IB DP Maths: AA HL复习笔记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?vertical?translation of?a?units?in the?positive direction?(up) is denoted by
- For the function?y = sin (x)
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)
- For the function?y = sin (x)
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)
- For the function?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 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
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