to produce sketches by addition of ordinates.

The advice I was eventually given was to have students draw and label the non-linear asymptotes and hope for the best.

Would definitely like to know (for genuine curiosity more than anything else) what the convention is here and (less importantly) what VCAA is and isn’t prepared to mark as correct.

Thanks Number 8. Still curious about current hard copies. It is impossible to tell from the Study Design what is expected from students. Whatever is expected, the above excerpt from the text is a mess.

My hard copy is 2016 – admittedly old – and has part(f). It’s possible the digital platform my school uses only has digital copies of the older text. But I have had students from this year (2019) using pdf copies asking about part(f). Whatever, it’s a mess – incompetent meddling in things outside the scope of the course that leaves a trail of destruction for years to come.

I can’t find copies printed since 2016, but a lot of the students I work with are using digital copies which means I haven’t been asked about this…

…yet.

OK – I’ll attempt to make a start (be gentle with me…)

But as x approaches infinity, surely y also approaches infinity?

The fact that y approaches sqrt(x) is true, but since sqrt(x) approaches infinity, isn’t this a bit redundant?

I understand your point, but then why is a big deal made of the diagonal asymptotes of something like x^2 – y^2 = 1?

Because straight line asymptotes on graphs of non functions are a key feature of ALL graphs of this type?

There also seems to be a (perhaps implicit) idea that asymptotes are straight lines – in that the graph of y=1/x has horizontal and vertical asymptotes and then a rotation by Pi/4 clockwise creates the two diagonal asymptotes.

No combination of dilations, reflections, rotations make straight asymptotes into non-straight line asymptotes and because sheers are (as far as I can tell) absent from the VCE curriculum…

…but I really don’t know.

I’d add that the general hyperbola (x-h)^2/a^- (y-k)^2/b^2 = 1 can be obtained from y = 1/x using not just rotations but also translations and dilations. So the argument could be made that the general diagonal asymptotes y – k = +/- b/a (x – h) emerge naturally via transformations from the vertical and horizontal asymptotes of y = 1/x and this validates them.

But it would appear that a diagonal asymptote is also called an oblique asymptote ….

Thomas and Finney (Calculus, 9th ed. pp 227 – 228, 231) says:

“If the degree of numerator of a rational function is one greater than the degree of the denominator, the graph has an oblique asymptote, that is, a linear asymptote that is neither vertical nor horizontal.”

and talks about graphing with asymptote and dominant terms. For example, it talks about the dominant terms of y = (x^3 + 1)/x and says “For |x| large, y approx x^2. For x near zero, y approx 1/x …. [and y has] a vertical asymptote at x = 0 …” It’s exercises include graphing of rational functions where ” … the graphs and equations of the asymptotes and dominant terms” are required. Thomas and Finney says dominant terms “… are the key to predicting a function’s behavior.”

So Thomas and Finney appear to define oblique asymptotes as linear asymptotes that are neither vertical or horizontal. So something like f(x) = x + 1/x would have an oblique asymptote y = x, whereas f(x) = x^2 + 1/x would have the dominant term x^2 for large |x| and the graph of y = x^2 would be included as a feature in a sketch graph of y = f(x) but not referred to as any sort of asymptote.

Stewart also talks about curves that have “… asymptotes that are oblique, that is, neither horizontal nor vertical.” and actually gives a mathematical definition in terms of limits. There is no mention of, for want for a better word, curved asymptotes. However, one of the exercises talks about f(x) = (x^3 + 1)/x, asks the reader to show that

lim[f(x) – x^2, x –> +/- oo] = 0

and then says that y = f(x) is asymptotic to the parabola y = x^2 ….

So Evans has opened a real can of worms, as has past questions on VCAA exams and the current Study Design ….

Number 8, you are spot on with both remarks. Whatever the text is attempting to do in Example 31, to write y → √x is at best vague, and at worst meaningless. And, you are correct that there is a fundamental distinction one should make between linear and non-linear “asymptotes”. The text not only fails to make that distinction, it frames the discussion so that this distinction is almost impossible to make.

I understand what Number 8 is saying and it makes a lot of sense (I too think you’re on the money).

