On the axes below, sketch the graph of \(y = \frac{1}{{f(x)}}\) , clearly showing the coordinates of the images of the points A, B and D, labelling them \({{\text{A}'}}\), \({{\text{B}'}}\), and \({{\text{D}'}}\) respectively, and the equations of any vertical asymptotes.

On the axes below, sketch the graph of the derivative \(y = f'(x)\) , clearly showing the coordinates of the images of the points  A and D, labelling them \({{\text{A}}}''\) and \({{\text{D}}}''\) respectively.

     A1A1A1 

Note: Award A1 for correct shape.

Award A1 for two correct asymptotes, and \(x = 1\) and \(x = 3\) .

Award A1 for correct coordinates, \({\text{A}'}\left( { - 1,\frac{1}{4}} \right),{\text{ B}'}\left( {0,\frac{1}{3}} \right){\text{ and D}'}\left( {2, -\frac{1}{3}} \right)\).

[3 marks]

     A1A1A1

Note: Award A1 for correct general shape including the horizontal asymptote.
          Award A1 for recognition of 1 maximum point and 1 minimum point.
          Award A1 for correct coordinates, \({\text{A}}''( - 1,0)\) and \({\text{D}}''(2,0)\) .

[3 marks]

Solutions to this question were generally disappointing. In (a), the shape of the graph was often incorrect and many candidates failed to give the equations of the asymptotes and the coordinates of the image points. In (b), many candidates produced incorrect graphs although the coordinates of the image points were often given correctly.

Solutions to this question were generally disappointing. In (a), the shape of the graph was often incorrect and many candidates failed to give the equations of the asymptotes and the coordinates of the image points. In (b), many candidates produced incorrect graphs although the coordinates of the image points were often given correctly.