Transforming graphs

Changing the equation of a graph moves or stretches it in predictable ways. Changes outside the function act on y; changes inside act on x, and they work the opposite way to what you might expect.

Key facts

  • y = f(x) + a: up a.
  • y = f(x + a): left a (the opposite way).
  • y = af(x): stretch vertically by a.
  • y = f(ax): stretch horizontally by 1/a.
  • y = −f(x): reflect in the x-axis. y = f(−x): reflect in the y-axis.

Worked examples

Example 1

y = x² has its turning point at (0, 0). Where is the turning point of y = (x − 3)² + 2?

  1. (x − 3): right 3.
  2. + 2: up 2.
  3. (0, 0) moves to (3, 2).

Answer: (3, 2)

Example 2

The point (2, 5) is on y = f(x). Where does it go on y = f(2x)?

  1. f(2x) squashes towards the y-axis by a factor of ½.
  2. x-coordinates halve: (1, 5).

Answer: (1, 5)