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**Graphing Sine and Cosine Functions**

In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are positive constants and x is in radian measure. The graphs of all sine and cosine functions are related to the graphs of y = sin x and y = cos x which are shown below. y = sin x y = cos x

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**Graphing Sine and Cosine Functions**

The functions y = sin x and y = cos x have the following characteristics: The domain of each function is all real numbers. The range of each function is –1 £ y £ 1. Each function is periodic, which means that its graph has a repeating pattern that continues indefinitely. The shortest repeating portion is called a cycle. The horizontal length of each cycle is called the period. The graphs of y = sin x and y = cos x each has a period of 2π. The maximum value of y = sin x is M = 1 and occurs when x = n π where n is any integer. The maximum value of y = cos x is also M = 1 and occurs when x = 2 n π where n is any integer. π 2

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**Graphing Sine and Cosine Functions**

The functions y = sin x and y = cos x have the following characteristics: The minimum value of y = sin x is m = –1 and occurs when x = n π where n is any integer. The minimum value of y = cos x is also m = –1 and occurs when x = (2n + 1)π where n is any integer. 3π 2 The amplitude of each function’s graph is (M – m) = 1. 1 2

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**Graphing Sine and Cosine Functions**

CHARACTERISTICS OF Y = A SIN BX AND Y = A COS BX The amplitude and period of the graphs of y = a sin bx and y = a cos bx, where a and b are nonzero real numbers, are as follows: amplitude = |a| and period = 2π |b| The graph of y = 2 sin 4x has amplitude 2 and period = 2π 4 π 2 Examples The graph of y = cos 2 π x has amplitude and period = 1. 2π 1 3

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**Graphing Sine and Cosine Functions**

CHARACTERISTICS OF Y = A SIN BX AND Y = A COS BX The amplitude and period of the graphs of y = a sin bx and y = a cos bx, where a and b are nonzero real numbers, are as follows: amplitude = |a| and period = 2π |b| The graph of y = 2 sin 4x has amplitude 2 and period = 2π 4 π 2 Examples 1 3 1 3 2π The graph of y = cos 2 π x has amplitude and period = 1. For a > 0 and b > 0, the graphs of y = a sin bx and y = a cos bx each have five key x-values on the interval 0 £ x £ : the x-values at which the maximum and minimum values occur and the x-intercepts. 2π b

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**( ) ( ) ( ) ( ) ( ) Graphing Sine and Cosine Functions**

Graph the function. y = 2 sin x SOLUTION The amplitude is a = 2 and the period is = = 2π. The five key points are: 2π b 1 Intercepts: (0, 0); (2π, 0); = (π, 0) ( ) • 2, 0 1 2 Maximum: ( ) • 2, 2 1 4 ( ) , 2 2 = Minimum: ( ) • 2, –2 3 4 ( ) , –2 3 2 =

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**( ) ( ) ( ) ( ) ( ) Graphing Sine and Cosine Functions**

Graph the function. y = cos 2x SOLUTION The amplitude is a = 1 and the period is = = π. The five key points are: 2π b 2 Intercepts: = ; = , 0 ( ) 4 ( ) • , 0 1 3 ( ) 3 Maximums: (0, 1); (π, 1) Minimum: = ( ) • , –1 1 2 ( ) , –1

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**( ) ( ) ( ) ( ) ( ) Graphing a Cosine Function Graph y = cos π x. 1 3**

SOLUTION The amplitude is a = and the period is = = 2. 1 3 2π b π The five key points are: Intercepts: = ; = ( ) • 2, 0 1 4 ( ) , 0 2 3 Maximum: ; ( ) 0, 1 3 2, Minimum: = ( ) • 2, 1 2 3 – ( ) 1,

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**Graphing Sine and Cosine Functions**

The periodic nature of trigonometric functions is useful for modeling oscillating motions or repeating patterns that occur in real life. Some examples are sound waves, the motion of a pendulum or a spring, and seasons of the year. In such applications, the reciprocal of the period is called the frequency. The frequency gives the number of cycles per unit of time.

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**Modeling with a Sine Function**

MUSIC When you strike a tuning fork, the vibrations cause changes in the pressure of the surrounding air. A middle-A tuning fork vibrates with frequency f = 440 hertz (cycles per second). You would strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. Write a sine model that gives the pressure P as a function of time t (in seconds). Then graph the model. SOLUTION In the model P = a sin b t, the maximum pressure P is 5, so a = 5. You can use the frequency to find the value of b. Frequency = 1 period 440 = b 2π 880π = b The pressure as a function of time is given by P = 5 sin 880π t.

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**( ) ( ) ( ) ( ) ( ) ( ) Modeling with a Sine Function**

MUSIC When you strike a tuning fork, the vibrations cause changes in the pressure of the surrounding air. A middle-A tuning fork vibrates with frequency f = 440 hertz (cycles per second). You would strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. Write a sine model that gives the pressure P as a function of time t (in seconds). Then graph the model. SOLUTION The amplitude is a = 5 and the period is = 1 440 f The five key points are: Intercepts: (0, 0); ; = ( ) 1 440 , 0 ( ) • , 0 2 880 Maximum: = ( ) • , 5 1 4 440 ( ) 1760 , 5 Minimum: = ( ) 3 1760 , –5 ( ) • , –5 4 1 440

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**Graphing Tangent Functions**

The graph of y = tan x has the following characteristics. The domain is all real numbers except odd multiples of At odd multiples of , the graph has vertical asymptotes. π 2 π2 The range is all real numbers. The graph has a period of π. CHARACTERISTICS OF Y = A TAN BX If a and b are nonzero real numbers, the graph of y = a tan bx has these characteristics: The period is π | b | There are vertical asymptotes at odd multiples of π 2| b | Example The graph of y = 5 tan 3x has period and asymptotes at x = (2n + 1) = where n is any integer. π 3 6 n π 2(3)

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**Graphing Tangent Functions**

The graph of y = tan x has the following characteristics. The graph at the right shows five key x-values that can help you sketch the graph of y = tan x for a > 0 and b > 0. These are the x-intercept, the x-values where the asymptotes occur, and the x-values halfway between the x-intercept and the asymptotes. At each halfway point, the function’s value is either a or – a.

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**( ) ( ); ( ) ( ) Graphing a Tangent Function**

Graph the function y = tan 4x. 3 2 SOLUTION The period is = π b 4 Intercept: (0, 0) x = • , or x = ; π 8 1 2 4 Asymptotes: x = – • , or x = – π 8 1 2 4 , ( ) ( ); • = 1 4 π 3 2 16 Halfway points: ( ) ( ) – • – = – – 1 4 3 2 π 16 ,

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6.4 Amplitude and Period of Sine and Cosine Functions.

6.4 Amplitude and Period of Sine and Cosine Functions.

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