Modeling Data with Polynomials

1. SECTION 11-10
MODELING DATA WITH POLYNOMIALS

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. 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. 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. 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. 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. EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.

8. EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
2
d = at + bt + c

9. EXAMPLE 1
B. WRITE A FORMULA TO MODEL THE DATA.
2
d = at + bt + c d = distance, t = time

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. EXAMPLE 1

25. EXAMPLE 1

26. EXAMPLE 1

27. EXAMPLE 1

28. EXAMPLE 1

29. EXAMPLE 1

30. EXAMPLE 1

31. EXAMPLE 1

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. 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. 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. 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. 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. EXAMPLE 2

38. EXAMPLE 2
3 2
y = ax + bx + cx + d

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. 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. 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. 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. 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. 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. 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. 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. EXAMPLE 2
37 = 37a + 7b + c

 19 = 19a + 5b + c
 7 = 7a + 3b + c


48. EXAMPLE 2
37 = 37a + 7b + c
37 = 37a + 7b + c

−19 = −19a − 5b − c
 19 = 19a + 5b + c
 7 = 7a + 3b + c


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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. EXAMPLE 2

63. EXAMPLE 2

64. EXAMPLE 2

65. EXAMPLE 2

66. EXAMPLE 2

67. EXAMPLE 2

68. EXAMPLE 2

69. EXAMPLE 2

70. HOMEWORK
P. 734 #1-17
“ACTION MAY NOT ALWAYS BRING HAPPINESS,
BUT THERE IS NO HAPPINESS WITHOUT ACTION.”
- BENJAMIN DISRAELI