2025 MI P2 Q1

The diagram shows the curve with equation \(y = \mathrm{f}(x)\). The curve passes through the points \(P(p, 0)\), \(Q(q, 0)\) and \(R(0, r)\).

1. The curve \(y = \mathrm{f}(x)\) is transformed onto the curve with equation \(y = \mathrm{f}(2x - 1)\). Find the coordinates of the points on the graph of \(y = \mathrm{f}(2x - 1)\) which correspond to the points \(P\), \(Q\) and \(R\) on curve \(y = \mathrm{f}(x)\).[3]
2. It is given that \(I = \displaystyle \displaystyle\int_p^q \mathrm{f}(x) \, \mathrm{d}x\). Find, in terms of \(I\), the area of the finite region bounded by
   1. the curve with equation \(y = -\mathrm{f}(x)\) and the \(x\)-axis,[1]
   2. the curve with equation \(y = \frac{1}{2}\mathrm{f}(x + 3)\) and the \(x\)-axis.[1]
3. Find the value of \(\displaystyle \displaystyle\int_0^q \mathrm{f}'(x) \, \mathrm{d}x\), justifying your answer.[1]