AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 21
2 marks · Medium difficulty · Short Answer
Draw the transformed graphs of $y = -f(x)$ and $y = f(x) + 1$ on the provided coordinate grids given the graph of $y = f(x)$ for $0 \le x \le 3$.
Practise this questionQuestion
Question text
21 The graph of y = f(x) for 0 ⩽ x ⩽ 3 is shown.
21 (a) On the grid below, draw the graph of y = –f(x) for 0 ⩽ x ⩽ 3
The graph of y = f(x) is shown to help you.
[1 mark]
21 (b) On the grid below, draw the graph of y = f(x) + 1 for 0 ⩽ x ⩽ 3
The graph of y = f(x) is shown to help you.
[1 mark]
Mark scheme
Show the mark scheme
Q Answer Mark Comments
Fully correct graph mark intention
accept horizontal line drawn joining the
two lines
condone dashed lines
ignore shading
21(a) B1
Fully correct graph mark intention
accept horizontal line drawn joining the
two lines
condone dashed lines
ignore shading
21(b) B1
How to answer it
Transforming Functions: Reflections and Translations
This question assesses your ability to apply graphical transformations to a given function y = f(x) on a coordinate grid:
- Part (a): Vertical reflection in the x-axis represented by y = −f(x) .
- Part (b): Vertical translation represented by y = f(x) + a .
- Tracking key coordinate points (vertices and endpoints) accurately to sketch transformed graphs.
Drawing y = −f(x) for 0 ≤ x ≤ 3
Total marks: 1
✅ Correct Answer
A reflected "V-shape" (inverted peak) drawn with straight lines connecting:
- Start: (0, 0)
- Peak (minimum): (1, −3)
- End: (3, 0)
[B1] Fully correct graph. Examiner condones dashed lines and marks intention (slight wobbles accepted).
📐 Step-by-Step Coordinate Mapping
When the negative sign is outside the bracket, every y-coordinate is multiplied by −1. The x-coordinates stay identical:
| Original (x, y) | Rule: (x, −y) | New Point |
|---|---|---|
| (0, 0) | (0, −0) | (0, 0) |
| (1, 3) | (1, −3) | (1, −3) |
| (3, 0) | (3, −0) | (3, 0) |
💡 Key Knowledge
- Outside the function: Affects y (vertical changes).
- y = −f(x) is a reflection in the x-axis (vertical flip).
- Points on the x-axis where y = 0 are invariant (they do not move).
❌ Common Errors
- Reflecting horizontally: Confusing y = −f(x) with y = f(−x) (reflection in the y-axis), which would place the shape on the negative x-axis.
- Drawing beyond domain: Extending lines beyond x = 0 or x = 3 .
Drawing y = f(x) + 1 for 0 ≤ x ≤ 3
Total marks: 1
✅ Correct Answer
The original shape translated vertically upwards by 1 unit, connecting:
- Start: (0, 1)
- Peak: (1, 4)
- End: (3, 1)
[B1] Fully correct graph. Examiner condones dashed lines, ignores shading, and marks intention.
📐 Step-by-Step Coordinate Mapping
Adding 1 outside the function means add 1 to every y-coordinate. The x-coordinates do not change:
| Original (x, y) | Rule: (x, y + 1) | New Point |
|---|---|---|
| (0, 0) | (0, 0 + 1) | (0, 1) |
| (1, 3) | (1, 3 + 1) | (1, 4) |
| (3, 0) | (3, 0 + 1) | (3, 1) |
💡 Key Knowledge
- y = f(x) + a represents a vertical translation by vector (0, a) .
- Because +1 is outside the bracket, shift the entire shape up by 1 unit.
- The shape, width, and steepness remain identical to the original graph.
🧠 Exam Technique
- Plot key points first: Always plot the vertices/corners and endpoints as discrete dots with a sharp pencil before drawing lines.
- Use a ruler: The original lines are straight segments; use a clear ruler to join the new points cleanly.
- Inside vs. Outside Rule:
• Inside brackets f(x + a) affects x (does the opposite: left/right).
• Outside brackets f(x) + a affects y (does what it says: up/down).
Topics
Algebra · 3.2.2 Graphs
Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.