To help you to construct the graphic yourself, it is useful to find the roots of the equation (the values of x that make y=0). In this case, because we have a equation of grade three, we will find three roots.
It is very simple to find the roots of this equation, just ask yourself: which values of x makes y=0?
To find the three roots, we should equal y to zero and see what values of x respect the equality: [tex]y=0=x^3-x[/tex]⇒[tex]x^3=x[/tex].
[tex]x^3=x[/tex] will be true only under three values of x: x=1 ([tex]1^3=1[/tex], x=(-1)([tex](-1)^3=-1[/tex]) and x=0 ([tex]0^3=0[/tex].
Finally, to know how to finish our graph, we should marked the roots in the x axis, we have four important intervals to analyse defined by our roots: (-∞, -1); (-1,0); (0;1) and (1,∞). We should check if the function takes possitive or negative values within this intervals, and this will give us an idea of how to join the different points defined by the roots.
As an example, if we take x=-10 ( a number of the first interval), the function will take the value of y=-900. If we do this with others values of this interval we will notice that they are all negative, wich means that the function increases to reach y=0 at the point where x=-1.
You can check points in the others intervals to see how you finally unite all the points and get the graph below.
