ECO 108
Math and Graph Appendix
Spring 2011
A.
Linear Functions
A linear function is a function of the form
f(x) = y = a + b*x
(#)
(# is simply a reference for this functional form.)
where a and b are constants.
The graph of a linear function is a straight line.
The slope of a linear function
is found by picking any two points on the line, say
1
1
2
2
( x
, y
) and ( x
, y
),
and determining the ratio of
the vertical distance and horizontal distance between the points.
slope =
2
1
2
1
y
y
vertical distance
.
horizontal distance
x
x

=

From the function, we can derive information about the line it generates:
the slope, the vertical intercept
and the horizontal intercept of the line.
For the function (#),
b = the slope
a = the vertical intercept
a
b

= the horizontal intercept
The vertical intercept is the value of y at which the line intersects the vertical axis.
The horizontal intercept
is the value of x at which the line intersects the horizontal axis.
When the line intersects the horizontal
axis, the value of the function is 0, i.e., the ycoordinate is zero.
Thus, to solve for the horizontal intercept:
y = a + b*x and y = 0
⇒
a + b*x
= 0.
So all have to do is solve a + b*x = 0 for x as a function of and b.
b*x = – a
x =
a
b

and as stated above,
a
b

is the horizontal intercept of the line.
For example, consider y = 10 – (1/2)*x.
If
1
1
2
2
( x
, y
)
( 8, 6 ) and ( x
, y
)
( 12, 4),
=
=
the
slope of the line can be calculated as
2
1
2
1
y
y
4
6
2
x
x
12
8
4



=
=
=


– 1/2.
Moreover, the vertical intercept is
10, and the horizontal intercept is 20.
The following diagram shows a graph of this function as well as the
points
( 8, 6 ) and ( 12, 4)
and a rise – run illustration.
Diagram A
0
2
4
6
8
10
12
14
16
18
20
22
0
2
4
6
8
10
12
y
x
( 8, 6 )
( 12, 4 )
+4

2
1