Amplitude, Period, and Phase Shift

Section 4-5

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.

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Section 4.5: Figure 4.49, Key Points in the Sine and Cosine

Curves

4

Radian versus Degree

• We will use the following to graph or write equations:– “x” represents radians– “” represents degrees– Example: sin x versus sin

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sin ( )a b x ps d Amplitude

Period:

2π/b Phase Shift:

Left (+)

Right (-)Vertical Shift

Up (+)

Down (-)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

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|

period = 2 /B.

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|

period = 2 /B.

x

y

Amplitude: | A|

Period: 2/B

y = A sin Bx

Starting point: x = C

The Graph of y = AsinB(x - C)

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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.

2

32

4

y

x

4

2

y = – 4 sin xreflection of y = 4 sin x y = 4 sin x

y = 2 sin x

2

1y = sin x

y = sin x

8

y

x

2

sin xy period: 2 2sin y

period:

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 .b

2

For b 0, the period of y = a cos bx is also .b

2

If 0 < b < 1, the graph of the function is stretched horizontally.

If b > 1, the graph of the function is shrunk horizontally.y

x 2 3 4

cos xy period: 2

2

1cos xy

period: 4

9

y

x

2y = cos (–x)

Use basic trigonometric identities to graph y = f (–x)Example 1: Sketch the graph of y = sin (–x).

Use the identity sin (–x) = – sin x

The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis.

Example 2: Sketch the graph of y = cos (–x).

Use the identity cos (–x) = 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)

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ExampleDetermine the amplitude, period, and phase shift of

y = 2sin (3x - )Solution:First factor out the 3y = 2 sin 3(x - /3)Amplitude = |A| = 2period = 2/B = 2/3phase shift = C/B = /3 right

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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:

Phase shift:

Period:

4 2 1

NA NA )(4

Right

360 7202

12

y

1

123

2

x 32 4

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.

maxx-intminx-intmax

30-303y = 3 cos x20x 2

2

3

(0, 3)

2

3( , 0)( , 0)

2

2( , 3)

( , –3)

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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 = • • rewrite the eq. • y = 3 sin4• 45 degrees left means +45

• Answer: y = 3sin4( + 45)

490

360,90

360 Bso

B

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Writing Equations• Write an equation for a positive cosine curve with an

amplitude of 1/2, period of 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 = • • 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 - )

84

1

2,

4

2

BsoB

4

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Homework

• Trig Value Table

• WS 7-1

• Quiz next class