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.
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...