AS June 2020 Paper 1 Q12

12 The mean value of the function \(\mathrm{f}\) over the interval \(1 \leqslant x \leqslant 5\) is \(m\).

The graph of \(y = \mathrm{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathrm{f}(x)\).

The graph of \(y = \mathrm{h}(x)\) is a translation of \(y = \mathrm{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\)

Determine, in terms of \(m\), the mean value of the function \(\mathrm{h}\) over the interval \(4 \leqslant x \leqslant 8\) [2 marks]