graphs of sine and cosine functions · 2020. 2. 5. · graphs of sine and cosine functions period...
TRANSCRIPT
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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21T
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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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i
0
1 g Sink2 y fsin x
90T EACHy VALUE IN
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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DOMAIN P A
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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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