AQA AS Level Mathematics Paper 1, June 2025: Question 8

4 marks · Medium difficulty · Multi-step Problem

Sketch the graph of y = (x - k)^2(x + 2k) for a positive constant k, labelling all axis intercepts.

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Question

Question 8 asks to sketch the graph of y = (x - k)^2(x + 2k), where k is a positive constant, and to label the coordinates of the points where the graph meets the axes. A blank set of Cartesian axes labelled x and y with origin O is provided below the text. The question is worth 4 marks.
Question text

8 Sketch the graph of

y = (x – k)2(x + 2k)

where k is a positive constant.

Label the coordinates of the points where the graph meets the axes.

[4 marks]

y

O x

Mark scheme

Show the mark scheme Mark scheme for Question 8 listing four marks: M1 for drawing a correctly orientated cubic graph with two turning points; A1 for cutting the negative x-axis; M1 for touching the x-axis at a turning point; A1 for a fully correct sketch with intercepts labelled at (-2k, 0), (0, 2k^3), and touching at (k, 0).

Q Marking instructions AO Marks Typical solution

8 Draws a correctly orientated 1.1a M1

cubic graph with two turning

points

Draws a cubic graph cutting 1.1b A1

negative x-axis

Draws a cubic graph with two 2.2a M1

turning points that just touches

the x-axis

Draws a correct graph with all 1.1b A1

points correctly labelled

Question 8 Total 4

How to answer it

Sketching Cubic Graphs with Unknown Constants

AQA AS Mathematics • Pure Maths • Algebra and Functions

What this question tests

  • Recognising the overall shape and end-behaviour of a positive cubic polynomial.
  • Understanding repeated roots: factor of multiplicity 2 means the curve touches the x-axis (tangent) rather than crossing it.
  • Finding axis intercepts in terms of a positive constant k .
  • Clear curve sketching technique, including turning point placement relative to axes.

Question 8 Walkthrough

Question: Sketch the graph of y = (x − k)²(x + 2k) where k is a positive constant. Label the coordinates of the points where the graph meets the axes. [4 marks]

📐 Step-by-Step Mathematical Analysis

  1. Identify graph type and orientation: Expanding the leading terms gives x² × x = +x³ . A positive cubic enters from the bottom-left (Quadrant 3 as x → −∞, y → −∞) and exits to the top-right (Quadrant 1 as x → +∞, y → +∞).
  2. Find the x-intercepts (set y = 0): (x − k)²(x + 2k) = 0
    • x − k = 0 ⇒ x = k (repeated root of order 2)
    • x + 2k = 0 ⇒ x = −2k (single root)
    Since k > 0 , −2k < 0 (lies on the negative x-axis) and k > 0 (lies on the positive x-axis).
  3. Determine root behavior: • At x = −2k , the curve crosses the x-axis.
    • At x = k , because of the squared bracket, the curve touches the x-axis and turns around (a local minimum at (k, 0)).
  4. Find the y-intercept (set x = 0): y = (0 − k)&sup2;(0 + 2k) = (−k)&sup2;(2k) = k&sup2; × 2k = 2k&sup3; .
    Since k > 0 , the y-intercept is positive: (0, 2k&sup3;) .

✅ Accurate Curve & Label Requirements

Exact Visual Layout to Draw:
  • Shape: A smooth positive cubic curve featuring one local maximum in the second quadrant and one local minimum in the first quadrant.
  • Left root: Crosses upwards through the negative x-axis at (−2k, 0) .
  • Peak (local max): Reaches a peak in the top-left region (x < 0, y > 0), higher than the y-intercept.
  • y-axis intersection: Crosses the positive y-axis on its way downward at (0, 2k&sup3;) .
  • Right root (tangent): Hits the positive x-axis at (k, 0) , smoothly turns around without crossing into negative y, and shoots upward into Quadrant 1.
Mark Breakdown:
• M1 (AO 1.1a): Correct orientation of a cubic graph with two turning points.
• A1 (AO 1.1b): Cubic curve explicitly cuts the negative x-axis.
• M1 (AO 2.2a): Cubic curve has two turning points and touches (tangent to) the x-axis at a positive value.
• A1 (AO 1.1b): Completely correct sketch with all three intercept coordinates fully labelled: (−2k, 0) , (k, 0) , and (0, 2k&sup3;) .

💡 Key Knowledge

  • Multiplicity rule: If a factor is linear (x − a) , the graph crosses straight through. If squared (x − a)&sup2; , it turns at the axis (tangent). If cubed (x − a)&sup3; , it forms a point of inflection that flattens as it crosses.
  • Constants are values: Treat k as a real fixed number. Given k > 0 , sign rules strictly apply: −2k < 0 , k > 0 , and 2k&sup3; > 0 .
  • Turning Point vs Intercept: The local maximum is to the left of the y-axis (at x = −k), meaning the curve must cross the y-axis while already sloping downwards!

🧠 Exam Technique & Sketching Tips

  • Coordinate format: The question asks to "Label the coordinates of the points where the graph meets the axes". Always write coordinates as pairs: (−2k, 0) , (k, 0) , and (0, 2k&sup3;) to leave zero ambiguity.
  • Draw the curve first: It is often much easier to draw a smooth cubic curve with the minimum touching an axis, and only then draw and label the coordinate axes around it.
  • No flat bottoms or curls: Examiners penalise turning points that look flat like a table or curl back inwards on themselves. Keep the curve smooth and continuously opening outward.

❌ Common Traps & Student Errors

  • Crossing at x = k: Forgetting that (x − k)&sup2; means the graph is tangent to the axis, drawing the curve crossing straight through.
  • Sign confusion on root: Solving x + 2k = 0 incorrectly as +2k , placing the single intercept on the positive axis.
  • Incorrect expansion for y-intercept: Forgetting that (−k)&sup2; = +k&sup2; , resulting in an erroneous negative y-intercept of −2k&sup3; .
  • Placing maximum on y-axis: Incorrectly drawing the crest of the curve directly on the y-axis rather than in the second quadrant.

Topics

Pure Mathematics · B: Algebra and functions

Question and mark scheme from the AQA AS Level Mathematics examination, Paper 1, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.