Sketch with inequalities or regions

Questions asking to sketch a factored polynomial curve and then use the sketch to solve inequalities or identify solution regions.

7 questions · Moderate -0.3

1.02n Sketch curves: simple equations including polynomials
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Edexcel P1 2021 January Q4
8 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-10_583_866_260_539} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The points \(P\) and \(Q\), as shown in Figure 2, have coordinates ( \(- 2,13\) ) and ( \(4 , - 5\) ) respectively. The straight line \(l\) passes through \(P\) and \(Q\).
  1. Find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found. The quadratic curve \(C\) passes through \(P\) and has a minimum point at \(Q\).
  2. Find an equation for \(C\). The region \(R\), shown shaded in Figure 2, lies in the second quadrant and is bounded by \(C\) and \(l\) only.
  3. Use inequalities to define region \(R\). \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-11_2255_50_314_34}
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OCR H240/01 2018 June Q6
9 marks Standard +0.3
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).
  1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
  3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
  4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.
Edexcel AS Paper 1 2022 June Q7
7 marks Standard +0.8
  1. (a) Factorise completely \(9 x - x ^ { 3 }\)
The curve \(C\) has equation $$y = 9 x - x ^ { 3 }$$ (b) Sketch \(C\) showing the coordinates of the points at which the curve cuts the \(x\)-axis. The line \(l\) has equation \(y = k\) where \(k\) is a constant.
Given that \(C\) and \(l\) intersect at 3 distinct points,
(c) find the range of values for \(k\), writing your answer in set notation. Solutions relying on calculator technology are not acceptable.
OCR C1 2013 June Q9
12 marks Moderate -0.8
  1. Sketch the curve \(y = 2x^2 - x - 6\), giving the coordinates of all points of intersection with the axes. [5]
  2. Find the set of values of \(x\) for which \(2x^2 - x - 6\) is a decreasing function. [3]
  3. The line \(y = 4\) meets the curve \(y = 2x^2 - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(PQ\). [4]
Edexcel C1 Q6
8 marks Moderate -0.8
  1. Sketch on the same diagram the curve with equation \(y = (x - 2)^2\) and the straight line with equation \(y = 2x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Find the set of values of \(x\) for which $$(x - 2)^2 > 2x - 1.$$ [3]
AQA AS Paper 1 2019 June Q5
5 marks Moderate -0.8
  1. Sketch the curve \(y = g(x)\) where $$g(x) = (x + 2)(x - 1)^2$$ [3 marks]
  2. Hence, solve \(g(x) \leq 0\) [2 marks]
AQA AS Paper 1 2021 June Q5
6 marks Moderate -0.3
  1. Sketch the curve $$y = (x - a)^2(3 - x) \quad \text{where } 0 < a < 3$$ indicating the coordinates of the points where the curve and the axes meet. [4 marks] \includegraphics{figure_5}
  2. Hence, solve $$(x - a)^2(3 - x) > 0$$ giving your answer in set notation form. [2 marks]