2019 YIJC P1 Q3

The function \(\mathrm{f}\) is defined by

\({\mathrm{f}(x)=\begin{cases} (x-2a)^2 & \text{for}\hspace{0.5em} 0 \leq x < 2a, \\ 2ax-4a^2 & \text{for}\hspace{0.5em} 2a \leq x < 4a, \end{cases}}\)

Where \(a\) is a positive real constant and that \(\mathrm{f}\left( x+4a \right)=\mathrm{f}\left( x \right)\) for all real values of \(x\).

1. Sketch the graph of \(y=\mathrm{f}\left( x \right)\) for \(-3a\le x\le 8a\).[3]
2. Hence find the value of \(\displaystyle\displaystyle\int\limits_{-2a}^{8a}{\mathrm{f}\left( \left| x \right| \right)}\mathrm{ d}x\) in terms of \(a\).[3]