A13: Graph Transformations
Sketch translations and reflections of a given function
📋 Key Concepts
Transformations: Moving or reflecting a graph without changing its basic shape.
| Transformation | Notation | Effect |
|---|---|---|
| Translation up | y = f(x) + a | Move up by a units |
| Translation down | y = f(x) - a | Move down by a units |
| Translation right | y = f(x - a) | Move right by a units |
| Translation left | y = f(x + a) | Move left by a units |
| Reflection in x-axis | y = -f(x) | Flip vertically |
| Reflection in y-axis | y = f(-x) | Flip horizontally |
📝 Translations
Translation: Moving a graph without rotation or reflection.
Vertical translations (outside the function):
- y = f(x) + a: shift UP by a units
- y = f(x) - a: shift DOWN by a units
Example 1
The graph of y = x² is translated to give y = x² + 3. Describe the transformation.
Solution:
The "+3" is outside the function, so this is a vertical translation.
Answer: Translation of 3 units UP
Horizontal translations (inside the function):
- y = f(x - a): shift RIGHT by a units
- y = f(x + a): shift LEFT by a units
Warning: The direction is opposite to what you might expect!
Example 2
The graph of y = x² is translated to give y = (x - 2)². Describe the transformation.
Solution:
The "-2" is inside the function. The minus means shift RIGHT.
Answer: Translation of 2 units RIGHT
Example 3
Describe the transformation from y = f(x) to y = f(x + 4)
Solution:
The "+4" is inside the function. The plus means shift LEFT.
Answer: Translation of 4 units LEFT
📝 Combined Translations
Example 4
Describe the transformation from y = x² to y = (x - 3)² + 2
Solution:
Inside: (x - 3) means shift 3 units RIGHT
Outside: +2 means shift 2 units UP
Answer: Translation by vector 3⁄2 (right 3, up 2)
Example 5
The graph of y = sin x is transformed to y = sin(x - 45°) - 1. Describe the transformation.
Solution:
x - 45°: shift 45° RIGHT
- 1: shift 1 unit DOWN
Answer: Translation: right by 45°, down by 1 unit
📝 Reflections
Reflection in the x-axis: y = -f(x)
- All y-coordinates change sign
- Points above x-axis move below and vice versa
- Graph is flipped vertically
Example 6
y = f(x) has points (0, 2), (1, 3), (2, 1). Find the points on y = -f(x).
Solution:
Multiply each y-coordinate by -1:
(0, -2), (1, -3), (2, -1)
Reflection in the y-axis: y = f(-x)
- All x-coordinates change sign
- Points to the right of y-axis move left and vice versa
- Graph is flipped horizontally
Example 7
y = f(x) has points (1, 2), (2, 3), (3, 1). Find the points on y = f(-x).
Solution:
Multiply each x-coordinate by -1:
(-1, 2), (-2, 3), (-3, 1)
📝 Describing Transformations
Example 8
The graph of y = x³ is transformed to y = -(x + 1)³ + 2. Describe fully the transformation.
Solution:
- Negative outside: reflection in x-axis
- (x + 1): shift LEFT by 1
- + 2: shift UP by 2
Answer: Reflection in x-axis, then translation left by 1 and up by 2
🧮 Worked Examples — Exam Style
Transformations move graphs predictably: OUTSIDE the bracket acts vertically (as written); INSIDE acts horizontally (backwards!).
The point (4, −2) lies on y = f(x). Where does it move under y = f(x) + 3 and y = f(x + 1)?
y = f(x) + 3: vertical shift → (4, 1)
y = f(x + 1): horizontal shift LEFT 1 (inside = backwards) → (3, −2)
B1 each. The x-shift reversing direction is THE examined trap — x + 1 moves LEFT.
Describe the single transformation mapping y = f(x) onto y = 2f(x).
Vertical stretch, scale factor 2 — every y-value doubles, x-values unchanged.
B1 with the scale factor named. f(2x) would be a horizontal stretch factor ½ — different!
Exam wording to expect: "Describe fully the transformation", "Sketch y = …", or tracing a known point through the transformation.
❓ Practice Questions
Q1: y = f(x) is transformed to y = f(x) + 5. Describe the transformation.
Q2: y = f(x) is transformed to y = f(x - 2). Describe the transformation.
Q3: y = f(x) is transformed to y = -f(x). Describe the transformation.
Q4: y = f(x) is transformed to y = f(-x). Describe the transformation.
