Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

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.

Objectives

Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

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.

Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

Use finite differences to determine the degree of the polynomial that best describes the data.

Example 1A: Using Finite Differences to Determine Degree

The x-values increase by a constant 2. Find the differences of the y-values.

x 4 6 8 10 12 14

y –2 4.3 8.3 10.5 11.4 11.5

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

The third differences are constant. A cubic polynomial best describes the data.

Third differences: 0.5 0.5 0.5 Constant

Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

Use finite differences to determine the degree of the polynomial that best describes the data.

Example 1B: Using Finite Differences to Determine Degree

The x-values increase by a constant 3. Find the differences of the y-values.

x –6 –3 0 3 6 9

y –9 16 26 41 78 151

y –9 16 26 41 78 151

First differences: 25 10 15 37 73 Not constant Second differences: –15 5 22 36 Not constant

The fourth differences are constant. A quartic polynomial best describes the data.

Third differences: 20 17 14 Not constant Fourth differences: –3 –3 Constant

Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

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 29 31 43

Holt McDougal Algebra 2

3-9 Curve Fitting with Polynomial Models

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

Homework

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1-3, 6-8