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Graphs and transformations

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 173

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 68 parts, 173 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q1(a) · 1 marks
1. The point P(6, –4) lies on the curve with equation y = f(x), x ∈ ℝ Find the point to which P is mapped when the curve with equation y = f(x) is transformed to the curve with equation (a) y = f(x + 2) (1)
2. June 2025 · Paper 1 · Q1(b) · 1 marks
–1(x) (b) y = f (1)
3. June 2025 · Paper 1 · Q1(c) · 2 marks
(c) y = 2½f(x)½ – 3 (2)
4. June 2025 · AS Paper 1 · Q1(i) · 3 marks
1. y B (2, 7) y = 3 A (–1, 0) x Figure 1 Figure 1 shows a curve with equation y = f (x) The curve • passes through the point A (–1, 0) • has a maximum turning point at B (2, 7) • has a horizontal asymptote with equation y = 3 On separate diagrams, sketch the curve with equation (i) y = f (x + 2) (3)
5. June 2025 · AS Paper 1 · Q1(ii) · 3 marks
(ii) y = –f (x) (3)
6. June 2025 · Paper 1 · Q7(b) · 3 marks
(b) Find the equation of C. You may leave your answer in factorised form. (3)
7. June 2025 · Paper 1 · Q7(c) · 2 marks
The curve C1 has equation y = f(x) + k, where k is a constant. Given that the graph of C1 intersects the x‑axis at exactly four places, (c) find the range of possible values for k. (2)
8. June 2025 · Paper 1 · Q12(a) · 2 marks
12. y l C x Q Figure 3 Figure 3 shows a sketch of the curve C with equation x x ≠ – 3 y = 2 x ≠ 3 x x ( )( ) and the straight line l with equation y = 2x – 10 (a) Verify that C and l intersect where x = 6 (2)
9. June 2025 · AS Paper 1 · Q12(b) · 2 marks
The gradient function for C has one of the following equations y y y y d d d d x x x O 3 x x x x ( )1 ( ) x x x x d d d d (b) State the correct equation for the gradient function, fully justifying your answer. (2)
10. June 2025 · Paper 1 · Q12(b) · 2 marks
The curve and line also intersect at the point Q shown in Figure 3. (b) Show that the x coordinate of Q is a solution of 4x 3 – 26x 2 – 3x + 90 = 0 (2)
11. June 2025 · Paper 2 · Q8(c) · 4 marks
The equation f (x) = k, where k is a constant, has 4 distinct solutions. (c) Using algebra and showing all stages of your working, find the range of values of k. Give the answer using set notation. (Solutions relying on calculator technology are not acceptable.) (4)
12. June 2025 · Paper 2 · Q12(a) · 2 marks
12. y x Figure 3 Figure 3 shows a sketch of the graph with equation y = f (x) where f (x) = 4 ½x – 3½ – 5 x ∈ ℝ Given that a is a constant and ½a½ = 1 (a) find the possible values of f (a) (2)
13. June 2025 · Paper 2 · Q12(b) · 2 marks
The function g is defined by g (x) = 2x + 17 x ∈ ℝ (b) Find the range of g f (x) (2)
14. June 2025 · Paper 2 · Q12(c) · 4 marks
The function h is defined by h (x) = kx x ∈ ℝ where k is a constant. Given that the equation f (x) = h (x) has no solutions, (c) find the range of values of k. (4)
15. June 2024 · AS Paper 1 · Q2(c) · 2 marks
(c) Hence, (i) state the number of real roots of the equation f (x) = 0 < (ii) write down the largest root of the equation f 1 3 x 87= 0 ;: (2)
16. June 2024 · AS Paper 1 · Q5(a) · 3 marks
