Edexcel IAL · Pure 1
Transformations of Functions
When you change a function's equation, its graph moves in a predictable, geometric way.
Master the three transformations — translations, stretches, and reflections —
and you can sketch any transformed graph with confidence.
3 Topics
6 Worked Examples
8 Practice Questions
Works Offline
Quick-Reference: All Six Transformations
| Function |
Transformation Type |
Effect on x |
Effect on y |
Vector / SF |
| y = f(x) + a |
Vertical Translation |
Unchanged |
+a added |
(0, a) |
| y = f(x + a) |
Horizontal Translation |
−a added |
Unchanged |
(−a, 0) |
| y = af(x) |
Vertical Stretch |
Unchanged |
× a |
S.F. = a |
| y = f(ax) |
Horizontal Stretch |
× 1/a |
Unchanged |
S.F. = 1/a |
| y = −f(x) |
Reflection in x-axis |
Unchanged |
× −1 (sign flips) |
x-axis |
| y = f(−x) |
Reflection in y-axis |
× −1 (sign flips) |
Unchanged |
y-axis |
The golden rule: Whatever you do outside the function (to the whole output) affects the y-coordinates.
Whatever you do inside the function (to the input x) affects the x-coordinates — and often in the opposite way you'd expect.
1A — Vertical Translation: f(x) + a
Imagine the graph of y = f(x) as a sticker sitting on a piece of graph paper.
Writing y = f(x) + a slides that sticker straight up or down
— the shape doesn't change at all, it just moves vertically.
Every point (x, y) on the original becomes (x, y + a)
on the new graph. Since we're adding to the output (the result of f), only the y-coordinates change.
The "outside" rule: f(x) + a adds to the whole function's result
after x has been plugged in. That's why it shifts output values — i.e. y-coordinates.
Worked Example
The graph of y = f(x) passes through (−2, 6) and (1, −3). Describe the graph of y = f(x) − 4 and state the new coordinates of these points.
1
Identify the transformation: y = f(x) − 4 is a vertical translation by vector (0, −4). The graph shifts down 4 units.