A2 June 2025 Paper 2 Q10

AQACurrent spec13 marksGraphs & Inequalities

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

\[\mathrm{f}(x) = \frac{2(x^2 - 4)}{x^2 + 6x + 9}\]

(a) Write down the equations of the asymptotes to the graph of \(y = \mathrm{f}(x)\) [2 marks]

(b) Without using calculus, show that the range of \(\mathrm{f}\) is \(\left\{k : k \geqslant -\dfrac{8}{5}\right\}\) [4 marks]

Without using calculus, find the coordinates of this stationary point. [3 marks]

(d) Sketch the graph of \(y = \mathrm{f}(x)\) on the axes below. [4 marks]

Blank axes: x-axis and y-axis crossing at O

Scheme	Marks	AO
Obtains \(x = -3\) or \(y = 2\)	M1	1.1a
Obtains \(x = -3\) and \(y = 2\) and no other asymptotes.	A1	1.1b
	(2)

Mark scheme (b)

Scheme	Marks	AO
Forms a quadratic equation in \(x\) involving \(k\) or equivalent.	M1	3.1a
Obtains a quadratic equation with \(k\) in a coefficient and sets the discriminant to zero, either in an equation or inequality.	M1	1.1a
Solves their equation or inequality in \(k\)	M1	1.1a
Completes a reasoned argument, without using calculus, to state that the range is \(\left\{k : k \geqslant -\dfrac{8}{5}\right\}\) AG Allow letters other than \(k\) used.	R1	2.1
	(4)

Typical solution

Let \(y = k\)

\[k(x^2 + 6x + 9) = 2x^2 - 8\]\[(k - 2)x^2 + 6kx + (9k + 8) = 0\]\[\Delta \geqslant 0\]\[\therefore (6k)^2 - 4(k - 2)(9k + 8) \geqslant 0\]\[36k^2 - 36k^2 + 40k + 64 \geqslant 0\]\[40k + 64 \geqslant 0\]\[k \geqslant -\frac{8}{5}\]

The range of \(\mathrm{f}\) is \(\left\{k : k \geqslant -\dfrac{8}{5}\right\}\)

Mark scheme (c)

Scheme	Marks	AO
Obtains a quadratic equation by substituting \(k = -\dfrac{8}{5}\)	M1	3.1a
Solves their quadratic equation	M1	1.1a
Deduces \(\left(-\dfrac{4}{3}, -\dfrac{8}{5}\right)\)	A1	2.2a
	(3)

Typical solution

\[k = -\frac{8}{5}\]\[\frac{-18}{5}x^2 + \frac{-48}{5}x + \frac{-32}{5} = 0\]\[9x^2 + 24x + 16 = 0\]\[(3x + 4)^2 = 0\]\[x = -\frac{4}{3}\]\[\text{Stationary point} = \left(-\frac{4}{3}, -\frac{8}{5}\right)\]

Sketch of y = f(x) with asymptotes x = −3 and y = 2; the left branch rises from y = 2 towards +infinity at x = −3; the right branch comes down from +infinity, crosses the x-axis at −2, has a minimum at (−4/3, −8/5), crosses the y-axis at −8/9 and the x-axis at 2, then approaches y = 2 from below — Mark scheme (d)

Scheme	Marks	AO
Draws approximately correct shape of the right-hand branch	B1	1.1b
Draws both branches of the graph approaching the correct asymptotes.	B1	1.1b
Shows at least two correctly labelled axis intercepts.	B1	1.1b
Draws completely correct graph including correctly labelled asymptotes, a stationary point in the third quadrant, and all three correctly labelled axis intercepts. Condone no labelling of stationary point.	B1	1.1b
	(4)
	(13 marks)