We are given:
If I were going to sketch the graph of the given function, I would first note that the function is even, which means it is reflected across the y-axis.
It has a positive root at (1,0), and so must have one at (-1,0). It has a y-intercept at (0,-1). The roots are of odd multiplicity, so they will pass through the x-axis, rather than just being tangent to it.
There are no asymptotes.
Now, differentiating, we find:
The root is of odd multiplicity, so we can expect an extremum there, but the other two roots are of even multiplicity, so they will not be at extrema.
We then find that the function is decreasing on the intervals and increasing on .
By the first derivative test, we can state then that there is a global minimum at (0,-1).
Differentiating again, we find:
Here, all of the roots of of odd multiplicity so we can expect all of the roots to be at points of inflection.
The critical numbers are
Testing the intervals made by these numbers, we find:
We now have enough information to make a reasonable sketch of the graph.