In mathematics, functions representing linear relationships (relationships that can be graphed as lines) between sets of two variables have what is called a "slope" as a trait. The slope represents how the dependent variable (often labeled "y") changes for each one-unit change in the independent variable (often labeled "x"). In short, the larger the slope, the faster the rate of change. In essence, you can tell from a slope how sensitive the dependent variable is to changes in the independent variable.
Double slope theory states that linear relationships need not only have a single slope, but can have two slopes. While this concept may be difficult to understand at first, it becomes clear after gaining an understanding of piecewise functions, which are functions that describe how relationships between variables change at different degrees. Double slope theory's double slope comes into play when the linear relationship between the independent and dependent variable changes fundamentally after crossing a certain point on the independent variable's axis (the x-axis).
The absolute value function is a common example of how the double slope theory describes linear relationships. The absolute value function is a linear function that changes slope in terms of sign at a certain point. For example, the standard (unmodified) absolute value function, |x|, has a slope of 1 after x = 0 but -1 before x = 0. The result is a graph that looks like the letter "v."
There are many applications of the double slope theory in the real world, since many situations have fundamental changes at certain points. One example of the application of double slope theory is John Gottman's mathematical models of marriage. In these models, Gottman applies double slope functions to describe how spouses influence each other through positive and negative actions. The double slope comes into play because positive actions will influence spouses in different ways (with different slopes) than negative actions.