The gradient of a function

• For the linear function, \(y = f(x) = mx+c\), the gradient is represented by \(m\), and is constant.
• The figure (below) shows how the gradient is calculated for a linear function.
  
  

For example, \( f(x) = 0.25(x+5)(x+2)(x+1)(x-1)(x-3) \) is shown in the figure below. The average rate of change of the curve between two points \(x = -5\) and \(x = -4\), can be approximated by finding the gradient of the straight line connecting these two points.

The average rate of change, \(m\) for the curve between the points \((-5, 0)\) and \((-4, 52.5)\) will be: 
\[ m = \frac{\Delta f}{\Delta x} = \frac{f(x_{2}) - f(x_{1})}{x_{2}- x_{1}} = \frac{52.5 - 0}{-4-(-5)}=52.5\]
Therefore, between \(-5\) and \(-4\) the curves changes at an average rate of \(52.5\) units of \(y\) for each unit of \(x\).