31
Graphing Rational Functions
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Graphing Rational Functions
Example (1)
For the following function: determine the intervals of increase/decrease, the intervals of upward/downward concavity , the points of extremum and inflection, the intersections with the axes and the asymptotes and then graph it.
2
2
)1(
3)(
x
xxxf
2
2
11
2
2
11
2
2
22
2
2
2
2
22
2
)1(
3lim)(lim
)1(
3lim)(lim
:,0)1(
,0431
1,
,1
:
.1
1001
011
lim2
lim1lim
3lim1lim
121
31
lim
12
3lim
)1(
3lim)(lim
:
)1(
)3(
)1(
3)(
11
11
x
xxxf
x
xxxf
getwepositivekeepingwhilexwhile
xxleftthefromapproachesxasSince
fofasymptoteverticalaisxthennumeratoritsofzeroanotisthat
ffunctionrationaltheofinatoromdentheofzeroaisxSince
AsymptotesVertical
asymptotehorizontaltheisyxx
x
xx
x
xx
xx
x
xxxf
AsymptoteHorizontal
x
xx
x
xxxf
xx
xx
xxx
xx
x
xxx
Asymptotes
min)16
9,
5
3(0
)1(57
53
10)
5
3(
))1(,1(
)16
9,
5
3())
5
3(,
5
3(
)1(5
3
:
.))0(,0()0,0(
)0,3()0,0(:
30
:
)1(57
10
)1(
1410)(
)1(53
5
)1(
)35()(
)1(
)3(
)1(
3)(
4
4433
22
2
localoftinpoaisx
f
undefinedalsoisfincesfoftinpocriticalanotisxthatNote
foftinpocriticalonlytheisf
DNEfandfofzeroonlytheisx
Extremum
axisythewithectionrsnteithealsoisfthatNote
andareaxisxthewithectionrsteinoftsinpothesoand
natordenomithenotandnumeratortheofzeroesarexandx
axesthewithectionrsInte
x
x
x
xxfand
x
x
x
xxf
x
xx
x
xxxf
Intersection with the Axes & Extremum
See Appendix
tervalsintheseongadecreisfandonnegativeisxf
tervalinthisongaincreisfonpositiveisxf
andtervalsintheonxftheofsigntheingInvestigat
undefinedalsoisfncesitinpocriticalanotisxthatNote
undefinedisfandfhaveWe
x
x
x
xxf
sin),1()5
3,()(
sin)1,5
3()(
),1()1,5
3(),
5
3,()(
))1(1(
)1(0)5
3(
)1(53
5
)1(
)35()(
33
Intervals of increase/decrease
min5
3
)1,5
3()
5
3,()(
localoftinpoaisx
onpositiveandonnegativeisxf
haveWe
Another Way to show extremumFirst Derivative Test
tervalinthisondownwardconcaveisfonnegativeisxf
tervalsintheseonupwardconcaveisfandonpositiveisxf
andtervalsintheonxftheofsigntheingInvestigat
undefinedalsoisfncesiflectioninforpoincandidatenotisxthatNote
flectioninforpoincandidateaisx
undefinedisfandfhaveWe
x
x
x
xxf
)5
7,()(
),1()1,5
7()(
),1()1,5
7(),
5
7,()(
))1(1(5
7
.)1(0)5
7(
)1(57
10
)1(
1410)(
44
Concavity
Graph
Example (2)
For the following function: determine the intervals of increase/decrease, the intervals of upward/downward concavity , the points of extremum and inflection, the intersections with the axes and the asymptotes and then graph it.
