

Linear
function

The
linear function f
(x)
= mx + c 
The graph of
the linear function 
Properties
of the linear function 





The linear function
f
(x) = mx + c 
The expression,
f (x) =
mx
+ c or
y
= a_{1}x + a_{0},
we call linear function
or a first degree polynomial function of one variable. Where, the constant m is called the
slope or gradient
and the constant c is the
yintercept. 
The slope
of a line is the ratio of its vertical change to its horizontal
change, or it is the tangent of the angle between the direction
of the line and the xaxis. 
The yintercept is the point of intersection between
the graph of the function and the yaxis. 
The zero
of a function or the xintercept
is the value of the
independent variable x
at which the value of the function is zero. 

The graph of
the linear function 

this
expression, in the Cartesian coordinate plane, represents the translation
of the source linear function 
y =
mx
in the direction of the yaxis
by y_{0} =
c
or the translation in the direction of the xaxis
by x_{0} =

c/m 
as is shown
in the figure above. 
Therefore,
the linear expression we also write as 
y
= a_{1}x + a_{0}
or y
= a_{1}(x

x_{0})
or y
 y_{0}
= a_{1}x,
where y_{0}
= a_{0} 
To find
the zero or the xintercept
of the linear function set y
= 0 and solve
the equation for x,
i.e., 
y
= 0 =>
0
= a_{1}x + a_{0} 


Example:
Find the equation of the line that passes through the
origin and the point A(3,
2).
Translate the line in the direction of the xaxis
by x_{0} =

3, then find its equation. 
Calculate
the slope m by
plugging the coordinates of the point A
into the equation y =
mx, 





By
plugging x_{0} =

3 into y
= a_{1}(x

x_{0}) 



Properties
of the linear function

We examine the behavior of a function
y =
f (x)
by moving from left to right in the direction of xaxis by inspecting its graph. 
The linear
function f (x)
= mx
+ c,
m
> 0 is increasing, the graph rises from left to right, that is, 
f (x_{1})
< f (x_{2}) for all
x_{1
}< x_{2} 


If
m
< 0 linear function decreasing,
f
(x_{1})
> f (x_{2}) for all
x_{1
}< x_{2 }i.e., the graph falls from left to right. 

Linear
functions
have a constant rate of increase or decrease. 

A
linear function changes the sign at the root or zero point. 
Thus,
if 
m > 0, 
then 
f
(x)
< 0 
for all 

while

f
(x) > 0 
for all 



That
is, f
(x)
= mx
+ c,
m
> 0 is negative for all x
less than the root,
positive for all x
greater than the root, 
and at the root
f
(x)
= 0. 
If 
m < 0, 
then 
f
(x)
> 0 
for all 

while

f
(x)
< 0 
for all 











College
algebra contents C 



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