Graphs

50 maths questions with worked answers with answers for A8–A16 Graphs in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

1. Question 1 · Coordinates · A8

   Find the midpoint of (2, 7) and (8, −1).

   Answer:

   Average the x-values: 2 + 82 = 5. Average the y-values: 7 + (−1)2 = 3.
   Midpoint (5, 3).

2. Question 2 · Coordinates · A8

   The midpoint of AB is (4, 6). A is (1, 2). Find B.

   Answer:

   From A to the midpoint: +3 in x and +4 in y.
   B = (4 + 3, 6 + 4) = (7, 10).

3. Question 3 · Gradient · A9

   Find the gradient of the line through (1, 3) and (5, 11).

   Answer:

   Gradient = change in ychange in x = 11 − 35 − 1.
   84 = 2.

4. Question 4 · Gradient · A9

   Find the gradient of the line through (−2, 5) and (4, −7).

   Answer:

   −7 − 54 − (−2) = −126.
   −2.

5. Question 5 · Straight lines · A9

   Write the gradient and y-intercept of y = 5x − 3.

   Answer:

   Compare with y = mx + c.
   Gradient 5, y-intercept (0, −3).

6. Question 6 · Straight lines · A9

   Find the gradient and y-intercept of 2y + 6x = 8.

   Answer:

   Rearrange: 2y = −6x + 8, so y = −3x + 4.
   Gradient −3, y-intercept (0, 4).

7. Question 7 · Equation of a line · A9

   Find the equation of the line with gradient 4 through (2, 5).

   Answer:

   y = 4x + c. Substitute: 5 = 8 + c, so c = −3.
   y = 4x − 3.

8. Question 8 · Equation of a line · A9

   Find the equation of the line through (1, 2) and (3, 8).

   Answer:

   Gradient = 8 − 23 − 1 = 3. 2 = 3 + c, so c = −1.
   y = 3x − 1.

9. Question 9 · Parallel lines · A9

   Write the equation of a line parallel to y = 2x + 1 passing through (0, −4).

   Answer:

   Parallel lines have the same gradient, 2.
   y = 2x − 4.

10. Question 10 · Perpendicular lines · A9 · Higher only

    Find the gradient of a line perpendicular to y = 23x + 1.

    Answer:

    Perpendicular gradients multiply to −1.
    −32.

11. Question 11 · Perpendicular lines · A9 · Higher only

    Find the equation of the line perpendicular to y = 4x − 2 through (8, 1).

    Answer:

    Gradient = −14. 1 = −14 × 8 + c, so 1 = −2 + c, c = 3.
    y = −14x + 3.

12. Question 12 · Graphs · A9

    Where does the line y = 3x − 6 cross the x-axis?

    Answer:

    On the x-axis, y = 0: 3x − 6 = 0.
    x = 2, so (2, 0).

13. Question 13 · Roots and intercepts · A10

    Write the coordinates of where y = x² − 9 crosses the axes.

    Answer:

    y-axis: x = 0 gives y = −9. x-axis: x² = 9.
    (0, −9), (3, 0) and (−3, 0).

14. Question 14 · Turning points · A11 · Higher only

    Write x² − 6x + 11 in the form (x − a)² + b.

    Answer:

    (x − 3)² = x² − 6x + 9.
    (x − 3)² + 2.

15. Question 15 · Turning points · A11 · Higher only

    Use completing the square to find the turning point of y = x² + 8x + 3.

    Answer:

    (x + 4)² − 16 + 3 = (x + 4)² − 13.
    Turning point (−4, −13), a minimum.

16. Question 16 · Quadratic graphs · A11

    Describe the shape of the graph of y = −x² + 4.

    Answer:

    The coefficient of x² is negative.
    An n-shaped parabola with a maximum at (0, 4).

17. Question 17 · Quadratic graphs · A11

    The roots of a quadratic are x = 1 and x = 5. Find the x-coordinate of the turning point.

    Answer:

    The turning point lies halfway between the roots.
    x = 1 + 52 = 3.

18. Question 18 · Graph shapes · A12

    What shape is the graph of y = x³?

    Answer:

    A cubic curve.
    It rises from bottom left, flattens through the origin and rises to the top right.

19. Question 19 · Graph shapes · A12

    Describe the graph of y = 1x.

    Answer:

    A reciprocal graph.
    Two separate curves in opposite quadrants; it never touches either axis.

20. Question 20 · Graph shapes · A12 · Higher only

    Sketch the key features of y = 2ˣ.

    Answer:

    It passes through (0, 1).
    It rises steeply for positive x and approaches the x-axis for negative x without touching it.

21. Question 21 · Trig graphs · A12 · Higher only

    State the maximum value and period of y = sin x.

    Answer:

    sin x varies between −1 and 1.
    Maximum 1; period 360°.

22. Question 22 · Trig graphs · A12 · Higher only

    For 0° ≤ x ≤ 360°, where does y = cos x cross the x-axis?

    Answer:

    cos x = 0.
    At x = 90° and x = 270°.

23. Question 23 · Trig graphs · A12 · Higher only

    Given sin 30° = 0.5, find another angle between 0° and 360° with sine 0.5.

    Answer:

    Use the symmetry of the sine graph: 180° − 30°.
    150°.

24. Question 24 · Transformations · A13 · Higher only

    Describe the transformation that maps y = f(x) to y = f(x) + 3.

    Answer:

    3 is added to every y-value.
    Translation by 03.

25. Question 25 · Transformations · A13 · Higher only

    Describe the transformation that maps y = f(x) to y = f(x − 2).

    Answer:

    The graph moves 2 units in the positive x-direction.
    Translation by 20.

