2016 A-Level H2 Mathematics

The curve \(y={{x}^{4}}\) is transformed onto the curves with equation \(y=\mathrm{f}\left( x \right)\). The turning point on \(y={{x}^{4}}\) corresponds to the point with coordinates \(\left( a,b \right)\) on \(y=\mathrm{f}\left( x \right)\). The curve \(y=\mathrm{f}\left( x \right)\) also passes through the point with coordinate \(\left( 0,c \right)\). Given that \(\mathrm{f}\left( x \right)\) has the form \(k{{\left( x-l \right)}^{4}}+m\) and that \(a\), \(b\) and \(c\) are positive constants with \(c>b\), express \(k\), \(l\) and \(m\) in terms of \(a\), \(b\) and \(c\).[2]

By sketching the curve \(y=\mathrm{f}\left( x \right)\), or otherwise, sketch the curve \(y=\frac{1}{\mathrm{f}\left( x \right)}\). State, in terms of \(a\), \(b\) and \(c\), the coordinates of any points where \(y=\frac{1}{\mathrm{f}\left( x \right)}\) crosses the axes and of any turning points.[4]