So what do we do with, say, f(x) = x + 1/x? Do we talk about a straight line asymptote in this case (even though there is no the same motivation as there is for rotated hyperbolas)? And if we do, what do we say about the asymptotes of, say, f(x) = x^2 + 1/x … What’s so special about straight line asymptotes? Is it OK for asymptotes to be polynomial but not non-polynomial (in which case, what do we say about f(x) = e^x + e^-x, say?)

The crazy thing is that we’re all in the dark.

The Study Design is singularly uninformative on this (as it is on many things) and as far as I know there has been no advice given in any VCAA Bulletin. In such cases one might look to past exams and Reports for additional insight. So it’s extremely interesting to note the 2000 Exam 1 Q1 Part 2 and subsequent comments in the Examiners Report (I don’t think the Study Design has changed all that much over 15 years with this part of the course), as well as the 2008 Exam 2 Q1.

That, Sir John, is the right question to be asking, because it is (a) common enough a type of question and (b) rare enough (non existent) in official VCAA publications. I would assume even raising it at a “meet the examiners” session may prove pointless? I don’t get to go to these, so cannot offer to raise it.

I will confess that teaching this part of the course prompts some anxiety in me. I can only imagine what it does to the students. I feel that it all stems from the study design, which offers very, very little in the way of direction. I am less anxious about rational functions, because I feel on (fairly) safe ground with assuming that both the numerator and denominator are polynomial functions. As curvilinear asymptotes have been marked in on previous exam questions (as noted by John) I feel fairly confident about telling students that they should include these on their graph along with the equation. However, at the bottom of the Functions and Graphs section of the Study Design sits the dot point: graphs of simple quotient functions. I do not feel that it is at all obvious what a ‘simple quotient function’ actually is, and the level of detail that students would be expected to include with these graphs. My anxiety is not helped by there being no mention of ‘quotient functions’ in any textbooks that I have looked at. I think that Example 31 is an attempt to address this dot point, although I can’t be sure. In any case, I now feel sick, so I am going to take whatever I can scrounge up in my drawer and go and lie down in the storage cupboard.

Also, with Example 31, and going by how curvilinear asymptotes have been calculated earlier in the text, for this question should it not be y=sqrt(x-1)?

Thanks very much, Damo. Your question is excellent and goes to the heart of this tricky WitCH: what, if anything, is wrong with the answer y = √(x-1) to 31(c)?

Underlying your question, what precisely does one need to show to prove that a certain “line or curve” is a “non-vertical asymptote”?

I’m not sure, but let me have a go. With polynomial functions a(x) and b(x), if a(x)/b(x)=q(x)+r(x)/b(x) then the graph of y=a(x)/b(x) will have a “non-vertical asymptote” at y=q(x) because as x goes to infinity (or neg infinity) r(x)/b(x) will go to 0 and the distance between the curve and q(x) will thus go towards 0. With the above example, we can do something similar: (x+1)/√(x-1)=(x+1)√(x-1)/(x-1)=(x-1+2)√(x-1)/(x-1)=√(x-1)+2√(x-1)/(x-1)

The second part of this equation goes towards 0 as x goes towards infinity, so the graph goes towards √(x-1). I feel that this is a more consistent extension of the approach taken with polynomial functions, as opposed to the logic that the textbook uses to give an asymptote of y=√x. But, well, really?

It leaves me feeling deeply uneasy, and if students really can expect to get this type of question, then I don’t know where it stops, and I don’t know how to prepare them adequately to tackle such questions. In particular, Number 8 refers below to “quotient functions, or rational functions as some IB textbooks have taken to calling them”, but my question is: Are quotient functions the same thing as rational functions? Because in the Functions and Graphs section of the Study Design, one dot point refers specifically to graphs of rational functions (with mention of asymptotes and stationary points) and another dot point refers to “graphs of simple quotient functions” (the dot point in its entirety). I don’t know what they mean by that. I’m just going to have a quick lie down in my son’s pillow fort. It feels safe in there.

That’s very good, Damo! I hadn’t thought of that: y = Sqrt[x-1] + 2/Sqrt[x-1] so y –> Sqrt[x-1] as x –> +oo.

So there’s at least *two* ‘plausible’ non-linear asymptotes.

