1. 4.4 Graphs of Sine and Cosine Functions

2. These will be key points on the graphs of y = sin x and y = cos x
Fill in the chart. x
Sin x
Cos x
These will be key points on the graphs of y = sin x and y = cos x

3. Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1
sin x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = sin x

4. Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1
sin x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = sin x

5. Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1
cos x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = cos x

6. Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1
cos x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = cos x

7. The graphs of y = sin x and y = cos x have similar properties:

8. The graphs of y = sin x and y = cos x have similar properties:

9. The graphs of y = sin x and y = cos x have similar properties:

10. The graphs of y = sin x and y = cos x have similar properties:

11. The graphs of y = sin x and y = cos x have similar properties:

12. The graphs of y = sin x and y = cos x have similar properties:

13. The graphs of y = sin x and y = cos x have similar properties:

14. The graphs of y = sin x and y = cos x have similar properties:

15. The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.

16. Amplitude – Constant that gives vertical stretch or shrink.
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.

17. Amplitude – Constant that gives vertical stretch or shrink.y x
y = 2sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x

18. Amplitude – Constant that gives vertical stretch or shrink.
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.

19. Example 1: Sketch the graph of y = 3 cos x on the interval [0, 2].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: y = 3 cos x

20. Example 1: Sketch the graph of y = 3 cos x on the interval [0, 2].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)

21. Example 2: Sketch the graph of y = 3 cos x on the interval [0, 2].
max
x-int
min
y = -3 cos x 2  x
( , –3) y x
( , 0)
( , 0)
( , 3)
(0, -3)

22. Example 2: Sketch the graph of y = 3 cos x on the interval [0, 2].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min
y = -3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)

23. Example: Sketch the graph of y = 1/3 cos x on the interval [0, 2].
max
x-int
min
y = 1/3 cos x 2  x y x
( , 0)
( , 0)
( , 1/3)
(0, 1/3)
( , –1/3)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: y = 3 cos x

24. HOMEWORK
Pgs
1-4 ALL

26. For b  0, the period of y = a sin bx is .
The period of a function is the x interval needed for the function to complete one cycle.
For b  0, the period of y = a sin bx is
For b  0, the period of y = a cos bx is also
Interval – Divide period by 4
Critical points – You need 5.(max., min., intercepts.)

27. Amplitudes and Periods
The graph of y = a sin bx or a cos bx
amplitude = | a |
period =
To get your critical points (max, min, and intercepts) just take your period and divide by 4.
Example:
Interval

28. If b > 1, the graph of the function is shrunk horizontally.
The period of a function is the x interval needed for the function to complete one cycle.
If b > 1, the graph of the function is shrunk horizontally. y x
period: 2
period:

29. The period of a function is the x interval needed for the function to complete one cycle.
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period: 4
period: 2

30. Example 1 x

31. Sketch the graph of y = sin 2x
Example 2
Sketch the graph of y = sin 2x
y = sin 2x x

32. Example 3 x

33. Example: Sketch the graph of y = -2 sin (3x).
period: 2 3 =
amplitude: |a| = |-2| = 2
Calculate the five key points. 2 -2
y = 2 sin 3x x y x
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: y = 2 sin(-3x)

34. HOMEWORK
Pgs
5-8 ALL

35. Example 4

36. Example 6 y = Sin x Amplitude PERIOD /INT CRITICAL POINTS Phase shift
Amplitude
PERIOD /INT
CRITICAL POINTS
 Phase shift
ADD to x
Vertical Shift
ADD to y

37. Example 6

38. Example 6 y = Cos x 1 Amplitude PERIOD /INT CRITICAL POINTS
Amplitude
PERIOD /INT
CRITICAL POINTS
 Phase shift
ADD to x
Vertical Shift
ADD to y