How to draw a function like this (without the help of calculator)

Question

$y = 3x^2-x^3$

How should I draw a graph like this? any efficient way to draw this (not just plotting x and y values on the coordinate system)

Answer 1

It completely depends on what you're looking for. If you want a graph that it's actually decently accurate and you can estimate values from, then I suggest just making a table with values, plot those as points in your graph and drawing a smooth line through it. If you're just looking for a general shape, see that it has three roots of which two are same so it probably looks something like

(and yes, I drew that with MS Paint).

Conclusion is, assess what you want from your graph and adjust accordingly. Compute enough values for your graph to be accurate enough, but don't overdo it as it's very time consuming (without calculator).

Answer 2

First check the first derivative.

$y'=6x-3x^2=3x(2-x)$ Hence for $x\in (-\infty, 0)$ the graph is decreasing. For $x\in (0,2)$ the graph is increasing and for $x\gt 2$ the graph is again decreasing.

You can also check the general shape or concavity and community of graph using second derivative. Indeed $$y''=6-6x=6(1-x)$$ . Hence the graph changes the concavity at $x=1$ . Moreover the roots are pretty easily noticeable and they are $0,0,3$. I guess using these information you will be pretty easily drawing the graph. Moreover you can check the limits of the function as $x\to \pm \infty$ to get better overlook at behaviour of function at $\pm\infty$.