Transformations of Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Transformations of Trigonometric Functions

What transformations of trigonometric functions do I need to know?

• As with other graphs of functions, trigonometric graphs can be transformed through translations, stretches and reflections

• Translations can be either horizontal (parallel to the x-axis) or vertical (parallel to the y-axis)

  • For the function y = sin (x)

    • A vertical translation of a units in the positive direction (up) is denoted by
      y = sin (x) + a

    • A vertical translation of a units in the negative direction (down) is denoted by
      y = sin (x) - a

    • A horizontal translation in the positive direction (right) is denoted by y = sin (x - a)

    • A horizontal translation in the negative direction (left) is denoted by y = sin (x + a)

• Stretches can be either horizontal (parallel to the x-axis) or vertical (parallel to the y-axis)

  • For the function y = sin (x)

    • A vertical stretch of a factor a units is denoted by y = a sin (x)

    • A horizontal stretch of a factor a units is denoted by y = sin ()

• Reflections can be either across the x-axis or across the y-axis

  • For the function y = sin (x)

    • A reflection across the x-axis is denoted by y = - sin (x)

    • A reflection across the y-axis is denoted by y = sin (-x)

What combined transformations are there?

• Stretches in the horizontal and vertical direction are often combined

• The functions a sin(bx) and a cos(bx) have the following properties:

  • The amplitude of the graph is |a |

  • The period of the graph is ° (or rad)

• Translations in both directions could also be combined with the stretches

• The functions a sin(b(x - c )) + d and a cos(b(x - c )) + d have the following properties:

  • The amplitude of the graph is |a |

  • The period of the graph is ° (or )

  • The translation in the horizontal direction is c

  • The translation in the vertical direction is d

    • d represents the principal axis (the line that the function fluctuates about)

• The function a tan(b(x - c )) + d has the following properties:

  • The amplitude of the graph does not exist

  • The period of the graph is ° (or )

  • The translation in the horizontal direction is c

  • The translation in the vertical direction (principal axis) is d

How do I sketch transformations of trigonometric functions?

• Sketch the graph of the original function first

• Carry out each transformation separately

  • The order in which you carry out the transformations is important

  • Given the form y = a sin(b(x - c )) + d carry out any stretches first, translations next and reflections last

    • If the function is written in the form y = a sin(bx - bc ) + d factorise out the coefficient of x before carrying out any transformations

  • Use a very light pencil to mark where the graph has moved for each transformation

• It is a good idea to mark in the principal axis the lines corresponding to the maximum and minimum points first

  • The principal axis will be the line y = d

  • The maximum points will be on the line y = d + a

  • The minimum points will be on the line y = d - a

• Sketch in the new transformed graph

• Check it is correct by looking at some key points from the exact values