The Sine Graph: Introduction and Transformations

1. The Sine Graph: Introduction and Transformations 26 April 2011

2. The Sine Graph – A Review • sin(t) = y

4. Key Features of y = sin(t) 2 • Maximum: • Minimum: • Domain: • Range: 0 – 2

5. Multiple Revolutions

6. Trigonometric Graphs Repeat!!! Range:Domain:

7. 2 Periodicity Period: π • Trigonometric graphs are periodic because the pattern of the graph repeats itself • How long it takes the graph to complete one full wave or revolution is called the period 0 1 Period 1 Period –2

8. Periodicity, cont.

9. Your Turn: • Complete problems 1 – 3 on the Identifying Key Features of Sine Graphs Handout

10. Calculating Periodicity • If f(t) = sin(bt), then period = • Period is always positive 1. f(t) = sin(–6t) 2. 3.

11. Your Turn: • Calculate the period of the following graphs: 1. f(t) = sin(3t) 2. f(t) = sin(–4t) 3. 4. f(t) = 4sin(2t) 5. 6.

12. Amplitude • Amplitude is a trigonometric graph’s greatestdistance from the x-axis. • Amplitude is always positive. • If f(t) = a sin(t), then amplitude = | a |

13. Calculating Amplitude Examples 1. f(t) = 6sin(4t) 2. f(t) = –5sin(6t) 3. 4.

14. Your Turn: • Complete problems 4 – 9 on the Identifying the Key Features of Sine Graphs handout

15. Sketching Sine Graphs – Single Smooth Line!!!

16. Transformations Investigation – Investigation #1

17. Refection Questions 3. What transformations did you see? • A. B. 5. A. B.

18. Transformations f(t) = a sin(bt – c) + k Vertical Shift Pay attention to the parentheses!!!

19. Investigation #2!

20. Reflection Questions 4. What transformation did you see? • Stretch = coefficient is a whole # • Compression = coefficient is a fraction 5. A. B. C. 6. A. B. C.

21. Transformations f(t) = a sin(bt – c) + k Stretch or Compression “Amplitude Shift” Vertical Shift Pay attention to the parentheses!!!

22. Reflection Questions 4. What transformation did you see? 5. A. B. C. 6. A. B. C.

23. Transformations f(t) = a sin(bt – c) + k Stretch or Compression “Amplitude Shift” Vertical Shift Period Shift Pay attention to the parentheses!!!

24. Reflection Questions 4. What transformation did you see? 4. A. B. C. 6. A. B. C.

25. Transformations f(t) = a sin(bt – c) + k Stretch or Compression “Amplitude Shift” Vertical Shift Phase Shift Period Shift Pay attention to the parentheses!!!

26. f(t) = 2 sin(4t – π) – 3 “Amplitude Shift”: Period Shift: Phase Shift: Vertical Shift: “Amplitude Shift”: Period Shift: Phase Shift: Vertical Shift: Identifying Transformations

27. Your Turn: • Identify the transformations of the following sine graphs: 1. f(t) = 3 sin(t) + 2 2. f(t) = –sin(t – 4) + 1 3. 4.

28. Sketching Transformations • Step 1: Identify the correct order of operations for the function • Period Shifts • Phase Shifts • Trig Function • “Amplitude Shifts” (Stretches or Compressions) • Vertical Shifts

29. Sketching Transformations, cont. • Step 2: Make a table that follows the order of operations for the function (Always start with the key points!) • Step 3: Complete the table for the key points (0, , , , ) • Step 4: Plot the key points • Step 5: Connect the key points with a smooth line

30. Example 1: y = –sin(t) + 1

31. Example 1: y = –sin(t) + 1 • Domain: • Range:

32. Example 2: y = 2 sin(t)– 3

33. Example 2: y = 2 sin(t) – 3 • Domain: • Range:

34. Review – Solving for Coterminal Angles • If an angle is negative or greater than 2π, then we add or subtract 2π until the angle is between 0 and 2π. • –5π + 2π = –3π + 2π = –π + 2π = π

35. Your Turn: • On a separate sheet of paper (or in the margin of your notes), find a coterminal angle between 0 and 2π for each of the following angles: 1. 2. 3π 3. 4π 4. 5. 3π

36. Problem 6:

37. Problem 6: • Domain: • Range:

38. Problem 7: