1. 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.

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 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.

4. 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.

5. 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.

6. 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

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. Homework
p , 6-8