Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions 6-9 Curving Fitting with Polynomial Functions

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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.

Objectives

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Use finite differences to determine the degree of the polynomial that best describes the data.

Example 1A: 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 11.5

y –2 4.3 8.3 10.5 11.4 11.5

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

The third differences are constant. A cubic polynomial best describes the data.

Third differences: 0.5 0.5 0.5 Constant

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Use finite differences to determine the degree of the polynomial that best describes the data.

Example 1B: Using Finite Differences to Determine Degree

The x-values increase by a constant 3. Find the differences of the y-values.

x –6 –3 0 3 6 9

y –9 16 26 41 78 151

y –9 16 26 41 78 151

First differences: 25 10 15 37 73 Not constant Second differences: –15 5 22 36 Not constant

The fourth differences are constant. A quartic polynomial best describes the data.

Third differences: 20 17 14 Not constant Fourth differences: –3 –3 Constant

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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.

x 12 15 18 21 24 27

y 3 23 29 29 31 43

y 3 23 29 29 31 43

Second differences: –14 –6 2 10 Not constant

The third differences are constant. A cubic polynomial best describes the data.

Third differences: 8 8 8 Constant

First differences: 20 6 0 2 12 Not constant

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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.

Year 1960 1970 1980 1990 2000

Population (thousands)

4,267 5,185 6,166 7,830 10,812

Second differences: 63 683 1318 Third differences: 620 635 Close

First differences: 918 981 1664 2982

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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) ≈ 104.58x3 – 283.85x2 + 1098.34x + 4266.79

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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.

Speed 25 30 35 40 45 50 55 60

Gas (gal) 23.8 25 25.2 25 25.4 27 30.6 37

Second differences: –1 –0.4 0.6 1.2 2 2.8Third differences: 0.6 1 0.6 0.8 0.8Close

First differences: 1.2 0.2 –0.2 0.4 1.6 3.6 6.4

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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.

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Example 3: Curve Fitting Polynomial ModelsThe 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.

Year 1994 1995 1996 1997 1998 1999

Price ($) 683 652 948 1306 863 901

cubic: R2 ≈ 0.5833 quartic: R2 ≈ 0.8921The quartic function is more appropriate choice.

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Step 2 Write the polynomial model. The data can be modeled by f(x) = 32.23x4 – 339.13x3 + 1069.59x2 – 858.99x + 693.88

Step 3 Find the value of the model corresponding to 2000.

2000 is 6 years after 1994. Substitute 6 for x in the quartic model.

Example 3 Continued

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.

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

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.

Year 1994 1995 1996 2000 2003 2004

Price ($) 3754 3835 5117 11,497 8342 10,454

cubic: R2 ≈ 0.8624 quartic: R2 ≈ 0.9959The quartic function is more appropriate choice.

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Check It Out! Example 3 Continued

Step 2 Write the polynomial model. The data can be modeled by f(x) = 19.09x4 – 377.90x3 + 2153.24x2 – 2183.29x + 3871.46

Step 3 Find 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.

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

Lesson Quiz: Part I

1. Use finite differences to determine the degree of the polynomial that best describes the data.

cubic

x 8 10 12 14 16 18

y 7.2 1.2 –8.3 –19.1 –29 –35.8

Holt Algebra 2

6-9 Curve Fitting with Polynomial Functions

2.

Lesson Quiz: Part II

f(x) = 7.08x4 – 126.92x3 + 595.95x2 – 241.81x + 2780.54; about $3003.50

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