Sketching Rational Functions with Oblique Asymptote

7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + p x + 1 } { x - 2 }\), where \(p\) is a constant. Given that \(C\) has two asymptotes, find the equation of each asymptote. Find the set of values of \(p\) for which \(C\) has two distinct turning points. Sketch \(C\) in the case \(p = - 1\). Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.

9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 3 } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(- 1 < y < 3\). Find the coordinates of the turning points of \(C\). Sketch \(C\).

9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9

1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
   Fully justify your answer.
   9
2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
   \section*{Question 9 continues on the next page} 9
4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470} \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
   1. Fird \(\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}\)

7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$

1. Find the equations of the asymptotes of \(C\).
2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.