BUILDING LINEAR FUNCTIONS FROM DATA

1
SECTION 2.2

• BUILDING LINEAR FUNCTIONS FROM DATA

2
LINEAR CURVE FITTING

• 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 3 Find an equation which describes this
  relation.

3
FINDING AN EQUATION FOR LINEARLY RELATED DATA

• A farmer collected the following data, which
  shows crop yields for various amounts of
  fertilizer used.

4
Fertilizer (X lbs) Yield (Y bushels) 0 4
0 6 5 10 5 7 10 12 10
10 15 15 15 17 20 18 20
21 25 23 25 22
5
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
1Edit Turn on a Statistics Plotter and set the
desired parameters 2nd Y Push Zoom and
choose ZoomStat.
6
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.
7
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.
8
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 Y1. 3. Type in
(17).
9
MAKING A PREDICTION
Our prediction is that the crop yield for 17
Pounds of fertilizer per 100 ft2 will be 17
Bushels
10

• CONCLUSION OF SECTION 2.2

11
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.
12
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
13
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/m2 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