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I recall as a student when I asked TA's for help on problems (calculus and analysis), they often had an idea of how to approach problems (usually regarding limits, sequences, integration) before even really thinking about the problem because they had some idea of what the particular function looked like.

So I thought it might be worth asking what graphs of functions a mathematician should be familiar with. (I wasn't able to find a duplicate.) Also, how should I ask it? I'm not quite sure how to phrase the question properly.

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  • $\begingroup$ I think using a function plotter to learn on the go reduces the amount of function graphs one "has to know", which is anyways somewhat ambiguous. You smay narrow done the purpose you learn this for and get a better and clearer question. $\endgroup$
    – Michael Greinecker
    May 11, 2013 at 6:00

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I am personally not a fan of these "[blank] every mathematician should ..." type questions. Their history shows that eventually most will be closed:

My advice would be to not ask such a question.

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  • $\begingroup$ I hear what you're saying, but oddly enough those questions get large number of upvotes and favorites -- perhaps implying that they are beneficial still to many other community members (?). $\endgroup$
    – TheRealFakeNews
    May 11, 2013 at 0:06
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    $\begingroup$ @AlanH Then seeing the Batman curve was incredibly beneficial... $\endgroup$
    – 75064
    May 12, 2013 at 22:12
  • $\begingroup$ @75064 Touche, so not an entirely accurate characterization, but probably not totally inaccurate either. $\endgroup$
    – TheRealFakeNews
    May 12, 2013 at 22:52
  • $\begingroup$ @75064 I hope this background process is not programmed to eventually self-delete. Good to have you here. $\endgroup$
    – Michael Greinecker
    May 14, 2013 at 15:57
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I think the premise of the question is false. When asked how the graph of $y=x^6-2x^2$ looks like, one does not reach into the catalog of "known graphs of functions". Instead, one parses the formula and identifies two summands: $x^6$ and $-2x^2$. Three observations are made then:

  • Both summands are even powers, therefore the function is even
  • For small values of $x$ ($x\approx 0$) the first summand is much smaller than the second
  • When $x$ is large, the first summand is much greater than the second.

The above is enough to picture the graph, which is flat at $x=0$, moves down symmetrically in both directions as upside-down parabola, but then goes upward (also symmetrically) roughly like a parabola but steeper. If one wishes for the picture to be on accurate scale, then looking for intercepts $y=0$ and critical points $y'=0$ is the next logical step.

So, instead of a catalog of graphs one should know, the question might be about the skills one needs to develop for this purpose. But I don't see how to put this latter question in SE format: it looks too broad for it. Qualitative understanding of functions cannot be acquired separately from other skills, such as the ability to see that $10\cdot 10-1.78\approx 100$ without reaching for a calculator.

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Specifically for function graphs - you should be able to have a rough idea of any $\mathbb R\to\mathbb R$ function you encounter, especially any combinations of elementary functions. Of course, the rough idea may be refined when taking a closer look, but you definitely should know what to look for and how (e.g. that $\frac 1x\sin\frac1x$ has an accumulation point of zeroes and is unbounded).

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From Robert Cartaino's Good Subjective, Bad Subjective blog post, we have a list of Guidelines for Great Subjective Questions:

  1. Great subjective questions inspire answers that explain “why” and “how”.
  2. Great subjective questions tend to have long, not short, answers.
  3. Great subjective questions have a constructive, fair, and impartial tone.
  4. Great subjective questions invite sharing experiences over opinions.
  5. Great subjective questions insist that opinion be backed up with facts and references.
  6. Great subjective questions are more than just mindless social fun.

I'd say the proposed question could not meet 1, 2, 4, and 5 and depending on how it's worded might not meet 3 also. The intention seems to be educational, so it does meet 6. Nevertheless, it's not a good score.

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This is not a bad idea, in my opinion. It is useful to be able to identify certain curves at a glance, and the MSE community might be able to provide some uncommon curves. Like Arthus Fischer says, though, I don't think "_ every mathematician should know" questions are a good idea. This makes it sound more like a cracked.com article than a real question.

Instead my suggestion would be to refine what you want enough to put it into write a CW/big-list question. For example, one could request a list of curves with easily identifiable graphs that appear frequently in higher mathematics, while specifically requesting that these be "non-trivial" examples, so you wouldn't have $\sin$/$\cos$, parabolas,etc. (If I were answering this question I might submit $x^3-y^2=0$.) In other words, refine your question so it is more definitively answerable, and the community may be more receptive to the idea.

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