Transcript Linear Functions

Linear Functions and
Modeling
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.
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?

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:
Rise
Slope 
Run
Change in y
Rate of Change 
Change in x
Our definition, then, of a linear function is a
relationship that has a fixed or constant rate of
change.
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
11
31
Example: Value of Computer
Depreciated over 5 years
Number of Years
t
Value of Computer ($)
V
0
1200
1
960
2
720
3
480
4
240
5
0
Is it linear? If so find the equation for V
(Value) as a function of t (Time).
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
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
Practice: for the following, determine whether the
function is linear and if so, write the equation for the
function.
x
y
x
y
x
y
x
y
5
-4
1
1
2
1
2
20
10
-1
2
3
7
5
4
13
15
2
5
9
9
11
6
6
20
5
7
13
12
17
8
-1