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Irva Alg2 Lesson 3.9.notebook
1
November 06, 2014
warmup
39 Curve Fitting with Polynomials
Relax! You will do fine today!
We will review for quiz!!!(which is worth 10 points, has 20 questions, group, graphing calculator allowed, and will not be on your first quarter grades)
Learning Target
39 Curve Fitting with Polynomials
1. Use finite differences to determine the degree of a polynomial that will fit a given set of data.
2. Use technology to find models.
To predictWe will also review for quiz!!!
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
Regression Eq.
How to do a regression equation on your calculator:1. Note: if you want to find “r”, the correlation
coefficient, you will need to turn your Diagnostics On.
This is a onetime thing. To this, press “2nd” and the “on”
button which will get you to the catalog. Then scroll down
to the “DiaGnosticOn” option. Push enter, then enter again.
2. Enter in your data:
a. Push the “STAT” button
b. Select “1:Edit” and push enter
c. Enter your data into L1 and L2
3. Calculate your line/parabola/etc.:
a. Push the “STAT” button again
b. Arrow over to “CALC”
c. Choose the appropriate regression (#4. LinReg if you want to find a linear regression equation.)
Graph Data
How to Graph Data:
1. Enter in your data:
a. Push the “STAT” button
b. Select “1:Edit” and push enter
c. Enter your data into L1 and L2
2. Set up your graph:
a. Push the “STAT PLOT” button (2nd Y=)
b. Select “1:Plot 1…” and push enter
c. Turn on by pushing enter on the word “On”
d. Select type (dotted)
e. Check Xlist: L1,
f. Check Ylist: L2
3. Set up your window:
a. Push the “WINDOW” button
b. Set your xvalues to include all of your data values in the xtable
c. Set your yvalues to include all of your data values in the ytable
d. Xscl and Yscl should be appropriate for the data
4. Graph your data:
Just push “GRAPH”
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
Insert a regression
How to insert a regression equation into Y1
1. Enter data into list 1 and list 2 (see “how to calculate a regression equation”)
2. Calculate a regression equation (see “how to calculate a regression equation”)
Note: your Homescreen will now have the regression equation
3. Select the “Y=” button
4. At the Y1 screen, select “VARS”
5. Choose “5. Statistics”
6. Arrow over to EQ
7. Select “1: RegEQ”
You should now have the regression equation you calculated in your Y1
Finding a value
How to find a value of a function that is in Y1 given the xvalue
1. You could graph and use trace until you get around the xvalue then look at y
2. You could go to the table and search the xvalues to find the yvalue
3. You could go to table set “TBLSET” and set the TblStart to the xvalue, then look at the table
4. OR… Go to the homescreen
5. Press “VARS”
6. Arrow over to “YVARS”
7. Select “1:Function”
8. Select “1: Y1” or wherever your function resides
9. Now Y1 shows up on the homescreen.
10. Type a “(“ and the xvalue of choice. You could finish it with a “)”
11. Push enter
Example: Y1 = 8x + 3
Y1(5) enter
43
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
1
I. Using Finite Differences
1. Use finite differences to determine the degree of the polynomial that best describes the data.
The xvalues increase by a constant 2. Find the differences of the yvalues.
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.
You try
2. Use finite differences to determine the degree of the polynomial that best describes the data.
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 xvalues increase by a constant 3. Find the differences of the yvalues.
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
2
II. Use Differences for Real Data
4. The table below shows the population of a city from 1960 to 2000. Write a polynomial function for the data.
First differences: 918 981 1664 2982Second differences: 63 683 1318 Third differences: 620 635 Close
f(x) ≈ 0.10x3 – 2.84x2 + 109.84x + 4266.79
you try
5. 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.
First differences: 1.2 0.2 –0.2 0.4 1.6 3.6 6.4Second differences: –1 –0.4 0.6 1.2 2 2.8
Third differences: 0.6 1 0.6 0.8 0.8 Close
f(x) ≈ 0.001x3 – 0.113x2 + 4.134x + 24.867
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
Helpful Hint
Often, realworld 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 R2value is to 1, the better the function fits the data.
Helpful Hint
3
III. Using the Regression Feature6. 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.
cubic: R2 ≈ 0.5833 quartic: R2 ≈ 0.8921. The quartic function is more appropriate choice.
f(x) = 32.23x4 – 339.13x3 + 1069.59x2 – 858.99x + 693.88
Step 1: enter data in the calculatorStep 2: graph data to help with guessesStep 3: compute different regressions, check R2
Step 4: Answer the question
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.
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November 06, 2014
Homework
3.9 p.213 #1 5, 7 17 odd, 18, 20 22*15 problems
Review for Quiz Next Slide...
snapshot of chapter
3.1 Polynomials add/sub/identify3.2 Multiplying Polynomials & Pascal3.3 Dividing Polynomials synthetic/long division3.4 Factoring Polynomials3.5 Finding Real Roots (solving)3.6 Fundamental Theorem of Algebra3.7 Graphing Polynomials3.8 Transforming Polynomial Functions*3.9 Curve Fitting (regression)
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
3.13.3
3.1 Polynomials add/sub/identify3.2 Multiplying Polynomials & Pascal3.3 Dividing Polynomials synthetic/long division
3.43.6
3.4 Factoring Polynomials3.5 Finding Real Roots (solving)3.6 Fundamental Theorem of Algebra
Irva Alg2 Lesson 3.9.notebook
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November 06, 2014
3.73.9
3.7 Graphing Polynomials3.8 Transforming Polynomial Functions*3.9 Curve Fitting (regression)