Linear functions
The simplest function, which at the same time has extremely important applications, is the polynomial of the first degree.
where
a and b
are given constant numbers. We shall
see that the graph of
this function is a straight line. It is called a linear function. We
shall first consider the case of b
equal to zero. The function then has the form:
This
expresses the fact that the variable
y is directly proportional to the
variable x, the constant coefficient a being
called the coefficient of proportionality.
Turning
to the figure (Fig. 5), we see that
equation (3) expresses the following geometric property of the graph
in question: whatever point M
we take on it, the ratio of the ordinate
y = NM of this point to its abscissa x = ON is equal to the constant a. Since, on the other hand, this
ratio is equal to the tangent of
the angle a between the segment OM
and the axis OX, it is evident that
the geometrical locus of M
is a
straight line, passing through the origin
of coordinates O at
an angle a( or
)
to the axis OX. Angle
a is reckoned counter-clockwise from the direction OX. The
geometrical importance of the coefficient a
in equation (3) is simultaneously
revealed: a is the tangent of the angle a between the axis
OX and the straight line corresponding to this equation, a being therefore
called the slope of the straight line. I t may be noted that if
a is a negative number, the angle a is obtuse, and the corresponding line is as
shown in Fig.6. Let
us now return to the general case of a linear function, viz, to equation
(2). The ordinate y of the graph of this equation differs from the
corresponding ordinate of the graph of equation (3) by the constant amount
a. Thus, we immediately obtain the graph of equation (2), if the graph
of equation (3) shown in Fig. 5 (for a > 0) is displaced parallel
to the axis OY
through a distance b: upwards for b
positive, downwards
for b negative. We obtain a straight line, parallel to the initial line,
and cutting off a segment OM0 = b on the axis OY (Fig. 7).
Thus,
the graph of function (2) is a
straight line, the coefficient a being equal to the tangent of
the angle that the line makes with the axis OX, and the constant term b
equal to the segment cut off by the line on the axis OY, measured from the
origin 0. Conversely,
given any straight line L,
not parallel to the axis OY, its
equation can easily be written in the form (2). In accordance with
the above, it is sufficient to take the coefficient a equal to the tangent of the angle of inclination of this
line to the axis OX, and b
equal to the segment that it cuts off
on the axis OY.
the source:
A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.
By: Fady tarek
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