Transcript Linear Functions and Models

Linear Functions and
Models
Lesson 2.1
Problems with Data
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Real data recorded
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Problems
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Experiment results
Periodic transactions
Data not always recorded accurately
Actual data may not exactly fit theoretical
relationships
In any case …
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Possible to use linear (and other) functions to
analyze and model the data
Fitting Functions
to Data
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Temperature
Viscosity
(lbs*sec/in2)
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28
170
26
180
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190
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200
16
210
13
220
11
230
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Consider the data
given by this example
Viscosity (lbs*sec/in2)
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Note the plot of
the data points
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Close to being
in a straight line
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25
20
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10
5
0
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200
210
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230
240
Finding a Line to Approximate the Data
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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
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5
0
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Graphs of Linear Functions
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For the moment, consider the first option
Given the graph with tic marks = 1
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Determine
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Slope
Y-intercept
A formula for the function
X-intercept (zero of the function)
Graphs of Linear Functions
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Slope – use difference quotient
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Y-intercept – observe
Write in form
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change in y y
m

change in x x
y  m x b
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Zero (x-intercept) – what value of x gives a
value of 0 for y?
Modeling with Linear
Functions
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Linear functions will model data when
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The constant rate is the slope of the function
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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
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Or the b in y = mx + b
Modeling with Linear
Functions
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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?
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Determine the slope of the function
Determine the y-intercept
Write in the form of y = mx + b
Creating a Function from a
Table
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Determine slope by using
change in y y
m

change in x x
x
y
3
7
4
8.5
x  5  3  2
5
10
y 3
  1.5  m
y 2
6
11.5
Answer:
y  10  7  3
x
y
Creating a Function from a
Table
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Now we know slope m = 3/2
Use this and one of
x
the points to determine
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y-intercept, b
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Substitute an
3
10


5

b
5
ordered
2
pair into
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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
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Double check results
Substitute a different ordered pair into the
formula
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Should give a true statement
3
5
solution : y  x 
2
2
x
y
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4
8.5
5
10
6
11.5
Piecewise Function
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Function has different behavior for different
portions of the domain
Greatest Integer Function
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f ( x)  x = the greatest integer less than or
equal to x
6.7  6
3 3
2.5  3
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Examples
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Calculator – use the floor( ) function
Assignment
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Lesson 2.1A
Page 88
Exercises 1 – 65 EOO
Finding a Line to Approximate
the Data
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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
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35
30
25
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10
5
0
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15240
You Try It
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Consider table of ordered pairs
showing calories per minute
as a function of body weight
Enter data into data matrix of
calculator
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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,
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Using Regression On
Calculator
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Choose F5 for
Calculations
Choose calculation
type (LinReg for this)
Specify columns where x and y values will come
from
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Using Regression On
Calculator
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It is possible to store the Regression EQuation
to one of the Y= functions
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Using Regression On
Calculator
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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
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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
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170
4.6
200
5.4
220
5.9
Note: It is dangerous to extrapolate beyond the
existing data
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Consider C(1500) or C(-100) in the context of the
problem
The function gives a value but it is not valid
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Interpolation
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Use given data
Determine
proportional
“distances”
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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
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Interpolation vs. Extrapolation
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Which is right?
Interpolation
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Between values with ratios
Extrapolation
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Uses modeling functions C ( x)  0.027 x  0.0169
Remember do NOT go beyond limits of known data
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Correlation Coefficient
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A statistical measure of how well a modeling
function fits the data
-1 ≤ corr ≤ +1
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|corr| close to 1  high correlation
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|corr| close to 0  low correlation
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Note: high correlation does NOT imply cause
and effect relationship
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Assignment
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Lesson 2.1B
Page 94
Exercises 85 – 93 odd