Combining transformations of graphs

P3 builds on the single transformations from P1. You now combine two or more, such as y = 2f(3x) or y = 3 + sin 2x.

The single transformations

y = f(x) + a: translate up by a
y = f(x + a): translate left by a
y = af(x): vertical stretch, scale factor a
y = f(ax): horizontal stretch, scale factor 1/a
y = -f(x): reflect in the x-axis; y = f(-x): reflect in the y-axis

Combining

Changes inside the bracket affect x; changes outside affect y. For y = af(x) + b, stretch first, then translate, following the order you would calculate in. For y = 3 + sin 2x: horizontal stretch scale factor 1/2, then translate up 3.

Tracking points

A point (p, q) on y = f(x) moves to (p/b, aq + c) on y = af(bx) + c. Asymptotes move the same way: x = 6 on y = f(x) becomes x = 2 on y = f(3x).

Worked example

y = f(x) has a maximum at (4, 3). Find the maximum of y = 2f(3x).

Inside: f(3x) divides x-coordinates by 3: 4/3.
Outside: 2f multiplies y-coordinates by 2: 6.

Answer: (4/3, 6)

Key idea: Inside changes act on x in the opposite way to what you might expect; outside changes act on y directly. Move key points one transformation at a time.

Check yourself

The point (6, -2) lies on y = f(x). Which point lies on y = 3f(2x)?
- (12, -6)
- (3, -2/3)
- (3, -6)
- (6, -6)
Show answer
(3, -6). f(2x) halves x: 6 becomes 3. 3f multiplies y by 3: -2 becomes -6.
The curve y = f(x) passes through A(-4, 5). Where is A mapped to on y = f(-x) + 1?
- (4, 6)
- (-4, 6)
- (4, 4)
- (-4, -4)
Show answer
(4, 6). f(-x) reflects in the y-axis: x = -4 becomes 4. Then + 1 moves up: 5 becomes 6.
Which describes how to get y = 3 + sin 2x from y = sin x?
- Horizontal stretch, scale factor 2, then translation 3 units up
- Vertical stretch, scale factor 2, then translation 3 units up
- Horizontal stretch, scale factor 1/2, then translation 3 units up
- Horizontal stretch, scale factor 1/2, then translation 3 units left
Show answer
Horizontal stretch, scale factor 1/2, then translation 3 units up. sin 2x squashes the graph horizontally by a factor of 1/2; adding 3 outside moves it up.
Find the maximum value of y = 3 - 2cos 4x.
Show answer
5. The largest value of -2cos 4x is 2, when cos 4x = -1. So the maximum is 3 + 2 = 5.
The curve y = f(x) has a vertical asymptote x = 6. The curve y = f(3x) has a vertical asymptote x = k. Find k.
Show answer
2. f(3x) divides x-coordinates by 3, so x = 6 becomes x = 2.
(1, 4) lies on y = f(x). Which point lies on y = 2f(x) + 1?
- (1, 10)
- (1, 9)
- (1/2, 9)
- (2, 9)
Show answer
(1, 9). Multiply y by 2, then add 1: 2(4) + 1 = 9. The x-coordinate is unchanged.

Practise with instant feedback

Sources

Last updated 7 November 2026. Spotted a mistake? Tell us.