1. Linear Functions and Modeling

2. What is a function?
1.  It is a relationship between two variables or two quantities.
2.  It has a domain and a range.  The practical domain consists of all input values that make sense.  The practical range consists of all output values that correspond to the values in the practical domain.
3.  Functions can be represented by a data table, a graph or an equation.
4.  It 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. So what then is a linear function?
Most people would say it is a straight line or that it fits the equation y = mx + b. 
They are correct, but what is true about a function that when graphed yields a straight line? 
What is the relationship between the variables in a linear function? 

4. In a linear function, for a fixed change in one variable, there is fixed change in the other variable.  That change is called rate of change or slope.  (Slope is a graphical term while rate of change is a more mathematical or algebraic term.)
The formulas are as follows:
Our definition, then, of a linear function is a relationship that has a fixed or constant rate of change.

5. Does the data represent a linear function?
The first thing we want to do is be able to determine whether a table of values for 2 variables represents a linear function: x y 3 11 5 16 7 21 9 26 31

6. Example: Value of Computer Depreciated over 5 years
Number of Years t
Value of Computer ($) V
1200 1
960 2
720 3
480 4
240 5
Is it linear? If so find the equation for V (Value) as a function of t (Time).

7. Warning: Not all graphs that look like lines represent linear functions:
Graph the following data where t is the years and P is the population of Mexico (in millions) t P
1980
67.38
1981
69.13
1982
70.93
1983
72.77
1984
74.67
1985
76.61
1986
78.60

8. What if you were given the population for every ten years
What if you were given the population for every ten years? Would the graph no longer appear to be linear?
Graph the following data t P
1980
67.38
1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32

9. Practice: for the following, determine whether the function is linear and if so, write the equation for the function. x y 5 -4 1 2 20 10 -1 3 7 4 13 15 9 11 6 12 17 8