Graphs of rational functions: AQA A‑Level Further Maths questions

AQA A-Level Further Maths past-paper questions on D.5 Graphs of rational functions, 2019–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 16 questions on Graphs of rational functions, as a list

June 2025, Paper 2, Q8 (5 marks): The function f is defined by f ( ) = + − 3 2 12 x x x x ( ) x∈ Figure 1 shows the graph of ( ) f y x = Figure 1 x O y 8 (a) The graph of ( ) f y x = is transfo
June 2025, Paper 2, Q10 (13 marks): The function f is defined by ( ) f( ) − = + + 2 2 2 4 6 9 x x x x 10 (a) Write down the equations of the asymptotes to the graph of f ( ) y x = [2 marks] 10 (b)
June 2025, AS Paper 1, Q3 (1 mark): Find the equations of the asymptotes of the curve with equation y = (x – 1)(x + 2) (x + 1)(x – 2) Tick () one box. [1 mark] x = 1, x = –2, y = 1 x = 1, x = –2,
June 2024, Paper 2, Q15 (7 marks): The diagram shows the line y = 5 – x O y x 15 (a) On the diagram above, sketch the graph of y = │x2 – 4x│, including all parts of the graph where it intersects
June 2024, Paper 2, Q16 (9 marks): The function f is defined by f(x) = ax + 5 x + b where a and b are constants. The graph of y = f(x) has asymptotes x = –2 and y = 3 16 (a) Write down the value
June 2024, AS Paper 1, Q10 (6 marks): The curve C has equation y = 2x – 10 3x – 5 Figure 1 shows the curve C with its asymptotes. Figure 1 x 5 y O 2 10 (a) Write down the equations of the asymptotes
June 2023, AS Paper 1, Q10 (9 marks): The curve C has equation y = 3x2 + mx + p x2 + px + m where m and p are integers. The vertical asymptotes of C are x = – 4 and x = –1 The curve C is shown in th
June 2022, Paper 2, Q7 (8 marks): The function f is defined by f (x) = ax − 5 2x + b x 2 R, x 6= 9 2 where a and b are integers. The graph of y = f (x) has asymptotes x = 9 2 and y = 3 7 (a) Fin
June 2022, AS Paper 1, Q13 (10 marks): A curve C1 has equation y = 2x + 7 3x + 5 13 (a) Write down the equations of the asymptotes of curve C1 [2 marks] 13 (b) On the axes below, sketch the graph of
June 2022, AS Paper 1, Q14 (15 marks): The function f is defined by f (x) = x2 − 3 x2 + px + 7 x 2 R where p is a constant. The graph of y = f (x) has only one asymptote. 14 (a) Write down the equati
November 2021, Paper 1, Q7 (7 marks): The diagram below shows the graph of y = f (x) ( − 4 ≥ x ≥ 4) The graph meets the x-axis at x = 1 and x = 3 The graph meets the y-axis at y = 2 x 4 – 4 3 1 2 y
November 2021, AS Paper 1, Q16 (8 marks): Curve C has equation y = ax x + b where a and b are constants. The equations of the asymptotes to C are x = −2 and y = 3 x y y = ax x + b O 16 (a) Write down th
November 2020, Paper 1, Q9 (13 marks): The function f is defined by f (x) = x(x + 3) x + 4 (x 2 R, x 6= −4) 9 (a) Find the interval (a, b) in which f (x) does not take any values. Fully justify your
November 2020, AS Paper 1, Q10 (8 marks): (a) Show that the equation y = 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] 10 (b) Write dow
June 2019, Paper 2, Q2 (1 mark): Which of the straight lines given below is an asymptote to the curve y = ax2 x − 1 where a is a non-zero constant? Circle your answer. [1 mark] y = ax + a y = a
June 2019, AS Paper 1, Q11 (8 marks): (a) Curve C has equation y = x2 + px − q x2 − r where p, q and r are positive constants. Write down the equations of its asymptotes. [2 marks] 11 (b) Find the s

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