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4-4 Graphing Sine and Cosine Chapter 4 Graphs of Trigonometric Functions.

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Presentation on theme: "4-4 Graphing Sine and Cosine Chapter 4 Graphs of Trigonometric Functions."— Presentation transcript:

1 4-4 Graphing Sine and Cosine Chapter 4 Graphs of Trigonometric Functions

2 Warm-up Find the exact value of each expression. 1.sin 315 ° 2. 3.cot 510 ° 4. 5.

3 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

4 http://www.univie.ac.at/future.media/moe/gal erie/fun2/fun2.html http://www.univie.ac.at/future.media/moe/gal erie/fun2/fun2.html

5 Recreate the sine graph. Domain and Range x- and y-intercepts symmetry

6 Recreate the cosine graph. Domain and Range x- and y-intercepts symmetry

7 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|

8 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.

9 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.

10 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) =

11 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.

12 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 )

13 Example 4 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.

14 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) =

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

16 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)

17 Example 6 State the amplitude, period, frequency, phase shift, and vertical shift of y = sin (x + π ) + 1. Then graph two periods of the function.

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


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