4.6 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

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

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

Concept: Using Finite Difference to find degree Before we can use finite differences the x-values must have a constant difference between them. 4 –6 –2 6 –8 –2 8 –10 –2 10 –12 –2 12 –14 –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

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. –2 . – 4.3 –6.3 4.3 . –8.3 . –4 8.3. – 10.5 2–2.2 10.5. – 11.4 . … ––0.9 11.4. – 11.5 . … ––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

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

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

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

Concept: Real World Applications 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: 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

Homework PM 4.6A Holt McDougal Algebra 2 Holt Algebra 2