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**Curving Fitting with 6-9 Polynomial Functions Warm Up**

Lesson Presentation Lesson Quiz Holt Algebra 2

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Warm Up 1. For f(x) = x3 + 5, write the rule for g(x) = f(x – 1) – 2 and sketch its graph. g(x) = (x – 1)3 + 3 2. Write a function that reflects f(x) = 2x3 + 1 across the x-axis and shifts it 3 units down. h(x) = –2x3 – 4

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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.

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**Example 1A: Using Finite Differences to Determine Degree**

Use finite differences to determine the degree of the polynomial that best describes the data. x 4 6 8 10 12 14 y –2 4.3 8.3 10.5 11.4 11.5 The x-values increase by a constant 2. Find the differences of the y-values. y –2 4.3 8.3 10.5 11.4 11.5 First differences: Not constant Second differences: –2.3 –1.8 –1.3 – Not constant Third differences: Constant The third differences are constant. A cubic polynomial best describes the data.

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**Example 1B: Using Finite Differences to Determine Degree**

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 The x-values increase by a constant 3. Find the differences of the y-values. y –9 16 26 41 78 151 First differences: Not constant Second differences: – Not constant Third differences: Not constant Fourth differences: –3 – Constant The fourth differences are constant. A quartic polynomial best describes the data.

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Check It Out! Example 1 Use finite differences to determine the degree of the polynomial that best describes the data. x 12 15 18 21 24 27 y 3 23 29 31 43 The x-values increase by a constant 3. Find the differences of the y-values. y 3 23 29 31 43 First differences: Not constant Second differences: –14 – Not constant Third differences: Constant The third differences are constant. A cubic polynomial best describes the data.

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**Example 2: Using Finite Differences to Write a Function**

The table below shows the population of a city from 1960 to Write a polynomial function for the data. Year 1960 1970 1980 1990 2000 Population (thousands) 4,267 5,185 6,166 7,830 10,812 Step 1 Find the finite differences of the y-values. First differences: Second differences: Third differences: Close

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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) ≈ x3 – x x

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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. Speed 25 30 35 40 45 50 55 60 Gas (gal) 23.8 25.2 25.4 27 30.6 37 Step 1 Find the finite differences of the y-values. First differences: – Second differences: –1 – Third differences: Close

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**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 – 0.113x x

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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.

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**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. Year 1994 1995 1996 1997 1998 1999 Price ($) 683 652 948 1306 863 901 Step 1 Choose the degree of the polynomial model. Let x represent the number of years since 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 ≈ quartic: R2 ≈ The quartic function is more appropriate choice.

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

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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. Year 1994 1995 1996 2000 2003 2004 Price ($) 3754 3835 5117 11,497 8342 10,454 Step 1 Choose the degree of the polynomial model. Let x represent the number of years since 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 ≈ quartic: R2 ≈ The quartic function is more appropriate choice.

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**Check It Out! Example 3 Continued**

Step 2 Write the polynomial model. The data can be modeled by f(x) = 19.09x4 – x x2 – x Step 3 Find the value of the model corresponding to 1999. 1999 is 5 years after Substitute 5 for x in the quartic model. f(5) = 19.09(5)4 – (5) (5)2 – (5) Based on the model, the opening value was about $11, in 1999.

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**x 8 10 12 14 16 18 y 7.2 1.2 –8.3 –19.1 –29 –35.8 Lesson Quiz: Part I**

1. Use finite differences to determine the degree of the polynomial that best describes the data. x 8 10 12 14 16 18 y 7.2 1.2 –8.3 –19.1 –29 –35.8 cubic

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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. Year 1994 1996 1998 2000 2001 2004 Price ($) 2814 3603 5429 3962 4117 3840 f(x) = 7.08x4 – x x2 – x ; about $

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