
Graphing
Linear Equation, Linear Function (First Degree Polynomial)



Lines
parallel to the axes, horizontal and vertical lines

The
pointslope form of a line

Parallel
and perpendicular lines






Lines
parallel to the axes, horizontal and vertical lines

If
m =
0, the function does not depend of
x. To every number
x
associated is the same constant value y =
c.

A
line parallel to the xaxis
is called a horizontal line (or constant). 
If
the x
value never changes a line is parallel to the yaxis.
A line parallel to the yaxis
is called a vertical line. 



The 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 




The
pointslope form of a line

The equation of the line
y
= mx
+ c
is defined by the slope m
and by one of its points P_{1}(x_{1},
y_{1}). 
As
coordinates of the point P_{1
}must
satisfy
the equation of the line, we calculate the yintercept,
by
substituting them into the equation of the line, 
P_{1}(x_{1},
y_{1})
=>
y
= mx
+ c 
y_{1}
= mx_{1}
+ c
or
c
= y_{1} 
mx_{1}

Therefore, the line that passes through the given point
P_{1}(x_{1},
y_{1})
and given is its slope m, has the equation 

y
= mx
+ y_{1} 
mx_{1} 
or 
y 
y_{1}
= m(x
 x_{1}) 



Example:
Find the equation of the line that is parallel with the line
y
=  x
 2 and passes through the point
P_{1}( 2,
1) . 


Parallel
and perpendicular lines

Two lines having slopes
m_{1}
and m_{2
}are parallel if 

m_{1}
= m_{2} 

that is, if
they have the same slope. 


To acquire the criteria when two lines,
y
= m_{1}x
and y
= m_{2}x
are
perpendicular or orthogonal we can use the principle of similar triangles,
OA'A
and OB'B in the picture. 
Therefore, 
m_{1
}: 1
= 1_{
}: m_{2} 
=> 






This relation will stay
unchanged if we translate the perpendicular lines, that is, when
lines 
y
= m_{1}x
+ c_{1
} and y
= m_{2}x
+ c_{2
}are written in the slopeintercept form. 
Two lines are perpendicular if the slope of one line is the negative
reciprocal of the other. 

Example:
Find the equation of the line that is perpendicular to the line 

and passes through 

the point A(2,
5). 









Beginning
Algebra Contents C 



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