26. Question 26 · Transformations · A13 · Higher only

    Describe the transformation that maps y = f(x) to y = −f(x).

    Answer:

    Every y-value changes sign.
    Reflection in the x-axis.

27. Question 27 · Transformations · A13 · Higher only

    Describe the transformation that maps y = f(x) to y = f(−x).

    Answer:

    Every x-value changes sign.
    Reflection in the y-axis.

28. Question 28 · Transformations · A13 · Higher only

    y = x² has turning point (0, 0). State the turning point of y = (x + 5)² − 1.

    Answer:

    Translation by −5−1.
    (−5, −1).

29. Question 29 · Real-life graphs · A14

    On a distance–time graph, what does the gradient represent?

    Answer:

    Distance ÷ time.
    Speed.

30. Question 30 · Real-life graphs · A14

    On a distance–time graph, what does a horizontal line show?

    Answer:

    Distance is not changing.
    The object is stationary.

31. Question 31 · Real-life graphs · A14

    A taxi charges £3 plus £2 per mile. Write a formula for the cost C of m miles.

    Answer:

    Fixed charge £3, plus 2 for each mile.
    C = 2m + 3.

32. Question 32 · Real-life graphs · A14

    In the formula C = 2m + 3, what does the 3 represent on the graph?

    Answer:

    It is the y-intercept.
    The fixed charge of £3 before any miles are travelled.

33. Question 33 · Real-life graphs · A14

    A car travels 120 km in the first 2 hours. Find its speed from the graph's gradient.

    Answer:

    Gradient = 1202.
    60 km/h.

34. Question 34 · Velocity–time graphs · A15 · Higher only

    On a velocity–time graph, what does the area under the graph represent?

    Answer:

    Velocity × time.
    Distance travelled.

35. Question 35 · Velocity–time graphs · A15 · Higher only

    On a velocity–time graph, what does the gradient represent?

    Answer:

    Change in velocity ÷ time.
    Acceleration.

36. Question 36 · Velocity–time graphs · A15 · Higher only

    A car accelerates uniformly from 0 to 20 m/s in 8 seconds. Find the distance travelled.

    Answer:

    The area under the graph is a triangle.
    12 × 8 × 20 = 80 m.

37. Question 37 · Rates of change · A15 · Higher only

    How do you estimate the gradient of a curve at a point?

    Answer:

    Draw a tangent to the curve at that point.
    Find the gradient of the tangent: change in y ÷ change in x.

38. Question 38 · Area under a curve · A15 · Higher only

    How can you estimate the area under a curve?

    Answer:

    Split the area into trapezia (or strips).
    Add the areas of the shapes.

39. Question 39 · Circles · A16 · Higher only

    Write the equation of the circle with centre (0, 0) and radius 6.

    Answer:

    x² + y² = r².
    x² + y² = 36.

40. Question 40 · Circles · A16 · Higher only

    Find the radius of x² + y² = 50.

    Answer:

    r² = 50.
    r = √50 = 5√2.

41. Question 41 · Circles · A16 · Higher only

    Show that (3, 4) lies on x² + y² = 25.

    Answer:

    3² + 4² = 9 + 16.
    = 25, so the point lies on the circle.

42. Question 42 · Circles · A16 · Higher only

    Find the gradient of the tangent to x² + y² = 25 at (3, 4).

    Answer:

    Gradient of radius = 43. The tangent is perpendicular.
    Gradient = −34.

43. Question 43 · Circles · A16 · Higher only

    Find the equation of the tangent to x² + y² = 25 at (3, 4).

    Answer:

    Gradient −34: y − 4 = −34(x − 3).
    y = −34x + 254 (or 3x + 4y = 25).

44. Question 44 · Solving graphically · A10 · Higher only

    How can you use a graph to solve x² − 2x − 3 = 0?

    Answer:

    Plot y = x² − 2x − 3.
    Read off the x-values where the curve crosses the x-axis (x = −1 and x = 3).

45. Question 45 · Solving graphically · A10

    How do you use graphs to solve a pair of simultaneous equations?

    Answer:

    Draw both lines on the same axes.
    The solution is the point where they intersect.

46. Question 46 · Straight lines · A9

    Does the point (3, 10) lie on y = 4x − 2?

    Answer:

    4 × 3 − 2 = 10.
    Yes.

47. Question 47 · Graph shapes · A12

    Which graph is a straight line: y = 3x + 1, y = x² or y = 1x?

    Answer:

    Only linear equations give straight lines.
    y = 3x + 1.

48. Question 48 · Coordinates · A8

    Which quadrant contains the point (−3, 5)?

    Answer:

    Negative x, positive y.
    The second quadrant (top left).

49. Question 49 · Straight lines · A9

    What is the equation of the x-axis?

    Answer:

    Every point on it has y-coordinate 0.
    y = 0.

50. Question 50 · Quadratic graphs · A11

    Find the y-intercept of y = x² − 5x + 6.

    Answer:

    Substitute x = 0.
    (0, 6).

More Edexcel GCSE Maths topics

• N1–N9 Structure and calculation
• N10–N12 Fractions, decimals and percentages
• N13–N16 Measures and accuracy
• A1–A7 Notation, vocabulary and manipulation
• A8–A16 Graphs
• A17–A22 Solving equations and inequalities
• A23–A25 Sequences
• R1–R16 Ratio, proportion and rates of change
• G1–G15 Properties and constructions
• G16–G23 Mensuration and calculation
• G24–G25 Vectors
• P1–P9 Probability
• S1–S6 Statistics

Other Maths courses: AQA GCSE Maths, OCR GCSE Maths.