Function transformations
OCR J560 Higher (J560/04–06) tests four transformations of y = f(x). Examiners give a sketch of f(x) and ask for the sketch of a transformed version.
The four standard transformations
| Transformation | Effect | Direction |
|---|---|---|
| y = f(x) + a | Translation by (0, a) | Up if a > 0, down if a < 0 |
| y = f(x + a) | Translation by (−a, 0) | Left if a > 0, right if a < 0 — note the sign flip |
| y = −f(x) | Reflection in the x-axis | Flip vertically |
| y = f(−x) | Reflection in the y-axis | Flip horizontally |
The "inside the bracket" transformations affect x and behave opposite to intuition: f(x − 3) shifts the graph 3 to the right, not left.
Strategy for sketching
- Start by tracing the original f(x), noting its key features: roots, turning point(s), y-intercept, asymptotes.
- Apply the transformation to each key feature.
- Sketch the new curve passing through the transformed key points.
✦Worked example
f(x) is a parabola with roots at x = 0 and x = 4 and minimum at (2, −4). Sketch y = f(x) + 3.
This shifts the whole graph up by 3.
- Roots: solve f(x) + 3 = 0, i.e. f(x) = −3. The minimum at (2, −4) becomes (2, −1) — still below x-axis but shifted up. New roots can be re-read or computed; the curve still crosses the x-axis but at different x-values closer to x = 2.
- New minimum: (2, −1).
- New y-intercept: original was (0, 0), now (0, 3).
Combinations
OCR sometimes asks y = −f(x) + 2: reflect first, then translate up. The minimum (2, −4) → (2, 4) by reflection → (2, 6) by translation up 2.
OCR mark scheme conventions
- B1 for the correct shape (e.g. still a parabola the right way up or upside down).
- B1 for the correct turning point coordinates.
- B1 for at least one labelled intercept on the new curve.
⚠Common mistakes
- Shifting f(x − 3) the wrong way (must be RIGHT, not left).
- Confusing y = −f(x) with y = f(−x) — different reflections.
- Forgetting to label new key points after the transformation.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves