2025 EJC Promo Q4

The diagram below shows the curve \(y=\mathrm{f}(x)\). The curve crosses the \(x\)-axis at point \(A(a,0)\), where \(a<-1\) and has a maximum point \(B\left(-\frac{1}{2},-\frac{10}{9}\right)\). The curve also has a horizontal asymptote \(y=-1\) and vertical asymptotes \(x=-1\) and \(x=0\).

On separate diagrams, sketch the graphs of

1. \(y=3\mathrm{f}(x-1),\)[2]
2. \(y=\dfrac{1}{\mathrm{f}(x)},\)[3]

showing clearly in each case, the equations of any asymptotes, the coordinates of any stationary points and axial intercepts in terms of \(a\).

The curve \(y=\mathrm{f}(x)\) undergoes the following transformations in succession:

\(P\): Reflection in the \(y\)-axis

\(Q\): Scaling parallel to the \(x\)-axis by a scale factor of \(k\)

4. Determine the coordinates of the \(x\)-intercept of the transformed curve.[2]