1
2)(
2
2
x
xxf
)1)(1(
2lim)(lim
)1)(1(
2lim)(lim
)1)(1(
2lim)(lim
)1)(1(
2lim)(lim
11,
,11
:
.2
201
21
lim1lim
2lim
11
2lim
1
2lim)(lim
:
)1)(1(
2
1
2)(
2
1)1(
2
1)1(
2
11
2
11
22
2
2
2
2
2
xx
xxfand
xx
xxf
xx
xxfand
xx
xxf
fofasymptoteverticalaarexandxthennumeratoritsofzeroanotarethat
ffunctionrationaltheofinatoromdentheofzeroesarexandxSince
AsymptotesVertical
asymptotehorizontaltheisyxx
x
xxf
AsymptoteHorizontal
xx
x
x
xxf
xxxx
xxxx
xx
x
x
xx
Asymptotes
max)0,0(04)10(
)10(4)0(
))1()1(,11(
)0,0())0(,0(
,)1()1(0
:
.))0(,0()0,0(
)0,0(:
0
:
)1(
)13(4)(
)1(
4)(
)1)(1(
2
1
2)(
3
32
2
22
2
2
2
localoftinpoaisf
undefinedalsoarefandfincesfoftsinpocriticalnotarexandxthatNote
foftinpocriticalonlytheisf
DNEfandfwhilefofzeroonlytheisx
Extremum
axisythewithectionrsnteithealsoisfthatNote
areaxisxthewithectionrsteinoftsinpothesoand
fofnatordenomithenotandnumeratortheofzeroaisx
axesthewithectionrsInte
x
xxfand
x
xxf
xx
x
x
xxf
Intersection with the Axes & Extremum
See Appendix
tervalsintheseongadecreisfandonnegativeisxf
tervalinthisongaincreisfandonpositiveisxf
andtervalsintheon
xftheofsigntheingInvestigat
undefinedalsoarefncesistinpocriticalanotarexthatNote
undefinedisfandfhaveWe
x
xxfand
xx
x
x
xxf
sin),1()1,0()(
sin)0,1()1,()(
),1()1,0(,)0,1(),1,(
)(
))1(1(
)1(0)0(
)1(
4)(
)1)(1(
2
1
2)(
22
2
2
2
Intervals of increase/decrease
0max
)1,0()0,1()(
xatlocalahasf
onnegativeandonpositiveisxfSince
haveWe
Another Way to show extremumFirst Derivative Test
tervalinthisondownwardconcaveisfonnegativeisxf
tervalsintheseonupwardconcaveisfandonpositiveisxf
andtervalsintheonxftheofsigntheingInvestigat
undefinedalsoarefncesi
flectioninforpoincandidatenotarexthatNote
flectioninforpoincandidateNo
undefinedarefandwhyzeroneverisfhaveWe
x
xxfand
xx
x
x
xxf
)1,1()(
),1()1,()(
),1()1,1(),1,()(
))1(
1(
.)1(?)(
)1(
)13(4)(
)1)(1(
2
1
2)(
32
2
2
2
2
Concavity
Graph
Example (3)
For the following function: determine the intervals of increase/decrease, the intervals of upward/downward concavity , the points of extremum and inflection, the intersections with the axes and the asymptotes and then graph it.
