# Linear Functions and Models

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```					Linear Functions and
Models
Lesson 2.1
Problems with Data
l   Real data recorded
l   Experiment results
l   Periodic transactions
l   Problems
l   Data not always recorded accurately
l   Actual data may not exactly fit theoretical
relationships
l   In any case …
l   Possible to use linear (and other) functions to
analyze and model the data
Fitting Functions                            Viscosity

to Data                      Temperature
160
(lbs*sec/in2)
28
170             26
180             24

l   Consider the data           190             21
200             16
given by this example       210             13
220             11
230             9

l   Note the plot of
the data points
l   Close to being
in a straight line
Finding a Line to Approximate the Data

l   Draw a line “by eye”
l   Note slope, y-intercept
l   Statistical process (least squares method)
l   Use a computer program
such as Excel
Graphs of Linear Functions
l   For the moment, consider the first option
Given the graph with tic marks = 1

l   Determine
l   Slope
l   Y-intercept
l   A formula for the function
l   X-intercept (zero of the function)
Graphs of Linear Functions
l   Slope – use difference quotient

l   Y-intercept – observe
l   Write in form

l   Zero (x-intercept) – what value of x gives a
value of 0 for y?
Modeling with Linear
Functions
l   Linear functions will model data when
l   Physical phenomena and data changes at a constant
rate
l   The constant rate is the slope of the function
l   Or the m in y = mx + b
l   The initial value for the data/phenomena is the
y-intercept
l   Or the b in y = mx + b
Modeling with Linear
Functions
l   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?
l   Determine the slope of the function
l   Determine the y-intercept
l   Write in the form of y = mx + b
Creating a Function from a
Table
l   Determine slope by using

x    y
3    7
4   8.5
5   10
6   11.5
Creating a Function from a
Table
l   Now we know slope m = 3/2
l   Use this and one of
x    y
the points to determine
3    7
y-intercept, b
4   8.5
l   Substitute an
ordered                    5   10
pair into                   6   11.5
y = (3/2)x + b
Creating a Function from a
Table
l   Double check results
l   Substitute a different ordered pair into the
formula
l   Should give a true statement       x       y
3       7
4       8.5
5       10
6   11.5
Piecewise Function
l   Function has different behavior for different
portions of the domain
Greatest Integer Function
l             = the greatest integer less than or
equal to x

l   Examples

l   Calculator – use the floor( ) function
Assignment

l   Lesson 2.1A
l   Page 88
l   Exercises 1 – 65 EOO
Finding a Line to Approximate
the Data
l   Draw a line “by eye”
l   Note slope, y-intercept
l   Statistical process (least squares method)
l   Use a computer program
such as Excel

15
You Try It                               Weight   Calories
100       2.7
l   Consider table of ordered pairs       120       3.2

showing calories per minute           150       4.0
170       4.6
as a function of body weight          200       5.4
l   Enter data into data matrix of        220       5.9

calculator
l   APPS, Date/Matrix Editor, New,

16
Using Regression On
Calculator
l   Choose F5 for
Calculations
l   Choose calculation
type (LinReg for this)
l   Specify columns where x and y values will come
from

17
Using Regression On
Calculator
l   It is possible to store the Regression EQuation
to one of the Y= functions

18
Using Regression On
Calculator

l   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
l   Resulting function:
l   Use function to find Calories           Weight   Calories
100       2.7
for 195 lbs.                             120       3.2

l   C(195) = 5.24                            150       4.0
170       4.6
This is called extrapolation             200       5.4
220       5.9

l   Note: It is dangerous to extrapolate beyond the
existing data
l   Consider C(1500) or C(-100) in the context of the problem
l   The function gives a value but it is not valid       20
Interpolation
l   Use given data
Weight   Calories
l   Determine              100       2.7
proportional           120       3.2
“distances”            150       4.0
170       4.6
25                       x
30          195       ??               0.8
200       5.4
220       5.9

different from the
extrapolation results   21
Interpolation vs. Extrapolation
l   Which is right?
l   Interpolation
l   Between values with ratios
l   Extrapolation
l   Uses modeling functions
l   Remember do NOT go beyond limits of known data

22
Correlation Coefficient
l   A statistical measure of how well a modeling
function fits the data
l   -1 ≤ corr ≤ +1
l   |corr| close to 1 ó high correlation
l   |corr| close to 0 ó low correlation

l   Note: high correlation does NOT imply cause
and effect relationship
23
Assignment

l   Lesson 2.1B
l   Page 94
l   Exercises 85 – 93 odd

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