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y
x- intercepts
x
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x
x- intercepts
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x
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y
x
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x
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y
x
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Draw one full period of y2sin(3xp/2)
Period 2p/b 2p/3
2
Amplitude a 2
Phase shift -c/b p/6
2p/3
5p/6
p/2
p/6
p/3
-2
This is the graph of 2sin(3x). Now click to see
the phase shift and to get 2sin(3xp/2)
26
Graph one full period of sin(xp /2) 1/2
a 1, b 1,c p/2 and d 1/2 Amplitude a
1 Period 2p/b 2p Phase shift c/b
p/2 Vertical translation 1/2 units down
Section 5.7 Question 43
Vertical translation ½ units down
1
y sin(xp /2)
1/2
1
y sin(x)
3/2
y sin(xp /2) 1/2
Phase shift p/2 units right
27
Graph one full period of 2sin(3xp /2) 1
y
a 2, b 3,c p/2 and d 1 Amplitude a
2 Period 2p/b 2p/3 Phase shift c/b
p/6 Vertical translation 1 unit up
3
2
1
p/2
2p/3
p/6
x
p/3
1
2
This is the graph of 2sin(3x). Now click to see
the phase shift , vertical translation and to get
2sin(3xp/2)1
28
Graph one full period of sin(xp /6)
y
a 1, b 1,c p/6 Amplitude a 1 Period
2p/b 2p Phase shift c/b p/6
Section 5.7 Question 18
1
3p/2
2p
p/2
p
x
p/6
1
This is the graph of sin(x). Now click to see the
phase shift and to get sin(xp/6)
29
Graph one full period of cos(2xp/3)
y
Section 5.7 Question 20
a 1, b 2,c p/3 Amplitude a 1 Period
2p/b p Phase shift c/b p/6
1
p/2
x
p
p/6
p/4
3p/4
7p/6
1
This is the graph of cos(2x). Now click to see
the phase shift and to get cos(2x-p/3)
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Graph one full period of y(1/2)sin(px/3)
y
a 1/2, b p/3 Amplitude a 1/2 Period
2p/b 6
1/2
9/2
6
3
3/2
x
1/2
31
y
x
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Graph one full period of y2sinx and y sinx
In 0 , p ,
2
0 sinx 2sinx
In p , 2p ,
1
2sinx sinx 0
1
2
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x
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x
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x
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x
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Draw one full period of y 2tan(x/2)
y
a 2 and b 1/2 , 4b 2
Asymptotes
x 2p/4b 2p/2 p
Lets draw asymptotes
2
Mark 2 and 2 on the y-axis and p/4b p/2 on
the x-axis
x
2
Now we can draw the graph
Section 5.6 Question 29
40
Graph one full period of (3/2)csc(3x)
a 3/2, b 3 Period 2p/b 2p/3
3/2
p/2
p/6
p/3
2p/3
3/2
Section 5.6 Question 34
41
Graph one full period of (1/3)tanx
a 1/3, b 1?4b 4 Period p/b p
1/3
p/4
p/4
p/2
p/2
1/3
Section 5.6 Question 22
42
Graph one full period of 2cscx
y
a 2, b 1 Period 2p/b 2p
2
3p/2
p/2
x
p
2p
2
Section 5.6 Question 28
43
Graph one full period of -3sec(2x/3)
y
a 3 , b 2/3 Period 2p/b 3p
3
x
3p/4
3p/2
9p/4
3p
3
Section 5.6 Question 36
44
y
Draw one full period of y 3tan(3x)
a 3 and b 3 , 4b 12
Asymptotes
x 2p/4b 2p/12 p/6 Period p/b p/2
Lets draw asymptotes
3
Mark 3 and 3 on the y-axis and p/4b p/12 on
the x-axis
x
3
Now we can draw the graph
45
y
Draw one full period of y (1/2)cot(2x)
a 1/2 and b 2 , 4b 8
Asymptotes
x p/b p/2 and x 0(y-axis) Period p/b
p/2
Lets draw asymptotes
1/2
Mark 1/2 and 1/2 on the y-axis and p/8, 2p/8,
3p/8 and 4p/8 on the x-axis
x
1/2
Now we can draw the graph
46
Graph one full period of y3/2sin(x /43p /4)
y
a 3/2, b 1/4,c 3p/4 Amplitude a 3/2
Period 2p/b 8p Phase shift c/b 3p
3/2
6p
8p
x
3p
4p
2p
3/2
This is the graph of y3/2sin(x/4). Now click to
see the phase shift and to get y3/2sin(x/43p/4)
47
Graph one full period of y sec(x - p/2 )1
y
a 1 , b 1, c -p/2, d 1 Period 2p/b
2p Phase shift -c/b p/2 Vertical
translation (d ) 1 unit up
sec(x)
2
1
cos(x)
p




x
3p/2
2p
p/2
1
Click to shift p/2 unit to right
Click to shift 1 unit up
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Graph one full period of y csc(x/3-p/12)4
y
a 1, b 1/3, c p/3, d 4 Period 2p/b
6p Phase shift -c/b p/4 Vertical translation
4 unit up
5
3
sin(x/3)
1
9p/2




x
3p
6p
3p/2
p/4
1
csc(x/3)
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Sketch the graph of y (1/2)sin(3x)
1/2sin(3x) 0
1/2sin(3x) 0
1/2
2p/3
0
p/3
-2p/3
-p/3
-1/2
1/2sin(3x) 0
1/2sin(3x) 0
Section 5.5 Question 48
50
Sketch the graph of y cos2(x)
1
1/2
-p
-p/4
p/2
p
3p/2
-p/2
p/4
-3p/2
Section 5.5 Question 65
51
Sketch the graph of y sinx
sin(x) if x 0
y sinx
-sin(x) if x 0
1
0
-2p
p
-p
2p
-1
Section 5.5 Question 68