
Graphing
Linear Equation, Linear Function (First Degree Polynomial)



Linear
function f (x) = mx + c 
The
graph of the linear function 
Slope
or gradient, yintercept and xintercept 





Linear function f
(x)
= mx
+ c 
The expression 
y =
mx
+ c
or f
(x)
= ax + b

we call linear function,
where x
is the argument or the independent variable, f
(x) or
y
is the dependent variable or the function, numbers, a and
b
or m and
c are
constants. 
If the coefficient
m =
0, then y does not depend of
x. 
Linear
function f
(x)
= mx
+ c,
m is
not 0, to
each value of the argument
x associates a unique value
of y. 

The graph of the linear function 
A function
y =
f
(x)
can be considered as the set of ordered pairs (x,
y) where each pair
represents a point in a Cartesian coordinate system xOy. 
When plotted on an
x,
y graph linear function forms a
straight line. 
The constant
m is called the
slope or gradient
and the constant c is the
yintercept.
This is the point of intersection between the graph of the
function and the yaxis. 
If the coefficient
m is
not 0 and c =
0
then, the linear function has the form y
=
f
(x) =
mx. 
To the graph of the
function f (x) =
mx belong the points
P (x,
mx
) of a coordinate plane of which the abscissas are values of
argument x
taken arbitrarily, and the ordinates are the calculated values
of the function.

Thus, for
x =
0 =>
f (x) =
f
(0)
=
m
· 0
=
0,
this means that the graph of the function f
(x) =
mx
is running through the origin
O(0,
0) or
the xintercept
is at the origin. 



As the above
right triangles OAP_{1
}and OBP
are similar the ratios of the ordinates and the abscissas of the
points
P and
P_{1}
are equal: 

or 


The
value of the ratio 



determines the rate at which the ycoordinate of a straight line
changes with respect to the xcoordinate. 
On a line graph, changing
m
makes the line steeper or more gentle (shallower). 
If m
is negative, y
decreases
as x
increases. 

Example:
Find the equation of the line that passes through the
origin and the point A(3,
2). 
By
plugging the coordinates of the point A
into the equation y =
mx we
determine the slope m. 









Beginning
Algebra Contents C 



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