Graph transformations are a Higher-tier topic that appears regularly on AQA, Edexcel and OCR papers — often worth four or five marks. You need to understand how changing the equation of a function affects its graph. This includes translations (shifts), reflections, and stretches. The key is learning the four core transformations using f(x) notation and knowing which direction the graph moves. Once you master the rules, every question follows the same logic.
What Are Graph Transformations?
A graph transformation takes an existing graph y = f(x) and moves, reflects or stretches it to create a new graph. There are four transformations you need to know:
Translations (shifts)
y = f(x) + a — translates the graph a units up (or down if a is negative).
y = f(x + a) — translates the graph a units to the left (or right if a is negative).
Note the counter-intuitive direction: f(x + 2) moves the graph 2 to the left, not the right.
Reflections
y = −f(x) — reflects the graph in the x-axis.
y = f(−x) — reflects the graph in the y-axis.
Stretches (less commonly examined at GCSE)
y = af(x) — stretches the graph vertically by scale factor a.
y = f(ax) — stretches the graph horizontally by scale factor 1/a.
Step-by-Step Method
How to describe a transformation
- Compare the new equation to the original y = f(x).
- Identify what has changed: is there a number added inside the bracket, outside the bracket, or is there a negative sign?
- State the transformation type (translation, reflection or stretch).
- Give the details: the vector for a translation, the mirror line for a reflection, or the scale factor and direction for a stretch.
How to apply a transformation to a graph
- Identify key points on the original graph (turning points, intercepts, endpoints).
- Apply the rule to each point's coordinates.
- Plot the transformed points and draw the new curve.
Quick reference for coordinate changes
| Transformation | What happens to (x, y) |
|---|---|
| y = f(x) + a | (x, y + a) |
| y = f(x + a) | (x − a, y) |
| y = −f(x) | (x, −y) |
| y = f(−x) | (−x, y) |
Worked Example 1 — Translation
Question: The graph of y = f(x) has a turning point at (3, −2). Write down the coordinates of the turning point of y = f(x) + 5.
Step 1: y = f(x) + 5 is a translation of 5 units upward.
Step 2: Every y-coordinate increases by 5.
Step 3: The turning point moves from (3, −2) to (3, 3).
Follow-up: What about the turning point of y = f(x − 4)?
y = f(x − 4) is f(x + (−4)), so it translates 4 units to the right. The turning point moves to (7, −2).
Worked Example 2 — Reflection
Question: The curve y = f(x) passes through the points (−1, 4), (0, 1), (2, −3) and (5, 0). Find the coordinates of the corresponding points on y = f(−x).
Step 1: y = f(−x) is a reflection in the y-axis.
Step 2: The x-coordinates change sign; the y-coordinates stay the same.
| Original | Transformed |
|---|---|
| (−1, 4) | (1, 4) |
| (0, 1) | (0, 1) |
| (2, −3) | (−2, −3) |
| (5, 0) | (−5, 0) |
Note that (0, 1) stays the same because it lies on the y-axis (the mirror line).
Common Mistakes
- Getting the horizontal direction wrong. y = f(x + 2) moves the graph left 2, not right 2. This is the most common error on exam papers. Remember: the change is opposite to what you might expect.
- Confusing f(x) + a with f(x + a). If the number is inside the bracket, it is a horizontal shift. If it is outside, it is a vertical shift.
- Mixing up −f(x) and f(−x). −f(x) reflects in the x-axis (flips vertically). f(−x) reflects in the y-axis (flips horizontally).
- Forgetting to transform all key points. When sketching, make sure you move every labelled point, not just the turning point.
Exam Tips
- Use the vector notation when describing translations. Examiners on AQA and Edexcel specifically look for "translation by vector (a, b)" and award a mark for the correct notation.
- Practise with specific graphs. Try transforming y = x², y = sin x and y = 1/x — these are the functions most commonly used in exam questions.
- If in doubt, try a point. Pick a point on the original curve, apply the transformation to its coordinates, and check it lies on the new curve.
- For combined transformations, apply them one at a time, in the correct order. For example, y = f(x + 1) + 3 means shift left 1, then up 3.
Practice Questions
Question 1: The graph of y = f(x) passes through (4, 6). Write down the coordinates of the corresponding point on y = f(x) − 3.
Question 2: Describe fully the single transformation that maps y = f(x) onto y = f(x + 5).
Question 3: The curve y = f(x) has a maximum point at (2, 7). Write down the maximum point of y = −f(x) and state the type of transformation.
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Related Topics
- Quadratic Graphs
- Other Graphs: Cubic, Reciprocal, Exponential
- Functions and Function Notation
- Linear Graphs and Equation of a Line
Summary
- y = f(x) + a shifts the graph up by a (translation by vector (0, a)).
- y = f(x + a) shifts the graph left by a (translation by vector (−a, 0)) — note the opposite direction.
- y = −f(x) reflects the graph in the x-axis.
- y = f(−x) reflects the graph in the y-axis.
- Always apply transformations to key points (turning points, intercepts) and re-plot.
- Use vector notation when describing translations in your exam answers.
- For combined transformations, apply each step in order and track the coordinates through each stage.