1.2 Slopes and equations of lines 11and the absolute values of the coordinates tell us how far we go. The
quadrant (or subset) of the plane consisting of points having nonnegative
coordinates is called the closed first quadrant. The quadrant (or subset)
of the plane consisting of points having positive coordinates is called the
open first quadrant. The Roman numerals of Figure 1.24 show us how the
quadrants are numbered.
Figure 1.25 shows a line L which slopes upward to the right. The line
L does not necessarily pass through the origin, but we suppose that it0 x
Figure 1.251passes through a given point (xl,yl). The angle 0 (theta) lies between 0
and a/2 (that is, between 0° and 90°), and tan 0 is called the slope of the
line and is denoted by the letter m, so that(1.252) m = slope = tan 0 = y - yl
x - xl
when (xl,yi) and (x,y) are two different points on L.
Figure 1.251 shows a line L which slopes downward
to the right. This time tan 0 is negative, but it is
still called the slope of the line. We must always
remember that lines which slope upward to the right
have positive slopes and lines which slope downward
to the right have negative slopes. For horizontalJ
lines, we have 0 = 0, so tan 8 = 0 and in = 0; thus, m-
horizontal lines have slope zero. For vertical lines,
we have 0 = it/2 (or 0 = 90°), so tan 0 does not
exist; thus, vertical lines do not have slopes. To
locate a second point on a line which passes through
a given point and has slope m, we start at the given
point, go 1 unit to the right and then goin units in
the direction of the positive y axis. When m < 0, a
journey of m units in the direction of the positivey
axis is always interpreted to be a journey of Iml
m=-Figure 1.253units in the direction of the negativey axis. Everyone should look at
Figure 1.253 and think about this.
Theorem 1.26 -4 point P(x,y) lies on the line which contains the
Point Pi(xi,yi) and has slope m if and only if its coordinates satisfy the point-