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SECTION 2.2 BUILDING LINEAR FUNCTIONS FROM DATA BUILDING LINEAR FUNCTIONS FROM DATA

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LINEAR CURVE FITTING STEP 1: Ask whether the variables are related to each other. STEP 1: Ask whether the variables are related to each other. STEP 2: Obtain data and verify a relation exists. Plot the points to obtain a scatter diagram. STEP 2: Obtain data and verify a relation exists. Plot the points to obtain a scatter diagram. STEP 3: Find an equation which describes this relation. STEP 3: Find an equation which describes this relation.

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FINDING AN EQUATION FOR LINEARLY RELATED DATA A farmer collected the following data, which shows crop yields for various amounts of fertilizer used. A farmer collected the following data, which shows crop yields for various amounts of fertilizer used.

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Fertilizer (X lbs)Yield (Y bushels) 04 06 510 57 1012 1010 1515 1517 2018 2021 2523 2522

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GETTING A SCATTER PLOT OF THE DATA Ensure that all equations in the Y= menu are cleared out or disabled. Input the data into the lists in the statistics editor: STAT 1:Edit Turn on a Statistics Plotter and set the desired parameters: 2nd Y= Push Zoom and choose ZoomStat.

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GETTING A LINE OF BEST FIT Verify by the scatter plot that the data has a linear relationship. Go to the home screen, press STAT, arrow to CALC, and choose LinReg. A linear regression equation will appear in the home screen.

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GRAPHING THE REGRESSION EQUATION To put the regression equation in the Y= menu: 1. Push Y= 2. Push VARS, choose Statistics, arrow to EQ, and choose RegEQ. Now push GRAPH.

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MAKING A PREDICTION Use the Linear Regression Equation to Estimate the Yield if the farmer uses 17 pounds of fertilizer. 1. Go to home screen 2. Go into YVARS, choose Function, Choose Y 1. 3.Type in (17).

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MAKING A PREDICTION Our prediction is that the crop yield for 17 Pounds of fertilizer per 100 ft 2 will be 17 Bushels

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CONCLUSION OF SECTION 2.2 CONCLUSION OF SECTION 2.2

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VARIATION Relationships between variables are often described in terms of proportionality. For Example: Force is proportional to acceleration. Pressure and volume of an ideal gas are inversely proportional.

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DIRECT VARIATION Let x and y denote two quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero number k such that y = kx constant of proportionality

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EXAMPLE For a certain gas enclosed in a container of fixed volume, the pressure P (in newtons per square meter) varies directly with temperature T (in kelvins). If the pressure is found to be 20 newtons/m 2 at a temperature of 60 K, find a formula that relates pressure P to temperature T. Then find the pressure P when T = 120 K.

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SOLUTION First, we know that P varies directly with T. P = k T And, we know P = 20 when T = 60. Thus, 20 = k(60)

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SOLUTION The formula, then, is Now, we must find P when T = 120K P = 40 newtons per square meter

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