Name: Transformations of Trigonometric Functions (3.5.2) | IB DP Maths: AA HL Revision Notes 2021 Membership
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3.5.2 Transformations of Trigonometric Functions

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 ( $\frac{x}{a}$ )
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 $\frac{360}{b}$ ° (or $\frac{2 π}{b}$ 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 $\frac{360}{b}$ ° (or $\frac{2 π}{b}$ )
- 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 $\frac{180}{b}$ ° (or $\frac{2 π}{b}$ )
- 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

Worked Example

Sketch the graph of $y = 2 \sin (3 (x - \frac{π}{4})) - 1$ for the interval -2π ≤ x ≤ 2π. State the amplitude, period and principal axis of the function.

aa-sl-3-5-2-transformations-of-trig-functions-we-solution-1

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IB DP Maths: AA HL

Revision Notes

3.5.2 Transformations of Trigonometric Functions

Transformations of Trigonometric Functions

What transformations of trigonometric functions do I need to know?

What combined transformations are there?

How do I sketch transformations of trigonometric functions?

Worked Example

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