Sheet № 149 · Higher only · AQA · Edexcel · OCR
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 app
§Key definitions
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.
Answer:
(0, 5), (3, 8), and (−1, 3)
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.
§Formulas to memorise
y = f(x) + a → translation (0, a) — moves up by a
y = f(x + a) → translation (−a, 0) — moves left by a
y = −f(x) → reflection in the x-axis
y = f(−x) → reflection in the y-axis
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.
Identify the transformation — from the function notation (is it +a outside, +a inside, a minus outside, or a minus inside?).
Recall the rule — changes outside f affect y (vertical), changes inside f affect x (horizontal and in the opposite direction).
Worked example
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:
⚠ 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.