Graphs of sine and cosine curves
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Graphs of sine and cosine curves. Sections 10.1 – 10.3. 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

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Graphs of sine and cosine curves
Graphs of sine and cosine curves

Sections 10.1 – 10.3


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 periodof the sine curve is 2.


Take a look at the graph of y = sin x:

(one cycle)

Some points on the graph:


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.


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 periodof the cosine curve is also 2.


Take a look at the graph of y = cos x:

(one cycle)

Some points on the graph:


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.


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 ________.


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)



II.Vertical Stretchingor Compression (Amplitude change)

Example


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.



y

x


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.




IV. Phase Shifts (continued)

Example


Example:

Determine the amplitude, period, and phaseshift of the function. Then sketch the graph of the function by hand.


y

x

Example:


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.


y

x


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.


y

x


Modeling using a sinusoidal function
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.


Modeling using a sinusoidal function1
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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