Multiple transformations including squared

Questions that sketch a rational function and then require sketching both y = |f(x)| and y² = f(x) or other multiple transformations.

5 questions · Challenging +1.1

1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives
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CAIE Further Paper 1 2023 June Q7
13 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } + 1 } { \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the turning points on \(C\).
  3. Sketch \(C\).
  4. Sketch the curves with equations \(y = \left| \frac { x ^ { 2 } + 2 x + 1 } { x - 3 } \right|\) and \(y ^ { 2 } = \frac { x ^ { 2 } + 2 x + 1 } { x - 3 }\) on a single diagram, clearly identifying each curve. If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2020 November Q6
13 marks Standard +0.8
6 Let \(a\) be a positive constant.
  1. The curve \(C _ { 1 }\) has equation \(\mathrm { y } = \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } }\). Sketch \(C _ { 1 }\). The curve \(C _ { 2 }\) has equation \(\mathrm { y } = \left( \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right) ^ { 2 }\). The curve \(C _ { 3 }\) has equation \(\mathrm { y } = \left| \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right|\).
    1. Find the coordinates of any stationary points of \(C _ { 2 }\).
    2. Find also the coordinates of any points of intersection of \(C _ { 2 }\) and \(C _ { 3 }\).
  3. Sketch \(C _ { 2 }\) and \(C _ { 3 }\) on a single diagram, clearly identifying each curve. Hence find the set of values of \(x\) for which \(\left( \frac { x - a } { x - 2 a } \right) ^ { 2 } \leqslant \left| \frac { x - a } { x - 2 a } \right|\).
CAIE Further Paper 1 2023 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\), giving your answers correct to 1 decimal place.
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
  5. Find the set of values for which \(\frac { 1 } { \mathrm { f } ( x ) } < \mathrm { f } ( x )\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2023 November Q7
16 marks Challenging +1.2
7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } } { x + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\).
  4. Find the coordinates of any stationary points on the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
  5. Sketch the curve with equation \(y = \frac { 1 } { f ( x ) }\) and find, in exact form, the set of values for which $$\frac { 1 } { \mathrm { f } ( x ) } > \mathrm { f } ( x ) .$$ If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2024 November Q6
13 marks Challenging +1.2
The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).
  1. Find the equations of the asymptotes of \(C\). [2]
  2. Find the coordinates of any stationary points on \(C\). [4]
  3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
  4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]