Introduction

IntroductionAs we saw in the previous lesson, we can use a process called regression to fit the best equation with the data in an attempt to draw conclusions and make predictions. We have learned how to interpret and analyze regression models and how to come up with a regression model using what we know about quadratic equations. In this lesson, we will learn how to easily find a quadratic regression with a graphing calculator.

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5.9.2: Fitting a Function to Data

Key Concepts• A graphing calculator can graph a scatter plot of given

data and find a regression equation that models the given data.

• Begin by entering the data into lists on your calculator, as outlined in the following steps.

• Decide whether the data would best be modeled with a quadratic regression, linear regression, or exponential regression. In this lesson, we focus on quadratic regressions, but you can choose the type of regression that is most appropriate for the situation.

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Key Concepts, continued• To decide which regression model is best, look at the

scatter plot of the data you entered. • After you find the appropriate model using your

calculator, you can graph this equation on top of the scatter plot to verify that it is a reasonable model.

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Key Concepts, continuedGraphing Equations Using a TI-83/84:

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Step 1: Press [STAT].

Step 2: Press [ENTER] to select Edit.

Step 3: Enter x-values into L1.

Step 4: Enter y-values into L2.

Step 5: Press [2nd][Y=].

Step 6: Press [ENTER] twice to turn on the Stat Plot.

Step 7: Press [ZOOM][9] to select ZoomStat and show the scatter plot.

Key Concepts, continuedGraphing Equations Using a TI-Nspire:

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Step 1: Press the [home] key and select the Lists & Spreadsheet icon.

Step 2: Name Column A “x” and Column B “y.” Step 3: Enter x-values under Column A. Step 4: Enter y-values under Column B. Step 5: Select Menu then 3: Data, and then 6: Quick

Graph.Step 6: Press [enter]. Step 7: Move the cursor to the x-axis and choose x. Step 8: Move the cursor to the y-axis and choose y.

Key Concepts, continued• Your graphing calculator can help you to find a

quadratic regression model after you input the data.

Entering Lists Using a TI-83/84:

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Step 1: Press [STAT]. Step 2: Arrow to the right to select Calc. Step 3: Press [5] to select QuadReg. Step 4: At the QuadReg screen, enter the

parameters for the function (Xlist: L1, Ylist: L2, Store RegEQ: Y1). To enter Y1, press [VARS] and arrow over to the right to “Y-VARS.” Select 1: Function. Select 1: Y1.

Key Concepts, continued

Entering Lists Using a TI-Nspire:

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Step 1: Select Menu, then 4: Analyze, then 6: Regression, and then 4: Show Quadratic.

Step 2: Move the cursor over the equation and press the center key in the navigation pad to drag the equation for viewing, if necessary.

Step 5: Press [ENTER] twice to see the quadratic regression equation.

Step 6: Press [ZOOM][9] to view the graph of the scatter plot and the regression equation.

Common Errors/Misconceptions• not identifying the correct function for the data

• using the incorrect commands or sequence of commands on the graphing calculator

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Guided Practice

Example 1The students in Ms. Swan’s class surveyed people of all ages to find out how many people in each of several age groups exercise on a regular basis. Their data is shown in the table on the next slide. Use your calculator to make a scatter plot of the data in the table and to find a quadratic regression of this data. Use the “Group number” column in the table to represent the age group, the x-values. Graph the regression equation on top of your scatter plot.

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Guided Practice: Example 1, continued

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Age range11–20

21–30

31–40

41–50

51–60

61–70

Group number (x)

1 2 3 4 5 6

Number of people who exercise (y)

21 38 43 41 26 11


1. Make a scatter plot of the data. On a TI-83/84:








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Guided Practice: Example 1, continuedOn a TI-Nspire:


Step 2: Name Column A “group” and Column B “people.”

Step 3: Enter x-values under Column A.

Step 4: Enter y-values under Column B.

Step 5: Select Menu, then 3: Data, and then 6: Quick Graph.

Step 6: Press [enter].

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Guided Practice: Example 1, continuedStep 7: Move the cursor to the x-axis and choose

“group.”

Step 8: Move the cursor to the y-axis and choose “people.”

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2. Find the quadratic regression model that best fits this data. On a TI-83/84:


Step 2: Arrow to the right to select Calc.

Step 3: Press [5] to select QuadReg.

Step 4: At the QuadReg screen, enter the parameters for the function (Xlist: L1, Ylist: L2, Store RegEQ: Y1). To enter Y1, press [VARS] and arrow over to the right to “Y-VARS.” Select 1: Function. Select 1: Y1. 15


Guided Practice: Example 1, continuedStep 5: Press [ENTER] twice to see the quadratic

regression equation.


On a TI-Nspire:



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A quadratic regression model for this problem is

y = –4.286x2 + 27.486x – 1.2.

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http://www.walch.com/ei/00279

Guided Practice

Example 3Doctors recommend that most people exercise for 30 minutes every day to stay healthy. To get the best results, a person’s heart rate while exercising should reach between 50% and 75% of his or her maximum heart rate, which is usually found by subtracting your age from 220. The peak rate should occur at around the 25th minute of exercise. Alice is 30 years old, and her maximum heart rate is 190 beats per minute (bpm). Assume that her resting rate is 60 bpm.

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Guided Practice: Example 3, continuedThe table below shows Alice’s heart rate as it is measured every 5 minutes for 30 minutes while she exercises. Interpret the model.

Make a scatter plot of the data. Use a graphing calculator to find a quadratic regression model for the data. Use your model to extrapolate Alice’s heart rate after 35 minutes of exercise.

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Time (minutes) 0 5 10 15 20 25 30

Heart rate (bpm) 60 93 113

134

142

152

148


1. Make a scatter plot of the data. On a TI-83/84:








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Guided Practice: Example 3, continuedOn a TI-Nspire:


Step 2: Name Column A “time” and Column B “rate.”Step 3: Enter x-values under Column A.Step 4: Enter y-values under Column B.Step 5: Select Menu, then 3: Data, and then 6:

Quick Graph.Step 6: Move the cursor to the x-axis and choose

“time.”Step 7: Move the cursor to the y-axis and choose

“rate.”22



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2. Find a quadratic regression model using your graphing calculator. On a TI-83/84:


Step 2: Arrow to the right to select Calc.

Step 3: Press [5] to select QuadReg.

Step 4: At the QuadReg screen, enter the parameters for the function (Xlist: L1, Ylist: L2, Store RegEQ: Y1). To enter Y1, press [VARS] and arrow over to the right to “Y-VARS.” Select 1: Function. Select 1: Y1. 24


Guided Practice: Example 3, continuedStep 5: Press [ENTER] twice to see the quadratic

regression equation.


On a TI-Nspire:



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A quadratic regression model for this problem is y = –0.1243x2 + 6.6643x + 60.7143.


3. Use your model to extrapolate Alice’s heart rate after 35 minutes of exercise.

Substitute 35 for x in the regression model.

y = –0.1243(35)2 + 6.6643(35) + 60.7143

y ≈ 141.70

After 35 minutes of exercise, we can expect Alice’s heart rate to be approximately 141.7 beats per minute.

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4. Interpret the model. Alice appears to be reducing her heart rate and, therefore, reducing her exercise intensity after a peak at approximately 27 minutes. If she continues the trend, her heart rate will be back to her resting heart rate at approximately 54 minutes. Heart rates below her resting heart rate can be ignored.

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http://www.walch.com/ei/00280

Documents

Introduction