3
23)(
x
xxf
?)(23
lim)(lim23
lim)(lim
0,
,0
:
.0
01
00
1lim
2lim
3lim
1
23
lim
23lim)(lim
:
32
323
)(
310300
3232
3
33
Whyx
xxfand
x
xxf
fofasymptoteverticalaisxthennumeratoritsofzeroanotisthat
ffunctionrationaltheofinatoromdentheofzeroareaisxSince
AsymptotesVertical
asymptotehorizontaltheisy
xxxx
x
xxf
AsymptoteHorizontalx
x
x
xxf
xxxx
x
xx
x
xx
Asymptotes
max)1,1(061
)43(6)1(
))0(,0(
)1,1())1(,1(
)0(1
:
)0(
)0,3
2(:
3
2
:
)43(6)(
)1(6)(
32
323
)(
54
33
localoftinpoaisf
undefinedalsoisfincesfoftsinpocriticalnotisxthatNote
foftinpocriticalonlytheisf
DNEfwhilefofzeroonlytheisx
Extremum
axisytheectsrsteinnotdoesfsoandDNEfthatNote
isaxisxthewithectionrsteinoftinpothesoand
fofnatordenomithenotandnumeratortheofzeroaisx
axesthewithectionrsIntex
xxfand
x
xxf
x
x
x
xxf
Intersection with the Axes & Extremum
See Appendix
tervalsintheseongadecreisfonnegativeisxf
tervalinthisongaincreisfandonpositiveisxf
andtervalsintheon
xftheofsigntheingInvestigat
undefinedalsoisfncesistinpocriticalanotisxthatNote
undefinedisfandfhaveWex
xxf
x
x
x
xxf
sin),1()(
sin)1,0()0,()(
),1()1,0(,)0,(
)(
))0(0(
)0(1)1(
)1(6)(
32
323
)(
4
33
Intervals of increase/decrease
0max
),1()1,0()(
xatlocalahasf
onnegativeandonpositiveisxfSince
haveWe
Another Way to show extremumFirst Derivative Test
tervalinthisondownwardconcaveisfonnegativeisxf
tervalsintheseonupwardconcaveisfandonpositiveisxf
andtervalsintheonxftheofsigntheingInvestigat
undefinedalsoisfncesi
flectioninforntoipcandidatenotisxthatNote
flectioninforntoipcandidateaisfx
undefinedisfandxatzeroisfhaveWe
x
x
x
xxfand
x
x
x
xxf
)3
4,0()(
),3
4()0,()(
),3
4()
3
4,0(),0,()(
))0(
0(
)32
27,
3
4())
3
4(,
3
4(
.)0(3
4
)34
(9)43(6)(
32
323
)(
55
33
Concavity
Graph
Appendix
Differentiating & Simplifying
33
3
22
3
2
4
22
4
22
2
2
)1(
)35(
)1(
35
)1(
)62()32(
)1(
2)3()32)(1(
)1(
)1(2)3()32()1(
)1(
)1(2)3()32()1()(
)1(
3)(
x
x
x
x
x
xxxx
x
xxxx
x
xxxxx
x
xxxxxxf
x
xxxf
Example(1)
444
4
4
6
23
3
)1(
1410
)1(
)1410(
)1(
1410
)1(
)915()55(
)1(
3)35(5)1(
)1(
)1(3)35(5)1()(
)1(
35)(
x
x
x
x
x
x
x
xx
x
xx
x
xxxxf
x
xxf
Example(2)
2222
22
22
22
22
22
22
2
2
)1(
4
)1(
]1[4
)1(
]1[4
)1(
])1[(4
)1(
224)1()(
1
2)(
x
x
x
x
x
xxx
x
xxx
x
xxxxxf
x
xxf
32
2
32
2
32
2
32
22
32
22
32
2
42
222
22
)1(
)13(4
)1(
)13(4
)1(
)13(4
)1(
]41[4
)1(
]4)1[(4
)1(
2244)1(
)1(
2)1(244)1()(
)1(
4)(
x
x
x
x
x
x
x
xx
x
xx
x
xxx
x
xxxxxf
x
xxf
Example(3)
4
4
4
4
6
23
3
)1(6
66
)69(3
3)23(3
3)23(3)(
23)(
x
xx
xx
xxx
xxx
xxxxf
x
xxf
Example(3)
4
4
43
32
3
)1(6
)1(6
66)(
23
23)(
:
x
x
xx
xxxf
xx
x
xxf
methodAnother
5
5
5
5
8
34
44
)43(6
)436
)44(6
4)1(6
4)1(6)(
16
)1(6)(
x
xx
xx
xxx
xxx
xxxxf
x
x
x
xxf
55
54
434
)43(6)43(6
)43(6)(
)(6)1(6
)(
:
x
xxx
xxxf
xxx
xxf
MethodAnother
