WARM UP Describe the end behavior of: Graph the function Determine the interval(s) on which the function is increasing and on which it is decreasing. 12 5 17 3 6 ) ( 2 3 4 x x x x x f 1 ) 3 ( ) ( 2 x x f
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WARM UP
Describe the end behavior of:
Graph the function
Determine the interval(s) on which the
function is increasing and on which it is
decreasing.
1251736)( 234 xxxxxf
1)3()( 2 xxf
LESSON 3-7 GRAPHS OF RATIONAL FUNCTIONS
Objective: 1. To graph rational functions.
2. To determine vertical, horizontal and slant
asymptotes
RATIONAL FUNCTIONS
Rational functions are the
quotient of 2 polynomial functions
where h(x) ≠ 0.
The parent function is
)(
)()(
xh
xgxf
xxf
1)(
ASYMPTOTES
Asymptotes are lines that a
graph approaches, but does not
intersect.
We will look at vertical,
horizontal and slant asymptotes.
ASYMPTOTES
A rational function can have
more than one vertical
asymptote, but it usually has
one horizontal asymptote at
most.
VERTICAL ASYMPTOTES
If g(x) and h(x) have no common factors, then f(x) has vertical asymptote(s) when h(x) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator. (where the denom. is undefined.)
Rational functions are the quotient of 2 polynomial functions
where h(x) ≠ 0.)(
)()(
xh
xgxf
VERTICAL ASYMPTOTES
x=a is a vertical asymptote for f(x) if or as from either the left or the right.
)(xf
)(xf ax
EXAMPLE
For , the vertical
asymptote is x = 0
xxf
1)(
EXAMPLE
Find the vertical asymptote of
Since
the function is undefined at 1 and
-1. Thus the vertical asymptotes are
x = 1 and x = -1.
.1
2)(
2 x
xxf
)1)(1(1)( 2 xxxxh
HORIZONTAL ASYMPTOTE
To determine or prove the horizontal
asymptote:
Find the highest degree variable in
the denominator.
Divide each term in the function by
this.
HORIZONTAL ASYMPTOTE
3
13
x
xy
xxx
xxx
y3
13
highest degree variable is x
0togoesthis
x
EXAMPLE CONT’D
01
03
3y is the horizontal asymptote
SHORT CUTS
a. If the degree of g(x) is less than the degree of h(x), then the horizontal asymptote is y = 0. (this could be y= some vertical shift also)b. If the degree of g(x) is equal to the degree
of h(x), then the horizontal asymptote is
.)( oft coefficien leading
)( oft coefficien leading
xh
xgy
c. If the degree of g(x) is greater than the degree of h(x), then there is no horizontal asymptote.
)(
)()(
xh
xgxf
HORIZONTAL ASYMPTOTES Example:Find the horizontal asymptote:
.1
3)(
2 x
xxf
Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote is y = 0.
Degree of numerator = 1Degree of denominator = 2
HORIZONTAL ASYMPTOTES Example:Find the horizontal asymptote: .
12
13)(
x
xxf
Degree of numerator = 1Degree of denominator = 1
Since the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is . 2
3y
Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
HORIZONTAL ASYMPTOTES
Example:Find the horizontal asymptote: .
12
13)(
2
x
xxf
Degree of numerator = 2Degree of denominator = 1
EXCEPTIONS
If there are 2 vertical asymptotes, the
horizontal asymptote may or may not hold.
vert. asymp at x=.562 and -3.562
horiz. aymp at y = 1
23)(
2
2
xx
xxf
EXCEPTIONS
VERTICAL & HORIZONTAL ASYMPTOTES Practice:Find the vertical and horizontal asymptotes:
1
12)(
x
xxf
SLANT ASYMPTOTES
Slant asymptotes occur when the degree
of the numerator of a rational function is 1
more than the degree of the denominator.
Find by using polynomial long division.
ex: 2
3 1)(
x
xxf
SLANT ASYMPTOTES
Using polynomial long division will
yield
As , therefore
the slant asymptote is y = x.
xx
1
x 01
x
PRACTICE
Find the slant asymptote for
2
132)(
x
xxxf
4
3764)(
2
x
xxxf
EXCEPTIONS
Some rational functions will only have
point discontinuity instead of an
asymptote.
This occurs whenever the numerator and
the denominator share a common factor.
EXCEPTIONS
Find the asymptotes for 2
2)(
2
x
xxxf
No vertical asymptoteNo horizontal asymptoteBecause numer. & denom. have (x-2) in common there is also no slant asymptote – but a hole(point ofdiscontinuity) at x=2.
SOURCESteachers.henrico.k12.va.us/math/hcpsalgebra2/.../AII7_E_asymptotes.ppt, Sept 17,2013
WolframAlpha. Wolfram Alpha LLC, 2013. Web. 20 Sept. 2013. <http://www.wolframalpha.com>.
Find the horizontal asymptote:
x
. f xx
2 11
2
x. f x
x
3
2
12
x
. f xx x2
23
20
Exponents are the same; divide the coefficients
Bigger on Top; None
Bigger on Bottom; y=0
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