1. Curving Fitting with 6-9 Polynomial Functions Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. Warm Up
Find a line of best fit for the data. 1. x 2 8 15 21 24 y 70 62 80
190
160
y = 5.45x 2. x 38 42 44 35 49 y 92 80 75 81 68
y = –1.28x

3. Objectives
Use finite differences to determine the degree of a polynomial that will fit a given set of data.
Use technology to find polynomial models for a given set of data.

4. 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 5-8, 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.

6. Example 1A: Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data. x 4 6 8 10 12 14 y –2
4.3
8.3
10.5
11.4
11.5
The x-values increase by a constant 2. Find the differences of the y-values. y –2
4.3
8.3
10.5
11.4
11.5
First differences: Not constant
Second differences: –2.3 –1.8 –1.3 – Not constant
Third differences: Constant
The third differences are constant. A cubic polynomial best describes the data.

7. Example 1B: Using Finite Differences to Determine Degree
Use finite differences to determine the degree of the polynomial that best describes the data. x –6 –3 3 6 9 y –9 16 26 41 78
151
The x-values increase by a constant 3. Find the differences of the y-values. y –9 16 26 41 78
151
First differences: Not constant
Second differences: – Not constant
Third differences: Not constant
Fourth differences: –3 – Constant
The fourth differences are constant. A quartic polynomial best describes the data.

8. Check It Out! Example 1
Use finite differences to determine the degree of the polynomial that best describes the data. x 12 15 18 21 24 27 y 3 23 29 31 43
The x-values increase by a constant 3. Find the differences of the y-values. y 3 23 29 31 43
First differences: Not constant
Second differences: –14 – Not constant
Third differences: Constant
The third differences are constant. A cubic polynomial best describes the data.

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

10. Example 2: 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.
Year
1960
1970
1980
1990
2000
Population (thousands)
4,267
5,185
6,166
7,830
10,812
Step 1 Find the finite differences of the y-values.
First differences:
Second differences:
Third differences: Close

11. Example 2 Continued
Step 2 Determine the degree of the polynomial.
Because the third differences are relatively close, a cubic function should be a good model.
Step 3 Use the cubic regression feature on your calculator.
f(x) ≈ 0.10x3 – 2.84x x

12. Check It Out! Example 2
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.
Speed 25 30 35 40 45 50 55 60
Gas (gal)
23.8
25.2
25.4 27
30.6 37
Step 1 Find the finite differences of the y-values.
First differences: –
Second differences: –1 –
Third differences: Close

13. Check It Out! Example 2 Continued
Step 2 Determine the degree of the polynomial.
Because the third differences are relatively close, a cubic function should be a good model.
Step 3 Use the cubic regression feature on your calculator.
f(x) ≈ 0.001x3 – 0.113x x

14. 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.

15. Example 3: 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.

16. Example 3 Continued
Step 2 Write the polynomial model. The data can be modeled by f(x) = 32.23x4 – x x2 – x
Step 3 Find the value of the model corresponding to 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.

17. Check It Out! Example 3
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.

18. Check It Out! Example 3 Continued
Step 2 Write the polynomial model. The data can be modeled by f(x) = 19.09x4 – x x2 – x
Step 3 Find the value of the model corresponding to 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.

19. x 8 10 12 14 16 18 y 7.2 1.2 –8.3 –19.1 –29 –35.8 Lesson Quiz: Part I
1. Use finite differences to determine the degree of the polynomial that best describes the data. x 8 10 12 14 16 18 y
7.2
1.2
–8.3
–19.1
–29
–35.8
cubic

20. Lesson Quiz: Part II 2.
The table shows the opening value of a stock index on the first day of trading in various years. Write a polynomial model for the data and use the 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
f(x) = 7.08x4 – x x2 – x ; about $