Curve Fitting with Polynomial Models Essential Questions 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 Holt Algebra 2

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. x 8 10 12 14 16 18 y 7.2 1.2 –8.3 –19.1 –29 –35.8 1. First differences: – 6 –9.5 –10.8 –9.9 –6.8 Not constant Second differences: –3.5 –1.3 0.9 3.1 Not constant Third differences: 2.2 2.2 2.2 Constant The third differences are constant. A cubic polynomial best describes the data.

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.

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 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.

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 2 Write the polynomial model. f(x) = 32.23x4 – 339.13x3 + 1069.59x2 – 858.99x + 693.88 Step 3 Estimate the value at 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.

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 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 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.

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 1999. Year 1994 1995 1996 2000 2003 2004 Price ($) 3754 3835 5117 11,497 8342 10,454 Step 1 Write the polynomial model. f(x) = 19.09x4 – 377.90x3 + 2153.24x2 – 2183.29x + 3871.46 Step 3 Estimate the value at 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.

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 2002. Year 1994 1996 1998 2000 2001 2004 Price ($) 2814 3603 5429 3962 4117 3840 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 ≈ 0.6663 quartic: R2 ≈ 0.8199 The quartic function is more appropriate choice.

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 2002. Year 1994 1996 1998 2000 2001 2004 Price ($) 2814 3603 5429 3962 4117 3840 Step 1 Write the polynomial model. f(x) = 7.08x4 – 126.92x3 + 595.95x2 – 241.81x + 2780.54 Step 3 Estimate the value at 2002. 2002 is 8 years after 1994. Substitute 8 for x in the quartic model. f (8) = 7.08(8)4 – 126.92(8)3 + 595.95(8)2 – 241.81(8) + 2780.54 Based on the model, the opening value was about $ 3003.50 in 2002.

Lesson 15.1 Practice B