A functional form is an algebraic expression that corresponds to a particular functional relationship. For example, there are various ways to describe a direct relationship between two variables:

Function: y = f(x) where dy/dx > 0
Functional forms: y = a + bx where b > 0 (straight line)
y = a + b*ln(x) where b > 0 (curve line with decreasing slope)
y = a + b*x*x* where b > 0 (curve line with increasing slope)

All of the above functional forms express the same idea as the functional relationship given by y = f(x) where dy/dx > 0, but each functional form produces a graph that appears differently.