What about re-writing as y = (Sqrt[x] + 1/Sqrtx])/Sqrt[1 – 1/x] and then taking x –> oo. What, if anything, is wrong with that (it also gives y = Sqrt[x]).

Or expanding y = (x+1)/Sqrt[x-1] at infinity (ask WolframAlpha: y = Sqrt[x] + (x to a negative power terms)) and then taking the limit?

I totally agree with Marty’s question:

“what precisely does one need to show to prove that a certain “line or curve” is a “non-vertical asymptote”?”

And just to throw the cat among the pigeons, could y = Sqrt[x-a] where a e R+ represent plausible equations for the non-linear asymptote? An infinite number of non-linear asymptotes …?!

Is the equation for the non-linear asymptote unique? (This question can only be answered after answering Marty’s question). And if it’s NOT unique, does it even make sense to talk about a non-linear asymptote? Under what circumstances *is* it unique? Does f(x) = x^2 + 1/x have a unique non-linear asymptote?

So many questions! The only things I feel sure about right now are:

An oblique (diagonal/slant) asymptote is a linear asymptote that is not parallel or vertical to the axes.

A curvilinear asymptote is an asymptote that is not linear.

Cambridge originally had a question 17: f(x) = (x^2 + x + 7)/Sqrt[2x+1] with a part (f) that asked for the equation of the “… other asymptote.” (the previous part asked for the vertical asymptote). This question appears to have been removed some time after 2016 ….

John, I have been trying to craft a reply for a while now (I actually have already made one attempt at posting a reply, but for some reason it didn’t appear. Boo). However, you have summed up much of what I’m thinking. So many questions, precious few answers. To add to the uncertainty, Number 8 commented that:

Quotient functions, or “rational functions” as some IB textbooks have taken to calling them…

However, in the Study Design under Functions and Graphs there is one dot point that says:

• graphs of rational functions of low degree, their asymptotic behaviour and nature and location of stationary points

And another dot point which says simply

• graphs of simple quotient functions.

So I am assuming that VCAA intend for quotient functions to be different to rational functions (so they contain non-polynomial functions?). However, there is no elaboration on what these functions may look like, no direction to how detailed such graphs would need to be and no explicit reference to them in any textbooks I’ve seen. I’m assuming that it means that questions like Example 31 are fair game, but, well. Yeah.

I am now going to crawl into my son’s pillow fort. It feels safe in there.

Damo, I hear and share your anxiety with this part of the SM34 course. Quotient functions, or “rational functions” as some IB textbooks have taken to calling them bring with them all sorts of questions about what to label – not helped by a number of the issues you raise above. But the worst of them all (in my opinion anyway) is the example you mentioned of an examiners report where non linear asymptotes are labelled but there is no help from the report nor the study design about whether this is required.

Students ask (rightfully so) about this every year and I am still at a loss about what examiners want to see. Which, unfortunately is the driving motivation here, not education.

Was the author trying to follow a VCAA guide perhaps? (Some sarcasm intended, but not sure where it is directed)

The author was trying, and failing, to impress.

Trying to impress who? The general consensus (from this website and other places) is that no-one really reads anything except the exercises in these books – possibly because it doesn’t make sense to a lot of students (whom it is presumably written for…?)

Whom, not who, but good points. I was just being snarky. There’s no question the authors try to impress on occasion, but here they’re just faking it. However, the fact that teachers don’t read this text doesn’t mean they’re not unduly “impressed” by it. Yes the text is unreadable for many teachers (and most students), but my sense is that teachers typically blame their own perceived shortcomings for this, rather than placing the blame where it belongs, with the authors.

Apologies for the double up comments above, I think my posting today reflects my feelings about this topic. I don’t know what I’m doing.

Damo, you appear to know MUCH more of what you’re doing than the authors of the textbook. You have given a perfect argument that the function y = (x+1)√(x-1) is asymptotic to the “curve” y = √(x-1).

(Sorry about the delay: your comment was hiding in spam.)

“Rational function” means a ratio of polynomials (analogous to “rational number”). The term “quotient function”, similar to “hybrid function”, doesn’t have any precise meaning. Still, one can get a sense of “simple quotient” as a ratio of polynomials, rooty guys and the like.