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# AA Section 11-10

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Modeling Data with Polynomials

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• ### AA Section 11-10

1. 1. SECTION 11-10 MODELING DATA WITH POLYNOMIALS
2. 2. EXAMPLE 1 MATT MITARNOWSKI ROLLED A BALL DOWN AN INCLINED PLANE IN A PHYSICS LAB. HE ACCURATELY MEASURED THE TOTAL DISTANCE TRAVELED BY THE BALL AS A FUNCTION OF TIME AND OBTAINED THE FOLLOWING DATA: Time(sec) 1 2 3 4 5 6 7 8 Distance 3 12 27 48 75 108 147 192 (cm)
3. 3. EXAMPLE 1 A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE? Time(sec) 1 2 3 4 5 6 7 8 Distance 3 12 27 48 75 108 147 192 (cm)
4. 4. EXAMPLE 1 A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE? Time(sec) 1 2 3 4 5 6 7 8 Distance 3 12 27 48 75 108 147 192 (cm) 9 15 21 27 33 39 45
5. 5. EXAMPLE 1 A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE? Time(sec) 1 2 3 4 5 6 7 8 Distance 3 12 27 48 75 108 147 192 (cm) 9 15 21 27 33 39 45 6 6 6 6 6 6
6. 6. EXAMPLE 1 A. DOES A POLYNOMIAL MODEL OF DEGREE LESS THAN 5 EXIST FOR THIS DATA? IF SO, WHAT DEGREE? Time(sec) 1 2 3 4 5 6 7 8 Distance 3 12 27 48 75 108 147 192 (cm) 9 15 21 27 33 39 45 6 6 6 6 6 6 YES, THERE IS A QUADRATIC MODEL FOR THIS DATA
7. 7. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA.
8. 8. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c
9. 9. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time
10. 10. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 3 = a + b + c   12 = 4a + 2b + c  27 = 9a + 3b + c 
11. 11. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 3 = a + b + c 27 = 9a + 3b + c   12 = 4a + 2b + c −12 = −4a − 2b − c  27 = 9a + 3b + c 
12. 12. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 3 = a + b + c 27 = 9a + 3b + c   12 = 4a + 2b + c −12 = −4a − 2b − c  27 = 9a + 3b + c 15 = 5 a + b 
13. 13. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c  27 = 9a + 3b + c 15 = 5 a + b 
14. 14. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b 
15. 15. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  15 = 5 a + b −9 = −3a − b
16. 16. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  15 = 5 a + b −9 = −3a − b 6 = 2a
17. 17. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  15 = 5 a + b −9 = −3a − b 6 = 2a a=3
18. 18. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  15 = 5(3) + b 15 = 5 a + b −9 = −3a − b 6 = 2a a=3
19. 19. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  15 = 5(3) + b 15 = 5 a + b 15 = 15 + b −9 = −3a − b 6 = 2a a=3
20. 20. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  15 = 5(3) + b 15 = 5 a + b 15 = 15 + b −9 = −3a − b b=0 6 = 2a a=3
21. 21. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  3= 3+0+c 15 = 5(3) + b 15 = 5 a + b 15 = 15 + b −9 = −3a − b b=0 6 = 2a a=3
22. 22. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  3= 3+0+c 15 = 5(3) + b 15 = 5 a + b c=0 15 = 15 + b −9 = −3a − b b=0 6 = 2a a=3
23. 23. EXAMPLE 1 B. WRITE A FORMULA TO MODEL THE DATA. 2 d = at + bt + c d = distance, t = time 12 = 4a + 2b + c 3 = a + b + c 27 = 9a + 3b + c  −3 = −a − b − c  12 = 4a + 2b + c −12 = −4a − 2b − c 9 = 3a + b  27 = 9a + 3b + c 15 = 5 a + b  3= 3+0+c 15 = 5(3) + b 15 = 5 a + b c=0 15 = 15 + b −9 = −3a − b b=0 6 = 2a 2 d = 3t a=3
24. 24. EXAMPLE 1
25. 25. EXAMPLE 1
26. 26. EXAMPLE 1
27. 27. EXAMPLE 1
28. 28. EXAMPLE 1
29. 29. EXAMPLE 1
30. 30. EXAMPLE 1
31. 31. EXAMPLE 1
32. 32. EXAMPLE 2 FIT A POLYNOMIAL MODEL TO THE DATA. 1 2 3 4 5 6 7 8 X 8 15 34 71 132 223 350 519 Y
33. 33. EXAMPLE 2 FIT A POLYNOMIAL MODEL TO THE DATA. 1 2 3 4 5 6 7 8 X 8 15 34 71 132 223 350 519 Y 7 19 37 61 91 127 169
34. 34. EXAMPLE 2 FIT A POLYNOMIAL MODEL TO THE DATA. 1 2 3 4 5 6 7 8 X 8 15 34 71 132 223 350 519 Y 7 19 37 61 91 127 169 12 18 24 30 36 42
35. 35. EXAMPLE 2 FIT A POLYNOMIAL MODEL TO THE DATA. 1 2 3 4 5 6 7 8 X 8 15 34 71 132 223 350 519 Y 7 19 37 61 91 127 169 12 18 24 30 36 42 6 6 6 6 6
