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4-4 Graphing Sine and CosinePowerPoint Presentation

4-4 Graphing Sine and Cosine

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- 6-3 Objective: Use the graphs of sine and cosine (sinusoidal) functions
- 6-4 Objectives:
- Find amplitude and period for sine and cosine functions, and
- Write equations of sine and cosine functions given the amplitude and period.
- Graph transformations of the sine and cosine functions

- http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.htmlhttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

Recreate the sine graph.http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- Domain and Range
- x- and y-intercepts
- symmetry

Recreate the cosine graph.http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- Domain and Range
- x- and y-intercepts
- symmetry

Keyhttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.htmlConcepts: Transformations of Sine and Cosine Functions

For y = a sin (bx + c) + d and y = a cos (bx + c) + d,

Amplitude (half the distance between the maximum and the minimum values of the function or half the height of the wave) = |a|

Example 1 http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- Describe how the graphs of
f(x) = sin x and g(x) = 2.5 sin x

are related. Then find the amplitude of g(x). Sketch two periods of both functions.

Example 2 Reflectionshttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- Describe how f(x) = cos x and g(x) = -2cos x are related. Then find the amplitude of g(x). Sketch two periods of both functions.

Keyhttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.htmlConcepts: Transformations of Sine and Cosine Functions

For y = a sin (bx + c) + d and y = a cos (bx + c) + d,

Period (distance between any two sets of repeating points on the graph) =

Example 3http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- Describe how the graphs of f(x) = cos x and g(x) = cos are related. Then find the period of g(x). Sketch at least one period of both functions.

Keyhttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.htmlConcepts: Transformations of Sine and Cosine Functions

For y = a sin (bx + c) + d and y = a cos (bx + c) + d,

Frequency (the number of cycles the function completes in a one unit interval) =

(note that it is the reciprocal of the period or )

Example 4http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

A bass tuba can hit a note with a frequency

of 50 cycles per second (50 hertz) and

an amplitude of 0.75.

Write an equation for a

cosine function that

can be used to

model the initial

behavior of

the sound

wave associated

with the note.

Keyhttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.htmlConcepts: Transformations of Sine and Cosine Functions

For y = a sin (bx + c) + d and y = a cos (bx + c) + d,

Phase shift (the difference between the horizontal position of the function and that of an otherwise similar function) =

Example 5http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- State the amplitude, period, frequency, and phase shift of . Then graph two periods of the function.

Keyhttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.htmlConcepts: Transformations of Sine and Cosine Functions

For y = a sin (bx + c) + d and y = a cos (bx + c) + d,

Vertical shift (the average of the maximum and minimum of the function) = d

(Note the horizontal axis—the midline–is y = d)

Example 6http://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

- State the amplitude, period, frequency, phase shift, and vertical shift of y = sin (x + π) + 1. Then graph two periods of the function.

Assignmenthttp://www.univie.ac.at/future.media/moe/galerie/fun2/fun2.html

P. 264, 1, 3, 9, 15, 17, 19.

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