Sketching  Graph of Derived Function

At the stationary points, the value of f’(x) is always zero.
This means that the graph of f’(x) will cut the x –axis whenever
f(x) is a maximum, minimum or point of inflection.

   

To sketch the graph of a  derived function

1. Look for stationary points.
   Mark the x-axis underneath these points.
2. Investigate the nature of these stationary points
   • If the gradient is increasing, value of f’(x) is positive and will appear above the x-axis.
   • If the gradient is decreasing, value of f’(x) is negative and will appear below the x-axis.
3. Mark points on graph and lightly sketch shape.

 

Example

The graph of a function is given.

Sketch the graph of the derived function .

 

The function has a minimum turning point, so the derived function will cut the x-axis at this point.

The function is decreasing to the left of this turning point, since the slope is going downwards. The derived function will appear below the x-axis.

The function isincreasing to the right of the turning point, since the slope is going upwards. The derived function will appear above the x-axis.

 

 

Example

 

Example

The graph of a function f intersects the x-axis at
(-a, 0), (h,0) and ( k,0)  and the y-axis at ( 0,d) as shown.

There are minimum turning points at ( -b, c) and ( i,j)
and  a maximum turning point at ( f, g).

Sketch the graph of the derived function  f’

 

Inspecting gradients