4.1 the gift - utah education network · !mathematics!vision!project!|!mvp!!...

!Mathematics!Vision!Project!|!MVP!!

Licensed(under(the(Creative(Commons(Attribution4NonCommercial4ShareAlike(3.0(Unported(license.!(

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4.6$Features$of$Rational$Functions$A"Solidify!Understanding+Task!!

Part!I:!Identifying!features!of!rational!functions.!

Using!prior!knowledge!of!other!functions!you!have!worked!with,!determine!the!features!of!each!rational!function!and!then!sketch!a!graph.!(features!include!intercepts,!domain,!asymptotes,!end!behavior*,!and!any!other!features!you!notice!about!each!rational!function).!

!

1. ! ! = !!!!!! !!! !

!!xGintercept(s):!!

yGintercept:!

Vertical!aysmptote(s):!

!

Proper!or!Improper:!

End!behavior!asymptotes*:!

!

Sketch!the!features!on!the!graph!below:!

!

!

Create!a!Sign!Line,!then!complete!graph!above:!

!

!

2. ! ! = !!!(!!!)(!!!)!

!!

xGintercept(s):!!

yGintercept:!


!



!


!

!


!

!

3. !(!) = (!!!)(!!!)!!! !

!!xGintercept(s):!!

yGintercept:!


!



!


!

!


!

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4. Has!your!conjecture!about!the!end!behavior!of!rational!functions!changed?!Rewrite!your!

conjecture!(including!any!modifications/changes!you!may!have!made):!

!

!

5. Complete!the!sentence:!The!domain!of!a!rational!function!is!all!real!numbers!except!!

_________________________________________________________________________!.!

!

Part!II:!Sketch!a!graph!of!each!rational!function.!Start!by!graphing!the!features!of!the!function!(same!

features!from!Part!I),!then!‘fill!in’!the!rest!of!the!graph!using!a!sign!line!to!guide!you.!

6.!Asymptotes!and!sign!lines!

a.!f(x)!=!1

x2G9!! ! ! b.!f(x)!=!!

G3

x2G3x+2!! ! c.!f(x)!=!!

2

(xG1)2(x+2)!! !!!!!d.!f(x)!=!

1

x2!!!!!

! ! !

! ! ! !

!

7.!Roots,!asymptotes,!and!sign!lines!

a.!f(x)!=!!Gx

x2G9!! ! ! b.!f(x)!=!!

x+2

x2G2x+1!! ! c.!f(x)!=!

(xG1)(x+2)

(x3+4x2+3x)!!! ! !

! !

!

!

d.!f(x)!=!3x2

x2G9!

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4.6$Features$of$Rational$Functions–!Teacher'Notes!$A"Solidify!Understanding+Task!Purpose:$$The$purpose$of$this$task$is$for$students$to$solidify$their$understanding$of$the$features$of$rational$functions$and$to$‘curve$sketch’$graphs$of$rational$functions$using$the$features$and$a$sign$line.$$

Standards.Focus:..F.IF.4.For$a$function$that$models$a$relationship$between$two$quantities,$interpret$key$features$of$graphs$and$tables$in$terms$of$the$quantities,$and$sketch$graphs$showing$key$features$given$a$verbal$description$of$the$relationship.$Key!features!include:!intercepts,!intervals!where!the!function!is!increasing/decreasing,!positive!or!negative;!relative!maximums!and!minimums;!symmetries;!end!behavior;!and!periodicity.*!$.F.IF.7d.Graph$functions$expressed$symbolically$and$show$key$features$of$the$graph,$by$hand$in$simple$cases$and$using$technology$for$more$complicated$cases.*$d.$Graph$rational$functions,$identifying$zeros$when$suitable$factorizations$are$available,$and$showing$end$behavior.$$$Related:.F.IF.5,.F.IF.9,.A.CED.1.

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Launch!Part!I(Whole!Class): !Start!this!task!by!presenting!students!with!a!rational!function!

(make!one!up!or!have!them!look!at!question!1)!and!ask!them!how!they!would!find!the!roots,!

vertical!asymptote,!and!other!features.!Let!students!know!that!they!will!be!graphing!rational!

functions!in!this!task!by!determining!features!of!a!function!and!then!sketching!the!graph!as!a!‘curve!

sketch’.!In!your!example,!you!will!want!to!model!how!to!use!a!sign!line!to!determine!where!a!

function!is!positive!and!negative!before!students!work!on!the!task.!

