2025 SAJC P1 Q6

The graph of \(y = \mathrm{f}(x)\) cuts the \(x\)-axis at the point \((6,0)\) and has a maximum point and a minimum point at \((0,0)\) and \((3,-5)\) respectively. The graph of \(y = \mathrm{g}(x)\) cuts the \(x\)-axis at the point \((2,0)\), and has a minimum point and a maximum point at \((0,0)\) and \((1,5)\) respectively. The diagrams below show the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{g}(x)\).

1. Describe fully a sequence of two transformations which transforms the graph of \(y = \mathrm{f}(x)\) onto the graph of \(y = \mathrm{g}(x)\).[2]

It is given that \(\mathrm{g}(x)\) can be expressed in the form \(a\mathrm{f}(bx+c)+d\) where \(a\), \(b\), \(c\) and \(d\) are constants.

2. State the values of \(a\), \(b\), \(c\) and \(d\).[2]
3. The derivative of the function \(y = \mathrm{f}(x)\) with respect to \(x\) is denoted by \(\mathrm{f}'(x)\). It is given that \(\mathrm{f}'(6) = 4\). Find the exact value of \(\mathrm{g}'(2)\).[3]