1. 4.7 Curve Fitting with Finite Differences
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.
Holt McDougal Algebra 2
Holt Algebra 2

2. Concept: Handy Chart To Have
Holt McDougal Algebra 2
Holt Algebra 2

3. Concept: Definitions (again)
Finite – ending
Constant – the number without a variable
Even or Odd function – determined by the degree of the function
Polynomials are named by their degree:
0 degree: constant 1st degree: linear
2nd degree: a quadratic 3rd degree: cubic
4th degree: quartic 5th degree: quintic

4. Concept: Using Finite Difference to find degree
Before we can use finite differences the x-values must have a constant difference between them.
– –2
– –2
– –2
– –2
– –2 x 4 6 8 10 12 14 y –2
4.3
8.3
10.5
11.4
11.5 1.
This one is good to go!!!
Holt McDougal Algebra 2
Holt Algebra 2

5. Concept: Using Finite Difference to find degree cont…
Now do the same thing with the y–values. x 4 6 8 10 12 14 y –2
4.3
8.3
10.5
11.4
11.5 1.
– – –6.3
– –4
– –2.2
– … ––0.9
– … ––0.1
First differences:
The differences in these terms was not constant. So, we do it again. Find the 2nd differences by using the 1st differences.
Holt McDougal Algebra 2
Holt Algebra 2

6. The data describes a 3rd degree (cubic) polynomial.
Concept: Using Finite Difference
to find degree cont…
We continue doing this until we have a constant difference, or we run out of numbers. 1. x 4 6 8 10 12 14 y –2
4.3
8.3
10.5
11.4
11.5
1st differences:
–6.3 –4
–2.2
–0.9
–0.1
Not constant
2nd differences:
–2.3
–1.8
–1.3
–0.8
Not constant
3rd differences:
–0.5
–0.5
–0.5
Constant
The data describes a 3rd degree (cubic) polynomial.
Holt McDougal Algebra 2
Holt Algebra 2

7. The data describes a 4rd degree (quartic) polynomial.
Concept: Using Finite Difference
to find degree cont…
You Try:
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 2.
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 data describes a 4rd degree (quartic) polynomial.
Holt McDougal Algebra 2
Holt Algebra 2

8. Concept: Using Finite Difference to find degree cont…
Once you have determined the degree of the polynomial that best describes the data, you can use your calculator to create the function.
Holt McDougal Algebra 2
Holt Algebra 2

9. Concept: Real World Applications
The table below shows the population of a city from to Write a polynomial function for the data.
Step 1 Find the finite differences of the y-values.
Year
1960
1970
1980
1990
2000
Population (thousands)
4,267
5,185
6,166
7,830
10,812
First differences:
918
981
1664
2982
Second differences: 63
683
1318
Third differences:
620
635
Close
The third differences are constant A cubic polynomial best describes the data.
Holt McDougal Algebra 2
Holt Algebra 2

10. Homework
PM 4.6A
Holt McDougal Algebra 2
Holt Algebra 2