Graph Transformations: Translations and Reflections

Graph transformations allow you to move, stretch, or reflect a graph without plotting every single point. At GCSE Higher level, you need to understand four key transformations: vertical translations, horizontal translations, reflection in the x-axis, and reflection in the y-axis. These are described using function notation. This topic appears in every Higher exam and typically carries 2–4 marks.

What Are Graph Transformations?

A graph transformation changes the position or orientation of a graph. Starting from a known graph y = f(x), you apply changes to the function to produce a new graph. The four transformations you need are:

• y = f(x) + a — translates the graph up (a > 0) or down (a < 0).
• y = f(x + a) — translates the graph left (a > 0) or right (a < 0).
• y = −f(x) — reflects the graph in the x-axis.
• y = f(−x) — reflects the graph in the y-axis.

Key Formulas

Step-by-Step Method

1. Identify the transformation from the function notation (is it +a outside, +a inside, a minus outside, or a minus inside?).
2. Recall the rule: changes outside f affect y (vertical), changes inside f affect x (horizontal and in the opposite direction).
3. Apply the transformation to key points on the original graph.
4. Draw the new graph through the transformed points.
5. Label key coordinates on the new graph.

Worked Example 1 — Foundation Level

Question: The graph of y = f(x) passes through (0, 2), (3, 5), and (−1, 0). Write the coordinates of these points on the graph of y = f(x) + 3.

Working:

y = f(x) + 3 means every y-coordinate increases by 3 (vertical translation up 3).

(0, 2) → (0, 5). (3, 5) → (3, 8). (−1, 0) → (−1, 3).

Answer: (0, 5), (3, 8), and (−1, 3)

Worked Example 2 — Higher Level

Question: Describe the transformation that maps y = f(x) to y = f(x − 4).

Working:

The change is inside the function: x is replaced by (x − 4). This is y = f(x + a) where a = −4.

The translation is (−(−4), 0) = (4, 0). The graph moves 4 units to the right.

Remember: changes inside the bracket act in the opposite direction. Replacing x with (x − 4) moves right, not left.

Answer: Translation of 4 units to the right, or translation by vector (4, 0).

Worked Example 3 — Exam Style

Question: The curve y = f(x) has a maximum point at (2, 6). Write down the coordinates of the maximum point on: (a) y = −f(x), (b) y = f(−x), (c) y = f(x) − 4.

Working:

(a) y = −f(x): reflect in the x-axis. The x-coordinate stays the same, the y-coordinate changes sign. Maximum (2, 6) becomes minimum (2, −6).

(b) y = f(−x): reflect in the y-axis. The y-coordinate stays the same, the x-coordinate changes sign. Maximum (2, 6) becomes maximum (−2, 6).

(c) y = f(x) − 4: translate down 4. The x-coordinate stays the same, the y-coordinate decreases by 4. Maximum (2, 6) becomes maximum (2, 2).

Answer: (a) (2, −6), (b) (−2, 6), (c) (2, 2)

Common Mistakes

• Getting the horizontal direction wrong. y = f(x + 2) moves the graph 2 units to the left, not right. The direction is opposite to the sign inside the bracket. This is the most common error on this topic.
• Confusing −f(x) and f(−x). −f(x) reflects in the x-axis (y-values change sign). f(−x) reflects in the y-axis (x-values change sign). Mix these up and you lose all the marks.
• Forgetting to transform all key points. When sketching, make sure you apply the transformation to every important point (intercepts, turning points, endpoints).

Exam Tips

• Remember the rule: changes outside f affect y (up/down), changes inside f affect x (opposite direction).
• Use the phrase "opposite for x" to remind yourself that f(x + 2) moves left, not right.
• When describing a transformation, state the type (translation or reflection) and give the details (vector for translation, mirror line for reflection).
• On AQA and Edexcel, you may be asked to apply two transformations in sequence — apply them one at a time in the correct order.

Practice Questions

Q1 (Foundation): The graph y = f(x) passes through (1, 4). State the corresponding point on y = f(x) − 2.

Q2 (Higher): Describe the transformation from y = f(x) to y = f(x + 5).

Q3 (Higher): The curve y = x² has vertex (0, 0). Write down the vertex of y = −(x − 3)².

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Related Topics

• Graph Transformations
• Quadratic Graphs
• Cubic and Reciprocal Graphs

Summary

• y = f(x) + a translates the graph vertically by a units.
• y = f(x + a) translates the graph horizontally by −a units (opposite direction).
• y = −f(x) reflects the graph in the x-axis.
• y = f(−x) reflects the graph in the y-axis.
• Changes outside the function affect y; changes inside the function affect x in the opposite direction.
• Always state the type of transformation and its details when answering exam questions.