1. Translations of Trigonometric Graphs LESSON 12–8

2. Over Lesson 12–7 5-Minute Check 1 Find the amplitude of y = 3 cos . Then graph the function. A.1 B. C.3 D.9 __ 1 3

3. Over Lesson 12–7 5-Minute Check 2 Find the period of y = sin 3 . Then graph the function. A. B. C. D.3 

4. Over Lesson 12–7 5-Minute Check 3 Write the equation of a tangent function with a period of 40°. A.y = tan 40° B.y = 40 tan C.y = tan D.y = tan 40° +

5. Over Lesson 12–7 5-Minute Check 4 A.12,000 insects B.20,000 insects C.24,000 insects D.40,000 insects A population of insects can be modeled by the equation y = 12,000 + 8000 sin. What is the greatest number of insects you will find? __ x 6

6. Then/Now You translated exponential functions. Graph horizontal translations of trigonometric graphs and find phase shifts. Graph vertical translations of trigonometric graphs.

7. Vocabulary phase shift vertical shift midline

8. Note: Recall that a translation occurs when a figure is moved from one location to another on the coordinate plane without changing its orientation. A horizontal translation of a periodic function is called a phase shift.

9. Concept

10. Example 1 Graph Horizontal Translations State the amplitude, period, and phase shift for the function y = 2 sin (θ + 20°). Then graph the function. Since a = 2 and b = 1, the amplitude and period of the function are the same as y = 2 sin . However h = –20 , so the phase shift is –20 . Because h < 0, the parent graph is shifted to the left. To graph y = 2 sin (  + 20  ), consider the graph of y = 2 sin . Graph this function and then shift the graph 20  to the left.

11. Example 1 Graph Phase Shifts Answer: amplitude: 2; period: 360°; phase shift: 20° left

12. Example 1 State the amplitude, period, and phase shift for the function y = 3 sin (θ + 30°). Then graph the function. amplitude: 3; period: 360°; phase shift: –30°

13. Concept

15. Example 2 Graph Vertical Translations State the amplitude, vertical shift, and equation of the midline, for. Then graph the function. vertical shift: k = 3, so the midline is the graph of y = 3. amplitude: period:

16. Example 2 Graph Vertical Translations Since the amplitude of the function is, draw dashed lines parallel to the midline that are unit above and below the midline. Then draw the cosine curve. Answer: vertical shift: +3; midline: y = 3; amplitude: period: 2 π

17. Example 2 State the amplitude, period, vertical shift and equation of the midline for y = 3 sin θ – 2. Then graph the function. amplitude: 3; period: 2 π; vertical shift: –2; midline: y = –2

18. Concept Use the following steps to graph trigonometric functions involving phase shifts and vertical shifts.

19. Example 3 Graph Transformations State the amplitude, period, phase shift, and vertical shift for Then graph the function. The function is written in the form y = a cos [b(θ – h) + k]. Identify the values of a, b, and k. a = 3, so the amplitude is |3| or 3. b = 2, so the period is or π. h = so the phase shift is right. k = 4, so the vertical shift is 4 units up.

20. Example 3 Graph Transformations Graph the function. Step 1The vertical shift is 4. Graph the midline y = 4.

21. Example 3 Graph Transformations Step 2The amplitude is 3. Draw dashed lines 3 units above and below the midline at y = 1 and y = 7.

22. Example 3 Graph Transformations Step 3The period is π, so the graph is compressed. Graph y = 3 sin 2θ + 4 using the midline as a reference.

23. Example 3 Graph Transformations Step 4Shift the graph to the right. Answer:

24. Example 3 State the amplitude, period, phase shift and vertical shift for. Sketch the graph. ; vertical shift: –2

25. Translations of Trigonometric Graphs LESSON 12–8

26. Example 4 Represent Periodic Functions WAVE POOL The height of water in a wave pool oscillates between a maximum of 10 feet and a minimum of 6 feet. The wave generator pumps 3 waves per minute. Write a sine function that represents the height of the water at time t seconds. Then graph the function. Step 1Write the equation for the midline, and determine the vertical shift. The midline lies halfway between the maximum and the minimum. Since the midline is y = 8, the vertical shift is k = 8.

27. Example 4 Represent Periodic Functions Step 2Find the amplitude. |a| = |10 – 8| or 2Find the difference between the midline value and the maximum value. Step 3Find the period. Since there are 3 waves per minute, there is one wave every 20 seconds. So, the period is 20 seconds.

28. Example 4 Represent Periodic Functions Solve for |b|. Simplify. Step 4Write an equation for the function. h= a sin b (t – 0) + 8Write the equation for sine relating height h and time t. | b | =

29. Example 4 Represent Periodic Functions Simplify. Answer:

30. Example 4 WAVE POOL The height of water oscillates between a maximum of 8 feet and a minimum of 4 feet, and the wave generator pumps 5 waves per minute. Determine the correct sine function that represents the height of the water at time t seconds. A. B. C. D.