1. Transformations of Trigonometric Functions

2. Stretch v. Compression

3. Transformations f(x) = a sin(bx – c) + k
Stretch or Compression
Vertical Shift
Horizontal Shift,
Phase Shift
Period Shift
Pay attention to the parentheses!!!

4. Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

5. Sketching Transformations
Step 1: Graph the parent equation
Step 2: Identify the correct order of operations for the function
Step 3: Make a table that follows the order of operations for the function
Step 4: Complete the table for the key points (0, , , , )

6. *Memorize: y = sin(t) t
sin(t)

7. Example 1: y = -2 sin(t) – 2 t

8. Example 2: t

9. Cosine Graph
cos(t) = x t
cos t

10. *Memorize: y = cos(t) t
cos t

11. Transformations f(x) = a cos(bx – c) + k
Stretch or Compression
Vertical Shift
Horizontal Shift,
Phase Shift
Period Shift
Pay attention to the parentheses!!!

12. Cosine Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

13. Example 1: y = –cos(t) – 1 t

14. Example 2: t

15. Transformations f(x) = a tan(bx – c) + k
Stretch or Compression
Vertical Shift
Horizontal Shift,
Phase Shift
Period Shift
Pay attention to the parentheses!!!

16. Tangent Graph
tan(t) = x t
tan t

17. Tangent Examples Stretch/Compression: Horizontal Shift:
Stretch/Compression:
Horizontal Shift:
Vertical Shift:
Stretch/Compression:
Horizontal Shift:
Vertical Shift:

18. x
tan x

19. x
tan x

20. Transformations f(x) = a cot(bx – c) + k
Stretch or Compression
Vertical Shift
Horizontal Shift,
Phase Shift
Period Shift
Pay attention to the parentheses!!!

21. Cotangent Graph
cot(t) = x t
cot t

22. Cotangent Examples Stretch/Compression: Horizontal Shift:
Stretch/Compression:
Horizontal Shift:
Vertical Shift:
Stretch/Compression:
Horizontal Shift:
Vertical Shift:

23. x
tan x

24. x
tan x

25. Transformations f(x) = a sec(bx – c) + k
Stretch or Compression
Vertical Shift
Horizontal Shift,
Phase Shift
Period Shift
Pay attention to the parentheses!!!

26. Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

27. *Memorize: y = sec(t) t
sec (t)

28. Example 1: y = -3 sec(t) – 1 t

29. Example 2: t

30. Transformations f(x) = a csc(bx – c) + k
Stretch or Compression
Vertical Shift
Horizontal Shift,
Phase Shift
Period Shift
Pay attention to the parentheses!!!

31. Examples Stretch/Compression: Horizontal Shift: Vertical Shift:

32. *Memorize: y = csc(t) t
csc(t)

33. Example 1: y = -3 csc(πt) – 1 t

34. Example 2: t