1. Linear Functions and Modeling
Week 1
LSP 120
Joanna Deszcz

2. 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.

3. What is a linear 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

4. Linear Function QR Definition:
relationship that has a fixed or constant rate of change

5. Data Does this data represent a linear function?x y 3 11 5 16 7 21 9 26 31
Does this data represent a linear function?
We’ll use Excel to figure this out

6. (y2- y1) (x2-x1) Rate of Change Formula Example: (16-11) = 5 ( 5-3) 2
(16-11) = ( 5-3) x y 3 11 5 16 7 21 9 26 31

7. In Excel 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 5 16
=(B3-B2)/(A3-A2) 4 7 21 9 26 6 31

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

9. Derive the Linear Equation
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

10. Calculating Initial Value (b variable)
Choose one set of x and y values
We’ll use 3 and 11
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 A B C 1 x y
Rate of Change 2 3 11 5 16
2.5 4 7 21 9 26 6 31
So the linear equation for this data is:
y= 2.5x + 3.5

11. Practice – Which functions are linearx y 5 -4 10 -1 15 2 20 x y 1 2 3 5 9 7 13 x y 2 1 7 5 9 11 12 17 x y 2 20 4 13 6 8 -1

12. Graph the Line Select all the data points Insert an xy scatter plot
Data points should line up if the equation is linear

13. Be Careful!!! t P
1980
67.38
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.
t=year; P=population of Mexico

14. Try this data Does the line still appear straight? t P 1980 67.38 1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32

15. Exponential Models 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

16. Mathematical Modeling
Linear Modeling and Trendlines

17. Uses of Mathematical Modeling
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

18. What is a mathematical model?
Equation
Graph or
Algorithm
that fits some real data reasonably well
that can be used to make predictions

19. Predictions 2 types of predictions Extrapolations Interpolations
predictions outside the range of existing data
Interpolations
predictions made in between existing data points
Usually can predict x given y and vise versa

20. Extrapolations 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.

21. Let’s Try Some
Cell phones.xls
MileRecords.xls

22. 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

23. Guideline 2 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

24. Guideline 3 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

25. Guideline 4 Look for Outliers Often bad data points
Entered incorrectly
Should be corrected
Sometimes data is correct
Anomaly occurred
Can be removed from data if justified

26. Guideline 5 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