I began to think about the problem. The secant method is similar to Newton's method. The idea is that the tangent line at a point is a good approximation of the curve at that point. And we can find the root of the a line easily. The fundamental iteration of Newton's method is x1 = x0 F(x0)/F'(x0). Upon closer examination, I came to some interesting realizations.
I thought to myself, What is F(x) divided by it's derivative? Well, it's the number we subtract from the previous root approximation to get the new root approximation, a.k.a. Our delta-x. But, what is it really?, I thought. What does it mean? Well I notice that if F' is more horizontal, it approaches 1 and our delta-x will be large, more vertical and it will be small. Additionally, if the y-value, F(x) is large, then delta-x is large, and when F(x) is closer to zero, delta-x is small. This means that we tend to converge faster when far away from the root, or if we are approaching the root slowly, i.e. a horizontal sloping function.
This is good.
However, I could see a problem if our starting point were far away from the root, and the function was very vertical, then we might shoot past our root. So, we must pick good starting points.