Unit 6
MCR 3U1
Lesson 4: Exploring Transformations of Periodic Functions
The transformations that apply to algebraic functions also apply to trigonometric functions.
Vertical Stretches
The graphs of y  a sin x and y  a cos x can be summarized as follows.
If a  1 , vertical expansion by a factor of a occurs.
If 0  a  1 , a vertical compression by a factor of a occurs.
For both functions the amplitude of the function is represented by a.
Horizontal Stretches
The graphs of y  sin kx and y  cos kx can be summarized as follows.
1
occurs.
k
1
If 0  k  1 , then a horizontal expansion by a factor of
occurs.
k
If k  1 , then a horizontal compression by a factor of
For both functions, the period is
360
.
k
Sketching Sine and Cosine Functions
The five point method is a convenient way to sketch the graph of a sine or cosine function using
its amplitude and period. This method includes the fact that one cycle of a sine of cosine
function includes a maximum, a minimum, and three zeroes. These five key points are equally
spaced along the x-axis, so they divide the period into quarters. The five points to plot and the
order in which they get plotted is as follows:
Point 1: First zero
Point 2: First Maximum
Point 3: Second zero
Point 4: First Minimum
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  3 sin 2 x
Point 5: Third zero
Unit 6
b) y 
MCR 3U1
1
4
cos x
2
5
Example 2: Write the equation of each sine function described.
a) amplitude 7, period 90o.
b) amplitude 0.5, period 270o.
Example 3: Determine the equation of the cosine function shown.