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Fitting with Polynomial Functions

Learn how to determine the degree of a polynomial that fits a given set of data using finite differences, and use technology to find polynomial models for the data. Examples and practice problems included.

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Fitting with Polynomial Functions

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  1. Curving Fitting with Polynomial Functions 6-9 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

  2. Warm Up Find a line of best fit for the data. 1. y = 5.45x + 36.12 2. y = –1.28x + 132.66

  3. 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.

  4. 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 5-8, 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.

  5. Example 1A: Using Finite Differences to Determine Degree Use finite differences to determine the degree of the polynomial that best describes the data. The x-values increase by a constant 2. Find the differences of the y-values. First differences: 6.3 4 2.2 0.9 0.1 Not constant Second differences: –2.3 –1.8 –1.3 –0.8 Not constant Third differences: 0.5 0.5 0.5 Constant The third differences are constant. A cubic polynomial best describes the data.

  6. Example 1B: Using Finite Differences to Determine Degree Use finite differences to determine the degree of the polynomial that best describes the data. The x-values increase by a constant 3. Find the differences of the y-values. 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 fourth differences are constant. A quartic polynomial best describes the data.

  7. Check It Out! Example 1 Use finite differences to determine the degree of the polynomial that best describes the data. The x-values increase by a constant 3. Find the differences of the y-values. First differences: 20 6 0 2 12 Not constant Second differences: –14 –6 2 10 Not constant Third differences: 8 8 8 Constant The third differences are constant. A cubic polynomial best describes the data.

  8. Once you have determined the degree of the polynomial that best describes the data, you can use your calculator to create the function.

  9. Example 2: Using Finite Differences to Write a Function 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. First differences: 918 981 1664 2982 Second differences: 63 683 1318 Third differences: 620 635 Close

  10. Example 2 Continued Step 2 Determine the degree of the polynomial. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on your calculator. f(x) ≈ 0.10x3 – 2.84x2 + 109.84x + 4266.79

  11. Check It Out! Example 2 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: 1.2 0.2 –0.2 0.4 1.6 3.6 6.4 Second differences: –1 –0.4 0.6 1.2 2 2.8 Third differences: 0.6 1 0.6 0.8 0.8Close

  12. Check It Out! Example 2 Continued Step 2 Determine the degree of the polynomial. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on your calculator. f(x) ≈ 0.001x3 – 0.113x2 + 4.134x + 24.867

  13. Often, real-world data can be too irregular for you to use finite differences or find a polynomial function that fits perfectly. In these situations, you can use the regression feature of your graphing calculator. Remember that the closer the R2-value is to 1,the better the function fits the data.

  14. Example 3: Curve Fitting Polynomial Models The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 2000. Step 1 Choose the degree of the polynomial model. Let x represent the number of years since 1994. Make a scatter plot of the data. The function appears to be cubic or quartic. Use the regression feature to check the R2-values. cubic: R2≈ 0.5833 quartic: R2≈ 0.8921 The quartic function is more appropriate choice.

  15. Example 3 Continued Step 2Write the polynomial model. The data can be modeled by f(x) = 32.23x4 – 339.13x3 + 1069.59x2 – 858.99x + 693.88 Step 3Find the value of the model corresponding to 2000. 2000 is 6 years after 1994. Substitute 6 for x in the quartic model. f(6) = 32.23(6)4 – 339.13(6)3 + 1069.59(6)2 – 858.99(6) + 693.88 Based on the model, the opening value was about $2563.18 in 2000.

  16. Check It Out! Example 3 The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 1999. Step 1 Choose the degree of the polynomial model. Let x represent the number of years since 1994. Make a scatter plot of the data. The function appears to be cubic or quartic. Use the regression feature to check the R2-values. cubic: R2≈ 0.8624 quartic: R2≈ 0.9959 The quartic function is more appropriate choice.

  17. Check It Out! Example 3 Continued Step 2Write the polynomial model. The data can be modeled by f(x) = 19.09x4 – 377.90x3 + 2153.24x2 – 2183.29x + 3871.46 Step 3Find the value of the model corresponding to 1999. 1999 is 5 years after 1994. Substitute 5 for x in the quartic model. f(5) = 19.09(5)4 – 377.90(5)3 + 2153.24(5)2 – 2183.29(5) + 3871.46 Based on the model, the opening value was about $11,479.76 in 1999.

  18. Lesson Quiz: Part I 1.Use finite differences to determine the degree of the polynomial that best describes the data. cubic

  19. Lesson Quiz: Part II 2. The table shows the opening value of a stock index on the first day of trading in various years. Write a polynomial model for the data and use the model to estimate the value on the first day of trading in 2002. f(x) = 7.08x4 – 126.92x3 + 595.95x2 – 241.81x + 2780.54; about $3003.50

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