5. The curve C1 has equation y = 6 x + 3 (a) (i) Sketch C1 stating the coordinates of any points where the curve cuts the coordinate axes. (ii) State the equations of any asymptotes to the curve C1 (3)
17. June 2024 · AS Paper 1 · Q5(b) · 2 marks
The curve C2 has equation 2 – 4x – 10 y = 3x (b) Show that C1 and C2 intersect when 3x 3 – 4x 2 – 13x – 6 = 0 (2)
18. June 2024 · Paper 1 · Q6(a) · 2 marks
6. y P x Figure 1 Figure 1 shows a sketch of the graph with equation y x The vertex of the graph is at the point P, shown in Figure 1. (a) Find the coordinates of P. (2)
19. June 2024 · Paper 1 · Q6(b) · 2 marks
(b) Solve the equation x x (2)
20. June 2024 · Paper 1 · Q6(c) · 2 marks
A line l has equation y = kx + 4 where k is a constant. Given that l intersects y x 5 at 2 distinct points, (c) find the range of values of k. (2)
21. June 2024 · AS Paper 1 · Q8(b) · 2 marks
The line l cuts C again at the point B. (b) Verify that the x coordinate of B is –4 (2)
22. June 2024 · Paper 2 · Q3 · 4 marks
3. The point P(3, –2) lies on the curve with equation y = f(x), x ∈ ℝ Find the coordinates of the point to which P is mapped when the curve with equation y = f(x) is transformed to the curve with equation (i) y = f(x – 2) (ii) y = f(2x) (iii) y = 3f(–x) + 5 (4)
23. June 2023 · AS Paper 1 · Q4(a) · 2 marks
4. (a) Sketch the curve with equation k y x ; 0 x where k is a positive constant. (2)
24. June 2023 · AS Paper 1 · Q4(b) · 3 marks
(b) Hence or otherwise, solve x (3)
25. June 2023 · AS Paper 1 · Q15(a) · 2 marks
15. In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable. The curve C1 has equation The curve C2 has equation (a) Verify that when x = 1 the curves C1and C2intersect. (2)
26. June 2023 · AS Paper 1 · Q15(b) · 5 marks
The curves also intersect when x = k Given that k < 0 (b) use algebra to find the exact value of k. (5)
27. June 2023 · Paper 2 · Q12(a) · 2 marks
12. N NA NB t Figure 2 The number of subscribers to two different music streaming companies is being monitored. The number of subscribers, NA , in thousands, to company A is modelled by the equation NA = |t – 3| + 4 t 0 where t is the time in years since monitoring began. The number of subscribers, NB , in thousands, to company B is modelled by the equation NB = 8 – |2t – 6| t 0 where t is the time in years since monitoring began. Figure 2 shows a sketch of the graph of NA and the graph of NB over a 5-year period. Use the equations of the models to answer parts (a), (b), (c) and (d). (a) Find the initial difference between the number of subscribers to company A and the number of subscribers to company B. (2)
28. June 2023 · Paper 2 · Q12(b) · 2 marks
When t = T company A reduced its subscription prices and the number of subscribers increased. (b) Suggest a value for T, giving a reason for your answer. (2)
29. June 2023 · Paper 2 · Q12(c) · 5 marks
(c) Find the range of values of t for which NA > NB giving your answer in set notation. (5)
30. June 2022 · Paper 1 · Q1(a) · 1 marks
1. The point P (-2, -5) lies on the curve with equation y = f (x), x ∈ Find the point to which P is mapped, when the curve with equation y = f (x) is transformed to the curve with equation (a) y = f (x) + 2 (1)
31. June 2022 · Paper 1 · Q1(b) · 1 marks
(b) y = | f (x) | (1)
32. June 2022 · Paper 1 · Q1(c) · 2 marks
(c) y = 3f (x - 2) + 2 (2)
33. June 2022 · AS Paper 1 · Q2(d) · 1 marks
