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Past the usual memorized curves like $y=\sin(x), y=|x|, y=1/x, y=\ln(x), \ldots,$ is there a way to have an intuition about the shape of a curve from looking at an arbitrary function/term? (that is, other than for transformations of the usual memorized functions and asymptotes that result from prevention of division by zero)

I can’t think of any good examples right now, but a few times in classes the teachers have commented something like “you can see that this function’s/term’s/expression’s curve will be … (he/she draws on board)…” despite the fact that the function/term/expression is a conglomeration of the basic ones we have memorized the curves for.

George V. Williams
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ChrisC70
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  • Try to think about long-term behavior, roots, and at least one point with nonzero output: often the $y$-intercept. As en example: $y=(x-1)^2e^{-x}$. In the long term, the exponential outweighs the quadratic. So as $x\to\infty$, this curve's $y$-values approach $0$. And as $x\to-\infty$, this curve's $y$-values approach $\infty$. The only roots are at $x=1$, and locally the function is $\approx c(x-1)^2$ there, so it's graph resembles a parabola near $x=1$ touching the $x$-axis. The $y$-intercept (and in fact every output) is positive. This is all enough information to sketch the curve roughly. – 2'5 9'2 May 18 '13 at 16:32
  • You talk about curves, and this leads me to interpret you want to deal with maps $\alpha : \mathbb{R} \to \mathbb{R}^n$ with $n=2$ or $n=3$, but in your question it seems you only want to deal with graphs of functions $f:\mathbb{R} \to \mathbb{R}$. So, which of them you want to deal with? General curves in $2$ or $3$ space or graphs of one variable functions? – Gold May 18 '13 at 16:35

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The way I try to imagine curves is try to identify zeros of the function. And then try to see how the slope is changing by finding its first derivative. And if I want more accurate information I ll try to look for other properties like second derivative to see how the curvature of the graph looks like....

But these kind of things will be learnt only on practice...

Phani Raj
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