Transformations of Trigonometric Graphs

Graphs of ,

These graphs are obtained by vertically scaling the graphs of or by a factor of and horizontally scaling by a factor of .
  • Maximum Value:
  • Minimum Value:
  • Period:

Graphs of ,


  1. This graph is obtained by horizontally translating the graph of by and vertically translating it by .

  2. Similarly, this graph is obtained by horizontally translating the graph of by and vertically translating it by .
  • Maximum Value:
  • Minimum Value:
  • Period:

Graph of

This graph is obtained by horizontally translating the graph of by and vertically translating it by .
  • Maximum and Minimum Values: None (unbounded)
  • Period:
  • Equation of Asymptotes: The asymptotes of , given by , are shifted horizontally by , resulting in , where is an integer.

Key Points for Transformations:

  1. Vertical Scaling:
    Scaling or vertically changes the range but does not affect the period.
  2. Horizontal Scaling:
    Scaling or horizontally changes the period but does not affect the range.
  3. Translations:
    • Horizontal translation (by ) affects the graph’s position but not its range or period.
    • Vertical translation (by ) changes the range but not the period.

Parameters Summary: