Secondary Ill
Unit 3.5 Notes: Graphing Rational Functions
Recall from previous math courses that the reciprocal of any number xis~. Throughout this
X
course you have studied many connections between polynomial functions and real numbers.
Some questions to ask ourselves are; Does it follow then that polynomial functions have
reciprocals? Is the reciprocal also a polynomial? Is it a function?
Example 1: Consider the graph and table of values for f(x)=x. The domain of f(x) is all real
numbers. The points (-1, -1) and (1, 1) are shown. Complete the tables and graph f(x)= ~.
X
-6
-6
6
4
5
5
'''~
''s
\1 (a
X
1
2
3
4
f(x) =X
1
2
3
1
g(x) =-x
\
\;,_ 'I~
6
Y't'
I
~~-
I I 1/
I I /
1/
VI
I
I
-~
l
I
'"
JO::.
I.
.L
n
"J;
I V
I) '
a.
1/
v
v
X
-1
f{x) =X
-
'- 1/t
-\
g{x)=X
_,
'3
X
x-
-
gr()<0::;~
::::.o .
..
I
I
1/ i
VI I
1/ I I
I
I
I I
I
II
Describe the graph of g(x) . How is it similar to the graphs of other f unctions that we
stud ied and how is it different?
SiYV\iletv-:
s YV'loo-rh c
1.)
v
ve
<~ ?~~r4"1--l ~r+s
t\Sy vY\ ~+e 5
b.
f\
~ VVlcMf,~cf
I/
..,
I
1/
- 1
X-:/; 0
I
1/
I I
-2
-2
/I
7D
~
I
I I
......
-3
-3
f[x) ~
l/l
./
-l.f
s
I
I
I
In
-4
-4
_, _,
-
-'-
~
!
I
I
i
I
-5
-5
Describe the ending behavior of g(x).
X --:;;> .00
.( Cx) ~
X--3
.?(x)--=, 0
-Oo
()
.
Secondary Ill
Now that we know what a parent graph looks like for a rational equation let's look at the look
at the key parts of the graph.
~.e+ ~YI() ~ j 1-'tc:.f+oV'" .= ()
.
C
~ d) V.e
Example 1: Find the domain of each rational function:
a.
x 2 -9
f(x)= - -
b. g (x)
x+3
x-5
=- ==-----2
x +3x-4
()( +4 ) (X- 1)
~ : ><.~-3
rR : )( =f. - 'I.~
l - 00.1 - 3) (_- ~ 1 ~)
x
Vertical Asymptotes: Look for values where the denominator is zero(A V.A. is written as a line
A graph can never cross a vertical asymptote.
(Why not?}
Example 2: Find the vertical asymptotes:
a.
'}'
\\iJ
f(x) =-
5x
x+2
[\;A : )("' -~
c.
2x+ l
x -5x+6 '
h(x)= - 2 - -
(x-1.-)lK-3)
d.
x3 - 1
3x2+4
y= -
"::0
-'-! -'-1
'3->< "2.. -:;:.
-
L(
~0
(!D \).A]
x=a)
Seconda ry Ill
-
Odd and Even Vertical Asymptotes: Look at the power for each facto r in the denominator.
Odd behavior:
·.·
.·.
y=
1
(x + 2) 1
n
'
v.A
.X
-- -'2..
.:•
:-
,-
-;.
'I'
.1• •
•l
.:,
'
'1'
+
~
·i·
'1'
.:.
+
Even behavior:
y=
,.
+
+
.
+
1
(x+2)l
-
..
,.
!·
.
+
+
..
+
.,
'
•:·
... ,.
'
, ;.
+
·'·
-'i·
-i·
Horizontal Asymptotes: Look at the highest power in the numerator and
the denominator.
Top> Bottom
None.
Bottom> Top
y = O.
Top= Bottom
y = ratio of leading coefficients
*Note: The graph can cross a horizontal asymptote.
{Why?) ~lC'i<I'Sl fvnt:t110J.1
Example 3: Determine the horizonta l asym ptotes and t he range:
a.
f
5x'
(x) =lx'+ 2
··
,
b.
g (X ) =
2#+ 1
tn
9- 4.x-
- _)Y
'2...-
c.
15.
oe-11 ~cl
2.
-f..(
Secondary Ill
If there is not a Horizontal Asymptote check for a Slant Asymptote. If the degree of the numerator is
exactly 1 bigger t han the degree of the denominator, then a slant asymptote exists. Find it by dividing
the denominator into the numerator. Ignore the remainder and graph the slanted line with dots.
Example 4: Find the slanted asymptote for
f(x) =
/
2:~~3
X-3
no H. A.
L
/
Stan+
How would you graph this?
To graph a rational function you need to find the domain, y and x intercepts, and vertical and
horizontal asymptotes.
To find a y-intercept, set x = 0.
x-intercepts are found by setting the numerator of the reduced function equal to ~ero.
y ::;. 0
~~
Example 5: Find the y-intercept, and the x-intercepts of each function .
a.
Sx
f(x)=-
x+2
0
$(o) .... ,_.,..--0
0+"2-
y-i VJ-t-
- . -z..,
c. h x
( )
=
2x+l
x 2 -5x+6
Y-'wr+
x3 - 1
d. y = - - 3x2 +4
'/-i f'r+ •.
X-·~ rr+-
o::2x+ l
: ( t1o )
Seco ndary Ill
-
l
Holes: A rational function will have a hole in the graph (or missing point) when a factor will reduce
completely out of t he denominat or.
Example 6: Fin d t he hole in t he graph of
-l-1
_, -t2
Example 7: Find the domain of
f
(x)
•
= . .,
x 2 -4
.JX
graph, and ho rizontal asymptote for
2
-1: 6x
- -'2.
-
-
I
·
•
Then list the vertical asymptot e(s), hole(s) in the
·
f (x) .
Domain:
'/
~
:
X
1:
0./ ~ 2.
L..
.-('\ -Red uced
(x) =IX'-2... , x=t -2
3x·
V.A:
x -::::o
Range:
Additional Notes:
-- [_
-~/ x~
- 2-2
(
"3(-2.)
Ho le: [ -
J
2.) '2./3
HA:
y-:;... _L
3