2022 HCI J1 BT Q8

The function \(y=\mathrm{f}\left( x \right)\) is defined by

\(y=\frac{{{x}^{2}}+x+p}{x+2}\), where \(p\) is a constant.

1. Find the value of \(p\) if \(\mathrm{f}\) is a linear function.[1]
2. If \(p\) is greater than the value found in part (i), use an algebraic method to determine the range of values that \(y\) can take, leaving your answer in terms of \(p\).[3]

Take \(p=12\) for the rest of this question.

3. Sketch the graph of \(y=\frac{{{x}^{2}}+x+12}{x+2}\). Give the equation(s) of any asymptotes, the coordinates of any turning point(s) and the points of intersection with the axes.[3]
4. Find the number of real roots of the equation
   \({{\left( {{x}^{2}}+x+12 \right)}^{2}}-{{\left( x+2 \right)}^{2}}\left( 36-{{x}^{2}} \right)=0\)
   by adding a suitable graph on the sketch in part (iii).[2]
5. Describe a sequence of \(3\) transformations which transforms the graph of \(y=\frac{{{x}^{2}}+x+12}{x+2}\) onto \(y=-2x+3-\frac{7}{x+1}\).[3]