19. y
x- intercepts x

20. y x
x- intercepts

21. y x

22. y x

23. y x

24. y x

25. Draw one full period of y=2sin(3x–π/2)
Period = 2π/|b| = 2π/3 2
Amplitude = |a| = 2
Phase shift = -c/b = π/6
π/2
2π/3
5π/6
π/6
π/3 -2
This is the graph of 2sin(3x). Now click to see the phase shift and to get 2sin(3x–π/2)

26. Graph one full period of sin(x–π /2) –1/2
a =1, b =1,c = – π/2 and d = –1/2
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b = π/2
Vertical translation: 1/2 units down
Section 5.7 Question 43
Vertical translation
½ units down 1
y = sin(x–π /2)
1/2 –1
y = sin(x)
–3/2
y = sin(x–π /2) –1/2
Phase shift
π/2 units right

27. Graph one full period of 2sin(3x–π /2) +1y
a =2, b =3,c = – π/2 and d = 1
Amplitude = |a| =2
Period = 2π/b = 2π/3
Phase shift = – c/b = π/6
Vertical translation:
1 unit up 3 2 1
π/2
2π/3
π/6
π/3 x –1 –2
This is the graph of 2sin(3x). Now click to see the phase shift , vertical translation and to get 2sin(3x–π/2)+1

28. Graph one full period of sin(x+π /6)y
a =1, b =1,c = π/6
Amplitude = |a| =1
Period = 2π/b = 2π
Phase shift = – c/b = –π/6
Section 5.7 Question 18 1
3π/2 2π
–π/6
π/2 π x –1
This is the graph of sin(x). Now click to see the phase shift and to get sin(x+π/6)

29. Graph one full period of cos(2x–π/3)y
a =1, b = 2,c = – π/3
Amplitude = |a| =1
Period = 2π/b = π
Phase shift = – c/b = π/6
Section 5.7 Question 20 1
π/2
π/6
π/4
3π/4 π
7π/6 x –1
This is the graph of cos(2x). Now click to see the phase shift and to get cos(2x-π/3)

30. Graph one full period of y=(1/2)sin(πx/3)
a =1/2, b = π/3
Amplitude = |a| =1/2
Period = 2π/b = 6
1/2
9/2 6
3/2 3 x
–1/2

31. y x

32. Graph one full period of y=2sinx and y= sinx
0≤ sinx ≤ 2sinx 2
In [π , 2π] , 1
2sinx ≤ sinx ≤ 0 –1 –2

35. y x

36. y x

37. y x

38. y x

39. Draw one full period of y = 2tan(x/2)
a = 2 and b = 1/2 , 4b = 2
Asymptotes:
x = ±2π/4b = ± 2π/2 = ± π
Lets draw asymptotes 2
Mark 2 and –2 on the y-axis
and ±π/4b = ±π/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 = 2π/b = 2π/3
3/2
π/2
π/6
π/3
2π/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 = π/|b| = π
1/3
–π/4
–π/2
π/4
π/2
–1/3
Section 5.6 Question 22

42. Graph one full period of 2cscxy
a =2, b = 1
Period = 2π/b = 2π 2
3π/2
π/2 π 2π x –2
Section 5.6 Question 28

43. Graph one full period of -3sec(2x/3)y
a = –3 , b = 2/3
Period = 2π/b = 3π 3
3π/4
3π/2
9π/4 3π x –3
Section 5.6 Question 36

44. Draw one full period of y = –3tan(3x) y
a = –3 and b = 3 , 4b = 12
Asymptotes:
x = ±2π/4b = ± 2π/12 = ± π/6
Period = π/b = π/2
Lets draw asymptotes 3
Mark 3 and –3 on the y-axis
and ±π/4b = ±π/12 on the x-axis x –3
Now we can draw the graph

45. Draw one full period of y = (1/2)cot(2x)
a = 1/2 and b = 2 , 4b = 8
Asymptotes:
x = π/b = π/2 and x = 0(y-axis)
Period = π/b = π/2
Lets draw asymptotes
1/2
Mark 1/2 and –1/2 on the y-axis
and π/8, 2π/8, 3π/8 and 4π/8 on the
x-axis x
–1/2
Now we can draw the graph

46. Graph one full period of y=3/2sin(x /4+3π /4)
a =3/2, b = 1/4,c = 3π/4
Amplitude = |a| = 3/2
Period = 2π/b = 8π
Phase shift = – c/b = –3π
3/2 6π 8π
–3π 2π 4π x
–3/2
This is the graph of y=3/2sin(x/4). Now click to see the phase shift and to get y=3/2sin(x/4+3π/4)

47. Graph one full period of y= sec(x − π/2 )+1
a = 1 , b = 1, c = −π/2, d = 1
Period = 2π/|b| = 2π
Phase shift = −c/b = π/2
Vertical translation :
(d =) 1 unit up
sec(x) 2 1
cos(x) π | | | |
3π/2 2π x
π/2 –1
Click to shift π/2 unit to right
Click to shift 1 unit up

48. Graph one full period of y= csc(x/3-π/12)+4
a = 1, b = 1/3, c = π/3, d = 4
Period = 2π/|b| = 6π
Phase shift = -c/b = π/4
Vertical translation: 4 unit up 5 3
sin(x/3) 1
9π/2 | | | |
π/4
3π/2 3π 6π x –1
csc(x/3)

49. Sketch the graph of y = |(1/2)sin(3x)|
-2π/3
π/3
2π/3
-π/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
-3π/2 -π
-π/2
-π/4
π/4
π/2 π
3π/2
Section 5.5 Question 65

51. Sketch the graph of y = sin|x|
sin(x) if x ≥ 0
y = sin|x| = 1
−sin(x) if x ≤ 0
-2π -π π 2π -1
Section 5.5 Question 68