This one comes courtesy of frequent commenter, Red Five. It is a question from the CAS-free component of Western Australia’s 2023 Mathematics Specialist exam.
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This one comes courtesy of frequent commenter, Red Five. It is a question from the CAS-free component of Western Australia’s 2023 Mathematics Specialist exam.
‘Tis a sloppy picture. By virtue of the asymptotes and x intersections, one would deduce that a = 1, b = 4, c = 3 and k = 2. The trouble is that the function f(x) so produced has f(0) = -8/9, not f(0) = -1 as shown. To have f(0) = -1, you’d need c to be sqrt(8), which is clearly not what the diagram shows.
This seems to be the biggest issue.
Interestingly they didn’t put a closed circle on the y-intercept. So maybe there is some convention about integer intercepts and closed circles. A less than ideal thing to do. They should have picked better values.
Looks like c was meant to be 2 root{2}. But that would be difficult to see so someone moved the asymptote a little to x=3. It’s an example of where the more careful student is less likely to get full marks?
Very much so.
I don’t think you can just write down that
from the diagram.
It seems like a reasonable guess, and if the question said
is an integer, then fine, but it does not say this.
Minor nit-pick perhaps, but a serious one.
So can we be sure that the y intercept is -1? It looks like it but maybe it’s -8/9?
This type of question can test the basic properties of a rational function. Can we give some guidance to exam setters? As you note, specifying a, b, c, and k to be positive integers is needed. If the graph is then corrected it becomes a reasonable question.
Nah. It’s much closer to 1 than 8/9. But if that was the intention then it was a truly idiotic idea.
Three things are clear to me from this graph: There is an asymptote
. There is an axial intercept at 
and another at 
.
As to the other features, the graph does not make these clear at all and the “solutions” seem to suggest that a good-enough-guess is sufficient (even though it causes contradictions…)
I’m less worried about the fact that the question was written and more about how it got through the checking process.
Me too.
I think (-1,0) is also “clear”, and it was clearly meant to be clear, as indicated by the dot. But five conditions and four free choices smells trouble. Plus, the dots on the x-axis, and no-dot on the y-axis says to me that the writers and/or vetters were aware of the issue, and still they let it through. It’s nuts.
The big problem with the question has been pretty well noted.
I also think another problem is that the marking is unreasonable. They expect students to grab the four values from observing the asymptotes and x-intercepts on the graph (as Alasdair did in the first comment) but assign one mark for “Provides justification for at least one of the values…” Yes the exam instructions say questions worth more than 2 marks must have working but when the question says to fill in a table and doing so is trivially done by observation it’s an easy thing to not do.
Good point.
If the parameters are as intended, the max is around x=14..
Oh my goodness – what a shocker! Really poor communication with the graph. I have an issue with the boxes for writing the answers in too. They aren’t great for encouraging the examinees to show their working. I wonder if the problem happened during typesetting?