How does the formula for length of a graph relate to that of parametric description? So I know how to solve for the length of a regular graph and for a parametric description.

Regular graph length formula:
$$L = \int_{x=a}^{x=b}\sqrt[]{1 + (f'(x))^2} \,\,\,dx$$

Parametric description length formula:
$$\int_{t = a}^{t=b} \sqrt[]{(x'(t))^2+(y'(t))^2} \,\,\,\,dt$$

The form of these two are so radically different that I can't see how one relates to the other. The parametric description uses the form for path velocity but I don't understand why..
If more elaboration is needed, please ask. I'll supply.
Highly appreciated,
-Bowser

The basic idea is that the differential arc length is
$$
ds^2 = dx^2 + dy^2
$$
Your first formula then derives from
$$
y = f(x) \\
L = 
\int ds 
= \int \frac{\sqrt{dx^2 + dy^2}}{dx} \, dx
= \int \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
$$
and the second from 
$$
\left(\frac{ds}{dt}\right)^2 = \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2
$$
