Graphs of Sine and Cosine Functions
[Pages:13]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
L28 - 1
THINKABOUTTHE UNIT CIRCLE
The Graph of y = sin x
Domain:
ANY X VALUE YOU WANT
Range: 7,10
Period: 21T STARTS REPEATING THE CYCLE AFTER 21T The key points in one period:
12
oil
x
P
ooo 10
3
0
2
2
2
01
y = sin x 0
312 2
I
O
I
0
1
2
2
2
-1
L28 - 2
The Graph of y = cos x
Domain: C P P ANY X VALUE YOU WANT Range: C I I Period: 21T The key points in one period:
12 0,1
x
o 1110
3
0
2
2
2
oil
y = cos x I
0
I
3I
2
O
l
To1
F
Y
z
2
-1
Checkpoint: Lecture 28, problem 1
L28 - 3
Amplitude for Functions y = a sin x and
y = a cos x
i
Def. The amplitude of y = a sin x and y = a cos x represents half the distance between the maximum and minimum values of the function and is given by
Amplitude = |a|
maximum value - minimum value
=
20
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].
1 ex. Graph y = sin x
2
AMP _141 1
1
DOMAIN 0,0
ZANGE I I
-
-1
1 g Sink
2 y fsin x
90T EACH
y VALUE IN
HALF
y
2i
i
i
L28 - 4
MULTIPLY EACH Y VA
ex. Graph y = -2 cosBxYZ
y COSCX 7 2 y 2C0S X
3 y 2C X
i
1a l e al
oo
fol
r
r
I rr
-
i
2
-1
X
r
I
l
rr
og
o
AMP
21 2
MAX MIN
I1
OR AMP 2
42 2
DOMAIN P A
RANHE f 2,2
Checkpoint: Lecture 28, problem 2
L28 - 5
Period for Functions y = a sin(bx) and y = a cos(bx) The period of y = a sin(bx) and y = a cos(bx) is given by
2 Period =
b where b is a positive real number.
NOTE: 1. If 0 < b < 1, the period of y = a sin(bx) is greater than 2 and represents a horizontal stretching of the graph y = a sin x. 2. If b > 1, the period of y = a sin(bx) is less than 2 and represents a horizontal shrinking of the graph y = a sin x.
L28 - 6
ex. Find the amplitude and period of each function, and sketch its graph.
1) y = 4 cos(3x)
1
-
-1
L28 - 7
1 2) y = -2sin x
2
1
-
-1
Checkpoint: Lecture 28, problem 3
L28 - 8
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