Transformations of Sine and Cosine Functions
MHF4UI
Tuesday November 13th, 2012
Relating Trig Functions to Angles in
Standard Position
Last class we learned about the general shape and characteristics of
the six Trig Functions
For each angle 𝜃 that that was inputted into a trig function, the
resultant output was the trig ratio for that particular angle.
We can construct the graphs of each Trig Function with the help of
our calculators
OR
We are able to draw the angle in standard position to determine
what its corresponding Trig Ratios will be
For example, let’s choose the angle
𝜋
2
The General Form of Sinusoidal Functions
A sinusoidal function is a function can be produced by shifting,
stretching or compressing the sine function.
The general form of a Sinusoidal Function can be written as:
𝑦 = a • sin 𝑘 𝑥 − 𝑐 + 𝑑
OR
𝑦 = a • cos 𝑘 𝑥 − 𝑐 + 𝑑
Where,
a affects the Amplitude of the function
𝑘 affects the Period of the function
𝑐 affects the Phase Shift (or horizontal shift) of the function
𝑑 affects the vertical shift of the function
The General Form of Sinusoidal Functions
(Continued)
𝑦 = a • sin 𝑘 𝑥 − 𝑐 + 𝑑
OR
𝑦 = a • cos 𝑘 𝑥 − 𝑐 + 𝑑
The absolute value of a, or 𝑎 , is the amplitude of the graph, if a is
negative, it represents a reflection in the x-axis.
𝑘, will stretch or compress the Period of the Function.
The period of the function will be
2𝜋
.
𝑘
𝑐, will result in a Phase Shift to the right 𝑐 units for positive 𝑐. Conversely
the Phase Shift will be 𝑐 units to the left for a negative 𝑐.
𝑑, will result in a Vertical Shift up 𝑑 units for positive 𝑑. Conversely the
Vertical Shift will be 𝑑 units down for a negative 𝑑.
The Function 𝑦 = sin 𝑥
The Sine function that we discussed yesterday is in general form with :
𝑎=1
𝑐=0
𝑑=0
𝑘=1
The period was
2𝜋
1
= 2𝜋
It is important to realize that the characteristics of the Sine Function
will only apply if we do not apply any transformations.
When we apply any transformations we are potentially altering the
amplitude, period, Max/Min and intercepts of the function.
𝜋
Note that if we apply a Phase shift of − to the Sine function the
2
graph will look identical to the Cosine Function.
Applying Transformations Example 1
Given the following equation, state the amplitude, period, phase shift
and vertical shift.
𝜋
𝑦 = 3 sin 2 𝑥 +
−2
2
Applying Transformations Example 2
Given the following equation, state the amplitude, period, phase shift
and vertical shift.
𝑦 = −2 sin 4𝑥 − 2𝜋 + 2
Mapping Notation for Sinusoidal Functions
Recall that for Logarithmic Functions we used mapping notation to
map key points to their coordinates for the transformed function.
The formula for mapping notation of Sinusoidal Functions is identical
to the mapping notation of Logarithmic Functions:
1
(x, 𝑦) ( 𝑥 + 𝑐, 𝑎𝑦 + 𝑑)
𝑘
What we must determine now is what key points we will choose map
to our transformed function.
𝑦 = sin 𝑥
The Key Points for 𝑦 = sin 𝑥
As we just pointed out, the key points for
the sine function are:
𝑥, 𝑦 = (0,0)
𝜋
𝑥, 𝑦 = ( , 1)
2
𝑥, 𝑦 = (𝜋, 0)
3𝜋
𝑥, 𝑦 = ( , −1)
2
𝑥, 𝑦 = 2𝜋, 0
You MUST memorize the key points!
𝑦 = cos 𝑥
The Key Points for 𝑦 = cos 𝑥
As we just pointed out, the key points for
the cosine function are:
𝑥, 𝑦 = (0,1)
𝜋
𝑥, 𝑦 = ( , 0)
2
𝑥, 𝑦 = (𝜋, −1)
3𝜋
𝑥, 𝑦 = ( , 0)
2
𝑥, 𝑦 = 2𝜋, 1
You MUST memorize the key points!
Graphing Sine and Cosine Functions
When sketching a sine or cosine function, you must:
Label all 5 key points
(After you have applied the mapping notation)
Show the general shape of the graph.
Graphing Sine and Cosine Functions Example 1
Find the key points and sketch the function 𝑦 = 2 sin 2𝑥 − 1
𝑎 = 2, 𝑘 = 2, 𝑐 = 0, 𝑑 = −1
1
(x, 𝑦) ( 𝑥, 2𝑦 − 1)
2
(0,0)
𝜋
( 2 , 1)
(𝜋, 0)
3𝜋
( 2 , −1)
2𝜋, 0
1
2
0 , 2 0 − 1 = (0, −1)
1 𝜋
2 2
1
𝜋
, 2 1 − 1 = ( 4 , 1)
𝜋
(2 𝜋, 2(0) − 1) = ( 2 , −1)
1 3𝜋
3𝜋
(2 ( 2 ), 2(−1) − 1) = ( 4 , −3)
1
2
( 2𝜋, 2(0) − 1) = (𝜋, −1)
Graphing Sine and Cosine Functions Example 1
Find the key points and sketch the function 𝑦 = 2 sin 2𝑥 − 1
Graphing Sine and Cosine Functions Example 2
𝜋
2
Find the key points and sketch the function 𝑦 = −5 sin 2(𝑥 − ) + 2
𝜋
𝑎 = −5, 𝑘 = 2, 𝑐 = , 𝑑 = 2
2
1
𝜋
(x, 𝑦) ( 𝑥 + , −5𝑦 + 2)
2
2
(0,0)
𝜋
( 2 , 1)
(𝜋, 0)
3𝜋
( 2 , −1)
2𝜋, 0
1
2
𝜋
𝜋
0 + 2 , −5 0 + 2 = ( 2 , 2)
1 𝜋
2 2
𝜋
3𝜋
+ 2 , −5 1 + 2 = ( 4 , −3)
1
𝜋
(2 𝜋 + 2 , −5(0) + 2) = (𝜋, 2)
1 3𝜋
𝜋
5𝜋
(2 ( 2 ) + 2 , −5(−1) + 2) = ( 4 , 7)
1
2
𝜋
2
3𝜋
, 2)
2
( 2𝜋 + , −5(0) + 2) = (
Graphing Sine and Cosine Functions Example 2
𝜋
2
Find the key points and sketch the function 𝑦 = −5 sin 2(𝑥 − ) + 2
Graphing Sine and Cosine Functions Example 3
1
Find the key points and sketch the function 𝑦 = − cos (𝑥 + 𝜋)
2
1
𝑎 = −1, 𝑘 = , 𝑐 = −𝜋, 𝑑 = 0
2
(x, 𝑦) (2𝑥 − 𝜋, −𝑦)
(0,1)
𝜋
2 0 − 𝜋, − 1
𝜋
2
− 𝜋, − 0
= (−𝜋, −1)
( 2 , 0)
2
= (0,0)
(𝜋, −1)
(2𝜋 − 𝜋, −(−1)) = (𝜋, 1)
3𝜋
(2( 2 ) − 𝜋, −(0)) = (2𝜋, 0)
2𝜋, 1
(2(2𝜋) − 𝜋, −(1)) = (3𝜋, −1)
( 2 , 0)
3𝜋
Graphing Sine and Cosine Functions Example 3
Find the key points and sketch the function 𝑦 =
1
− cos (𝑥
2
+ 𝜋)
Homework Questions:
Complete the Transformations
Worksheet Handout