Newton's Formula is used to approximate the root of a function. The formula states that if a function is defined over real x, and f is diffrentiable, then if we start with an initial value x_0, close enough to the root we are looking for, we can approximate x_1 by
The process can be repeated until we reach a satisfactory estimation.
unfortunatly this method does fail sometimes, when the x_n do not converge as needed, one of the reason for it failure can be starting with an intial value that is not close enough to the root. Another reason can be the behavior of the first derivative in the neighboring of the root.
Regarding the graphing you requested, I can not provide it her, but I included a link that shows how the method works.