Download Linear Functions and Models

Linear Functions and
Models
Lesson 2.1
Problems with Data

Real data recorded



Problems



Experiment results
Periodic transactions
Data not always recorded accurately
Actual data may not exactly fit theoretical
relationships
In any case …

Possible to use linear (and other) functions to
analyze and model the data
Fitting Functions
to Data

Temperature
Viscosity
(lbs*sec/in2)
160
28
170
26
180
24
190
21
200
16
210
13
220
11
230
9
Consider the data
given by this example
Viscosity (lbs*sec/in2)

Note the plot of
the data points

Close to being
in a straight line
30
25
20
15
10
5
0
160
170
180
190
200
210
220
230
240
Finding a Line to Approximate the Data

Draw a line “by eye”
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

Note slope, y-intercept
Statistical process (least squares method)
Use a computer program
such as Excel
Use your TI calculator
Chart Title

35
30
25
20
15
10
5
0
160
170
180
190
200
210
220
230
240
Graphs of Linear Functions

For the moment, consider the first option
Given the graph with tic marks = 1

Determine
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Slope
Y-intercept
A formula for the function
X-intercept (zero of the function)
Graphs of Linear Functions

Slope – use difference quotient

Y-intercept – observe
Write in form

change in y y
m

change in x x
y  m x b

Zero (x-intercept) – what value of x gives a
value of 0 for y?
Modeling with Linear
Functions

Linear functions will model data when


The constant rate is the slope of the function


Physical phenomena and data changes at a constant
rate
Or the m in y = mx + b
The initial value for the data/phenomena is the
y-intercept

Or the b in y = mx + b
Modeling with Linear
Functions

Ms Snarfblat's SS class is very popular. It
started with 7 students and now, 18 months
later has grown to 80 students. Assuming
constant monthly growth rate, what is a
modeling function?



Determine the slope of the function
Determine the y-intercept
Write in the form of y = mx + b
Creating a Function from a
Table

Determine slope by using
change in y y
m

change in x x
Answer:
y  10  7  3
x  5  3  2
y 3
  1.5  m
y 2
x
x
y
3
7
4
8.5
5
10
6
11.5
y
Creating a Function from a
Table



Now we know slope m = 3/2
Use this and one of
x
the points to determine
3
y-intercept, b
4
Substitute an
3
10


5

b
5
ordered
2
pair into
6
20  3  5  2b
y = (3/2)x + b
5  2b
5
 b solution : y 
2
y
7
8.5
10
11.5
3
5
x
2
2
Creating a Function from a
Table


Double check results
Substitute a different ordered pair into the
formula

Should give a true statement
3
5
solution : y  x 
2
2
x
y
3
7
4
8.5
5
10
6
11.5
Piecewise Function

Function has different behavior for different
portions of the domain
Greatest Integer Function

f ( x)  x = the greatest integer less than or
equal to x
6.7  6
3 3
2.5  3

Examples

Calculator – use the floor( ) function
Assignment



Lesson 2.1A
Page 88
Exercises 1 – 65 EOO
Finding a Line to Approximate
the Data

Draw a line “by eye”



Note slope, y-intercept
Statistical process (least squares method)
Use a computer program
such as Excel
Use your TI calculator
Chart Title

35
30
25
20
15
10
5
0
160
170
180
190
200
210
220
230
15240
You Try It


Consider table of ordered pairs
showing calories per minute
as a function of body weight
Enter data into data matrix of
calculator

Weight
Calories
100
2.7
120
3.2
150
4.0
170
4.6
200
5.4
220
5.9
APPS, Date/Matrix Editor, New,
16
Using Regression On
Calculator



Choose F5 for
Calculations
Choose calculation
type (LinReg for this)
Specify columns where x and y values will come
from
17
Using Regression On
Calculator

It is possible to store the Regression EQuation
to one of the Y= functions
18
Using Regression On
Calculator

When all options are
set, press ENTER and
the calculator comes
up with an equation approximates your data
Note both the original x-y
values and the function which
approximates the data
19
Using the Function


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Resulting function: C ( x)  0.027 x  0.0169
Use function to find Calories Weight Calories
100
2.7
for 195 lbs.
120
3.2
150
4.0
C(195) = 5.24
This is called extrapolation

170
4.6
200
5.4
220
5.9
Note: It is dangerous to extrapolate beyond the
existing data


Consider C(1500) or C(-100) in the context of the
problem
The function gives a value but it is not valid
20
Interpolation


Use given data
Determine
proportional
“distances”
30
25
25
x

30 0.8
x  0.4167
C  4.6  0.4167  5.167
Weight
Calories
100
2.7
120
3.2
150
4.0
170
4.6
195
??
200
5.4
220
5.9
x
Note : This answer is
different from the
extrapolation results
0.8
21
Interpolation vs. Extrapolation


Which is right?
Interpolation


Between values with ratios
Extrapolation


Uses modeling functions C ( x)  0.027 x  0.0169
Remember do NOT go beyond limits of known data
22
Correlation Coefficient

A statistical measure of how well a modeling
function fits the data
-1 ≤ corr ≤ +1

|corr| close to 1  high correlation

|corr| close to 0  low correlation


Note: high correlation does NOT imply cause
and effect relationship
23
Assignment



Lesson 2.1B
Page 94
Exercises 85 – 93 odd