The curve \(C\) has equation \(y=\frac{4 x^{2}-3 x}{x^{2}+1}\). Verify that the equation of \(C\) may be written in the form \(y=-\frac{1}{2}+\frac{(3 x-1)^{2}}{2\left(x^{2}+1\right)}\) and also in the form \(y=\frac{9}{2}-\frac{(x+3)^{2}}{2\left(x^{2}+1\right)}\).

Hence show that \(-\frac{1}{2} \leqslant y \leqslant \frac{9}{2}\).

Without differentiating, write down the coordinates of the turning points of \(C\).

State the equation of the asymptote of \(C\).

Sketch the graph of \(C\), stating the coordinates of the intersections with the coordinate axes and the asymptote.
[Question 11 is printed on the next page.]