AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 23

2 marks · Medium difficulty · Reasoning

Given the quadratic formula for the height of a roller coaster, identify two criticisms of a student's plotted graph of the function.

Practise this question

Question

Question 23 presents a problem where a roller coaster starts at ground level, with height given by h = -(t - 7)^2 + 49 for 0 <= t <= 14. A sketch shows a graph of h against t, starting at an h-intercept marked 49, curving upwards to a peak above 49, and descending to touch the horizontal axis at t = 14. The horizontal axis marks the origin O, 7, and 14, with uneven spacing between 0 to 7 and 7 to 14. The question asks candidates to make two criticisms of Sam's graph.

Mark scheme

Show the mark scheme Mark scheme for question 23 worth 2 marks (B2 for any two valid criticisms, B1 for one). Valid criticisms include: the h-intercept should be 0, the maximum point should be (7, 49), or the scaling on the horizontal axis is non-linear/uneven. Includes lists of accepted and rejected phrasing.

How to answer it

Criticising a Quadratic Graph Model (Roller Coaster)

📋 What this question tests

This question tests your ability to interpret a real-world quadratic model in completed-square form, find key features of a parabola (the maximum turning point, initial value/vertical intercept, and horizontal roots), and critically evaluate whether a sketch accurately represents mathematical data and correct graphical conventions (such as linear axis scaling).

Question 23 • 2 Marks

Analysis of Sam's Quadratic Graph

Identifying two distinct mathematical errors in the given sketch

📐 Step-by-Step Mathematical Analysis

Equation given: h = -(t - 7)² + 49

  1. Initial Height (when t = 0):
    Substitute t = 0 :
    h = -(0 - 7)² + 49 = -(49) + 49 = 0
    The roller coaster starts at ground level: point (0, 0).
  2. Maximum Height (Turning Point):
    Since -(t - 7)² ≤ 0 , the maximum value occurs when t = 7 .
    At t = 7 , h = -(0)² + 49 = 49 .
    The maximum turning point must be at (7, 49).
  3. End Point (when t = 14):
    Substitute t = 14 :
    h = -(14 - 7)² + 49 = -(49) + 49 = 0 .
    The coaster returns to the ground at (14, 0).
  4. Axis Scaling Check:
    On the horizontal axis, the gap from 0 to 7 is drawn much larger than the gap from 7 to 14 , even though both intervals represent 7 seconds.

✅ Acceptable Criticisms (Pick Any Two)

  • Criticism about the h-intercept (y-intercept):
    "The graph should start at (0, 0) / ground level, not at 49."
  • Criticism about the maximum point:
    "The maximum point should be at (7, 49), but Sam drew the peak at a higher value than 49 (or after t = 7)."
  • Criticism about the horizontal axis scaling:
    "The horizontal axis is not drawn with a linear scale (the spacing between 0 to 7 and 7 to 14 is not equal)."

🧠 Exam Technique & Mark Breakdown

  • B1 : Any ONE correct, valid criticism clearly described.
  • B2 : Any TWO correct, independent valid criticisms.
  • Be specific: Always state what is wrong and what it should be (e.g. state "the intercept should be 0, not 49" rather than "the y-axis is wrong").
  • Using y instead of h or x instead of t is fully credited by examiners.

❌ Common Errors & Vague Answers

  • Vague phrasing (0 marks): Writing statements like "The y-axis is wrong", "Wrong graph", or "Needs more labels" scores 0 because they do not state the mathematical mistake.
  • Subjective aesthetic comments: Writing "It should be more symmetrical" or "Scales are not easy to read" is not accepted. The issue is mathematical correctness, not neatness.
  • Confusing coordinates: Mixing up the peak with the intercept. Sam plotted the peak above 49 and put 49 on the vertical axis at t = 0 .

💡 Visualising the Correct Sketch

If you were asked to correct the sketch on the axes:

  • Start point: Start the curve directly at the origin (0, 0) .
  • Peak (vertex): Draw a smooth curve reaching its highest point directly above t = 7 at level h = 49 .
  • End point: Drop symmetrically back down to touch the axis at t = 14 .
  • Spacing: Ensure the distance from 0 to 7 matches the distance from 7 to 14 along the time axis.
Examiner Insight: Questions requiring students to "criticise a graph" look for precision. Don't simply write that a feature is "wrong"; explain precisely why it is wrong by referencing the calculated values from the formula or comparing to the contextual problem (e.g. "it starts at ground level").

Topics

Algebra · 3.2.2 Graphs

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.