# transformations of the trigonometric graphs ```MCR3U1
U5L5
TRANSFORMATIONS OF THE TRIGONOMETRIC GRAPHS
(Combinations of all Types)
PART A ~ INTRODUCTION
A trigonometric graph can be transformed as follows:
𝑦 = asin[k(𝑥 − 𝑑)] + 𝑐
AMPLITUDE
PHASE SHIFT
 vertical stretch
(|a| &gt; 1)
 vertical compression
(0 &lt; |a| &lt; 1)
 reflection (x-axis)
(a &lt; 0)
VERTICAL TRANSLATIONS
 (+) move left
 (–) move right
 (+) move up
 (–) move down
PERIOD
360𝑂
|𝑘|
Transformations should be applied as follows:
1. all stretches/compressions/reflections (shape)
2. all translations (location)
OR
Use mapping of the 5 key points of each trigonometric graph.
k and d affect the x value
𝑦 = 𝑎 ∙ sin 𝑘(𝑥 − 𝑑) + 𝑐
𝑥
(𝑥, 𝑦) → ( + 𝑑, 𝑎𝑦 + 𝑐)
𝑘
𝑦 = 𝑎 ∙ cos 𝑘(𝑥 − 𝑑) + 𝑐
a and c affect the y value
Ex.
Describe the transformations/properties for each function:
a)
1
𝑦 = cos(𝑥 − 300 ) + 2
2
b)
1
𝑦 = 3sin⁡( 𝑥 + 300 )
2
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MCR3U1
U5L5
PART B ~ DETERMINING THE HORIZONTAL SCALE
Determine: &frac14;  period
Determine: phase shift
Determine: the GCF between these two values
Examples
a)
This determines the horizontal scale!!
The period is 1800 and the phase shift is 300.
 use a scale with increments of ____________
b)
The period is 3600 and the phase shift is 450.
 use a scale with increments of ____________
PART C ~ EXAMPLES
Ex.
Graph one cycle of each of the following functions. State the range, amplitude, period,
phase shift, and equation of axis.
a)
𝑦 = − sin⁡(𝑥 − 60&deg;)
1
2
y
x
range:
______________________________________________________________
phase shift:
____________________
amplitude: ____________________
equ’n of axis:
____________________
period:
____________________
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b)
U5L5
𝑦 = 3 cos(2𝑥 + 90&deg;) − 1
y
x
range:
______________________________________________________________
phase shift:
____________________
amplitude: ____________________
equ’n of axis:
____________________
period:
____________________
Using mapping…
(x, y)
(00, 1)
(900, 0)
(1800, –1)
(2700, 0)
(3600, 1)
HOMEWORK: p.383–384 #3, 5, 6abcf (omit domain), 7abcde
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