Chapter 2.3
Linear Functions
Graphing Linear Functions
We begin our study of specific functions by
looking at linear functions. The name “linear”
comes from the fact that the graph of every linear
function is a straight line.
If a  0, thedomain and range of a linear
functionare both - ,  .
If a  0, then theequation becomes
f(x) b.
In thiscase, thedomain is (-, )
and therange is [b].
In Section 2.1, we graphed lines by finding
ordered pairs and plotting them. Although only
two points are necessary to graph a linear
function, we usually plot a third point as a check.
The intercepts are often good points to choose for
graphing lines
y
Example 1 Graphing a Linear Function Using Intercepts
Graph
f(x) = -2x + 6
Give the
domain and
range.
x
y
Example 2 Graphing a Horizontal Line
Graph
f(x) = -3
Give the
domain and
range.
x
y
Example 3 Graphing a Vertical Line
Graph
x = -3
Give the
domain and
range.
x
Standard Form
Ax + By = C
Equations of lines are often written in the form
Ax + By = C, called standard form.
y
Example 4 Graphing Ax + By = C with C = 0
Graph
4x + 5y = 0
Give the
domain and
range.
x
Slope
An important characteristic of a straight line is its
slope, a numerical measure of the steepness of a
line.
Geometrically,
this may be
interpreted as
the ratio of rise
to run.
To find the slope of a line, start with two distinct
points (x1, y1) and (x2, y2) on the line, as shown in
Figure 25, where x1 ≠ x2.
As we movealong theline from(x1 , y1 ) to (x2 , y 2 )
thehorizontaldifferencex 2 - x1 is called the
changein x,denotedby x (read" delta x"), where
 is theGreek letterdelta.
In thesame way, the verticaldifference
y 2 - y1 called thechangein y, can be written
y  y 2  y1.
T heslope of a nonvertical line is defined
as thequotient(ratio)of
thechangein y and thechangein x
as follows.
y
Graph
x=3
x
y
Example 5 Finding Slopes with the Slope Formula
Find the slope
of the line
through the
given points
(-4,8) & (2,-3)
x
y
Example 5 Finding Slopes with the Slope Formula
Find the slope
of the line
through the
given points
(2,7) & (2,-4)
x
y
Example 5 Finding Slopes with the Slope Formula
Find the slope
of the line
through the
given points
(5,-3) & (-2,-3)
x
y
Example 6 Finding the Slope from an Equation
Find the slope
of the line
y = -4x - 3
x
y
Example 7 Finding a Line Using a Point and the Slope
Graph the line
passing
through (-1, 5)
and having a
slope
x
5

3
Example 8 Interpreting Slope as Average Rate of Change
In 1997, sales of VCR’s numbered 16.7 million.
In 2002, estimated sales of VCR’s were 13.3
million. Find the average rate of change in VCR
sales, in millions per year.
Linear Models
In Example 8, we used the graph of a line to
approximate real data. Recall from Section 1.2
this this is called mathematical modeling. Points
on the straight line graph model (approximate) the
actual points that correspond to the data.
A linear cost function has the form
C(x) = mx + b
where x represents the number of items produced,
m represents the variable cost per item, and b
represents the fixed cost.
The fixed cost is constant for a particular product
and does not change as more items are made.
The variable cost per item, which increases as
more items are produced, covers labor, packaging,
shipping and so on.
The revenue function for selling a product
depends on the price per item p and the number of
items sold x, and is given by
R(x) = px
Profit is described by the profit function defined
as
P(x) = R(x) – C(x).
Example 9 Writing Linear Cost, Revenue, and Profit Functions
Assume that the cost to produce an item is a linear
function and all items produced are sold. The
fixed cost is $1500, the variable cost per item is
$100, and item sells for $125.
Write linear functions to model
(a)
(b)
(c)
(d)
cost
revenue
profit
How many items must be sold for the company
to make a profit?
Example 9 Writing Linear Cost, Revenue, and Profit Functions
Revenue = R(x) = px = 125x
Example 9 Writing Linear Cost, Revenue, and Profit Functions
Cost = C(x) = mx + b
100x + 1500
Example 9 Writing Linear Cost, Revenue, and Profit Functions
Profit = P(x) = R(x) – C(x)
125x –(100x – 1500)
Example 9 Writing Linear Cost, Revenue, and Profit Functions
To make a profit, P(x) must be positive.
From part (c) P(x) = 25x - 1500
P(x) > 0
25x - 1500 > 0