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)).

8 questions · Standard +0.6

1.02n Sketch curves: simple equations including polynomials
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CAIE Further Paper 1 2020 June Q3
9 marks Standard +0.3
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
9 marks Standard +0.3
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
11 marks Standard +0.8
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 \(C\).
Pre-U Pre-U 9795/1 2010 June Q5
8 marks Standard +0.8
5 A curve has equation \(y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }\) for \(x \neq - 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points on the curve.
  2. Sketch the curve, justifying all significant features.
Pre-U Pre-U 9795/1 2011 June Q7
11 marks Challenging +1.2
Sketch the curve with equation \(y = \frac{x^2 + 4x}{2x - 1}\), justifying all significant features. [11]
Pre-U Pre-U 9795/1 2015 June Q5
11 marks Standard +0.8
A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).
  1. Determine the equations of the asymptotes. [3]
  2. Find the coordinates of the turning points. [5]
  3. Sketch the curve. [3]
Pre-U Pre-U 9795/1 2018 June Q2
10 marks Standard +0.3
  1. Determine the asymptotes and turning points of the curve with equation \(y = \frac{x^2+3}{x+1}\). [7]
  2. Sketch the curve. [3]
Pre-U Pre-U 9795 Specimen Q8
9 marks Standard +0.3
The curve \(C\) has equation $$y = \frac{x^2 - 2x - 3}{x + 2}.$$
  1. Find the equations of the asymptotes of \(C\). [4]
  2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]