(d) Write down the real root of the equation f(x – 5) = 0 (1)
34. June 2022 · Paper 1 · Q6(b) · 2 marks
The line with equation y = k, where k is a constant, intersects C at only one point. (b) Find the set of values of k, giving your answer in set notation. (2)
35. June 2022 · Paper 1 · Q6(c) · 3 marks
(c) Find the equation of C. You may leave your answer in factorised form. (3)
36. June 2022 · AS Paper 1 · Q7(b) · 2 marks
The curve C has equation y = 9x – x3 (b) Sketch C showing the coordinates of the points at which the curve cuts the x‑axis. (2)
37. June 2022 · AS Paper 1 · Q7(c) · 3 marks
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. (3)
38. June 2022 · Paper 1 · Q11(a) · 2 marks
11. y C1 C2 x Figure 4 Figure 4 shows a sketch of part of the curve C1 with equation y = 2x3 + 10 x > 0 and part of the curve C2 with equation y = 42x - 15x2 - 7 x > 0 (a) Verify that the curves intersect at x = 1 (2)
39. June 2022 · Paper 1 · Q11(b) · 5 marks
The curves intersect again at the point P (b) Using algebra and showing all stages of working, find the exact x coordinate of P (5)
40. June 2022 · Paper 2 · Q1 · 4 marks
1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. y y = | 3 - 2x | x Figure 1 Figure 1 shows a sketch of the graph with equation y = | 3 - 2x | Solve | 3 - 2x | = 7 + x (4)
41. Oct 2021 · Paper 1 · Q2(b) · 2 marks
The curve with equation y = f (x) • meets the y-axis at the point P • has a minimum turning point at the point Q (b) Write down (i) the coordinates of P (ii) the coordinates of Q (2)
42. Oct 2021 · AS Paper 1 · Q16(d) · 2 marks
(d) Hence deduce the coordinates of the points of intersection of the curve with equation y = f(0.2x) and the coordinate axes. (2)
43. Oct 2021 · Paper 2 · Q7(b) · 1 marks
(b) Hence verify that l meets C again on the y-axis. (1)
44. Oct 2021 · Paper 2 · Q11(a) · 4 marks
11. y x Figure 4 Figure 4 shows a sketch of the graph with equation y = | 2x – 3k | where k is a positive constant. (a) Sketch the graph with equation y = f (x) where f (x) = k – | 2x – 3k | stating • the coordinates of the maximum point • the coordinates of any points where the graph cuts the coordinate axes (4)
45. Oct 2021 · Paper 2 · Q11(b) · 4 marks
(b) Find, in terms of k, the set of values of x for which k – | 2x – 3k | > x – k giving your answer in set notation. (4)
46. Oct 2021 · Paper 2 · Q11(c) · 2 marks
(c) Find, in terms of k, the coordinates of the minimum point of the graph with equation < y = 3 – 5f 1 2 x ;: (2)
47. Oct 2020 · AS Paper 1 · Q9(b) · 2 marks
(b) State the coordinates of the point to which P is mapped by the transformation which transforms the curve with equation y = 3 cos x° to the curve with equation (i) y = 3 cos x° (ii) y = 3 cos (x – 36)° (2)
48. Oct 2020 · Paper 1 · Q9(c) · 3 marks
The function g and the function h are defined by g (x) = 2f (x) x h (x) = 2f (x) - 3 x 0 (c) Find (i) the range of g (ii) the range of h (3)
49. Oct 2020 · AS Paper 1 · Q10(c) · 4 marks
The finite region R is bounded by the curve with equation y = g(x) and the x‑axis, and lies below the x‑axis. (c) Find, using algebraic integration, the exact value of the area of R. (4)
50. Oct 2020 · Paper 2 · Q11(a) · 2 marks
