Sketch rational with quadratic numerator

Rational functions with quadratic numerator and linear or quadratic denominator, requiring polynomial division or algebraic manipulation to find oblique or horizontal asymptotes (e.g., y = x²/(2x+1), y = x²/(x-2), y = (x²-3x+6)/(1-x)).

3 questions

CAIE Further Paper 1 2020 June Q3
3 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { 2 \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\).
CAIE FP1 2012 June Q6
6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Find the coordinates of the turning points on \(C\). Draw a sketch of \(C\).
CAIE FP1 2017 June Q9
9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 6 } { 1 - x }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the turning points of \(C\).
  3. Find the coordinates of any intersections with the coordinate axes.
  4. Sketch \(C\).