Stretches of graphs: y = af(x) and y = f(ax)

International GCSE Higher also asks about stretching graphs. A number outside the function stretches up and down; a number inside stretches sideways, the opposite way.

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y = af(x): a vertical stretch

Multiply every y-coordinate by a. The x-coordinates stay the same.
The point (4, 6) on y = f(x) moves to (4, 18) on y = 3f(x).
y = 2 sin x has a maximum of 2 and a minimum of −2.

y = f(ax): a horizontal stretch

Divide every x-coordinate by a. The y-coordinates stay the same.
It is a stretch parallel to the x-axis with scale factor 1a.
The point (4, 6) on y = f(x) moves to (2, 6) on y = f(2x).
y = sin 2x repeats every 360° ÷ 2 = 180°.

All four together

y = f(x) + a: up a. y = f(x + a): left a.
y = af(x): vertical stretch, scale factor a.
y = f(ax): horizontal stretch, scale factor 1a.
Outside the bracket changes y; inside changes x the 'wrong' way.
Worked example

y = f(x) has a minimum point at (6, −4). Find the minimum point of y = f(3x).

  1. Inside the bracket: divide the x-coordinate by 3.
  2. 6 ÷ 3 = 2. The y-coordinate stays −4.

Answer: (2, −4)

Key idea

af(x) multiplies y-coordinates by a. f(ax) divides x-coordinates by a. Changes outside the bracket act on y; changes inside act on x, the opposite way.

Key words
Function
A rule that turns each input into one output, written f(x).
Scale factor
The number every length is multiplied by in an enlargement or between similar shapes.

All GCSE and International GCSE Mathematics key terms and definitions →

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  1. (5, 2) is on y = f(x). Which point is on y = 4f(x)?

    1. (20, 2)
    2. (5, 8)
    3. (54, 2)
    4. (5, 6)
    Show the answer

    Answer: (5, 8)

    Multiply the y-coordinate by 4: (5, 8).

  2. (9, 4) is on y = f(x). Which point is on y = f(3x)?

    1. (27, 4)
    2. (3, 4)
    3. (9, 43)
    4. (9, 12)
    Show the answer

    Answer: (3, 4)

    f(3x) squashes the graph: x-coordinates are divided by 3.

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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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