Period and Amplitude Changes
Objectives:1. Evaluate sine and cosine functions with
amplitude and period changes
Vocabulary:Amplitude, period
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.
0-1010sin x
0x2
2
32
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.
y
2
3
2
22
32
2
5
1
1
x
y = sin x
Remember that y = sin x is an odd function whose graph is symmetry to origin.
Transformations of Trig Functions
sin ( )y A B x h k
Amplitude |A|: A controls vertical stretching/shrinkingPeriod p can be in radians or degrees: p is the horizontal length to complete one cycleB can be in radians or degrees: B controls horizontal stretching/shrinking
pB
2
pB
360
-6 -4 -2 2 4 6 8
4
3
2
1
-1
-2
-3
-4
f x = sin x
Period =2π Amplitude = 1
4
3
2
1
-1
-2
-3
-4
-8 -6 -4 -2 2 4 6 8
f x = cos x
Period =2π Amplitude = 1
Vertical Stretching/Shrinking of Sine Functions
KEY TAKE-AWAY: x-intercepts are unchanged; multiply y-value of max/min by A.
y = sin(x)
y = sin(2x)
y = sin(x)
y = sin(x/2)
y
x
2y = cos (–x)
Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis.
Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
x
2y = sin x
y = sin (–x)
y = cos (–x)
Give the amplitude and period of the function
4sin 3 . Then sketch at least one cycle of its graph.y x
amplitude = 4 4 2 2
period = 3B
Key points:
x sin3x0
2
3
3
6
2
0
1
0
1
0
x 4sin 3x0
6
3
22
3
0
4
0
4
0
x 4sin3x0
6
3
22
3
0
4
0
4
0
Assignment:
p. 305 #1-9 odd
Period and Amplitude Changes
Objectives:1. Evaluate sine and cosine functions with
amplitude and period changes2. Identify Period and Amplitude from a
graph3. Write the equation or a trig function
from a graph
Vocabulary:Amplitude, period
3cos ( )3
y x
-6 -4 -2 2 4 6 8
4
3
2
1
-1
-2
-3
-4
q x = -3cos
3 x
Use the cosine curve, cos , with amplitude 3 and period 8.y A Bx2 2
88 4
BB
So the equation is 3cos .4
y x
2sin 4y x
-10 -8 -6 -4 -2 2 4 6 8 10
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
3sin2
y x
-10 -8 -6 -4 -2 2 4 6 8 10
6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
12cos
2y x
Assignment:
p. 305 #11-17 odd, 27, 29
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