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**4-4 Graphing Sine and Cosine**

Chapter 4 Graphs of Trigonometric Functions

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Warm-up Find the exact value of each expression. sin 315° cot 510°

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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

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**Recreate the sine graph.**

Domain and Range x- and y-intercepts symmetry

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**Recreate the cosine graph.**

Domain and Range x- and y-intercepts symmetry

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**KeyConcepts: 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|

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**Example 1 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.

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Example 2 Reflections 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.

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**KeyConcepts: 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) =

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Example 3 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.

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**KeyConcepts: 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 )

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A bass tuba can hit a note with a frequency of 50 cycles per second (50 hertz) and an amplitude of Write an equation for a cosine function that can be used to model the initial behavior of the sound wave associated with the note. Example 4

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**KeyConcepts: 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) =

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Example 5 State the amplitude, period, frequency, and phase shift of Then graph two periods of the function.

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**KeyConcepts: 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)

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Example 6 State the amplitude, period, frequency, phase shift, and vertical shift of y = sin (x + π) Then graph two periods of the function.

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

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Graphs of Trigonometric Functions. Properties of Sine and Cosine Functions 2 6. The cycle repeats itself indefinitely in both directions of the x-axis.

Graphs of Trigonometric Functions. Properties of Sine and Cosine Functions 2 6. The cycle repeats itself indefinitely in both directions of the x-axis.

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