Contributed by:
OBJECTIVES:
1. We can determine the amplitude, period, and phase shifts of trig functions
2. We can write trig equations given specific periods, phase shifts, and amplitude.
1.
Amplitude, Period, and
Phase Shift
2.
Objectives
• I can determine amplitude,
period, and phase shifts of trig
functions
• I can write trig equations given
specific period, phase shift, and
amplitude.
2
3.
Section 4.5: Figure 4.49, Key
Points in the Sine and Cosine
Curves
3
4.
Radian versus Degree
• We will use the following to graph or write
equations:
– “x” represents radians
– “” represents degrees
– Example: sin x versus sin
4
5.
a sin b( x ps ) d
Period:
2π/b Phase Shift:
Left (+)
Right (-)
Vertical Shift
Up (+)
Down (-)
5
6.
The Graph of y = AsinB(x - C)
The graph of y = A sin B(x – C) is obtained by horizontally shifting the graph
of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to
x = C. The number C is called the phase shift.
amplitude = | A| y
period = 2 /BB. y = A sin Bx Amplitude: | A|
x
Starting point: x = C
Period: 2/B
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
7.
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = 2 sin x 3
2 2 2 x
1
y= 2
sin x
y = – 4 sin x y = sin x
reflection of y = 4 sin x y = 4 sin x
4
7
8.
The period of a function is the x interval needed for the
function to complete one cycle.
For b 0, the period of y = a sin bx is 2 .
b
For b 0, the period of y = a cos bx is also 2 .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y sin 2 period: 2
period: y sin x x
2
If b > 1, the graph of the function is shrunk horizontally.
y y cos x
1
y cos x period: 2
2 2 3 4 x
period: 4
8
9.
Use basic trigonometric identities to graph y = f (–x)
Example 1: 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. y y = sin (–x)
Use the identity
sin (–x) = – sin x x
y = sin x 2
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
cos (–x) = cos x x
2
y = cos (–x)
9
10.
Example
Determine the amplitude, period, and phase shift of
y = 2sin (3x - )
First factor out the 3
y = 2 sin 3(x - /3)
Amplitude = |A| = 2
period = 2/B = 2/3
phase shift = C/B = /3 right
10
11.
Find Amplitude, Period, Phase Shift
• Amplitude (the # in front of the trig. Function
• Period (360 or 2 divided by B, the #after the trig function
but before the angle)
• Phase shift (the horizontal shift after the angle and inside
the parenthesis)
• y = 4sin y = 2cos1/2 y = sin (4x - )
Amplitude: 4 2 1
Phase shift: NA NA ( Right )
4
Period: 360 720
2
11
12.
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
3
x 0 2
2 2
y = 3 cos x 3 0 -3 0 3
max x-int min x-int max
y
(0, 3) (2, 3)
2
1 2 3 4 x
1 ( 3 , 0)
2 ( 2 , 0) 2
3 ( , –3)
12
13.
Writing Equations
• Write an equation for a positive sine curve with an amplitude
of 3, period of 90 and Phase shift of 45 left.
• Amplitude goes in front of the trig. function, write the eq.so
far:
• y = 3sin
• period is 90. use P = 360 90, so B 360 4
• B 90
• rewrite the eq.
• y = 3 sin4
• 45 degrees left means +45
• Answer: y = 3sin4( + 45)
13
14.
Writing Equations
• Write an equation for a positive cosine curve with an
amplitude of 1/2, period of 4 and Phase shift of right .
• Amplitude goes in front of the trig. function, write the eq.so
far:
• y = 1/2cos x
• period is /4. use P = 2 , so B 2 4 8
• B 4 1
• rewrite the eq.
• y = 1/2cos 8x
• right is negative, put this phase shift inside the parenthesis
w/ opposite sign.
• Answer: y = 1/2cos8(x - )
14
+1-315-636-4485
+91 98151-74050 
