What is Gradient of a Curve?

In line or curve plotting of a graph, we normally come across the term "Gradient". This is an important parameter that governs the profile of the plotted line or curve.

What is this "gradient"?

Gradient is defined as the ratio of the change in the vertical unit to the change in the horizontal unit (at a certain point on the curve).

From the definition, it implies that the larger the numerical value of the gradient, the steeper is the curve or line plotted.

There are two types of gradient:

1. Positive gradient


2. Negative gradient

The diagrams below illustrate the meaning.


Diag 1

Positive gradient (Diag 1) exhibits vertically upward increment for increase in horizontal direction towards positive "x" value.

How about Negative Gradient?


Diag 2

Negative gradient (Diag 2)
shows a downward tendency when moving towards the positive "x" direction.

For straight line equation of y = mx + c , "m" represents the gradient.

Example 1: y = 3 x + c means a positive gradient with upward-going line.

Example 2: y = -4 x + c means a negative gradient with downward-going line.

NOTE:
A) m (gradient) = 3 means a vertical upward change of 3 units for each horizontal unit of positive (right-going) change.

B) m = -4 means a vertical drop of 4 units for each unit of horizontal positive change.

C) m = 1/3 means an increase of 1 unit upwards for 3 units of horizontal positive change.

However, do note also that for straight line, the gradient is constant as the gradient does not change along the line.

For curve, the gradient is changing as we move along the profile of the curve. Its gradient is therefore not constant, and may vary from negative to positive in numerical value (or vice versa). Diagram 3 below illustrates the point.



Diagram 3: Varying gradient for curve (quadratic function)

In diagram 3, we can see that as we slide along the curve from left to right, the gradient (as shown by the red straight line) changes in its slanting direction. The line on the left is of negative gradient whereas the one on the right is positive in gradient.

The importance of understanding gradient lies in predicting future happening from the trend the curve is moving. An example in real-life engineering application is the termination of the charging process of battery through monitoring the charging profile and its gradient value.

Easy to understand? I hope with the introduction of cartoons in this post, maths is made interesting.

:-)