Reflection or vertical transformation

Questions involving sketching a polynomial and then applying a reflection (such as y → -y or x → -x) or vertical translation, requiring analysis of how the curve changes under these transformations.

4 questions

OCR C1 2006 June Q4
4
  1. By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 ,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C _ { 1 }\).
  3. On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve \(C _ { 2 }\).
OCR C1 2013 January Q3
3
  1. Sketch the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\), giving the coordinates of all points of intersection with the axes.
  2. Describe the transformation that transforms the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\) to the curve \(y = ( 1 - x ) ( 2 + x ) ( 3 - x )\).
SPS SPS SM 2022 February Q4
4.
  1. By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C _ { 1 }\).
  3. On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve \(C _ { 2 }\).
AQA Paper 1 2024 June Q11
11 It is given that $$f ( x ) = x ( x - a ) ( x - 6 )$$ where \(0 < a < 6\) 11
  1. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
    \includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-14_1207_1105_733_447} 11
  2. Sketch the graph of \(y = \mathrm { f } ( - 2 x )\) on the axes below.
    \includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-15_1207_1107_413_445}