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.

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.

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: 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.

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: 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.

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

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

Homework p. 213 1-3, 6-8