Q5: y = x² has vertex (0, 0). Where is the vertex of y = (x - 3)² + 4?
Q6: y = f(x) passes through (2, 4). Where does y = -f(x) pass through?
Q7: y = f(x) has a minimum at (−1, 2). State the minimum of y = f(x − 3) + 4
Q8: Describe the transformation from y = f(x) to y = f(−x).
✅ Answers
- Translation 5 units UP
- Translation 2 units RIGHT
- Reflection in the x-axis
- Reflection in the y-axis
- (3, 4)
- (2, -4)
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. +5 outside → translation 5 units up (vector (0, 5))
Q2. x − 2 inside → translation 2 units RIGHT (vector (2, 0))
Q3. y = −f(x) → reflection in the x-axis
Q4. y = f(−x) → reflection in the y-axis
Q5. (x − 3)² + 4: right 3, up 4 → (3, 4)
Q6. y flips sign: (2, 4) → (2, −4)
Q7. Right 3, up 4: (2, 6)
Q8. Reflection in the y-axis
📌 Key Facts to Memorise
- f(x) + a: up a (down if negative) · f(x + a): LEFT a (backwards!)
- −f(x): reflect x-axis · f(−x): reflect y-axis
- af(x): vertical stretch scale factor a · f(ax): horizontal stretch factor 1/a
- Vertex tracking: apply shifts to the known vertex, in order
- "Fully" = name the transformation + direction + amount/centre
🗺️ In the Exam
Transformations: 2–4 marks on Papers 1/2. Tracing a labelled point through the mapping is the reliable method when the whole-graph picture is unclear.
Links to: quadratic graphs (A11), other graphs (A12), geometry transformations (G7).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Inside the bracket: horizontal transformation (opposite direction)
- Outside the bracket: vertical transformation (same direction)
- Minus sign outside: reflection in x-axis
- Minus sign inside: reflection in y-axis
- x - a means right; x + a means left
- Always state the transformation type AND direction
🧠 Problem-Solving Strategies
Problem-Solving
Outside the function = vertical transformation (same direction). Inside the function = horizontal transformation (opposite direction). Negative outside = reflection in x-axis. Negative inside = reflection in y-axis. Always state the transformation type AND direction.
Multi-Step Problem
y = f(x) has a maximum at (2, 5) and passes through (0, 1). Find the corresponding points on: (a) y = f(x - 3) + 2 (b) y = -f(x) (c) y = f(-x)
Solution:
(a) Shift right 3, up 2: maximum (5, 7), point (3, 3)
(b) Reflection in x-axis: maximum (2, -5), point (0, -1)
(c) Reflection in y-axis: maximum (-2, 5), point (0, 1)
⚠️ Common Errors
Watch Out!
1. Wrong: y = f(x - 3) shifts 3 units LEFT Correct: y = f(x - 3) shifts 3 units RIGHT (opposite direction inside brackets)
2. Wrong: y = -f(x) reflects in the y-axis Correct: y = -f(x) reflects in the x-axis (y-values change sign)
3. Wrong: y = f(x) + 2 shifts 2 units right Correct: y = f(x) + 2 shifts 2 units UP (outside the function = vertical)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The graph of y = x² has vertex (0, 0) and passes through (2, 4) and (-1, 1). (a) Describe fully the transformation that maps y = x² to y = (x - 3)² + 4. (b) State the new vertex and the images of (2, 4) and (-1, 1). (c) The graph is then reflected in the x-axis. Write the new equation and find the vertex.
(a) Translation right by 3, up by 4. Vector [3⁄4]
(b) New vertex: (3, 4). (2, 4) → (5, 8). (-1, 1) → (2, 5)
(c) New equation: y = -(x - 3)² - 4. Vertex: (3, -4)
Mark scheme: (a) 2 marks for full description with direction. (b) 2 marks for all three points. (c) 2 marks for equation and vertex.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The height h of a tide over time t is h = 5 + 3sin(t). A new model shifts the peak tide 2 hours later and raises the mean by 1 metre.
(a) Write the new equation.
(b) How does the amplitude change?
(c) Explain why the new model might be more realistic.
Answers: (a) h = 6 + 3sin(t - 2) — shift right 2 (inside) and up 1 (outside). (b) Amplitude is still 3 — unchanged by translations. (c) Shifting the peak accounts for later high tide; higher mean accounts for rising sea levels or seasonal effects.
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