1.3
LINEAR FUNCTIONS
1
Constant Rate of Change
• A linear function has a constant rate of change.
• The graph of any linear function is a straight line.
2
• Population Growth
Mathematical models of population growth are used by city planners to
project the growth of towns and states. Biologists model the growth of
animal populations and physicians model the spread of an infection in the
bloodstream. One possible model, a linear model, assumes that the
population changes at the same average rate on every time interval.
• Financial Models
Economists and accountants use linear functions for straight-line
depreciation. For tax purposes, the value of certain equipment is considered
to decrease, or depreciate, over time. For example, computer equipment
may be state-of-the-art today, but after several years it is outdated. Straightline depreciation assumes that the rate of change of value with respect to
time is constant.
3
Constant Rate of Change
Population Growth
Example 1
A town of 30,000 people grows by 2000 people every year. Since the
population, P, is growing at the constant rate of 2000 people per
year, P is a linear function of time, t, in years.
(a) What is the average rate of change of P over every time interval?
(c) Find a formula for P as a function of t.
Solution
(a) The average rate of change of population with respect to time is
2000 people per year.
(c) Population Size = P = Initial population+ Number of new people
= 30,000 + 2000 people/year ・ Number of years,
so a formula for P in terms of t is P = 30,000 + 2000t.
4
Solution:
5
A General Formula for the Family
of Linear Functions
If y = f(x) is a linear function, then for some constants b and m:
y = mx + b.
m is called the slope, and gives the rate of change of y with respect
to x. Thus,
y
m
x
If (x0, y0) and (x1, y1) are any two distinct points on the graph of f,
y y1  y0
then
m

x x1  x0
b is called the vertical intercept, or y-intercept, and gives the value
of y for x = 0. In mathematical models, b typically represents an
initial, or starting, value of the output.
6
Tables for Linear Functions
A table of values could represent a linear function if the
rate of change is constant, for all pairs of points in the
table; that is,
Change in output
Rate of change of linear function 
 Constant
Change in input
Thus, if the value of x changes by equal steps in a table
for a linear function, then the value of y goes up (or
down) by equal steps as well.
7
Tables for Linear Functions
Example
The table gives values of two functions, p and q. Could either of these be linear?
x
p(x)
q(x)
50
5
10
55
15
12
60
25
16
65
35
24
70
45
40
Solution
x
50
p(x)
5
Δp
10
55
2
35
10
70
2
25
10
65
2
15
10
60
Δp/Δx
45
2
The value of p(x) goes up
by equal steps of 10, so
Δp/Δx is a constant. Thus,
p could be a linear
function.
In contrast, the value of
q(x) does not go up by
equal steps. Thus, q could
not be a linear function.
x
q(x)
50
10
55
60
65
70
Δq
Δq/Δx
2
0.4
4
0.8
8
1.6
16
3.2
12
16
24
40
8
9
(a)
p
3990
4110
4200
4330
Q(p)
49000
ΔQ
ΔQp/Δp
-6000
-6000/120 = -50
-4500
-4500/90 = -50
-6500
-6500/130 = -50
43000
38500
32000
(b)
The number of Yugos sold decreased by 50 each time the price increased by $1
10
Warning: Not All Graphs That Look
Like Lines Represent Linear Functions
Graph of P = 100(1.02)t over 5
years: Looks linear but is not
Graph of P = 100(1.02)t
over 60 years: Not linear
Population (predicted) of
Mexico: 2000-2005
140
120
100
80
60
40
20
0
P (in millions)
P (in millions)
Population (predicted) of
Mexico: 2000-2005
0
1
2
3
4
5
350
300
250
200
150
100
50
0
0
10
t (years since 2000)
20
30
40
50
60
t (years since 2000)
Region of graph on left
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