You may be asked questions concerning curved graphs where the process of determining the value of the gradient is made more difficult. However, it can still be useful to work out the gradient, particularly in questions with real-world context. For example, suppose there is a graph with time on the x-axis, and the value of a company on the y-axis. The gradient of any graph is a measure of how much y changes with respect to x (often referred to as the rate of change of y with respect to x), so in this example the gradient is a measure of how quickly a company’s value is changing at certain points in time.What you are required to know how to do is estimate the gradient of this curved line. This comes down to two possibilities:
- Calculate the instantaneous rate of change by drawing a tangent to the curve (a straight line just touching the curve) at the desired point, and then calculating the gradient of this tangent (which can be worked out using standard straight line methods). This will correspond to the gradient of the curve at that individual point.
- Calculate the average rate of change by drawing a chord (a straight line between two points on the curve), and then calculating the gradient of this chord. This will correspond to the average gradient of the curve between the chosen two points in time.
A question on this could be presented with many different contexts and you are expected to determine what the gradient means in that context by understanding the fact that it is a measure of the rate of change of whatever is on the y-axis with respect to whatever is on the x-axis.