Week 1LSP 120

Joanna Deszcz

Linear Functions and Modeling

What is a function?

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.

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

Linear FunctionQR Definition:

relationship that has a fixed or constant rate of change

Data

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

Rate of Change Formula

(y2- y1)(x2-x1)

Example:(16-11) = 5 ( 5-3) 2

x y

3 11

5 16

7 21

9 26

11 31

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  

Is the function Linear?If the rate of change is constant (the same) between data points

The function is linear

Derive the Linear EquationGeneral Equation for a linear function

y = 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

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

Practice – Which functions are linear

x y

5 -4

10 -1

15 2

20 5

x y

1 1

2 3

5 9

7 13

x y

2 1

7 5

9 11

12 17

x y

2 20

4 13

6 6

8 -1

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-v

alue

sX -values

Linear graph

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

Val

ue

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

Try this data

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?

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

Linear Modeling and Trendlines

Mathematical Modeling

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

What is a mathematical model?EquationGraph or Algorithm

that fits some real data reasonably well

that can be used to make predictions

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

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.

Let’s Try SomeCell phones.xlsMileRecords.xls

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

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 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

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

Guideline 4Look for Outliers

Often bad data pointsEntered incorrectly

Should be correctedSometimes data is correct

Anomaly occurredCan be removed from

data if justified

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