Transcript Linear Functions and Modeling

Week 1
LSP 120
Joanna Deszcz


Relationship between 2 variables or quantities
Has a domain and a range
 Domain – all logical input values
 Range – output values that correspond to domain


Can be represented by table, graph or
equation
Satisfies the vertical line test:
 If any vertical line intersects a graph in more than
one point, then the graph does not represent a
function.


Straight line represented by y=mx + b
Constant rate of change (or slope)
 For a fixed change in one variable, there is a fixed
change in the other variable
 Formulas
▪ Slope = Rise
Run
▪ Rate of Change = Change in y
Change in x

QR Definition:
 relationship that has a fixed or constant rate of
change


x
y
3
11
5
16
7
21
9
26
11
31
Does this data represent a linear function?
We’ll use Excel to figure this out
y2- y1
x2-x1
Example:
16-11 = 5
5-3
2
x
y
3
11
5
16
7
21
9
26
11
31
Input (or copy) the
data
 In adjacent cell begin
calculation by typing =
 Use cell references in
the formula
 Cell reference =
column letter, row
number (A1, B3, C5,
etc.)
A
B
C
1
x
y
Rate of Change
2
3
11
3
5
16
4
7
21
5
9
26
6
11
31

=(B3-B2)/(A3-A2)

If the rate of change is constant (the same)
between data points
 The function is linear

General Equation for a linear function
 y = mx + b
 x and y are variables represented by data point
values
 m is slope or rate of change
 b is y-intercept (or initial value)
▪ Initial value is the value of y when x = 0
▪ May need to calculate initial value if x = 0 is not a data
point
A
B
C
Rate of
Change
1
x
y
2
3
11
3
5
16
2.5
4
7
21
2.5
5
9
26
2.5
6
11
31
2.5

Choose one set of x
and y values
 We’ll use 3 and 11
So the linear equation for this
data is:
y= 2.5x + 3.5

Rate of change = m
 m=2.5

Plug values into
y=mx+b and solve for b
 11=2.5(3) + b
 11=7.5 + b
 3.5=b
x
5
10
15
20
x
2
7
9
12
y
-4
-1
2
5
x
1
2
5
7
y
1
3
9
13
y
1
5
11
17
x
2
4
6
8
y
20
13
6
-1
Select all the data
points
 Insert an xy scatter
plot
 Data points should line
up if the equation is
linear
Linear graph

35
30
Y- values
25
20
15
y = 2.5x + 3.5
R² = 1
10
5
0
0
5
10
X - values
15
t
P
80
1980
67.38
78
1981
69.13
1982
70.93
1983
72.77
1984
74.67
1985
76.61
1986
78.60
Not all graphs that look like
lines represent linear
functions! Calculate the rate of
change to be sure it’s constant.
P Value
76
74
72
70
68
66
1975
1980
1985
t value
1990
t=year; P=population of Mexico
t
P
1980
67.38
1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32

Does the line still
appear straight?
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Previous examples show exponential data
It can appear to be linear depending on how
many data points are graphed
Only way to determine if a data set is linear is
to calculate rate of change
Will be discussed in more detail later
Linear Modeling and Trendlines

Need to plan, predict, explore relationships
 Examples
▪ Plan for next class
▪ Businesses, schools, organizations plan for future
▪ Science – predict quantities based on known values
▪ Discover relationships between variables



Equation
Graph or
Algorithm
 that fits some real data reasonably well
 that can be used to make predictions

2 types of predictions

Extrapolations
 predictions outside the range of existing data

Interpolations
 predictions made in between existing data points

Usually can predict x given y and vise versa

Be Careful  The further you go from the actual data, the less
confident you become about your predictions.
 A prediction very far out from the data may end
up being correct, but even so we have to hold
back our confidence because we don't know if the
model will apply at points far into the future.
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Cell phones.xls
MileRecords.xls
5 Prediction Guidelines
 Guideline 1


Do you have at least 7 data points?
▪
▪
▪
Should use at least 7 for all class examples
more is okay unless point(s) fails another guideline
5 or 6 is a judgment call
▪
▪
▪
How reliable is the source?
How old is the data?
Practical knowledge on the topic
Does the R-squared value indicate a relationship?
▪
Standard measure of how well a line fits
R2
Relationship
=1
perfect match between line and data
points
=0
no relationship between x and y values
Between .7 and 1.0 strong relationship; data can be used to
make prediction
Between .4 and .7
moderate relationship; most likely okay
to make prediction
< .4
weak relationship; cannot use data to
make prediction

Verify that your trendline fits the shape of
your graph.
 Example: trendline continues upward, but the
data makes a downward turn during the last few
years
 verify that the “higher” prediction makes sense
 See Practical Knowledge

Average July Tempurature
Chicago 2000-2010
Look for Outliers
 Often bad data points
 Entered incorrectly
▪ Should be corrected
 Sometimes data is correct
▪ Anomaly occurred
100
90
80
70
60
50
40
30
20
10
0
1998
2000
2002
 Can be removed from data if justified
2004
2006
2008
2010
2012

Practical Knowledge
 How many years out can we predict?
 Based on what you know about the topic, does it
make sense to go ahead with the prediction?
 Use your subject knowledge, not your
mathematical knowledge to address this
guideline