1. 5.3 Trigonometric Graphs
Matthew 6:33 But seek first his kingdom and his
righteousness, and all these things will be added unto you.

2. y = sin x

3. y = sin x
}
}
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}
Q1 Q2 Q3 Q4
Unit Circle

4. y = sin x
Max: 1
Min: -1
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}
}
}
Q1 Q2 Q3 Q4
Unit Circle

5. y = sin x
Max: 1
Min: -1
max− min
amplitude = =1
2
}
}
}
}
Q1 Q2 Q3 Q4
Unit Circle

6. y = sin x
Max: 1
Min: -1
max− min
amplitude = =1
2
}
}
}
}
1 cycle occurs in 2π
Q1 Q2 Q3 Q4
∴ Period : 2π
Unit Circle

7. y = sin x
Max: 1
Min: -1
max− min
amplitude = =1
2
}
}
}
}
1 cycle occurs in 2π
Q1 Q2 Q3 Q4
∴ Period : 2π
Unit Circle
Domain : {x : x ∈R}
Range : {y : −1 ≤ y ≤ 1}

8. y = sin x
Max: 1
Min: -1
max− min
amplitude = =1
2
}
}
}
}
1 cycle occurs in 2π
Q1 Q2 Q3 Q4
∴ Period : 2π
Unit Circle
Domain : {x : x ∈R} Use the 5 key points
Range : {y : −1 ≤ y ≤ 1} to help you graph

9. y = cos x

10. y = cos x
}
}
}
}
Q1 Q2 Q3 Q4
Unit Circle
Use the 5 key points
to help you graph

11. y = cos x
Max: 1
Min: -1
amplitude: 1
Period: 2π
}
}
}
}
Q1 Q2 Q3 Q4
Unit Circle
Use the 5 key points
to help you graph

12. y = cos x
Max: 1
Min: -1
amplitude: 1
Period: 2π
}
}
}
}
Domain : {x : x ∈R} Q1 Q2 Q3 Q4
Range : {y : −1 ≤ y ≤ 1} Unit Circle
Use the 5 key points
to help you graph

13. y = cos x
Max: 1
Min: -1
amplitude: 1
Period: 2π
}
}
}
}
Domain : {x : x ∈R} Q1 Q2 Q3 Q4
Range : {y : −1 ≤ y ≤ 1} Unit Circle
π
If you shift cosθ right , Use the 5 key points
2
it looks just like sin θ . to help you graph
π
They are out of phase by .
2

14. y = cos x
Max: 1
Min: -1
amplitude: 1
Period: 2π
}
}
}
}
Domain : {x : x ∈R} Q1 Q2 Q3 Q4
Range : {y : −1 ≤ y ≤ 1} Unit Circle
π
If you shift cosθ right , Use the 5 key points
2
it looks just like sin θ . to help you graph
π
They are out of phase by .
2
⎛ π ⎞
sin θ = cos ⎜ θ − ⎟
⎝ 2 ⎠

15. Sinusoidal Functions

16. Sinusoidal Functions
y = asinb ( x − c ) + d

17. Sinusoidal Functions
y = asinb ( x − c ) + d
a is the amplitude
if a < 0 , the graph is reflected about the x-axis

18. Sinusoidal Functions
y = asinb ( x − c ) + d
a is the amplitude
if a < 0 , the graph is reflected about the x-axis
b is related to the period in this way:
normal period
period =
b

19. Sinusoidal Functions
y = asinb ( x − c ) + d
a is the amplitude
if a < 0 , the graph is reflected about the x-axis
b is related to the period in this way:
normal period
period =
b
c is the horizontal shift (or ‘phase shift’)
if c < 0 , shifted left
if c > 0 , shifted right

20. Sinusoidal Functions
y = asinb ( x − c ) + d
a is the amplitude
if a < 0 , the graph is reflected about the x-axis
b is related to the period in this way:
normal period
period =
b
c is the horizontal shift (or ‘phase shift’)
if c < 0 , shifted left
if c > 0 , shifted right
d is the vertical shift

