1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\).

1. 1. On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.
      • \(a = - 2\)
2. \(a = - 1\)
3. \(a = 0\)
   (ii) State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
   (iii) State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
4. 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
   2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).
Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).
(iii) With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which
• \(y \geqslant 0\) for all \(x \geqslant 0\)
• \(y \leqslant 0\) for all \(x \geqslant 0\)
• In the case of \(a = 0\), the curve \(\mathrm { y } = \sqrt [ 4 ] { \mathrm { f } ( \mathrm { x } ) }\) has a cusp.
Find its coordinates and fully justify that it is a cusp.