EXAMPLE 2 Graph a cosine function 1 Graph

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

EXAMPLE 3 Model with a sine function Audio Test A sound consisting of a single frequency is called a pure tone. An audiometer produces pure tones to test a person’s auditory functions. Suppose an audiometer produces a pure tone with a frequency f of 2000 hertz (cycles per second). The maximum pressure P produced from the pure tone is 2 millipascals. Write and graph a sine model that gives the pressure P as a function of the time t (in seconds).

EXAMPLE 3 Model with a sine function SOLUTION STEP 1 Find the values of a and b in the model P = a sin bt. The maximum pressure is 2, so a = 2. You can use the frequency f to find b. 1 frequency = period b 2000 = 2 π 4000 π = b The pressure P as a function of time t is given by P = 2 sin 4000πt.

EXAMPLE 3 Model with a sine function STEP 2 Graph the model. The amplitude is a = 2 and the period is 1 = 1 2000 f 1 Intercepts: (0 , 0); ( 2 1 Maximum: ( 4 Minimum: ( 3 4 1 1 1 2000 , 0) = ( 4000 , 0) ; ( 2000 , 0) 1 1 = ( 8000 , 2) 2000 , 2) 3 1 = ( 8000 , – 2) 2000 , – 2)

GUIDED PRACTICE Graph the function. 1 5. y = sin πx 4 SOLUTION for Examples 2 and 3

GUIDED PRACTICE Graph the function. 1 6. y = cos πx 3 SOLUTION for Examples 2 and 3

GUIDED PRACTICE Graph the function. 7. f (x) = 2 sin 3 x SOLUTION for Examples 2 and 3

GUIDED PRACTICE Graph the function. 8. g(x) = 3 cos 4 x SOLUTION for Examples 2 and 3

GUIDED PRACTICE 9. for Examples 2 and 3 What If ? In Example 3, how would the function change if the audiometer produced a pure tone with a frequency of 1000 hertz? SOLUTION The period would increase because the frequency is decreased p = 2 sin 2000πt