LESSON



Reteach

Curve Fitting with Polynomial Models

To use finite differences to determine the degree of a polynomial,

− check that the x-values increase by a constant value, and

− find successive differences of the y-values until the differences are

constant.

Example:

A fourth degree polynomial best describes the data.

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

1.

|x |(2 |(1 |0 |1 |2 |

|y |(5 |2 |3 |4 |11 |

|First Differences | |

|Second Differences | |

|Third Differences | |

2. Identify the degree of the polynomial.

Reteach

Curve Fitting with Polynomial Models (continued)

Use finite differences that are close to select a polynomial model to fit a data set. Then use your calculator to write the function.

Since the third differences are reasonably close, you can

use a cubic function to model the data.

Use the cubic regression feature on your calculator.

Use the coefficients a, b, c, and d to write the function.

f(x) ( 0.12x 3 ( 8.06x 2 ( 273.1x ( 584.6

Write a polynomial function for the data.

3.

|x |2 |4 |6 |8 |10 |12 |

|y |(12 |(15 |38 |190 |446 |773 |

|First Differences | |

|Second Differences | |

|Third Differences | |

|Fourth Differences | |

4. Write a polynomial function that best describes the data set.

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LESSON

6-9

|Finite Differences |

|Function Type |Linear |Quadratic |Cubic |Quartic |Quintic |

|Degree |1 |2 |3 |4 |5 |

|Constant Finite Differences |First |Second |Third |Fourth |Fifth |

|x |(3 |(2 |(1 |0 |1 |2 |

|y |78 |14 |0 |0 |2 |18 |

|First Differences |14 ( 78 |0 ( 14 |0 ( 0 |2 ( 0 |18 ( 2 |

| |(64 |(14 |0 |2 |16 |

|Second Differences |(14 ( ((64) |0 ( ((14) |2 ( 0 |16 ( 2 |

| |50 |14 |2 |14 |

|Third Differences |14 ( 50 |2 ( 14 |14 ( 2 |

| |(36 |(12 |12 |

|Fourth Differences |(12 ( ((36) |12 ( ((12) |

| |24 |24 |

The x-values increase by 1.

First differences are not constant.

Second differences are not constant.

Third differences are not constant.

Fourth differences are constant.

LESSON

6-9

|x |10 |20 |30 |40 |50 |

|y[?] ª ¬ |2633 |3812 |4862 |6529 |9552 |

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|K | | | | | |

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|ÇÈÊËÍÎÔÕîïðñòóðìæìæìð| | | | | |

|ìðìðìðìðìðìàìÙìÙìÓàìÙ| | | | | |

|ìÓÍìÓÍìÓÍìÓìÈðì»ìðìð| | | | | |

|ìæìæìæìæìæ°æ§ìh6Û0J6?| | | | | |

|]?h6Û0J6?EH[?] | | | | | |

|First Differences |3812 ( 2633 |4862 ( 3812 |6529 ( 4862 |9552 ( 6529 |

| |1179 |1050 |1667 |3023 |

|Second Differences |1050 ( 1179 |1667 ( 1050 |3023 ( 1667 |

| |(129 |617 |1356 |

|Third Differences |617 ( ((129) |1356 ( 617 |

| |746 |739 |

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