1 Graphs of sine and cosine curves Sections 10.1 – 10.3

2 The graph of y = sin x The graph of y = sin x is a cyclical curve that takes on values between –1 and 1. We say that the range of the sine curve is Each cycle (wave) corresponds to one revolution (360  or 2  radians) of the unit circle. We say that the period of the sine curve is 2 .

3 Take a look at the graph of y = sin x: (one cycle) Some points on the graph:

4 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 the intercepts, the maximum point, and the minimum point.

5 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 also 2 .

6 Take a look at the graph of y = cos x: (one cycle) Some points on the graph:

7 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.

8 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 ________.

9 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 shifts/displacement)

10 I. Reflections over x-axis Example:

11 II. Vertical Stretching or Compression (Amplitude change) Example

12 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.

13 III. Horizontal Stretching or Shrinking/Compression (Period change) Example

14 x y

15 Graphs of Examples State the amplitude and period for each function. Then graph one cycle of each function by hand. Verify using your graphing calculator.

16 Graphs of

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19 IV. Phase Shifts Example

20 IV. Phase Shifts (continued) Example

21 Example: Determine the amplitude, period, and phase shift of the function. Then sketch the graph of the function by hand.

22 Example: x y

23 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.

24 x y

25 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.

26 x y

27 Modeling using a sinusoidal function P. 299 #56 On a Florida beach, the tides have water levels about 4 m between low and high tides. The period is about 12.5 h. Find a cosine function that describes these tides if high tide is at midnight of a given day.

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29 Modeling using a sinusoidal function A region that is 30° north of the equator averages a minimum of 10 hours of daylight in December. Average hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year with 1 for January, 2 for February, 3 for March, …through 12 for December. If y represents the average number of hours of daylight in month x, use a sine function of the form y = a sin(bx + c) +d to model the daylight hours.

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31 End of Section