NJC Graphing Techniques Tutorial Q4

The diagram below shows the graph of the function \(y=\mathrm{f}\left( x \right)\), where

\(\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c+2}{x+d}\) and \(a\), \(b\), \(c\) and \(d\) are integer constants.

The curve has a vertical asymptote at the \(y\)-axis and an oblique asymptote which intersects the axes at \(x=1\) and \(y=-1\). Find the values of \(a\), \(b\) and \(d\).

1. By considering\(\frac{\mathrm{d}y}{\mathrm{d}x}\), show that \(c<-2\).
2. Using the values of \(a\), \(b\) and \(d\) found in (i), sketch \({{C}_{1}}\), the graph of \(y=\mathrm{f}\left( x \right)\) if \(c=-1\), labelling clearly the equation of asymptotes, coordinates of the turning points and axial intercepts (if any).

On the same diagram, sketch the curve \({{C}_{2}}\), defined by the parametric equations

\(x=t-\cos t\), \(y=2\sin t\), \(0\le t\le \pi \).

Deduce the number of intersection points between \({{C}_{1}}\) and \({{C}_{2}}\).