Solve |f(x)| > k using sketch

Questions that require sketching y = |f(x)| and then solving an inequality of the form |f(x)| > k or |f(x)| < k for a specific constant k.

13 questions · Challenging +1.1

1.02n Sketch curves: simple equations including polynomials1.02s Modulus graphs: sketch graph of |ax+b|1.07n Stationary points: find maxima, minima using derivatives
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CAIE Further Paper 1 2020 June Q1
6 marks Standard +0.8
1 Let \(a\) be a positive constant.
  1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }\).
  2. Sketch the curve with equation \(y = \left| \frac { a x } { x + 7 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }\).
CAIE Further Paper 1 2020 June Q6
15 marks Challenging +1.2
6 The curve \(C\) has equation \(\mathrm { y } = \frac { 10 + \mathrm { x } - 2 \mathrm { x } ^ { 2 } } { 2 \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no turning points.
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right| < 4\).
CAIE Further Paper 1 2021 June Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|\) and find the set of values of \(x\) for which \(2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2021 June Q7
14 marks Challenging +1.3
7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right| < 3\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2022 June Q5
12 marks Challenging +1.2
5 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 }\).
  1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k\) has 4 distinct real solutions.
CAIE Further Paper 1 2023 June Q6
15 marks Challenging +1.2
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no stationary points.
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
  5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).
CAIE Further Paper 1 2024 June Q6
15 marks Challenging +1.2
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no stationary points.
  3. Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
    1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
    2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
    3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).
CAIE Further Paper 1 2021 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(y = \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2021 November Q6
14 marks Challenging +1.2
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that there is no point on \(C\) for which \(0 < y < 12\).
  3. Sketch C.
    1. Sketch the graphs of \(y = \left| \frac { x ^ { 2 } } { x - 3 } \right|\) and \(y = | x | - 3\) on a single diagram, stating the coordinates of the intersections with the axes.
    2. Use your sketch to find the set of values of \(c\) for which \(\left| \frac { x ^ { 2 } } { x - 3 } \right| \leqslant | x | + c\) has no solution. [1]
CAIE Further Paper 1 2021 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { 4 \mathrm { x } + 5 } { 4 - 4 \mathrm { x } ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2022 November Q7
16 marks Standard +0.8
7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\).
  4. Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2022 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } - \mathrm { x } } { \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the exact coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x } { x + 1 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x } { x + 1 } \right| < 6\).
    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 Specimen Q7
17 marks Challenging +1.2
7 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\).
  1. State the equations of the asymptotes of \(C\).
  2. Show that \(y \leqslant \frac { 25 } { 12 }\) at all points on \(C\).
  3. Find the coordinates of any stationary points of \(C\).
  4. Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.
  5. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2\).