In Example 5(a),the slope of the line is the positivenumber The graph of the
line in FIGURE 18slants up (rises) from left to right. The line in Example 5(b)has
negativeslope As FIGURE 19shows, its graph slants down (falls) from left to
right. These facts illustrate the following generalization.
- 4.
23.
152 CHAPTER 3 Graphs, Linear Equations, and Functions
Orientation of a Line in the PlaneA positive slope indicates that the line slants up(rises) from left to right.
A negative slope indicates that the line slants down(falls) from left to right.
FIGURE 20shows lines of positive, 0, negative, and undefined slopes.
yxUndefined
slopePositive
slope0
slopeNegative
slope0FIGURE 20OBJECTIVE 4 Use slopes to determine whether two lines are parallel,
perpendicular, or neither.Recall that the slope of a line measures the steepness of
the line and that parallel lines have equal steepness.
Slopes of Parallel LinesTwo nonvertical lines with the same slope are parallel.
Two nonvertical parallel lines have the same slope.
Determining Whether Two Lines Are ParallelDetermine whether the lines through and and through
and are parallel.
Find the slope of. Find the slope of.
Because the slopes are equal, the two lines are parallel. NOW TRY
To see how the slopes of perpendicular lines are
related, consider a nonvertical line with slope If
this line is rotated 90°, the vertical change and the
horizontal change are interchanged and the slope is
since the horizontal change is now negative.
See FIGURE 21. Thus, the slopes of perpendicular
lines have product and are negative reciprocals
of each other.
- 1
- ba,
ab.
m 2 =
- 2 - 0
0 - 3
=
- 2
- 3
=
2
3
m 1 =
5 - 1
4 - 1 - 22
=
4
6
=
2
3
L 1 L 2
1 0, - 22 ,
L 1 , 1 - 2, 1 2 1 4, 5 2 , L 2 , 1 3, 0 2
EXAMPLE 6
xybb
–a^90 ° aSlope is –.Slope is.0a
bbaFIGURE 21NOW TRY
EXERCISE 6
Determine whether the line
through and is
parallel to the line through
1 2, 0 2 and. 1 - 1, - 22
1 2, 5 2 1 4, 8 2
NOW TRY ANSWER
- no