Holt McDougal Algebra 2 Curve Fitting with Polynomial Models Curve Fitting with Polynomial Models...
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Transcript of Holt McDougal Algebra 2 Curve Fitting with Polynomial Models Curve Fitting with Polynomial Models...
Holt McDougal Algebra 2
Curve Fitting with Polynomial ModelsCurve Fitting with Polynomial Models
Holt Algebra 2Holt McDougal Algebra 2
• How do we use finite differences to determine the degree of a polynomial that will fit a given set of data?
• How do we use technology to find polynomial models for a given set of data?
Holt McDougal Algebra 2
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 12-2, 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
Curve Fitting with Polynomial Models
Holt McDougal Algebra 2
Curve Fitting with Polynomial Models
Use finite differences to determine the degree of the polynomial that best describes the data.
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 11.5First differences:Second differences:
The third differences are constant. A cubic polynomial best describes the data.
Third differences:
6.3–2.3
0.5
4 2.2 0.9 0.1 Not constant –1.8 –1.3 –0.8 Not constant
0.5 0.5 Constant
1.
Holt McDougal Algebra 2
Curve Fitting with Polynomial Models
Use finite differences to determine the degree of the polynomial that best describes the data.
Using Finite Differences to Determine Degree
The x-values increase by a constant 3. Find the differences of the y-values.
First differences:Second differences:
The fourth differences are constant. A quartic polynomial best describes the data.
Third differences:
25–15
20
10 15 37 73 Not constant 5 22 36 Not constant
17 14 Not constant
x –6 –3 0 3 6 9
y –9 16 26 41 78 151
Fourth differences: – 3 – 3 Constant
2.
Holt McDougal Algebra 2
Curve Fitting with Polynomial Models
Use finite differences to determine the degree of the polynomial that best describes the data.
Using Finite Differences to Determine Degree
The x-values increase by a constant 3. Find the differences of the y-values.
First differences:Second differences:
The third differences are constant. A cubic polynomial best describes the data.
Third differences:
20–14
8
6 0 2 12 Not constant –6 2 10 Not constant
8 8 Constant
3. x 12 15 18 21 24 27
y 3 23 29 29 31 43
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
Curve Fitting with Polynomial ModelsUsing Finite Differences to Write a Function
4. The table below shows the population of a city from 1960 to 2000. 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:Second differences:Third differences:
91863
620
981 1664 2982683 1318
635 CloseThe third differences are constant.
A cubic polynomial best describes the data.
Holt McDougal Algebra 2
Curve Fitting with Polynomial ModelsUsing Finite Differences to Write a Function
4. The table below shows the population of a city from 1960 to 2000. 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
Step 2 Use the cubic regression feature on your calculator.
f(x) ≈ 0.10x3 – 2.84x2 + 109.84x + 4266.79
Holt McDougal Algebra 2
Curve Fitting with Polynomial ModelsUsing Finite Differences to Write a Function
5. The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values.
First differences:Second differences:Third differences:
1.21
0.6
0.2 0.2 0.40.4 0.6
1 CloseThe third differences are constant.
A cubic polynomial best describes the data.
Speed 25 30 35 40 45 50 55 60
Gas (gal) 23.8 25 25.2 25 25.4 27 30.6 37
1.6 6.4 2.8
0.6 0.8 0.8
Holt McDougal Algebra 2
Curve Fitting with Polynomial ModelsUsing Finite Differences to Write a Function
5. The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data. Step 1 Find the finite differences of the y-values.
Speed 25 30 35 40 45 50 55 60
Gas (gal) 23.8 25 25.2 25 25.4 27 30.6 37
Step 2 Use the cubic regression feature on your calculator.
f(x) ≈ 0.001x3 – 0.113x2 + 4.134x – 24.867
Holt McDougal Algebra 2
Curve Fitting with Polynomial Models
Lesson 15.1 Practice A