AQA AS Level Mathematics Paper 2, June 2025: Question 1

1 mark · Easy difficulty · Short Answer

Identify which of the four given sketches represents the curve with equation y = (x + a)^2, where a is a positive constant.

Practise this question

Question

Question 1 asks to identify which one of four graphs represents the curve C with equation y = (x + a)^2, where a is a positive constant. Four options are presented with a checkbox next to each: 1) A parabola with its minimum point on the positive y-axis at (0, a); 2) A parabola with its vertex touching the positive x-axis at (a, 0); 3) A parabola with its minimum point on the negative y-axis at (0, -a); 4) A parabola with its vertex touching the negative x-axis at (-a, 0).
Question text

1 A curve C has equation

y = (x + a)2

where a is a positive constant.

Identify which one of the graphs represents C

Tick ( ) one box.

[1 mark]

a

a

–a

–a

Mark scheme

Show the mark scheme Mark scheme for Question 1 indicates that the correct answer is ticking the fourth box, awarded 1 mark (B1, AO 1.1b). The typical solution shows the sketch of a parabola touching the negative x-axis at -a.

Q Marking instructions AO Marks Typical solution

1 Ticks fourth box 1.1b B1

Question 1 Total 1

How to answer it

Transformations of Quadratic Graphs: y = (x + a)²

📌 What this question tests

This question assesses understanding of single-variable graph transformations applied to the base quadratic curve y = x²:

  • Horizontal translations of the form y = f(x + a).
  • Locating the minimum (vertex) and x-intercept of a repeated-root quadratic curve.
  • Correct interpretation of algebraic conditions (where a is a positive constant, so a > 0).
Question 1 • 1 Mark (AO 1.1b)

Identifying the Correct Sketch for y = (x + a)²

Curve C with a > 0

✅ Correct Answer

Tick the fourth box (Bottom diagram).

Graph characteristics:
  • A standard upward-opening parabola ("U-shaped").
  • The minimum turning point sits exactly on the negative x-axis at (-a, 0).
  • The y-axis is positioned to the right of the vertex, crossing the curve at a positive value (y = a²).
Mark Scheme: 1 mark (B1) for ticking the fourth box only.

📐 Step-by-Step Deduction

  1. Identify the vertex/turning point:
    Since y = (x + a)², the minimum value is y = 0, which occurs when x + a = 0 ⇒ x = -a.
    The vertex is at the coordinate (-a, 0).
  2. Interpret the sign of the constant:
    We are told that a is a positive constant (a > 0). Therefore, the coordinate -a is strictly negative.
  3. Determine the position relative to axes:
    The curve touches the x-axis to the left of the origin (at x = -a < 0), ruling out any graph touching the positive x-axis or sitting on the y-axis.

💡 Key Knowledge: Function Transformations

  • f(x + a): A horizontal translation by vector (-a, 0) . When a > 0, this shifts the graph left by a units.
  • f(x - a): A horizontal translation by vector (a, 0) , shifting the graph right by a units.
  • f(x) + a: A vertical translation by vector (0, a) , shifting the graph up by a units.
  • f(x) - a: A vertical translation by vector (0, -a) , shifting the graph down by a units.

❌ Common Errors & Pitfalls

  • Confusing x- and y-translations: Ticking Box 1 or Box 3 assumes that adding/subtracting a moves the graph vertically along the y-axis (i.e. y = x² ± a).
  • The "Intuitive Sign" Trap: Ticking Box 2 assumes + a means moving in the positive x-direction (to the right). Remember: transformations inside the function bracket do the opposite of what is expected!
  • Misreading the constant: Forgetting that a > 0 means that -a lies strictly to the left of the origin.

🧠 Exam Technique & Examiner Commentary

  • Check specific coordinates: If unsure about abstract transformation rules, test a simple value! Let a = 2 , so the equation is y = (x + 2)² .
    • When y = 0 ⇒ x = -2 (must touch the negative x-axis).
    • When x = 0 ⇒ y = 4 (positive y-intercept).
    Only the 4th diagram satisfies both conditions.
  • Fast-start confidence: Question 1 in AQA AS papers is designed to be a quick, 1-mark multiple choice question. Avoid second-guessing yourself by quickly noting the vertex coordinates (-a, 0) in the margin before looking at the options.

Topics

Pure Mathematics · B: Algebra and functions

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