Holt McDougal Algebra 2 Curve Fitting with Polynomial Models Curve Fitting with Polynomial Models Holt Algebra 2Holt McDougal Algebra 2 How do we use

Holt McDougal Algebra 2

Curve Fitting with Polynomial ModelsCurve Fitting with Polynomial Models

Holt Algebra 2Holt McDougal Algebra 2

• How do we use finite differences to determine the degree of a polynomial that will fit a given set of data?

• How do we use technology to find polynomial models for a given set of data?


Curve Fitting with Polynomial Models

The table shows the closing value of a stock index on the first day of trading for various years.

To create a mathematical model for the data, you will need to determine what type of function is most appropriate. In Lesson 12-2, you learned that a set of data that has constant second differences can be modeled by a quadratic function. Finite difference can be used to identify the degree of any polynomial data.





Use finite differences to determine the degree of the polynomial that best describes the data.

Using Finite Differences to Determine Degree

The x-values increase by a constant 2. Find the differences of the y-values.

x 4 6 8 10 12 14

y –2 4.3 8.3 10.5 11.4 11.5First differences:Second differences:

The third differences are constant. A cubic polynomial best describes the data.

Third differences:

6.3–2.3

0.5

4 2.2 0.9 0.1 Not constant –1.8 –1.3 –0.8 Not constant

0.5 0.5 Constant

1.






First differences:Second differences:

The fourth differences are constant. A quartic polynomial best describes the data.

Third differences:

25–15

20

10 15 37 73 Not constant 5 22 36 Not constant

17 14 Not constant

x –6 –3 0 3 6 9

y –9 16 26 41 78 151

Fourth differences: – 3 – 3 Constant

2.






First differences:Second differences:

The third differences are constant. A cubic polynomial best describes the data.

Third differences:

20–14

8

6 0 2 12 Not constant –6 2 10 Not constant

8 8 Constant

3. x 12 15 18 21 24 27

y 3 23 29 29 31 43



Once you have determined the degree of the polynomial that best describes the data, you can use your calculator to create the function.


Curve Fitting with Polynomial ModelsUsing Finite Differences to Write a Function

4. The table below shows the population of a city from 1960 to 2000. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values.

Year 1960 1970 1980 1990 2000

Population (thousands)

4,267 5,185 6,166 7,830 10,812

First differences:Second differences:Third differences:

91863

620

981 1664 2982683 1318

635 CloseThe third differences are constant.

A cubic polynomial best describes the data.



4. The table below shows the population of a city from 1960 to 2000. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values.

Year 1960 1970 1980 1990 2000

Population (thousands)

4,267 5,185 6,166 7,830 10,812

Step 2 Use the cubic regression feature on your calculator.

f(x) ≈ 0.10x3 – 2.84x2 + 109.84x + 4266.79



5. The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values.

First differences:Second differences:Third differences:

1.21

0.6

0.2 0.2 0.40.4 0.6

1 CloseThe third differences are constant.

A cubic polynomial best describes the data.

Speed 25 30 35 40 45 50 55 60

Gas (gal) 23.8 25 25.2 25 25.4 27 30.6 37

1.6 6.4 2.8

0.6 0.8 0.8



5. The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values.

Speed 25 30 35 40 45 50 55 60

Gas (gal) 23.8 25 25.2 25 25.4 27 30.6 37

Step 2 Use the cubic regression feature on your calculator.

f(x) ≈ 0.001x3 – 0.113x2 + 4.134x – 24.867



Lesson 15.1 Practice A

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Holt McDougal Algebra 2 Curve Fitting with Polynomial Models Curve Fitting with Polynomial Models Holt Algebra 2Holt McDougal Algebra 2 How do we use