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.

3 questions · Moderate -0.6

1.02n Sketch curves: simple equations including polynomials
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OCR C1 2006 June Q4
8 marks Easy -1.2
  1. By expanding the brackets, show that $$(x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12.$$ [3]
  2. Sketch the curve $$y = x^3 - 6x^2 + 5x + 12,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve $$y = -x^3 + 6x^2 - 5x - 12.$$ Label this curve \(C_2\). [2]
AQA Further Paper 1 2023 June Q11
7 marks Standard +0.8
The function f is defined by $$f(x) = 4x^3 - 8x^2 - 51x - 45 \quad (x \in \mathbb{R})$$
    1. Fully factorise \(f(x)\) [2 marks]
    2. Hence, solve the inequality \(f(x) < 0\) [2 marks]
  2. The graph of \(y = f(x)\) is translated by the vector \(\begin{pmatrix} 7 \\ 0 \end{pmatrix}\) The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = g(x)\) Solve the inequality \(g(x) \leq 0\) [3 marks]
SPS SPS SM 2022 February Q4
8 marks Easy -1.3
  1. By expanding the brackets, show that \((x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12\). [3]
  2. Sketch the curve \(y = x^3 - 6x^2 + 5x + 12\), giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve \(y = -x^3 + 6x^2 - 5x - 12\). Label this curve \(C_2\). [2]