Graphs of Sine and Cosine

1. Graphs of Sine and Cosine Part 3

2. Amplitude Phase shift Vertical shift Period You can combine the transformations of trigonometric functions. Use the values of a, b, h, andkto identify the important features of a sine or cosine function. y = asinb(x–h) + k

3. The number of people, in thousands, employed in a resort town can be modeled by, where x is the month of the year. a = 1.5, b = , h = –2, k = 5.2 Example 1: Employment Application A. Graph the number of people employed in the town for one complete period. Step 1 Identify the important features of the graph.

4. Example 1 Continued Amplitude: Period: The period is equal to months or year. Phase shift: Vertical shift: Maxima: Minima:

5. Period: Example 1 Continued Amplitude: 1.5 The period is equal to 12 months or 1 full year. Phase shift: 2 months left Vertical shift: 5.2 Maxima: 5.2 + 1.5 = 6.7 at 1 Minima: 5.2 – 1.5 = 3.7 at 7

6. Example 1 Continued Step 2 Graph using all the information about the function. B. What is the maximum number of people employed? The maximum number of people employed is 1000(5.2 + 1.5) = 6700.

7. Suppose that the height H of a Ferris wheel can be modeled by, where t is the time in seconds. H(t) = –16cos + 24 a = –16, b = , k = 24 Example 2 a. Graph the height of a cabin for two complete periods. Step 1 Identify the important features of the graph.

8. Example 2 Continued Amplitude: Period: The period is equal to the time required for one full rotation. Vertical shift: Maxima: Minima:

9. Period: Example 2 Continued Amplitude: –16 The period is equal to the time required for one full rotation. Vertical shift: 24 Maxima: 24 + 16 = 40 Minima: 24 – 16 = 8

10. Height (ft) Time (min) Example 2 Continued