AQA A-Level Mathematics Paper 2, June 2025: Question 1

1 mark · Easy difficulty · Short Answer

Identify the single transformation that maps the curve y = ln x onto the curve y = 2 ln x.

Practise this question

Question

Question 1 asks to describe the single transformation mapping the curve y = ln x onto y = 2 ln x by ticking one box. The four multiple-choice options are: 'Stretch, scale factor 2, parallel to the y-axis', 'Stretch, scale factor 1/2, parallel to the x-axis', 'Translation by column vector [2, 0]', and 'Translation by column vector [0, 2]'.
Question text

1 Describe the single transformation which maps the curve with the equation

y = ln x

onto the curve with the equation

y = 2ln x

Tick ( ) one box.

[1 mark]

Stretch, scale factor 2, parallel to the y‑axis

Stretch, scale factor , parallel to the x‑axis

Translation

Translation

Mark scheme

Show the mark scheme Mark scheme for Question 1: gives 1 mark (R1, AO 2.2a) for ticking the first box, with typical solution 'Stretch, scale factor 2, parallel to the y-axis'.

Q Marking instructions AO Marks Typical solution

1 Ticks the first 2.2a R1 Stretch, scale factor 2, parallel to the

y-axis

Question 1 Total 1

How to answer it

Single Transformation of Logarithmic Functions

📌 What this question tests

This question assesses your understanding of function transformations (specification topic: Coordinate geometry and functions). Specifically, it checks whether you can:

  • Distinguish between vertical transformations of the form y = a f(x) and horizontal transformations of the form y = f(ax) .
  • Correctly state the type of transformation (stretch vs translation), the scale factor, and the direction/axis.
Question 1 (1 Mark)

Identifying the Transformation from y = ln x to y = 2ln x

AQA A-Level Mathematics • Paper 1 • Multiple Choice

✅ Correct Answer

First Box: Stretch, scale factor 2, parallel to the y-axis

Mark Scheme: Ticks the first box — 1 mark [AO 2.2a] (Reasoning 1)

💡 Key Knowledge

Let f(x) = ln x . Then y = 2ln x is of the form y = a f(x) where a = 2 .

Form Transformation
y = a f(x) Vertical stretch, scale factor a , parallel to the y-axis
y = f(ax) Horizontal stretch, scale factor 1/a , parallel to the x-axis
y = f(x) + c Translation by vector [0, c]
y = f(x - c) Translation by vector [c, 0]

📐 Step-by-Step Mathematical Analysis

  1. Identify the base function:
    Let f(x) = ln x .
  2. Express the new function in terms of f(x):
    The target curve is y = 2 ln x = 2 × f(x) .
  3. Analyze the effect on coordinates:
    For any point (x, y) on the original curve, the corresponding point on the new curve is (x, 2y) .
  4. Deduce the transformation:
    Since the x-coordinate is unchanged and the y-coordinate is multiplied by 2, this represents an expansion away from the x-axis, which is described as a stretch of scale factor 2 parallel to the y-axis (or in the vertical direction).

🧠 Exam Technique & Examiner Insight

  • Quick Rule of Thumb: Operations outside the function bracket affect the y-direction directly as written (e.g., ×2 multiplies y by 2). Operations inside the function affect the x-direction inversely.
  • Only tick ONE box: If you change your mind, cross out the incorrect tick completely and clearly place a new tick in the correct box.
  • Full Description Requirements: In non-multiple-choice questions, examiners require all three elements:
    • The word Stretch
    • The scale factor (e.g. 2)
    • The direction (e.g. parallel to the y-axis or in the y-direction)

❌ Common Misconceptions & Distractor Breakdown

  • Confusing with "Stretch, scale factor 1/2, parallel to the x-axis":
    A common trap arises from logarithm laws: 2 ln x = ln(x²) , NOT ln(2x) . Even if a student thought about horizontal stretching, f(2x) = ln(2x) gives a scale factor of 1/2 parallel to the x-axis, but that equals ln 2 + ln x , which is a vertical translation, not 2 ln x !
  • Confusing multiplication with translation vectors [2, 0] or [0, 2]:
    Translations occur when constants are added, e.g.:
    • y = ln(x − 2) is a translation by [2, 0] .
    • y = ln x + 2 is a translation by [0, 2] .
    Multiplication by 2 affects scale (stretch), not a rigid shift (translation).

Topics

Pure Mathematics · B: Algebra and functions · F: Exponentials and logarithms

Question and mark scheme from the AQA A-Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.