Download presentation

Published byClaud Heath Modified over 3 years ago

1
**Curving Fitting with 6-9 Polynomial Functions Warm Up**

Lesson Presentation Lesson Quiz Holt Algebra 2

2
Warm Up Find a line of best fit for the data. 1. x 2 8 15 21 24 y 70 62 80 190 160 y = 5.45x 2. x 38 42 44 35 49 y 92 80 75 81 68 y = –1.28x

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.

6
**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.

7
**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.

8
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.

9
Once you have determined the degree of the polynomial that best describes the data, you can use your calculator to create the function.

10
**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

11
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 – 2.84x x

12
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

13
**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

14
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.

15
**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.

16
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.

17
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.

18
**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.

19
**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

20
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 $

Similar presentations

OK

Holt Algebra 1 5-1 Identifying Linear Functions 5-1 Identifying Linear Functions Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation.

Holt Algebra 1 5-1 Identifying Linear Functions 5-1 Identifying Linear Functions Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation.

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google