Right now I'm studying the proof of the mean value theorem and ther's something about it that I am having trouble understanding.

I'm going to attemp to explain where I'm having trouble with the proof.

Refer to the attached figure during this discussion.

In the proof, the first thing that they do is say that the equation to the secant line is

$\displaystyle y=\left[\frac{f(b)-f(a)}{b-a}\right](x-a)-f(a)$

I understand that $\displaystyle \left[\frac{f(b)-f(a)}{b-a}\right]$ is the slope of the secant line, but I can't understand the rest. My mind wants to say that $\displaystyle (x-a)$ is x (in the slope intercept form of a line) and $\displaystyle -f(a)=b$.

Could someone justify this as best they can?