2025 VJC P1 Q8

1. A curve \(C\) with equation \(y = \mathrm{f}(x)\) undergoes in succession, the following transformations.
   A: A reflection in the \(x\)-axis.
   B: A stretch parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\), with the \(y\)-axis invariant.
   The resulting curve has equation \(y = ax^2 + \frac{b}{x}\), where \(a\) and \(b\) are real constants. Given that \((-1, \frac{1}{3})\) is a turning point of \(y = \frac{1}{\mathrm{f}(x)}\), find the values of \(a\) and \(b\) and state the equation of \(C\).[5]

The diagram below shows the curve of \(y = \mathrm{g}(x)\). The curve has a minimum point at \((-2, -3)\) and crosses the \(x\)-axis at \((-1, 0)\) and \((-4, 0)\). The line \(x = 2\) is the vertical asymptote and the line \(y = 3\) is the horizontal asymptote.

1. Sketch the graph of \(y = \mathrm{g}'(x)\), labelling the coordinates of all relevant point(s) and state the equations of any asymptotes.[2]
2. Find the area of the region bounded by the graph of \(y = \mathrm{g}'(x)\), the lines \(x = -4\), \(x = -2\) and the \(x\)-axis.[2]