1. MAT 204 FALL 2008
7.6 Graphs of the Sine and Cosine Functions 7.8 Phase shift; Sinusoidal Curve Fitting
In these sections, we will study the following topics:
The graphs of basic sine and cosine functions
The amplitude and period of sine and cosine functions
Transformations of sine and cosine functions
Sinusoidal curve fitting

3. MAT 204 FALL 2008
The graph of y = sin x
The graph of y = sin x is a cyclical curve that takes on values between –1 and 1. The range of y = sin x is _____________.
Each cycle (wave) corresponds to one revolution of the unit circle. The period of y = sin x is _______ or _______.
Graphing the sine wave on the x-y axes is like “unwrapping” the values of sine on the unit circle.

4. Take a look at the graph of y = sin x:
MAT 204 FALL 2008
Take a look at the graph of y = sin x:
(one cycle)
Points on the graph of y = sin x

5. For example, are on the graph of y = sin x.
MAT 204 FALL 2008
Notice that the sine curve is symmetric about the origin. Therefore, we know that the sine function is an ODD function; that is, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.
For example, are on the graph of y = sin x.

6. Using Key Points to Graph the Sine Curve
MAT 204 FALL 2008
Using Key Points to Graph the Sine Curve
Once you know the basic shape of the sine curve, you can use the key points to graph the sine curve by hand.
The five key points in each cycle (one period) of the graph are:
3 x-intercepts
maximum point
minimum point

7. MAT 204 FALL 2008
The graph of y = cos x
The graph of y = cos x is also a cyclical curve that takes on values between –1 and 1. The range of the cosine curve is ________________.
The period of the cosine curve is _______ or _______.

8. Take a look at the graph of y = cos x:
MAT 204 FALL 2008
Take a look at the graph of y = cos x:
(one cycle)
Points on the graph of y = cos x

9. For example, are on the graph of y = cos x.
MAT 204 FALL 2008
Notice that the cosine curve is symmetric about the y-axis. Therefore, we know that the cosine function is an EVEN function; that is, for every point (x, y) on the graph, the point (-x, y) is also on the graph.
For example, are on the graph of y = cos x.

10. Using Key Points to Graph the Cosine Curve
MAT 204 FALL 2008
Using Key Points to Graph the Cosine Curve
Once you know the basic shape of the cosine curve, you can use the key points to graph the cosine curve by hand.
The five key points in each cycle (one period) of the graph are:
maximum point
2 x-intercepts
minimum point

11. Characteristics of the Graphs of y = sin x and y = cos x
MAT 204 FALL 2008
Characteristics of the Graphs of y = sin x and y = cos x
Domain: ____________
Range: ____________
Amplitude: The amplitude of the sine and cosine functions is half the distance between the maximum and minimum values of the function.
The amplitude of both y= sin x and y = cos x is ______.
Period: The length of the interval needed to complete one cycle.
The period of both y= sin x and y = cos x is ________.

13. Transformations of the graphs of y = sin x and y = cos x
MAT 204 FALL 2008
Transformations of the graphs of y = sin x and y = cos x
Reflections over x-axis
Vertical Stretches or Shrinks
Horizontal Stretches or Shrinks/Compression
Vertical Shifts
Phase shifts (Horizontal)

14. I. Reflections over x-axis
MAT 204 FALL 2008
I. Reflections over x-axis
Example:

15. II. Vertical Stretching or Shrinking (Amplitude change)
MAT 204 FALL 2008
II. Vertical Stretching or Shrinking (Amplitude change)
Example

16. II. Vertical Stretching or Shrinking (Amplitude change)
MAT 204 FALL 2008
II. Vertical Stretching or Shrinking (Amplitude change)
*Note:
If the curve is vertically stretched
if the curve is vertically shrunk

17. MAT 204 FALL 2008
Example
The graph of a function in the form y = A sinx or y = A cosx is shown. Determine the equation of the specific function.

19. III. Horizontal Stretching or Shrinking/Compression (Period change)
MAT 204 FALL 2008
III. Horizontal Stretching or Shrinking/Compression (Period change)
Example

20. III. Horizontal Stretching or Shrinking/Compression (Period change)
MAT 204 FALL 2008
III. Horizontal Stretching or Shrinking/Compression (Period change)
*Note:
If the curve is horizontally stretched
If the curve is horizontally shrunk

21. MAT 204 FALL 2008
Graphs of
Examples
State the amplitude and period for each function. Then graph each of function using your calculator to verify your answers.
(Use radian mode and ZOOM 7:ZTrig)

22. MAT 204 FALL 2008
Graphs of

23. Graphing Sinusoidal Functions Using Key Points
MAT 204 FALL 2008
Graphing Sinusoidal Functions Using Key Points
We will start with the parent graphs:

24. MAT 204 FALL 2008

25. MAT 204 FALL 2008 x y

26. MAT 204 FALL 2008

27. MAT 204 FALL 2008

28. MAT 204 FALL 2008
V. Vertical Shifts
Example

29. MAT 204 FALL 2008
V. Phase Shifts
Example

31. MAT 204 FALL 2008
Example:
For , determine the amplitude, period, and phase shift. Then sketch the function by hand. x y

32. MAT 204 FALL 2008

33. MAT 204 FALL 2008
Example:
List all of the transformations that the graph of y = sin x has undergone to obtain the graph of the new function. Graph the function by hand.

34. MAT 204 FALL 2008 x y

35. MAT 204 FALL 2008
Example:
List all of the transformations that the graph of y = sin x has undergone to obtain the graph of the new function. Graph the function by hand.

36. MAT 204 FALL 2008 x y

37. MAT 204 FALL 2008

38. MAT 204 FALL 2008

39. MAT 204 FALL 2008

40. MAT 204 FALL 2008

41. MAT 204 FALL 2008

42. MAT 204 FALL 2008

43. MAT 204 FALL 2008 *
*NOTE: In 2005, summer solstice was on June 21 (172nd day of the year).

45. MAT 204 FALL 2008
Use a graphing calculator to graph the scatterplot of the data in the table below. Then find the sine function of best fit for the data. Graph this function with the scatterplot.

46. MAT 204 FALL 2008

47. MAT 204 FALL 2008
End of Sections 7.6 & 7.8