11. y x P Figure 2 Figure 2 shows a sketch of the graph with equation y = 2 | x + 4 | - 5 The vertex of the graph is at the point P, shown in Figure 2. (a) Find the coordinates of P. (2)
51. Oct 2020 · Paper 2 · Q11(b) · 2 marks
(b) Solve the equation 3x + 40 = 2 | x + 4 | - 5 (2)
52. Oct 2020 · Paper 2 · Q11(c) · 3 marks
A line l has equation y = ax, where a is a constant. Given that l intersects y = 2 | x + 4 | - 5 at least once, (c) find the range of possible values of a, writing your answer in set notation. (3)
53. June 2019 · Paper 1 · Q2(a) · 2 marks
2. y x Figure 1 Figure 1 shows a plot of part of the curve with equation y = cos x where x is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1. (a) Use Diagram 1 to show why the equation cos x x - - = has only one real root, giving a reason for your answer. (2)
54. June 2019 · Paper 1 · Q5(b) · 3 marks
(b) Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point. (3)
55. June 2019 · Paper 1 · Q5(c) · 4 marks
(c) (i) Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where g(x) = 2(x – 2)2 + 4x – 3 x  < (ii) Find the range of the function x  < h( ) x = x x + + (4)
56. June 2019 · Paper 1 · Q6(b) · 2 marks
(b) Deduce the smallest positive solution to the equation 5 sin (2x – 50°) = 9 tan (x – 25°) (2)
57. June 2019 · AS Paper 1 · Q7(a) · 3 marks
7. The curve C has equation y = k + 1 x  \, x  0 x where k is a constant. (a) Sketch C stating the equation of the horizontal asymptote. (3)
58. June 2019 · Paper 1 · Q10(ii) · 2 marks
(ii) “Given x  <, the value of mu 3x – 28 mu is greater than or equal to the value of (x – 9).” State, giving a reason, if the above statement is always true, sometimes true or never true. (2)
59. June 2019 · AS Paper 1 · Q11(c) · 2 marks
y y = f(x) Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x). (c) Deduce, giving reasons for your answer, the number of real roots of the equation 2x3 – 13x2 + 8x + 46 = 0 (2)
60. June 2019 · AS Paper 1 · Q11(d) · 2 marks
Given that k is a constant and the curve with equation y = f(x + k) passes through the origin, (d) find the two possible values of k. (2)
61. June 2019 · Paper 1 · Q12(b) · 2 marks
y x Figure 3 Figure 3 shows a sketch of part of the curve with equation y = f(x). (b) Sketch the graph of H against t where H(t) = Ň10e±௘0.25t sin tŇ = = t . 0 showing the long-term behaviour of this curve. (2)
62. June 2019 · Paper 1 · Q13(a) · 3 marks
13. The curve C with equation p - x x  <, x x ≠- ≠ y = , x - q x + ( )( ) ⎛ ⎞ where S and q are constants, passes through the point 3 1 ⎠⎟ and has two vertical asymptotes , ⎝⎜ with equations x = 2 and x = –3 (a) (i) Explain why you can deduce that q = 4 (ii) Show that S = 15 (3)
63. June 2019 · Paper 2 · Q7(a) · 1 marks
7. A small factory makes bars of soap. On any day, the total cost to the factory, £y, of making x bars of soap is modelled to be the sum of two separate elements: Ɣ a fixed cost Ɣ a cost that is proportional to the number of bars of soap that are made that day (a) Write down a general equation linking y with x, for this model. (1)
64. June 2018 · Paper 1 · Q7(b) · 3 marks
k ∫ (b) show that d is inversely proportional to k. x ( x - k ) (3) k
65. June 2018 · AS Paper 1 · Q9(c) · 3 marks
y y = g(x) x Figure 2 Figure 2 shows a sketch of part of the curve with equation y = g(x) (c) Use your answer to part (b), and the sketch, to deduce the values of x for which (i) g(x) - 0 (ii) g(2x) = 0 (3)
