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INCLINATION AND SLOPE The
angle of inclination, denoted by the Greek letter alpha (
), as
portrayed in figure 14, views A and B, is the angle formed by the line
crossing the X axis and the positively directed portion of the X axis, such
that 0 <
< 180'. The
slope of any line is equal to the tangent of its angle of inclination. Slope is
denoted by the letter m. Therefore,
If
the axes are in their conventional positions, a line, such as AB in figure 14, view A, that slopes
upward and to the right will have a positive slope. A line, such as CD in
figure 14, view B, that slopes downward and to the right will have a negative
slope. Since
the tangent of a is the ratio of PM to
P_{1}M, we can relate the slope of line AB to the points P_{1}
and P_{2} as follows:
Designating
the coordinates of P_{1} as (x_{1},y_{1}) and those of
P_{2} as (x2,y2), we recall
that
Figure 14.Angles of inclination. The
quantities (x_{2}  x_{1}) and (y_{2}  y_{1}) represent changes that
occur in the values of the x and y coordinates as a result of the change from P_{2}
to P_{1} on line AB. The
symbol used by mathematicians to represent an increment of change is the Greek
letter delta (
).
Therefore,
x means
"the change in x" and
y means "the
change in y." The amount of change in the x coordinate, as we change from
P_{2} to P_{1}, is x_{2} x_{1}. Therefore,
and likewise,
We use this notation to express the slope of line AB as
follows:
EXAMPLE: Find the slope of the line connecting P_{2}(7,6)
and P_{1}(1,4). SOLUTION:
Note that the choice of labels for P_{1} and P_{2}
is strictly arbitrary. In the previous example, if we had chosen the point
(7,6) to be P_{1} and the point (1,  4) to be P_{2}, the
following results would have occurred:
Notice that this solution yields the same result as the
last example. A slope of 5/4 means that a point moving along this line would
move vertically + 5 units for every horizontal movement of + 4 units. This
result is consistent with our definition of positive slope; that is, sloping
upward and to the right. If line AB in figure 14, view A, was parallel to the X
axis, y_{1} and y_{2} would be equal and the difference (y_{2}
 y_{1}) would be 0. Therefore,
We conclude that the slope of a horizontal line is 0. We
can also reach this conclusion by noting that angle a (fig. 14, view A) is 0 when the line is parallel to the X axis. Since
the tangent of 0 is 0, then m=tan0=0 The slope of a line that is parallel to the Y axis becomes
meaningless. The tangent of the angle a increases indefinitely as a approaches
90. We sometimes say that
(m approaches infinity) when a approaches
90. 