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Definition: A rational function is a function that can be written
where p(x) and q(x) are polynomials.
( )( )
( )
p xf x
q x
8) Graph
Steps to graphing a rational function.
1) Factor first
2) Find the domain.
3) Find any holes
4) Identify the vertical asymptote(s). Include any multiplicity.
5) Find the horizontal asymptote and check to see if the graph crosses it.
6) Find the y-intercept.
7) Find the x-intercept(s)
Domain
The domain is all real numbers except the values of x that make the denominator equal to zero.
Write in interval notation.
HolesIf there are nonconstant factors that cancel you will have a hole.To find the location of the hole, set the common factor (the one the you canceled) equal to zero and solve for x. This is the x-coordinate of the hole. Substitute the x-coordinate into the simplified form of the function to find the y-coordinate. Write the hole as a point.
Example 1
Example 1
Vertical AsymptoteAfter you have found any holes, the values of x that make the denominator zero are the location of your vertical asymptotes.
The graph cannot cross a vertical asymptote.Vertical asymptotes are vertical lines and should be written as x =
Vertical asymptotes can have multiplicity.
For single factors the graph goes up on one side of the asymptote and down on the other side
For double factors the graph in the same direction on both sides of the asymptote.
or or
Example 1
To find the horizontal asymptote compare the degree of the numerator with the degree of the denominator. There are three possibilities.
If the degree of the numerator is larger than the degree of the denominator there is no horizontal asymptote.
If the degree of the numerator is smaller than the degree of the denominator the horizontal asymptote is y = 0.
If the degree of the numerator is equal to the degree of the denominator the horizontal asymptote is the ratio of the leading coefficient of the numerator to the leading coefficient of the denominator.
For example: if the horizontal asymptote would be
2
2
2 3
4 1
x xy
x
2 1
4 2y
Horizontal asymptotes are horizontal lines and should be written as y =
Horizontal Asymptote
Horizontal asymptotes deal with the end behavior of the graph. That is what is happening as x gets really large and really small.
Horizontal Asymptote (continued)
Horizontal asymptotes may be crossed.
To find if and where the graph crosses the horizontal asymptote set the function equal to the y-value of the asymptote and solve for x
If the point where the graph crosses the horizontal asymptote is not between two vertical asymptotes, it will come back toward the asymptote once it crosses it.
For example:
Example 1
Y-intercept
To find the y-intercept let x= 0 and solve for y.
X-intercept
To find the x-intercept let y= 0 and solve for x.
Write as a point.
Write as a point.
Note: Since the denominator can never equal zero, you really only need to set the numerator equal to zero.
Example 1
Example 1
Example 1:
2
2
2 3 2( )
4
x xf x
x
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
(back to notes)
(back to notes)
(2 1)( 2)
( 2)( 2)
x x
x x
(-,-2) (-2, 2) (2, )
2,
2 1
2
x
x
x = -2
y = 2 does not cross
10,2
1,02
5
4
(back to notes)
(back to notes)
(back to notes)
Example 2:
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
2
2( )
( 1)f x
x
( , 1) ( 1, )
None
1 ( 2)x M
0y
0, 2
None
Example 3:
2
3( )
5
xf x
x x
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
3
5
x
x x
,0 0,5 5,
3
5x
0,30,5
None
None
5x 0y
Example 4:
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
2
4( )
2
xg x
x x
4
( 1)( 2)
x
x x
, 2 2,1 1,
None
2, 1x x 0y
0,2
4,0
Crosses at x = 4
Example 5:
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
2 2 3
1
x xy
x
( 1)( 3)
1
x x
x
3x
( , 1) ( 1, ) (1, )(1,4)
( 3,0)(0,3)
None
None
Example 6:
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
3 1
2
xy
x
( , 2) (2, )
None
2x
3y 1
0,2
1,03
D: ______________________
Hole: ____________________
VA: ____________________
HA: _____________________
y-int: ____________________
x-int: _____________________
