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AA Section 3-9

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AA Section 3-9

1. 1. Section 3-9 Step Functions
2. 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. 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. 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. 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. 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. 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. 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. 9. Greatest-Integer:
10. 10. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x
11. 11. Greatest-Integer: ⎢x ⎥ ⎣ ⎦ The greatest integer less than or equal to x Step Function:
12. 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. 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. 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. 15. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦
16. 16. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦ 4
17. 17. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦ 4 -8
18. 18. Example 1 Simplify. a. ⎢ 4 ⎥ ⎣ ⎦ b. ⎢ −7 2 5 ⎥ ⎣ ⎦ c. ⎢3.2 ⎥ ⎣ ⎦ 4 -8 3
19. 19. Greatest-Integer Function
20. 20. Greatest-Integer Function The function f where f (x ) = ⎢ x ⎥ ⎣ ⎦ for all real numbers x.
21. 21. Greatest-Integer Function The function f where f (x ) = ⎢ x ⎥ ⎣ ⎦ for all real numbers x. *Also known as the rounding-down function*
22. 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. 23. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦
24. 24. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table
25. 25. Example 2 Graph f (x ) = ⎢ x ⎥ +1. ⎣ ⎦ 1. Set up a table 2. Determine the length of each interval
26. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 39. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦
40. 40. Example 4 Graph f (x ) =1.5 −1.5 ⎢1− x ⎥ . ⎣ ⎦ x f (x ) =1.5 −1.5 ⎢1− x ⎥ ⎣ ⎦
41. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 56. Homework
57. 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