Section 4-7Scale-Change Images of Circular Functions

Warm-upGraph each of these equations on the same axes

for -π ≤ θ ≤ π

y = sin x y = 3sin x y = sin2x

�

y = 3sin2x

All together:

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.

Sine Wave:

Amplitude:

The graph of the sine or cosine function over a composite or translations and scale changes

Half the distance between the maximum and minimum output values

Example 1What is the period of sine? Cosine? Tangent?

Sine and cosine both have periods of 2π.Tangent has a period of π.

Example 2a. 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 2πFor the second graph, the amplitude is 5 and the

period stays the same.

Example 2c. 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 2πFor the second graph, the amplitude is 1 and the

period is π/3.

Theorem for Amplitude and Period

For the functions

�

y = bsin xa

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

y = bcos xa

⎛ ⎝ ⎜

⎞ ⎠ ⎟ and

amplitude is|b| and period is 2π|a|

***NOTICE: a is in the denominator, so be careful when working with it!!!

Frequency:

The number of cycles the curve completes per unit of the independent variable

Found by taking reciprocal of the period

Example 3Consider the graph for

�

y = 14 sin2x

Give the period, amplitude, and frequency

Period = Amplitude =

Frequency =

�

a = 12

π

�

14

1π

Example 4Suppose 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( )

Homework

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