66. June 2018 · Paper 1 · Q14(b) · 3 marks
(b) (i) Sketch the curve C. (ii) Explain briefly why C does not include all points of y = 6 - (x - 3)2, x ∈ ℝ (3)
67. June 2018 · Paper 1 · Q14(c) · 5 marks
The line with equation x + y = k, where k is a constant, intersects C at two distinct points. (c) State the range of values of k, writing your answer in set notation. (5)
68. June 2018 · Paper 2 · Q3(b) · 3 marks
(b) (i) Sketch the graph of y = |x| + 3 (ii) Explain why |x| + 3 |x + 3| for all real values of x. (3)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · Paper 1 · Q1
Mark scheme June 2025 Paper 1 Q1(c)
June 2025 · AS Paper 1 · Q1
Mark scheme June 2025 Paper 1 Q1(ii)
June 2025 · Paper 1 · Q7
Mark scheme June 2025 Paper 1 Q7(c)
June 2025 · Paper 1 · Q12
Mark scheme June 2025 Paper 1 Q12(b)
June 2025 · AS Paper 1 · Q12
Mark scheme June 2025 Paper 1 Q12(b)
June 2025 · Paper 2 · Q8
Mark scheme June 2025 Paper 2 Q8(c)
June 2025 · Paper 2 · Q12
Mark scheme June 2025 Paper 2 Q12(c)
June 2024 · AS Paper 1 · Q2
Mark scheme June 2024 Paper 1 Q2(c)
June 2024 · AS Paper 1 · Q5
Mark scheme June 2024 Paper 1 Q5(b)
June 2024 · Paper 1 · Q6
Mark scheme June 2024 Paper 1 Q6(c)
June 2024 · AS Paper 1 · Q8
Mark scheme June 2024 Paper 1 Q8(b)
June 2024 · Paper 2 · Q3
Mark scheme June 2024 Paper 2 Q3
June 2023 · AS Paper 1 · Q4
Mark scheme June 2023 Paper 1 Q4(b)
June 2023 · AS Paper 1 · Q15
Mark scheme June 2023 Paper 1 Q15(b)
June 2023 · Paper 2 · Q12
Mark scheme June 2023 Paper 2 Q12(c)
June 2022 · Paper 1 · Q1
Mark scheme June 2022 Paper 1 Q1(c)
June 2022 · AS Paper 1 · Q2
Mark scheme June 2022 Paper 1 Q2(d)
June 2022 · Paper 1 · Q6
Mark scheme June 2022 Paper 1 Q6(c)
June 2022 · AS Paper 1 · Q7
Mark scheme June 2022 Paper 1 Q7(c)
June 2022 · Paper 1 · Q11
Mark scheme June 2022 Paper 1 Q11(b)
June 2022 · Paper 2 · Q1
Mark scheme June 2022 Paper 2 Q1
Oct 2021 · Paper 1 · Q2
Mark scheme Oct 2021 Paper 1 Q2(b)
Oct 2021 · AS Paper 1 · Q16
Mark scheme Oct 2021 Paper 1 Q16(d)
Oct 2021 · Paper 2 · Q7
Mark scheme Oct 2021 Paper 2 Q7(b)
Oct 2021 · Paper 2 · Q11
Mark scheme Oct 2021 Paper 2 Q11(c)
Oct 2020 · AS Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(b)
Oct 2020 · Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(c)
Oct 2020 · AS Paper 1 · Q10
Mark scheme Oct 2020 Paper 1 Q10(c)
Oct 2020 · Paper 2 · Q11
Mark scheme Oct 2020 Paper 2 Q11(c)
June 2019 · Paper 1 · Q2
Mark scheme June 2019 Paper 1 Q2(a)
June 2019 · Paper 1 · Q5
Mark scheme June 2019 Paper 1 Q5(c)
June 2019 · Paper 1 · Q6
Mark scheme June 2019 Paper 1 Q6(b)
June 2019 · AS Paper 1 · Q7
Mark scheme June 2019 Paper 1 Q7(a)
June 2019 · Paper 1 · Q10
Mark scheme June 2019 Paper 1 Q10(ii)
June 2019 · AS Paper 1 · Q11
Mark scheme June 2019 Paper 1 Q11(d)
June 2019 · Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(b)
June 2019 · Paper 1 · Q13
Mark scheme June 2019 Paper 1 Q13(a)
June 2019 · Paper 2 · Q7
Mark scheme June 2019 Paper 2 Q7(a)
June 2018 · Paper 1 · Q7
Mark scheme June 2018 Paper 1 Q7(b)
June 2018 · AS Paper 1 · Q9
Mark scheme June 2018 Paper 1 Q9(c)
June 2018 · Paper 1 · Q14
Mark scheme June 2018 Paper 1 Q14(c)
June 2018 · Paper 2 · Q3
Mark scheme June 2018 Paper 2 Q3(b)