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- 1. Section 3-9 Step Functions
- 2. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 4. .7777 5. -101.1 6. 2+ 3
- 3. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 2 4. .7777 5. -101.1 6. 2+ 3
- 4. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 2 3 4. .7777 5. -101.1 6. 2+ 3
- 5. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 2 3 4 4. .7777 5. -101.1 6. 2+ 3
- 6. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 2 3 4 4. .7777 5. -101.1 6. 2+ 3 0
- 7. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 2 3 4 4. .7777 5. -101.1 6. 2+ 3 0 -102
- 8. Warm-up Name the greatest integer that is less than or equal to the following: 1. 2.99 2. π 3. 24 2 3 4 4. .7777 5. -101.1 6. 2+ 3 0 -102 3
- 9. Greatest-Integer:
- 10. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x
- 11. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x Step Function:
- 12. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x Step Function: A graph that looks like a series of steps, with each “step” being a horizontal line segment
- 13. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x Step Function: A graph that looks like a series of steps, with each “step” being a horizontal line segment *It is a function, so it must pass the Vertical-Line Test*
- 14. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x Step Function: A graph that looks like a series of steps, with each “step” being a horizontal line segment *It is a function, so it must pass the Vertical-Line Test* *Each step will include one endpoint*
- 15. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦
- 16. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦ 4
- 17. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦ 4 -8
- 18. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦ 4 -8 3
- 19. Greatest-Integer Function
- 20. Greatest-Integer Function The function f where f (x ) = ⎢ x ⎥ ⎣ ⎦ for all real numbers x.
- 21. Greatest-Integer Function The function f where f (x ) = ⎢ x ⎥ ⎣ ⎦ for all real numbers x. *Also known as the rounding-down function*
- 22. Greatest-Integer Function The function f where f (x ) = ⎢ x ⎥ ⎣ ⎦ for all real numbers x. *Also known as the rounding-down function* ...because we’re rounding down
- 23. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦
- 24. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table
- 25. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table 2. Determine the length of each interval
- 26. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table 2. Determine the length of each interval 3. Choose an integer for one endpoint and the next integer for the other
- 27. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table 2. Determine the length of each interval 3. Choose an integer for one endpoint and the next integer for the other 4. Determine which endpoint is included by testing a value in between
- 28. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table 2. Determine the length of each interval 3. Choose an integer for one endpoint and the next integer for the other 4. Determine which endpoint is included by testing a value in between 5. Finish your table and plot your graph
- 29. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies?
- 30. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢ ⎥ ⎣ 50 ⎦
- 31. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 50 ⎦ ⎣ 50 ⎦
- 32. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ ⎣ ⎦ ⎣ 50 ⎦ ⎣ 50 ⎦
- 33. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎣ ⎦ ⎣ 50 ⎦ ⎣ 50 ⎦
- 34. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎣ ⎦ ⎣ 50 ⎦ ⎣ 50 ⎦ 3 rolls
- 35. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎣ ⎦ ⎢ ⎥ ⎣ 50 ⎦ ⎣ 50 ⎦ ⎣ 50 ⎦ 3 rolls
- 36. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎢ ⎥ = ⎢15.72 ⎥ ⎣ 50 ⎦ ⎣ 50 ⎦ ⎣ ⎦ ⎣ 50 ⎦ ⎣ ⎦ 3 rolls
- 37. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎢ ⎥ = ⎢15.72 ⎥ =15 ⎣ 50 ⎦ ⎣ 50 ⎦ ⎣ ⎦ ⎣ 50 ⎦ ⎣ ⎦ 3 rolls
- 38. Example 3 Banks often put pennies in rolls of 50. How many full rows can be made from p pennies? From 150 pennies? From 786 pennies? ⎢p⎥ ⎢150 ⎥ ⎢ 786 ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢3 ⎥ = 3 ⎢ ⎥ = ⎢15.72 ⎥ =15 ⎣ 50 ⎦ ⎣ 50 ⎦ ⎣ ⎦ ⎣ 50 ⎦ ⎣ ⎦ 3 rolls 15 rolls
- 39. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦
- 40. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦
- 41. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2
- 42. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ ⎣ ⎦
- 43. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦
- 44. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2)⎥ ⎣ ⎦
- 45. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦
- 46. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦
- 47. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ −3 x − 2 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦
- 48. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦
- 49. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1
- 50. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1 -1.5
- 51. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1 -1.5 −1< x ≤ 0
- 52. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1 -1.5 −1< x ≤ 0 0
- 53. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1 -1.5 −1< x ≤ 0 0 0 < x ≤ −1
- 54. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1 -1.5 −1< x ≤ 0 0 0 < x ≤ −1 1.5
- 55. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦ 1.5 −1.5 ⎢1− (−3)⎥ = −4.5 ⎣ ⎦ 1.5 −1.5 ⎢1− (−2.5)⎥ = −3 ⎣ ⎦ −3 < x ≤ −2 1.5 −1.5 ⎢1− (−2)⎥ = −3 ⎣ ⎦ −2 < x ≤ −1 -1.5 −1< x ≤ 0 0 0 < x ≤ −1 1.5
- 56. Homework
- 57. Homework p. 189 #1-24, skip 12, 16 “There ain’t no free lunches in this country. And don’t go spending your whole life commiserating that you got raw deals. You’ve got to say, ‘ I think that if I keep working at this and want it bad enough I can have it.’” - Lee Iacocca

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