Lesson 18: Graphing
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![Page 1: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/1.jpg)
. . . . . .
Section4.3Graphingfunctions
Math1aIntroductiontoCalculus
March17, 2008
Announcements
◮ Thankyoufortheevaluations!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
..Image: Flickruser Cobalt123
![Page 2: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/2.jpg)
. . . . . .
Announcements
◮ Thankyoufortheevaluations!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
![Page 3: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/3.jpg)
. . . . . .
Outline
Thechecklist
Bigexample
Yourturn
![Page 4: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/4.jpg)
. . . . . .
GraphingChecklist
Tographafunction f, followthisplan:
0. Findwhen f ispositive, negative, zero, notdefined.
1. Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.
2. Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflection.
3. Puttogetherabigchart
4. Graph!
![Page 5: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/5.jpg)
. . . . . .
Outline
Thechecklist
Bigexample
Yourturn
![Page 6: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/6.jpg)
. . . . . .
Bigexample
Let f(x) =1x
+1x2. Wewilldoacompletedissectionof f.
![Page 7: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/7.jpg)
. . . . . .
Step0Findwhen f ispositive, negative, zero, notdefined.
Weneedtofactor f:
f(x) =1x
+1x2
=x + 1x2
.
Thismeans f is 0 at −1 andhastroubleat 0. Infact,
limx→0
x + 1x2
= ∞,
so x = 0 isaverticalasymptoteofthegraph. Wecanmakeasignchartasfollows:
. .x + 1..0.−1
.− .+
.x2..0.0
.+ .+
.f(x)..∞.0
..0.−1
.− .+ .+
![Page 8: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/8.jpg)
. . . . . .
Step0Findwhen f ispositive, negative, zero, notdefined. Weneedtofactor f:
f(x) =1x
+1x2
=x + 1x2
.
Thismeans f is 0 at −1 andhastroubleat 0. Infact,
limx→0
x + 1x2
= ∞,
so x = 0 isaverticalasymptoteofthegraph.
Wecanmakeasignchartasfollows:
. .x + 1..0.−1
.− .+
.x2..0.0
.+ .+
.f(x)..∞.0
..0.−1
.− .+ .+
![Page 9: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/9.jpg)
. . . . . .
Step0Findwhen f ispositive, negative, zero, notdefined. Weneedtofactor f:
f(x) =1x
+1x2
=x + 1x2
.
Thismeans f is 0 at −1 andhastroubleat 0. Infact,
limx→0
x + 1x2
= ∞,
so x = 0 isaverticalasymptoteofthegraph. Wecanmakeasignchartasfollows:
. .x + 1..0.−1
.− .+
.x2..0.0
.+ .+
.f(x)..∞.0
..0.−1
.− .+ .+
![Page 10: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/10.jpg)
. . . . . .
Forhorizontalasymptotes, noticethat
limx→∞
x + 1x2
= 0,
so y = 0 isahorizontalasymptoteofthegraph. Thesameistrueat −∞.
![Page 11: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/11.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.
Wehavef′(x) = − 1
x2− 2
x3= −x + 2
x3.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
![Page 12: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/12.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
![Page 13: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/13.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−
.↘ .↗ .↘.min .VA
![Page 14: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/14.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘
.↗ .↘.min .VA
![Page 15: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/15.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗
.↘.min .VA
![Page 16: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/16.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
![Page 17: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/17.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min
.VA
![Page 18: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/18.jpg)
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
![Page 19: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/19.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.
Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢ .⌣ .⌣
.IP .VA
![Page 20: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/20.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢ .⌣ .⌣
.IP .VA
![Page 21: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/21.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−−
.++ .++.⌢ .⌣ .⌣
.IP .VA
![Page 22: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/22.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++
.++.⌢ .⌣ .⌣
.IP .VA
![Page 23: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/23.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++
.⌢ .⌣ .⌣
.IP .VA
![Page 24: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/24.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢
.⌣ .⌣
.IP .VA
![Page 25: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/25.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢ .⌣
.⌣
.IP .VA
![Page 26: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/26.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢ .⌣ .⌣
.IP .VA
![Page 27: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/27.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢ .⌣ .⌣
.IP
.VA
![Page 28: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/28.jpg)
. . . . . .
Step2
Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave
f′′(x) =2x3
+6x4
=2(x + 3)
x4.
Thecriticalpointsof f′ are −3 and 0. Signchart:
. .(x + 3)..0.−3
.− .+
.x4..0.0
.+ .+
.f′(x)
.f(x)..∞.0
..0.−3
.−− .++ .++.⌢ .⌣ .⌣
.IP .VA
![Page 29: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/29.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA
![Page 30: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/30.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA
. ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA
![Page 31: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/31.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟
.IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA
![Page 32: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/32.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP
. ✡.min . ✠.0 . ✠.VA . ✡ .HA
![Page 33: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/33.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡
.min . ✠.0 . ✠.VA . ✡ .HA
![Page 34: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/34.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min
. ✠.0 . ✠.VA . ✡ .HA
![Page 35: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/35.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠
.0 . ✠.VA . ✡ .HA
![Page 36: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/36.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠.0
. ✠.VA . ✡ .HA
![Page 37: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/37.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠.0 . ✠
.VA . ✡ .HA
![Page 38: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/38.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA
. ✡ .HA
![Page 39: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/39.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡
.HA
![Page 40: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/40.jpg)
. . . . . .
Step3
Puttogetherabigchart!
. .sign/value..∞.0
..0.−1
.−1/4.−2/9
.−∞.0
.∞.0
.− .+ .+
.monotonicity..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.concavity..∞.0
..0.−3
.−− .++ .−−.⌢ .⌣ .⌣
.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA
![Page 41: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/41.jpg)
. . . . . .
Step4
-4 -3 -2 -1 1 2
H-3,-2����
9L H-2,-
1����
4L
H-1,0L
![Page 42: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/42.jpg)
. . . . . .
Outline
Thechecklist
Bigexample
Yourturn
![Page 43: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/43.jpg)
. . . . . .
Yourturn
Plotthesefunctions(Groupwork):
1. f(x) = x4 − 4x3 + 10
2. f(x) =34(x2 − 1)2/3
3. f(x) =x3
3x2 + 14. f(x) = (2− x2)3/2.
![Page 44: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/44.jpg)
. . . . . .
Graphof f(x) = x4 − 4x3 + 10
-2 -1 1 2 3 4 5
20
40
60
80
![Page 45: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/45.jpg)
. . . . . .
Graphof f(x) =34(x2 − 1)2/3
-2 -1 1 2
0.5
1.0
1.5
![Page 46: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/46.jpg)
. . . . . .
Graphof f(x) =x3
3x2 + 1
-2 -1 1 2
-0.6
-0.4
-0.2
0.2
0.4
0.6
![Page 47: Lesson 18: Graphing](https://reader033.fdocuments.us/reader033/viewer/2022052601/5593f3e91a28ab43278b46bf/html5/thumbnails/47.jpg)
. . . . . .
Graphof f(x) = (2− x2)3/2
-1.0 -0.5 0.5 1.0
0.5
1.0
1.5
2.0
2.5