2025 VJC Promo Q9

A curve \(C\) has equation \(y=\dfrac{x^2-4x+1}{x+1}\), \(x \ne -1\).

1. Prove, using an algebraic method, that \(y=\dfrac{x^2-4x+1}{x+1}\) cannot lie between two certain values, to be determined in exact form.[3]
2. Sketch \(C\), stating the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.[4]
3. The curve \(C\) undergoes, in succession, the following transformations.
   1. A translation of 1 unit in the positive \(x\)-direction,

   2. A stretch parallel to the \(x\)-axis, factor \(3\), \(y\)-axis invariant,

   3. A reflection in the \(y\)-axis.
   Find the equation of the new curve in the form \(y=\mathrm{f}(x)\).[3]