There is a mathematical pattern of a stretching spring, where if the load on the spring was doubled, the extension also doubled. Three times the load gave three times the extension, and so on.
This shows that the extension is proportional to the load.
At a certain point in the load-extension graph, the graph curves and the line slopes up less steeply. This point is called the limit of proportionality.
Limit of proportionality: Up to this limit, the extension on a spring is proportional to load.
Spring constant: The constant of proportionality, the measure of the stiffness of a spring.
x is the extension of the spring.
The spring constant is defined as the force per unit of extension, k=xF.
The spring constant is a measure of the stiffness of the spring: the stiffer the spring, the bigger the load required to change its length and the steeper the gradient when the load is plotted against extension (i.e., with extension on the horizontal axis).
The following can be observed if you carefully add weights to a rubber band and remove them without releasing the tension in the rubber:
The graph is not a straight line but rather a slightly S-shaped curve. This indicates that the extension is not exactly proportional to the load.
Eventually, increasing the load no longer produces any extension. The rubber would feel very stiff. When the load is removed, the graph does not come back exactly to zero.