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. 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 8 10 12 14 16 18 y
7.2
1.2
–8.3
–19.1
–29
–35.8 1.
First differences:
– 6
–9.5
–10.8
–9.9
–6.8
Not constant
Second differences:
–3.5
–1.3
0.9
3.1
Not constant
Third differences:
2.2
2.2
2.2
Constant
The third differences are constant A cubic polynomial best describes the data.

3. Often, real-world data can be too irregular for you to use finite differences or find a polynomial function that fits perfectly. In these situations, you can use the regression feature of your graphing calculator. Remember that the closer the R2-value is to 1, the better the function fits the data.

4. Curve Fitting Polynomial Models
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2000.
Year
1994
1995
1996
1997
1998
1999
Price ($)
683
652
948
1306
863
901
Step 1 Choose the degree of the polynomial model.
Let x represent the number of years since Make a scatter plot of the data.
The function appears to be cubic or quartic. Use the regression feature to check the R2-values.
cubic: R2 ≈ quartic: R2 ≈
The quartic function is more appropriate choice.

5. Curve Fitting Polynomial Models
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2000.
Year
1994
1995
1996
1997
1998
1999
Price ($)
683
652
948
1306
863
901
Step 2 Write the polynomial model.
f(x) = 32.23x4 – x x2 – x
Step 3 Estimate the value at 2000.
2000 is 6 years after Substitute 6 for x in the quartic model.
f(6) = 32.23(6)4 – (6) (6)2 – (6)
Based on the model, the opening value was about $ in 2000.

6. Curve Fitting Polynomial Models
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 1999.
Year
1994
1995
1996
2000
2003
2004
Price ($)
3754
3835
5117
11,497
8342
10,454
Step 1 Choose the degree of the polynomial model.
Let x represent the number of years since Make a scatter plot of the data.
The function appears to be cubic or quartic. Use the regression feature to check the R2-values.
cubic: R2 ≈ quartic: R2 ≈
The quartic function is more appropriate choice.

7. Curve Fitting Polynomial Models
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 1999.
Year
1994
1995
1996
2000
2003
2004
Price ($)
3754
3835
5117
11,497
8342
10,454
Step 1 Write the polynomial model.
f(x) = 19.09x4 – x x2 – x
Step 3 Estimate the value at 1999.
1999 is 5 years after Substitute 5 for x in the quartic model.
f (5) = 19.09(5)4 – (5) (5)2 – (5)
Based on the model, the opening value was about $ 11, in 1999.

8. Curve Fitting Polynomial Models
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2002.
Year
1994
1996
1998
2000
2001
2004
Price ($)
2814
3603
5429
3962
4117
3840
Step 1 Choose the degree of the polynomial model.
Let x represent the number of years since Make a scatter plot of the data.
The function appears to be cubic or quartic. Use the regression feature to check the R2-values.
cubic: R2 ≈ quartic: R2 ≈
The quartic function is more appropriate choice.

9. Curve Fitting Polynomial Models
The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2002.
Year
1994
1996
1998
2000
2001
2004
Price ($)
2814
3603
5429
3962
4117
3840
Step 1 Write the polynomial model.
f(x) = 7.08x4 – x x2 – x
Step 3 Estimate the value at 2002.
2002 is 8 years after Substitute 8 for x in the quartic model.
f (8) = 7.08(8)4 – (8) (8)2 – (8)
Based on the model, the opening value was about $ in 2002.

10. Lesson 15.1 Practice B