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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
l   Use your TI calculator
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
      Answer:
                               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
l   Use your TI calculator



                                                 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


                             Note : This answer is
                              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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posted:6/9/2014
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