Sheet № 28 · Higher only · AQA · Edexcel · OCR
Graph Transformations
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
§Key definitions
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).
y = −f(x)
reflects the graph in the x-axis.
y = f(−x)
reflects the graph in the y-axis.
y = af(x)
stretches the graph vertically by scale factor a.
§Formulas to memorise
Translation by vector (0, a)
Translation by vector (−a, 0)
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).
y = −f(x): — reflects the graph in the x-axis.
y = f(−x): — reflects the graph in the y-axis.
y = af(x): — stretches the graph vertically by scale factor a.
y = f(ax): — stretches the graph horizontally by scale factor 1/a.
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?
Worked example
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.
⚠ 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.