AS June 2020 Paper 1 Q10

AQACurrent spec8 marksGraphs & Inequalities

(a) Show that the equation\[y = \frac{3x - 5}{2x + 4}\]

can be written in the form

\[(x + a)(y + b) = c\]

where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]

(b) Write down the equations of the asymptotes of the graph of\[y = \frac{3x - 5}{2x + 4}\]

[2 marks]

[3 marks]

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

Mark scheme (a)

Scheme	Marks	AO
Selects a method to find the values of \(a, b, c\). e.g. by expanding \((x + a)(y + b) = c\) or by multiplying the equation by \((2x + 4)\) and expanding or by dividing the numerator of the equation by its denominator.	M1	3.1a
Expresses the original equation in a form that allows comparison with \((x + a)(y + b) = c\).	M1	1.1a
Completes a rigorous argument to show that \(y = \frac{3x - 5}{2x + 4}\) can be written as \((x + 2)\left(y - \frac{3}{2}\right) = -\frac{11}{2}\).	R1	2.1

Typical solution

\[y(2x + 4) = 3x - 5\]\[2xy + 4y = 3x - 5\]\[xy + 2y - \tfrac{3}{2}x + \tfrac{5}{2} = 0\]\[xy + 2y - \tfrac{3}{2}x - 3 = -\tfrac{11}{2}\]\[(x + 2)\left(y - \tfrac{3}{2}\right) = -\tfrac{11}{2}\]

Sketch of y = (3x − 5)/(2x + 4) with dashed asymptotes x = −2 and y = 3/2; the left branch lies above y = 3/2 and rises towards x = −2; the right branch rises from below, crossing the y-axis at −5/4 and the x-axis at 5/3, towards y = 3/2 — Mark scheme (c)

Scheme	Marks	AO
Obtains a correct asymptote.	B1	1.1b
Obtains the other correct asymptote and no incorrect asymptotes.	B1	1.1b

Scheme	Marks	AO
Sketches a curve asymptotic to \(x = -2\) or \(y = \frac{3}{2}\). FT their asymptotes.	B1F	1.1b
Sketches a curve with two branches, asymptotic to their asymptotes.	M1	1.1a
Deduces the shape of the curve and sketches it correctly with a root at \(\frac{5}{3}\) and \(y\)-intercept at \(-\frac{5}{4}\). FT their asymptotes.	A1F	2.2a
	(8 marks)