21. Discuss and Graph
1. y = 3cosθ

22. Discuss and Graph
1. y = 3cosθ
amp : 3

23. Discuss and Graph
1. y = 3cosθ
amp : 3
per : 2π

24. Discuss and Graph
1. y = 3cosθ
amp : 3
per : 2π
H .S. : none

25. Discuss and Graph
1. y = 3cosθ
amp : 3
per : 2π
H .S. : none
V.S. : none

26. Discuss and Graph
1. y = 3cosθ
amp : 3
per : 2π
H .S. : none
V.S. : none
Know how to graph on trig graph paper using the
5 key points. Verify with your calculator.

27. Discuss and Graph
⎛ π ⎞
2. y = −2sin ⎜ x + ⎟
⎝ 2 ⎠

28. Discuss and Graph
⎛ π ⎞
2. y = −2sin ⎜ x + ⎟
⎝ 2 ⎠
amp : 2

29. Discuss and Graph
⎛ π ⎞
2. y = −2sin ⎜ x + ⎟
⎝ 2 ⎠
amp : 2
per : 2π

30. Discuss and Graph
⎛ π ⎞
2. y = −2sin ⎜ x + ⎟
⎝ 2 ⎠
amp : 2
per : 2π
π
H .S. : left
2

31. Discuss and Graph
⎛ π ⎞
2. y = −2sin ⎜ x + ⎟
⎝ 2 ⎠
amp : 2
per : 2π
π
H .S. : left
2
V.S. : none

32. Discuss and Graph
⎛ π ⎞
2. y = −2sin ⎜ x + ⎟
⎝ 2 ⎠
amp : 2
per : 2π
π
H .S. : left
2
V.S. : none
Reflected about the x-axis

33. Discuss and Graph
3. y = sin ( 2x − π )

34. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2

35. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2
⎛ π ⎞
y = sin 2 ⎜ x − ⎟
⎝ 2 ⎠

36. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2
⎛ π ⎞
y = sin 2 ⎜ x − ⎟
⎝ 2 ⎠
amp : 1

37. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2
⎛ π ⎞
y = sin 2 ⎜ x − ⎟
⎝ 2 ⎠
amp : 1
per : π

38. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2
⎛ π ⎞
y = sin 2 ⎜ x − ⎟
⎝ 2 ⎠
amp : 1
norm. per.
per : π period =
b
2π
p= =π
2

39. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2
⎛ π ⎞
y = sin 2 ⎜ x − ⎟
⎝ 2 ⎠
amp : 1
norm. per.
per : π period =
b
π 2π
H .S. : right p= =π
2 2

40. Discuss and Graph
3. y = sin ( 2x − π ) Factor out the 2
⎛ π ⎞
y = sin 2 ⎜ x − ⎟
⎝ 2 ⎠
amp : 1
norm. per.
per : π period =
b
π 2π
H .S. : right p= =π
2 2
V.S. : none

41. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠

42. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2

43. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2
1
amp :
2

44. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2
1
amp :
2
per : 4π

45. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2
1
amp : norm. per.
2 period =
b
per : 4π 2π
p= = 4π
1
2

46. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2
1
amp : norm. per.
2 period =
b
per : 4π 2π
p= = 4π
H .S. : 2π left 1
2

47. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2
1
amp : norm. per.
2 period =
b
per : 4π 2π
p= = 4π
H .S. : 2π left 1
2
V.S. : 1 down

48. Discuss and Graph
1 ⎛ 1 ⎞
4. y = cos ⎜ x + π ⎟ − 1
2 ⎝ 2 ⎠
1 1
y = cos ( x + 2π ) − 1
2 2
1
amp : norm. per.
2 period =
b
per : 4π 2π
p= = 4π
H .S. : 2π left 1
2
V.S. : 1 down
Use the 5 key points to help you graph this!

49. HW #4
Unless you’re willing to have a go, fail miserably,
and have another go, success won’t happen.
Phillip Adams