Algebra
II Notes:
Graphing
Rational
Functions
A Rational Function is
polynomial
polynomial
To graph a rational function, you must determine if any of the following exist and if they do exist then identify them.
1.
2.
Intercepts (both x-and y-intercepts)
Asymptotes ( vertical, horizontal, and slant)
3.
Identify any holes if they exist.
To get started:
1st Factor the numerator and denominator.
If the function
Reduce if possible.
reduces then there will be a hole in the graph.
To find the coordinates of the hole:
(a) Set the factor{s) that cancelled equal to zero
(b) Substitute this value in for "x" in the reduced equation to find the y-value of the ordered pair.
IF THE FUNO"lON
REDUCES THEN USE THE REDUCED FUNCTION
TO FIND
THE FOLLOWING
2nd Find the intercepts if there are any.
To find the x-intercept:
Let
y = 0 and
solve for X • If this statement
is false then there is no x-intercept.
If you get imaginary roots then the graph does not cross the x-axis.
To find the y-intercept:
Let X = 0 and solve for
y.
If the resulting expression is undefined
then there is no y-intercept.
3,d Find the asymptotes.
To find the vertical asymptotes:
set the factored denominator
The resulting equations will be the vertical asymptotes.(Note:
Note:
=
0 and solve.
Graphs can NEVER cross vertical asymptotes)
Multiple equal factors in the denominators will define multiple vertical asymptotes.
Ex: If the same factor occurs twice, the equation has a double vertical asymptote.
•
If a vertical asymptote occurs an even number of times, then the function
and go in the same direction.
•
If a vertical asymptote occurs an odd number of times, then the function
and go in opposite directions.
To find the horizontal asymptote:
compare the degree of the numerator
will approach the asymptote
will approach the asymptote
from both sides
from both sides
to the degree of the denominator
•
If the degree of the numerator
is less than the degree of the denominator,
•
If the degree of the numerator is greater than the degree of the denominator,
see if there is a slant asymptote.)
•
If the degree of the numerator
is equal to the degree of the denominator,
the HA is the x-axis (y=O)
there is no HA. (If this happens check to
then the equation of the HA is
lead coefficient of numerator
y = lead coefficient of denominator
(Note: Graphs CAN cross horizontal asymptotes. To determine if the function crosses the HA, write the
function equal to the HA and solve. If you get a value(s) for x, then it crosses the HA at this x-value. If this
statement is false then it does not cross the HA)
To find a slant asymptote:
y = quotient
divide the numerator by the denominator.
The slant asymptote
will be
(Ignoring the remainder)
(Note: Graphs CAN cross a slant asymptote.
To determine
if the function
crosses the SA, write the function
equal to the SA and solve. If you get a value(s) for x, then it crosses the SA at this x-value. If this statement
false then it does not cross the SA)
is
Class Examples - Rational Functions Day 1
Determine the intercepts, asymptotes, and sketch each graph. Listeach multiplicity greater than 1. If the graph crosses the
horizontal asymptote, list the ordered pair(s) that describers) the location(s) where the graph crosses the asymptote.
1.
x+I
Y=x2+x-6
1(+1
(X+3)(.X
z.
2.)
2.
y=
-
X:: - 3,
__
..,.,lj/-=---e;O
vertical asymptotes
horizontal asymptote
(_-_1 ,,-0---=-) __
Crosses H.A. at
(Zo)
(0, ~,,)
y-inte rce pt
')C::;J.
(x;l)
(x+3) (x-2)
( -I, 0)
___ ('--_/...:./_°--'1'---_x-intercepts
- '~,)
( 0, ---,-/.=..
--"--'-+-F,••• .L...
2x+1
3. Y=---l
x-
2
'X:::
__
-3(2.)
2
y-intercept
1{=-).(z)vertical
-,'1":J-=-,O~__
( -
x-intercepts
horizontal asymptote
--~(--...!.I4-' -""0
.....•
)'----_
Crosses H.A. at
__
-~:-~
-z. )(+2.
~)
2 lOx-intercepts
( (), ( )
asymptotes
J~~t~
1C~_=_I
X-I
=-.X-I
Z:J. -I
y-intercept
vertical asymptotes
--J7-:::---'~--horizontal asymptote
Crosses H.A. at
_
~
:L
~
I
-
--
I
,
r~
t
\if
3x2 + 3x - 36
6.
Y
3(X+4)(X-3)
= (X2 -16)( X - 3) -: (x •.ltJ(X-I/KMJ
_ .J-
.
(- ~.z..0)
(-/. .•. 0 J
(0.---..:..•••
~)
:J.'-"
( - Z I 0) (- 5'. (/) x-intercepts
x-intercepts
. r)
( 0,
_'K=.-::::._-3'-1-1 ....;1(_=---=.(_verticalasymptotes
__
.,.q_::.--,.J...~__
Crosses H.A. at __
I
-L
=
-I
(,,--,-74-1_Z.1.)
Crosses H.A. at
f-
,
~
--1-+' _-=-_-....:.C __
I
I
1(= 1. I
horizontal asymptote
1t
)(-1/
_---<~'_'_-'--
x-intercepts
__
y-intercept
•
y-intercept
--:><"":::'r-I
-
_
y-intercept
vertical asymptotes
horizontal asymptote
_(=--_~_Yz::"//--_'-!:)
_
.".(...:.O+-I_-_l..:....V4:L.L.)_
___
1L..c::::..._=-_if-'---_ vertical asymptotes
__
---=;qf--=---'O"--_horizontalasym ptote
Crosses H.A. at
_