A2 June 2025 Paper 2 Q8

AQACurrent spec5 marksGraphs & Inequalities

8 The function \(\mathrm{f}\) is defined by

\[\mathrm{f}(x) = x^3 + x^2 - 12x \qquad (x \in \mathbb{R})\]

Figure 1 shows the graph of \(y = \mathrm{f}(x)\)

Figure 1: graph of the cubic y = f(x), with a local maximum to the left of the y-axis, passing through O, with a local minimum to the right of the y-axis, and crossing the x-axis once on each side of O — Figure 1

(a) The graph of \(y = \mathrm{f}(x)\) is transformed by a stretch, scale factor 2, parallel to the \(x\)-axis with the \(y\)-axis fixed, to give the graph of \(y = \mathrm{g}(x)\)

On Figure 1, sketch the graph of \(y = \mathrm{g}(x)\), showing the values of \(x\) where the graph crosses the \(x\)-axis. [3 marks]

(b) Find the set of values of \(x\) such that the conditions \(\mathrm{f}(x) \gt 0\) and \(\mathrm{g}(x) \lt 0\) are both satisfied. [2 marks]

Sketch showing the original cubic and the stretched cubic y = g(x), which passes through O and crosses the x-axis at −8 and 6, with turning points at about the same heights as the original — Mark scheme (a)

Scheme	Marks	AO
Draws a cubic graph of the correct shape. Condone any horizontal stretch.	B1	1.1b
Draws a cubic graph that crosses \(x\)-axis at \(-8\), 0 and 6 Accept no indication of \(x = 0\) at the origin.	B1	1.1b
Shows local maximum and minimum points which are approximately the same height as in the original graph.	B1	1.1b
	(3)

Mark scheme (b)

Scheme	Marks	AO
Obtains a set of values of the form \(\{x : a \lt x \lt 2a\}\) or \(\{x : 2a \lt x \lt a\}\) Condone non-strict inequality. Condone set notation not used.	M1	2.2a
Obtains \(3 \lt x \lt 6\) Condone set notation not used.	A1	2.2a
	(2)
	(5 marks)

Typical solution

\[\{x : 3 \lt x \lt 6\}\]