Transformations of graphs

Outside the function: y-values

• y = f(x) + a: translate up by a
• y = af(x): stretch vertically, scale factor a
• y = -f(x): reflect in the x-axis

These change the y-coordinates only.

Inside the function: x-values

• y = f(x + a): translate left by a
• y = f(ax): stretch horizontally, scale factor 1/a
• y = f(-x): reflect in the y-axis

Changes inside the bracket do the opposite of what you might expect.

Points and asymptotes

Apply the change to key points: under y = f(x - 3) the point (2, 5) moves to (5, 5). Asymptotes move too: y = 1/(x - 3) + 2 has asymptotes x = 3 and y = 2.

Worked example

y = f(x) has a maximum at (2, 5). Find the maximum of y = f(x - 3), y = 2f(x) and y = f(2x).

1. f(x - 3): move right 3 to (5, 5)
2. 2f(x): double the y-coordinate to (2, 10)
3. f(2x): halve the x-coordinate to (1, 5)

Answer: (5, 5), (2, 10) and (1, 5)

Check yourself

1. How is y = f(x + 4) related to y = f(x)?
   • A translation 4 units to the right
   • A translation 4 units up
   • A translation 4 units to the left
   • A translation 4 units down
   Show answer

   A translation 4 units to the left. Adding inside the bracket moves the graph left.

2. How is y = f(3x) related to y = f(x)?
   • A horizontal stretch, scale factor 1/3
   • A horizontal stretch, scale factor 3
   • A vertical stretch, scale factor 3
   • A vertical stretch, scale factor 1/3
   Show answer

   A horizontal stretch, scale factor 1/3. f(ax) is a horizontal stretch with scale factor 1/a.

3. The point (6, -2) lies on y = f(x). Find the x-coordinate of its image on y = f(2x).
   Show answer

   3. f(2x) halves the x-coordinates: 6/2 = 3.

4. The point (6, -2) lies on y = f(x). Find the y-coordinate of its image on y = 3f(x).
   Show answer

   -6. 3f(x) multiplies the y-coordinates by 3: -6.

5. What are the asymptotes of y = 1/(x - 3) + 2?
   • x = -3 and y = 2
   • x = 3 and y = -2
   • x = 2 and y = 3
   • x = 3 and y = 2
   Show answer

   x = 3 and y = 2. y = 1/x moved right 3 and up 2, so the asymptotes x = 0 and y = 0 become x = 3 and y = 2.

6. y = x² is transformed to y = (x - 1)² - 4. Describe the transformation.
   • Translation 1 left and 4 down
   • Translation 1 right and 4 down
   • Translation 1 right and 4 up
   • Translation 1 left and 4 up
   Show answer

   Translation 1 right and 4 down. (x - 1) moves right 1; -4 outside moves down 4.

7. y = sin x is transformed to y = 2 + 3 sin x. Find the maximum value of the new graph.
   Show answer

   5. sin x has maximum 1. Stretch by 3 to get 3, then move up 2 to get 5.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

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