Lecture 28: Section 4.5

Graphs of Sine and Cosine Functions

Period of sine and cosine

Amplitude of sine and cosine

Horizontal translation - phase shift

Vertical translations

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The Graph of y = sin x

Domain:

Range:

Period:

The key points in one period:

x 0π

2π

3π

22π

y = sin x

1

−1

π2π

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The Graph of y = cos x

Domain:

Range:

Period:

The key points in one period:

x 0π

2π

3π

22π

y = cos x

1

−1

π2π

Checkpoint: Lecture 28, problem 1

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Amplitude for Functions y = a sin x and

y = a cos x

Def. The amplitude of y = a sin x and y =a cos x represents half the distance between the max-imum and minimum values of the function and isgiven by

Amplitude = |a|

=maximum value - minimum value

2

NOTE:

1. If |a| > 1, the curve is stretched vertically, and if|a| < 1, the curve is shrunk vertically.

2. The range is [−a, a].

ex. Graph y =1

2sin x

π− π 2 π

−1

1

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0

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AMP _141 1 HALF

DOMAIN 0,0 y

ZANGE I I i

i i

ex. Graph y = −2 cos x

−1

1

π− π 2 π

Checkpoint: Lecture 28, problem 2

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Period for Functions y = a sin(bx) andy = a cos(bx)

The period of y = a sin(bx) and y = a cos(bx) isgiven by

Period =2π

b

where b is a positive real number.

NOTE:

1. If 0 < b < 1, the period of y = a sin(bx) is greaterthan 2π and represents a horizontal stretching of thegraph y = a sin x.

2. If b > 1, the period of y = a sin(bx) is less than2π and represents a horizontal shrinking of the graphy = a sin x.

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ex. Find the amplitude and period of each function,and sketch its graph.

1) y = 4 cos(3x)

−1

1

π− π

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2) y = −2sin

(

1

2x

)

π− π

−1

1

Checkpoint: Lecture 28, problem 3

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Horizontal Translations

y = a sin(bx) y = a sin(bx− c)

Amplitude A = A =

Period P = P =

One cycle 0 ≤ bx ≤ 2π 0 ≤ bx− c ≤ 2π

≤ x ≤ ≤ x ≤

NOTE: The graph of y = a sin(bx) is shifted byc

bunits to get the graph of y = a sin(bx − c). The

numberc

bis called the phase shift.

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ex. Find the amplitude, period and phase shift ofeach function, and sketch its graph.

1) y =3

2sin

(

2x−π

2

)

−1

1

π− π

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2) y =3

4cos

(

2x +2π

3

)

−1

1

π− π

Checkpoint: Lecture 28, problem 4

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Vertical Translations

ex. Find the amplitude, period and phase shift of

y = 1 +3

2sin

(x

2+π

2

)

, and sketch its graph.

π− π

−1

1

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ex. As a wave passes by an offshore piling, the heightof the water is modeled by the function

h(t) = 3 cos( π

10t)

where h(t) is the height in feet above mean sea levelat time t seconds.

1) Find the period of the wave.

2) Find the wave height, that is, the vertical distancebetween the trough and the crest of the wave.

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