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Linear Functions
Chapter1, Section3
Linear Functions
 A linear function is a function that can be written in the form
f (x) = ax + b
where a and b are constants. Notice that the exponent on the
variable is a 1, hence first degree (linear).
EX: Which of the following equations are linear functions?
a. 0 = 2t - s +1
b. y = 5
c. xy = 2
Intercepts
The points where the graph crosses or touches the x-axis and
the y-axis are called the x-intercept and the y-intercept,
respectively.
Graphically these points can be seen on the axes.
To find the y-intercept algebraically, set x = 0 and solve the
equation for y. If the solution is b, then the point on the graph is
(0, b).
To find the x-intercept algebraically, set y = 0 and solve for x. If a
is the solution, then the resulting point is (a, 0).
Example 1
Example 2
Find the x and y-intercepts for the
equation, 2x - 3y =12 .
A business property is purchased with
a promise to pay off a $60,000 loan
plus $16,500 interest on this loan by
making 60 monthly payments of
$1275. The amount of money, y
remaining to be paid on $76,500 is
reduced by $1275 each month. This
can be modeled the linear function:
What does the graph look like?
y = 76,500 – 1275x.
a.
Find the intercepts of this
equation.
b. Interpret the intercepts for this
model.
c. What limits should there be?
d. Use the intercept to sketch the
graph.
Slope of a Line
The slope of a line is defined as:
verticalchange
rise
slope =
=
horizontalchange run
To calculate the value of the slope, we use the formula as
follows. When two points are given, ( x1, y1 ) and ( x2, y2 ),
y2 - y1
m=
x2 - x1
Example 3
 Find the slope of the line through the points (-3,2) and (5,-
4). What does the slope mean?
 Find the slope of the line joining the x and y-intercept
points in the previous Example 2.
The Relationship Between Orientation of
a Line and its Slope
m>0
m=0
m<0
m is undefined
Slope and y-intercept
Example 4
The slope of the graph of the
equation y = mx + b is m and the yintercept is b, or the point on the
graph (0,b).
Using Example 2, y = 76,500 – 1275x.
From this form we get our linear
function: f(x) = mx + b.
a) What is the slope and y-intercept?
b) How does the amount owed on
the loan change as the number of
months increases?
Constant Rate of
Change
The rate of change of the linear
function y = mx + b is the constant
m, the slope of the graph of the
function.
Example 5
Use the graph of the function
y = 16.908x – 20.945 where x
is the number of years after
1990 and y is the sales in
billions of dollars.
a)
200
180
160
120
100
80
60
40
20
0
What is the slope of the
graph of the function?
b) What is the rate at which
the sales grew during this
period?
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Years after 1990
Revenue, Cost and Profit
 Revenue is the money from sales of goods or services.
 Cost is the expense generated in producing those good or
services.
 Profit is the difference between Revenue and Cost. Its what
is left over from the production and sale of goods and/or
services:
P(x) = R(x) – C(x)
When these functions are linear the rates of change are called
MARGINAL COST, MARGINAL REVENUE, AND MARGINAL
PROFIT. (It’s the slopes of the functions.)
Example:
Marginal Revenue and Marginal Profit
 A company produces and sells a product with revenue given
by R(x) = 89.50x dollars per unit x and cost given by
C(x) = 54.36x + 6790 dollars per unit x.
a)
What is the marginal revenue for this product and what does
it mean?
b) Find the profit function.
c)
What is the marginal profit for this product and what does it
mean?