Transforming graphs · Year 13

In Year 13 you learn: combining transformations.

If you know one graph, y = f(x), you can sketch a whole family of related graphs. Adding, multiplying or changing sign — outside the bracket or inside it — translates, stretches or reflects the original graph in predictable ways.

Key facts

  • y = f(x) + a: translate up by a, vector (0, a).
  • y = f(x + a): translate left by a, vector (−a, 0) — changes inside the bracket work the opposite way.
  • y = af(x): vertical stretch, scale factor a (multiply the y-coordinates).
  • y = f(ax): horizontal stretch, scale factor 1/a (divide the x-coordinates).
  • y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis.
  • Follow a key point: the turning point (2, −1) of f moves to (5, −1) on y = f(x − 3).

Worked examples

Example 1

(4, 6) is on y = f(x). Find the matching point on (a) y = 2f(x), (b) y = f(2x), (c) y = −f(x).

Answer: (a) (4, 12) (b) (2, 6) (c) (4, −6)

Example 2

State the maximum value and the period of y = 3 sin(2x), with x in degrees.

Answer: Maximum 3; period 180°

Other years