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Recently, I was bombarded by dozens to "sketch/graph the following function" exercises. Some such functions are $y=x\sqrt{\frac{x}{4-x}}$ (cissoid of Diocles), $y=e^{-x^2}$ (probability curve), $y=x+\frac{1}{x}$ (trident of Newton), and $y=\log{(\log x)}$.

Currently, I have 2 methods:

  • dumbly and tediously find $f(x)$ of several $x$ and plot the points; then interpolate the curve; or,
  • plug the functions into Wolfram|Alpha (which isn't really me sketching the graph).

Are there any methods that quicken the process of sketching a graph?

I was thinking of a few methods, but they seem almost as tedious as functions get more complicated. For example, regarding the trident of Newton (see above): sketch $y=x$ and $y= \frac{1}{x}$ on the same graph. Then add the $y$ values to assemble the curve $y = x + \frac{1}{x}$.

$^\text{Note: I added the soft-question tag because this question doesn't solve a particular problem; if this tag is not needed, feel free to remove it.}$

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    What type of class or area is this for? Do you know differential calculus? – 2017-01-31
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    For example, the first thing you'd notice when you see $y=x+\frac{1}{x}$ is that this is an odd function. So sketching for $x>0$ gives all the required info about the graph. Further, in asymptotic you can see that $y/x\to 1$ as $x\to \pm\infty$, this means that roughly for large $x$ we can expect graph to look pretty much like $y=x$. It's also very easy to see that $\lim_{x\to\pm 0}=\pm\infty$ so, for $x>0$, you'd have pretty good intuition about how the graph should look like. If further details are required, you could differentiate and see where $y'=0$ etc – 2017-01-31
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    But it all comes down to having quite a bit of exprience in what you should do when trying to understand shape of graph. There are some pretty standard things to do, and most cases those will be enough – 2017-01-31
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    @TheCount -- I'm self-studying, so I'm not sure which class it ought to belong to. However, these exercised come from Demidovich's book "Problems in Mathematical Analysis". Calculus is in the book, but I'm only on the first chapter (goes only as far as limits). – 2017-01-31
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    I'm not familiar with the book, but they probably want you to do a good bit of plotting points, and some of what @user160738 suggests, then. – 2017-01-31
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    @user160738 -- Thanks, I'll use your tips regarding even/oddness and asymptotes. I'm not at the point in the book where differentiation is taught, but will consider that tip when I get there. – 2017-01-31

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