1. EXAMPLE 2 Graph a cosine function SOLUTION Graph y = cos 2 π x. 1 2 The amplitude is a = 1 2 and the period is 2 b π = 2 π = 2π2π 1. Intercepts: ( 1, 0) 1 4 = (, 0); 1 4 ( 1, 0) 3 4 = (, 0) 3 4 Maximums: (0, ) ; 1 2 2 1 (1, ) Minimum: ( 1, – ) 1 2 1 2 = (, – ) 1 2 1 2

2. EXAMPLE 3 Model with a sine function 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). Audio Test

3. 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. frequency = period 1 2000 = b 2 π 4000 = b π The pressure P as a function of time t is given by P = 2 sin 4000πt.

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

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

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

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

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

9. GUIDED PRACTICE for Examples 2 and 3 9. 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