SECTION 11-10MODELING DATA WITH POLYNOMIALS

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 (cm) 3 12 27 48 75 108 147 192

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 (cm) 3 12 27 48 75 108 147 192

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 (cm) 3 12 27 48 75 108 147 192

9 15 21 27 33 39 45

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 (cm) 3 12 27 48 75 108 147 192

9 15 21 27 33 39 45

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 (cm) 3 12 27 48 75 108 147 192

9 15 21 27 33 39 45

6 6 6 6 6 6

YES, THERE IS A QUADRATIC MODEL FOR THIS DATA

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

15 = 5(3)+ b

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

15 = 5(3)+ b 15 = 15 + b

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

15 = 5(3)+ b 15 = 15 + b

b = 0

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

15 = 5(3)+ b 15 = 15 + b

b = 0

3 = 3 + 0 + c

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

15 = 5(3)+ b 15 = 15 + b

b = 0

3 = 3 + 0 + c c = 0

EXAMPLE 1

B. WRITE A FORMULA TO MODEL THE DATA.

d = at 2 + bt + c d = distance, t = time

3 = a + b + c

12 = 4a + 2b + c

27 = 9a + 3b + c

27 = 9a + 3b + c

−12 = −4a − 2b − c

15 = 5a + b

12 = 4a + 2b + c

−3 = −a − b − c

9 = 3a + b

15 = 5a + b

−9 = −3a − b

6 = 2a a = 3

15 = 5(3)+ b 15 = 15 + b

b = 0

3 = 3 + 0 + c c = 0

d = 3t 2

EXAMPLE 1

EXAMPLE 1

EXAMPLE 1

EXAMPLE 1

EXAMPLE 1

EXAMPLE 1

EXAMPLE 1

EXAMPLE 1

EXAMPLE 2

FIT A POLYNOMIAL MODEL TO THE DATA.

X 1 2 3 4 5 6 7 8

Y 8 15 34 71 132 223 350 519

EXAMPLE 2

FIT A POLYNOMIAL MODEL TO THE DATA.

X 1 2 3 4 5 6 7 8

Y 8 15 34 71 132 223 350 519

7 19 37 61 91 127 169

EXAMPLE 2

FIT A POLYNOMIAL MODEL TO THE DATA.

X 1 2 3 4 5 6 7 8

Y 8 15 34 71 132 223 350 519

7 19 37 61 91 127 169

12 18 24 30 36 42

EXAMPLE 2

FIT A POLYNOMIAL MODEL TO THE DATA.

X 1 2 3 4 5 6 7 8

Y 8 15 34 71 132 223 350 519

7 19 37 61 91 127 169

12 18 24 30 36 42

6 6 6 6 6

EXAMPLE 2

FIT A POLYNOMIAL MODEL TO THE DATA.

X 1 2 3 4 5 6 7 8

Y 8 15 34 71 132 223 350 519

7 19 37 61 91 127 169

12 18 24 30 36 42

6 6 6 6 6

A CUBIC MODEL WILL FIT.

EXAMPLE 2

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

37 = 37a + 7b + c

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

37 = 37a + 7b + c

34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

37 = 37a + 7b + c

34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d

19 = 19a + 5b + c

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

37 = 37a + 7b + c

34 = 27a + 9b + 3c + d −15 = −8a − 4b − 2c − d

19 = 19a + 5b + c 15 = 8a + 4b + 2c + d −8 = −a − b − c − d

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

37 = 37a + 7b + c

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

EXAMPLE 2

y = ax 3 + bx 2 + cx + d

8 = a + b +C + d

15 = 8a + 4b + 2c + d

34 = 27a + 9b + 3c + d

71 = 64a + 16b + 4c + d

71 = 64a + 16b + 4c + d

−34 = −27a − 9b − 3c − d

37 = 37a + 7b + c

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

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b b = 0

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b b = 0

7 = 7 + 0 + c

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b b = 0

7 = 7 + 0 + c

c = 0

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b b = 0

7 = 7 + 0 + c

c = 0

8 = 1 + 0 + 0 + d

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b b = 0

7 = 7 + 0 + c

c = 0

8 = 1 + 0 + 0 + d

d = 7

EXAMPLE 2

37 = 37a + 7b + c

19 = 19a + 5b + c

7 = 7a + 3b + c

37 = 37a + 7b + c

−19 = −19a − 5b − c

18 = 18a + 2b

19 = 19a + 5b + c

−7 = −7a − 3b − c

12 = 12a + 2b

18 = 18a + 2b −12 = −12a − 2b

6 = 6a a = 1

18 = 18 + 2b b = 0

7 = 7 + 0 + c

c = 0

8 = 1 + 0 + 0 + d

d = 7 y = x 3 + 7

EXAMPLE 2

EXAMPLE 2

EXAMPLE 2

EXAMPLE 2

EXAMPLE 2

EXAMPLE 2

EXAMPLE 2

EXAMPLE 2

HOMEWORK

P. 734 #1-17

“ACTION MAY NOT ALWAYS BRING HAPPINESS, BUT THERE IS NO HAPPINESS WITHOUT ACTION.”

- BENJAMIN DISRAELI