1. Curve Fitting with
Polynomial Models
Essential Questions
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?
Holt McDougal Algebra 2
Holt Algebra 2

2. 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.

4. Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data.
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
11.5 1.
First differences:
6.3 4
2.2
0.9
0.1
Not constant
Second differences:
–2.3
–1.8
–1.3
–0.8
Not constant
Third differences:
0.5
0.5
0.5
Constant
The third differences are constant A cubic polynomial best describes the data.

5. Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data.
The x-values increase by a constant 3. Find the differences of the y-values. x –6 –3 3 6 9 y –9 16 26 41 78
151 2.
First differences: 25 10 15 37 73
Not constant
Second differences:
–15 5 22 36
Not constant
Third differences: 20 17 14
Not constant
Fourth differences:
– 3
– 3
Constant
The fourth differences are constant A quartic polynomial best describes the data.

6. Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data.
The x-values increase by a constant 3. Find the differences of the y-values. x 12 15 18 21 24 27 y 3 23 29 31 43 3.
First differences: 20 6 2 12
Not constant
Second differences:
–14 –6 2 10
Not constant
Third differences: 8 8 8
Constant
The third differences are constant A cubic polynomial best describes the data.

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

8. Using Finite Differences to Write a Function
The table below shows the population of a city from 1960 to 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:
918
981
1664
2982
Second differences: 63
683
1318
Third differences:
620
635
Close
The third differences are constant A cubic polynomial best describes the data.

9. Using Finite Differences to Write a Function
The table below shows the population of a city from 1960 to 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 – 2.84x x

10. Using Finite Differences to Write a Function
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
25.2
25.4 27
30.6 37
First differences:
1.2
0.2
-0.2
0.4
1.6
3.6
6.4
Second differences: -1
-0.4
0.6
1.2 2
2.8
Third differences:
0.6 1
0.6
0.8
0.8
Close
The third differences are constant A cubic polynomial best describes the data.

11. Using Finite Differences to Write a Function
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
25.2
25.4 27
30.6 37
Step 2 Use the cubic regression feature on your calculator.
f(x) ≈ 0.001x3 – 0.113x x –

12. Lesson 15.1 Practice A