Download Using Transformations to Sketch Sinusoidal Functions

Unit 6
MCR 3U1
Lesson 5: Using Transformations to Sketch Graphs of Sinusoidal Functions
Vertical Translations
The graphs of y  sin x  c and y  cos x  c can be summarized as follows.
If c  0 , a vertical translation of c units upwards occurs.
If c  0 , a vertical translation of c units downwards occurs.
Vertical translations affect the range of the function.
The equation of the axis is y  c .
Horizontal Translations
For trigonometric functions, a horizontal translation is called the phase shift. The graphs of
y  sin( x  d ) and y  cos( x  d ) can be summarized as follows.
If d  0 , a horizontal translation of d units to the left occurs.
If d  0 , a horizontal translation of d units to the right occurs.
Reflections
If a  0 , then the function is reflected in the x-axis.
If k  0 , then the function is reflected in the y-axis.
Example 1: For each of the following:
i) Describe the transformations that occur.
ii) Sketch one cycle of the graph.
iii) State the period, amplitude, maximum, minimum and domain and range of one cycle for
each function.
a) y  2 cos x  2
Unit 6
b) y  0.5 sin( x  90)
MCR 3U1
c) y  2 sin( 0.5 x  90)  1
Example 2: Determine the equation of the sine function shown.