College Algebra 9th txtbk

SECTION 2–3 Equations of a Line 135
In general, the slope of a line may be positive, negative, 0, or not defined. Each of these
cases is interpreted geometrically as shown in Table 1.
Table 1 Geometric Interpretation of Slope
Line Slope Example
y
Rising as x moves from left to right
y values are increasing
Positive
x
y
Falling as x moves from left to right
y values are decreasing
Negative
x
y
Horizontal 0
y values are constant
x
y
Vertical
x values are constant
Not defined
x
In using the formula to find the slope of the line through two points, it doesn’t matter
which point is labeled P 1 or P 2 , because changing the labeling will change the sign in both
the numerator and denominator of the slope formula:
Z Figure 3
a
b
b b
m
a a
a
b
y 2 y 1
x 2 x 1
y 1 y 2
x 1 x 2
For example, the slope of the line through the points (3, 2) and (7, 5) is
5 2
7 3 3 4 3
4 2 5
3 7
In addition, it is important to note that the definition of slope doesn’t depend on the
two points chosen on the line as long as they are distinct. This follows from the fact that
the ratios of corresponding sides of similar triangles are equal (Fig. 3).
EXAMPLE 2 Finding Slopes
For each line in Figure 4, find the run, the rise, and the slope. (All the horizontal and vertical
line segments have integer lengths.)
y
y
5
5
5
5
x
5
5
x
5
5
Z Figure 4
(a)
(b)