1. MAT 204 SP09
7.6 Graphs of the Sine and Cosine Functions 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 SP09
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 SP09
Take a look at the graph of y = sin x:
(one cycle)
Some points on the graph of y = sin x

5. More about the graph of y = sin x
MAT 204 SP09
More about the graph of y = sin x
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 SP09
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 SP09
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 SP09
Take a look at the graph of y = cos x:
(one cycle)
Some points on the graph of y = cos x

9. More about the graph of y = cos x
MAT 204 SP09
More about the graph of y = cos x
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 SP09
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 SP09
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 SP09
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. MAT 204 SP09
I. Reflection in x-axis
Example:

15. II. Vertical Stretch or Compression
MAT 204 SP09
II. Vertical Stretch or Compression
(Amplitude change)
Example

16. II. Vertical Stretch or Compression
MAT 204 SP09
II. Vertical Stretch or Compression
*Note:
If the curve is vertically stretched
if the curve is vertically shrunk

17. MAT 204 SP09
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 Stretch or Compression (Period change)
MAT 204 SP09
III. Horizontal Stretch or Compression (Period change)
Example

20. III. Horizontal Stretch or Compression (Period change)
MAT 204 SP09
III. Horizontal Stretch or Compression (Period change)
*Note:
If the curve is horizontally stretched
If the curve is horizontally shrunk

21. MAT 204 SP09
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 SP09

23. MAT 204 SP09 x y

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26. MAT 204 SP09
V. Vertical Shifts
Example

27. MAT 204 SP09
V. Phase Shifts
Example

28. MAT 204 SP09
EXAMPLE
For , determine the amplitude, period, and phase shift. Then sketch at least one cycle of the function by hand. x y

29. MAT 204 SP09
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.

30. MAT 204 SP09
EXAMPLE (CONTINUED) x y

31. MAT 204 SP09
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.

32. MAT 204 SP09
EXAMPLE (CONTINUED) x y

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39. MAT 204 SP09 *
*NOTE: In 2005, summer solstice was on June 21 (172nd day of the year).

41. MAT 204 SP09
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.

42. MAT 204 SP09

43. MAT 204 SP09
End of Sections 7.6 & 7.8