1. Curve Fitting with 3-9 Polynomial Models Warm Up Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
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.

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

6. 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

7. 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

8. 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

9. 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.

10. 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.

11. 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.

12. 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

13. Homework pp