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# Notes 4-7

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Scale-Change Images of Circular Functions

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### Notes 4-7

1. 1. Section 4-7 Scale-Change Images of Circular Functions
2. 2. Warm-up Graph each of these equations on the same axes for -π ≤ θ ≤ π y = sin x y = 3sin x y = sin 2x y = 3sin2x All together:
3. 3. Recall that a periodic function has a value such that f(x + p) = f(x), where p is the period of the function. The trig functions (sine, cosine, and tangent) are all periodic, as they begin to trace the same output values after a certain amount of input values.
4. 4. Sine Wave: The graph of the sine or cosine function over a composite or translations and scale changes Amplitude: Half the distance between the maximum and minimum output values
5. 5. Example 1 What is the period of sine? Cosine? Tangent? Sine and cosine both have periods of 2π. Tangent has a period of π.
6. 6. Example 2 a. Graph y = sin x and y = 5sin x for -2π ≤ x ≤ 2π. b. What are the amplitude and period of these graphs? For y = sin x, the amplitude is 1 and the period is 2π For the second graph, the amplitude is 5 and the period stays the same.
7. 7. Example 2 c. Graph y = cos x and y = cos 6x for -2π ≤ x ≤ 2π. d. What are the amplitude and period of these graphs? For y = cos x, the amplitude is 1 and the period is 2π For the second graph, the amplitude is 1 and the period is π/3.
8. 8. Theorem for Amplitude and Period For the functions ⎛ x⎞ ⎛ x⎞ and y = bsin⎜ ⎟ y = bcos⎜ ⎟ ⎝ a⎠ ⎝ a⎠ amplitude is|b| and period is 2π|a| ***NOTICE: a is in the denominator, so be careful when working with it!!!
9. 9. Frequency: The number of cycles the curve completes per unit of the independent variable Found by taking reciprocal of the period
10. 10. Example 3 Consider the graph for y = sin2x 1 4 Give the period, amplitude, and frequency Period = π Amplitude = 1 4 a = 12 1 Frequency = π
11. 11. Example 4 Suppose a tuning fork vibrates with a frequency of 440 cycles per second. If the vibration displaces air molecules by a maximum of .2mm, give a possible equation for the sound wave produced. y = .2sin(880πx )
12. 12. Homework p. 275 # 1 - 18