linear functions and modeling
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Week 1LSP 120
Joanna Deszcz
Linear Functions and Modeling
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What is a function?Relationship between 2 variables or
quantitiesHas 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.
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What is a linear function?Straight line represented by y=mx + bConstant rate of change (or slope)
For a fixed change in one variable, there is a fixed change in the other variable
FormulasSlope = Rise
Run
Rate of Change = Change in y Change in x
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Linear FunctionQR Definition:
relationship that has a fixed or constant rate of change
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Datax y3 115 167 219 26
11 31
Does this data represent a linear function?We’ll use Excel to figure this out
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Rate of Change Formula(y2- y1)(x2-x1)
Example:(16-11) = 5 ( 5-3) 2
x y3 115 167 219 2611 31
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In ExcelInput (or copy) the
dataIn 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 =(B3-B2)/(A3-A2)
4 7 21
5 9 26
6 11 31
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Is the function Linear?If the rate of change is constant (the same) between data points
The function is linear
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Derive the Linear EquationGeneral Equation for a linear functiony = mx + bx and y are variables represented by data point values
m is slope or rate of changeb is y-intercept (or initial value)
Initial value is the value of y when x = 0May need to calculate initial value if x =
0 is not a data point
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Calculating Initial Value (b variable)
A B C
1 x yRate of Change
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 valuesWe’ll use 3 and 11
Rate of change = mm=2.5
Plug values into y=mx+b and solve for b11=2.5(3) + b11=7.5 + b3.5=b
So the linear equation for this data is:
y= 2.5x + 3.5
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Practice – Which functions are linear
x y5 -410 -115 220 5
x y1 12 35 97 13
x y2 17 59 1112 17
x y2 204 136 68 -1
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Graph the LineSelect all the data
pointsInsert an xy
scatter plot Data points
should line up if the equation is linear
y = 2.5x + 3.5R² = 1
0
5
10
15
20
25
30
35
0 5 10 15Y-
value
sX -values
Linear graph
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Be Careful!!!t P
1980 67.38
1981 69.13
1982 70.93
1983 72.77
1984 74.67
1985 76.61
1986 78.6066
68
70
72
74
76
78
80
1975 1980 1985 1990P
Value
t value
Not all graphs that look like lines represent linear functions! Calculate the rate of change to be sure it’s constant. t=year; P=population of
Mexico
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Try this datat 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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Exponential ModelsPrevious examples show exponential dataIt can appear to be linear depending on
how many data points are graphedOnly way to determine if a data set is linear
is to calculate rate of changeWill be discussed in more detail later
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Linear Modeling and Trendlines
Mathematical Modeling
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Uses of Mathematical ModelingNeed to plan, predict, explore
relationshipsExamples
Plan for next classBusinesses, schools, organizations plan for
futureScience – predict quantities based on
known valuesDiscover relationships between variables
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What is a mathematical model?EquationGraph or Algorithm
that fits some real data reasonably well
that can be used to make predictions
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Predictions2 types of predictions
Extrapolationspredictions outside the range of existing data
Interpolationspredictions made in between existing data
points
Usually can predict x given y and vise versa
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ExtrapolationsBe 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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Let’s Try SomeCell phones.xlsMileRecords.xls
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Is the trendline a good fit? 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
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Guideline 2Does 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
valuesBetween .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
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Guideline 3Verify 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
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Guideline 4Look for Outliers
Often bad data pointsEntered incorrectly
Should be correctedSometimes data is correct
Anomaly occurredCan be removed from
data if justified
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Guideline 5Practical 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