Rational functions with parameters: analysis depending on parameter sign/range

Questions where the curve behaviour (asymptotes, turning points, sketch) is analysed for different ranges or signs of the parameter, or where conditions on the parameter are derived (e.g., show no turning points if λ < 0, find values of p for two distinct turning points).

8 questions · Standard +0.8

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CAIE Further Paper 1 2022 June Q3
10 marks Challenging +1.2
3 A curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { a } \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\), where \(a\) is a positive constant.
  1. Find the equations of the asymptotes of \(C\).
  2. Show that there is no point on \(C\) for which \(1 < \mathrm { y } < 1 + 4 \mathrm { a }\).
  3. Sketch C. You do not need to find the coordinates of the intersections with the axes.
CAIE FP1 2009 June Q10
11 marks Standard +0.8
10 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant. Obtain the equation of each of the asymptotes of \(C\). In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\). In both cases the coordinates of the turning points must be indicated.
CAIE FP1 2012 November Q7
9 marks Standard +0.8
7 The curve \(C\) has equation $$y = \lambda x + \frac { x } { x - 2 }$$ where \(\lambda\) is a non-zero constant. Find the equations of the asymptotes of \(C\). Show that \(C\) has no turning points if \(\lambda < 0\). Sketch \(C\) in the case \(\lambda = - 1\), stating the coordinates of the intersections with the axes.
AQA Further AS Paper 1 2022 June Q14
15 marks Challenging +1.2
14 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 2 } + p x + 7 } \quad x \in \mathbb { R }$$ where \(p\) is a constant.
The graph of \(y = \mathrm { f } ( x )\) has only one asymptote.
14
  1. Write down the equation of the asymptote.
    14
  2. Find the set of possible values of \(p\) □
    14
  3. Find the coordinates of the points at which the graph of \(y = \mathrm { f } ( x )\) intersects the axes. \section*{Question 14 continues on the next page} 14
  4. \(\quad A\) curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 } { x ^ { 2 } - 3 x + 7 }$$ The curve \(C\) has a local minimum at the point \(M\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-24_371_835_587_605} The line \(y = k\) intersects curve \(C\) 14 (d) (i) Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14 (d) (ii) Hence, find the \(y\)-coordinate of point \(M\)
CAIE FP1 2018 November Q6
Standard +0.8
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) intersects the \(x\)-axis twice.
  3. Justifying your answer, find the number of stationary points on \(C\).
  4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.
Pre-U Pre-U 9795/1 Specimen Q9
13 marks Standard +0.8
9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant.
  1. Obtain the equation of each of the asymptotes of \(C\).
  2. Find the coordinates of the turning points of \(C\).
  3. In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\).
CAIE FP1 2018 November Q6
9 marks Standard +0.8
The curve \(C\) has equation $$y = \frac{x^2 + ax - 1}{x + 1},$$ where \(a\) is constant and \(a > 1\).
  1. Find the equations of the asymptotes of \(C\). [3]
  2. Show that \(C\) intersects the \(x\)-axis twice. [1]
  3. Justifying your answer, find the number of stationary points on \(C\). [2]
  4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. [3]
OCR FP2 2010 January Q8
10 marks Standard +0.3
The equation of a curve is $$y = \frac{kx}{(x-1)^2},$$ where \(k\) is a positive constant.
  1. Write down the equations of the asymptotes of the curve. [2]
  2. Show that \(y \geq -\frac{1}{4}k\). [4]
  3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve. [4]