36. 36. EXAMPLE 2 FIT A POLYNOMIAL MODEL TO THE DATA. 1 2 3 4 5 6 7 8 X 8 15 34 71 132 223 350 519 Y 7 19 37 61 91 127 169 12 18 24 30 36 42 6 6 6 6 6 A CUBIC MODEL WILL FIT.
37. 37. EXAMPLE 2
38. 38. EXAMPLE 2 3 2 y = ax + bx + cx + d
39. 39. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d   15 = 8a + 4b + 2c + d  34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 
40. 40. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 
41. 41. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  37 = 37a + 7b + c 34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 
42. 42. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  37 = 37a + 7b + c 34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 34 = 27a + 9b + 3c + d  −15 = −8a − 4b − 2c − d
43. 43. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  37 = 37a + 7b + c 34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 34 = 27a + 9b + 3c + d  −15 = −8a − 4b − 2c − d 19 = 19a + 5b + c
44. 44. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  37 = 37a + 7b + c 34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 34 = 27a + 9b + 3c + d  −15 = −8a − 4b − 2c − d 19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d
45. 45. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  37 = 37a + 7b + c 34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 34 = 27a + 9b + 3c + d  −15 = −8a − 4b − 2c − d 19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d 7 = 7a + 3b + c
46. 46. EXAMPLE 2 3 2 y = ax + bx + cx + d 8 = a + b + C + d 71 = 64a + 16b + 4c + d  −34 = −27a − 9b − 3c − d  15 = 8a + 4b + 2c + d  37 = 37a + 7b + c 34 = 27a + 9b + 3c + d   71 = 64a + 16b + 4c + d 34 = 27a + 9b + 3c + d  −15 = −8a − 4b − 2c − d 19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d 37 = 37a + 7b + c 7 = 7a + 3b + c   19 = 19a + 5b + c  7 = 7a + 3b + c 
47. 47. EXAMPLE 2 37 = 37a + 7b + c   19 = 19a + 5b + c  7 = 7a + 3b + c 
48. 48. EXAMPLE 2 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 
49. 49. EXAMPLE 2 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 
50. 50. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 
51. 51. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b 
52. 52. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  18 = 18a + 2b −12 = −12a − 2b
53. 53. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  18 = 18a + 2b −12 = −12a − 2b 6 = 6a
54. 54. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  18 = 18a + 2b −12 = −12a − 2b 6 = 6a a=1
55. 55. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  18 = 18a + 2b −12 = −12a − 2b 6 = 6a a=1 18 = 18 + 2b
56. 56. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  18 = 18a + 2b −12 = −12a − 2b 6 = 6a a=1 18 = 18 + 2b b=0
57. 57. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  7 = 7+0+c 18 = 18a + 2b −12 = −12a − 2b 6 = 6a a=1 18 = 18 + 2b b=0
58. 58. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  7 = 7+0+c 18 = 18a + 2b c=0 −12 = −12a − 2b 6 = 6a a=1 18 = 18 + 2b b=0
59. 59. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  7 = 7+0+c 18 = 18a + 2b c=0 −12 = −12a − 2b 6 = 6a 8 = 1+0+0+d a=1 18 = 18 + 2b b=0
60. 60. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  7 = 7+0+c 18 = 18a + 2b c=0 −12 = −12a − 2b 6 = 6a 8 = 1+0+0+d a=1 d=7 18 = 18 + 2b b=0
61. 61. EXAMPLE 2 19 = 19a + 5b + c 37 = 37a + 7b + c 37 = 37a + 7b + c  −19 = −19a − 5b − c −7 = −7a − 3b − c  19 = 19a + 5b + c  7 = 7a + 3b + c 18 = 18a + 2b 12 = 12a + 2b  7 = 7+0+c 18 = 18a + 2b c=0 −12 = −12a − 2b 6 = 6a 3 y= x +7 8 = 1+0+0+d a=1 d=7 18 = 18 + 2b b=0
62. 62. EXAMPLE 2
63. 63. EXAMPLE 2
64. 64. EXAMPLE 2
65. 65. EXAMPLE 2
66. 66. EXAMPLE 2
67. 67. EXAMPLE 2
68. 68. EXAMPLE 2
69. 69. EXAMPLE 2
70. 70. HOMEWORK P. 734 #1-17 “ACTION MAY NOT ALWAYS BRING HAPPINESS, BUT THERE IS NO HAPPINESS WITHOUT ACTION.” - BENJAMIN DISRAELI