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

Exam Tip

Be sure to apply transformations in the correct order – applying them in the wrong order can produce an incorrect transformation
When you sketch a transformed graph, indicate the new coordinates of any points that are marked on the original graph
Try to indicate the coordinates of points where the transformed graph intersects the coordinate axes (although if you don't have the equation of the original function this may not be possible)
If the graph has asymptotes, don't forget to sketch the asymptotes of the transformed graph as well

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

Transformations of Trigonometric Functions (DP IB Maths: AA HL)

Revision Note