Name _______________________________________ Date __________________ Per ___________________
Algebra 2 Notes Section 3-9X
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.
Finite Differences
Function Type
Linear
Quadratic
Cubic
Quartic
Quintic
1
2
3
4
5
First
Second
Third
Fourth
Fifth
Degree
Constant Finite
Differences
Example:
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
The x-values
increase by 1.
First differences are
not constant.
Second differences
are not constant.
Third differences
are not constant.
12  (36)
12  (12)
24
24
Fourth differences
are constant.
A fourth degree polynomial (quartic function) best describes the data.
Use finite differences to determine the degree of the polynomial that best describes the
data.
x
2
y
5
1
2
0
1
2
3
4
11
Use finite differences that are close to select a polynomial model to fit a data set. Then use
your calculator to write the function.
x
10
20
30
40
50
y
2633
3812
4862
6529
9552
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
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.
(2nd, CATALOG, DiagnosticON, ENTER, ENTER;
STAT, EDIT, enter data into lists L1 and L2; then STAT, CALC,
CubicReg, highlight CALCULATE, ENTER )
Use the coefficients a, b, c, and d to write the function.
f(x)  0.12x 3  8.06x 2  273.1x  584.6
Remember the closer that the value of R2 is to 1, the better the fit.
Write a polynomial function for the data. Round decimals to the nearest thousandth.
x
2
4
6
8
10
12
y
12
15
38
190
446
773
CC Alg2 3-9X notes