!

Explore!Part !I(small !groups):!Monitor!student!thinking!as!they!work!on!Part!I!in!small!groups.!

Help!groups!move!along!in!their!thinking,!and!move!to!the!whole!group!discussion!as!soon!as!you!

have!students!who!have!correctly!completed!questions!1!–!3.!!

!!

Discuss!Part!I(Whole!Class): !Once!several!students!have!completed!questions!1H3,!select!three!

students!to!present!these!problems,!leaving!them!visible!for!the!whole!class!to!compare/contrast.!

At!this!time,!solidify!student!understanding!of!end!behavior!asymptotes!(at!least!the!difference!

between!when!the!asymptote!is!horizontal!and!when!it!is!not)!and!how!to!create!a!sign!line,!as!well!

as!making!sure!students!understand!how!to!find!all!other!features!of!the!function.!!

Part!II!has!students!practicing!finding!the!features!as!well!as!graphing,!with!the!end!behavior!

asymptotes!all!being!horizontal.!Other!end!behavior!asymptotes!occur!in!the!next!task.!

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Explore!Part!II !(Small !Group):!Monitor!student!work!for!Part!II,!redirecting!students!who!

make!mistakes.!Choose!a!couple!of!problems!from!this!section!that!students’!struggle!with!and!have!

this!be!the!focus!of!your!discussion.!

Discuss!Part!II !(Whole!Class): !For!the!whole!group!discussion,!select!problems!that!seem!to!

stump!students!and!go!over!these,!highlighting!areas!of!concern!and!solidifying!student!

understanding.!In!this!case,!you!may!wish!to!have!students!who!correctly!answered!the!problem!

share.!Or…!you!may!choose!to!have!different!groups!do!different!problems!and!then!have!the!class!

compare!problems!(or!see!if!there!are!any!they!do!not!agree!with).!During!the!discussion,!it!would!

also!be!good!to!ask!if!there!are!any!observations!they!are!making!(for!example,!the!multiplicity!of!2!

for!the!root!at!zero!in!7d…!and!how!we!also!see!that!the!sign!does!not!change!when!the!vertical!

asymptote!is!caused!by!a!‘double!factor’!in!problems!like!6c!or!6d).!!

!

Aligned!Ready, !Set , !Go!Homework:!Rational!Functions!4.6!

Name%%%%%%%%%%%%%%%%%%%%%%Rational%Expressions%and%Functions% 4.6$!

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!Mathematics!Vision!Project!|!MVP!!!Licensed!under!the!Creative!Commons!Attribution4NonCommercial4ShareAlike!3.0!Unported!license!

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Ready,$Set,$Go!$Ready$Topic:%Geometry,%volume,%surface%area%and%right%triangles%%%Find%the%indicated%values%for%the%geometric%figures%below.%%1.%Volume:%%%%%%Surface%Area:%

2.%Volume:%%%%%%Surface%Area:%

%%%%%%%%%%%%%%%%%%%%rectangular%prism% %

regular%hexagonal%pyramid%%%

3.%Solve%the%right%triangle.%∠! =%!" =%!" =%

4.%Solve%the%right%triangle.%∠! =%∠! =%!" =%

%

% %5.%Area:%%%%%Perimeter:%

6.%Area:%%%%%Perimeter:%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%regular%pentagon%

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regular%octagon%

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!!

!

!!

7.%Provide%a%sketch%of%a%decomposition%of%a%circle%that%shows%how%the%area%of%a%circle%can%be%approximated%by%a%rectangle.%

%

%

$Set$Topic:%Features%of%Rational%Functions%%Provide%the%features%of%the%function,%be%sure%to%include%a%sketch%of%the%function%as%well%as%domain,%range,%increasing,%decreasing,%intercepts,%maximum/minima,%end%behavior,%asymptotes%8.% 9.%

! ! = !!! + 1%

%%%%%%%%%%%%%

! ! = !!! − 1%

10.% 11.%ℎ ! = ! − 2

! − 1%%%%%%%%%%%%

! ! = (2 − !)(! + 3)! − 1 %

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!


!

!!

!

!!

12.% 13.%! ! = (! + 3)(! + 2)(! − 2)(! − 3)(! + 10)

! − 100 %%%%%%%%%%%%%%%%%%

! ! = − 1!!%

%%Go$Topic:%Simplifying%Rational%Expressions%%Rewrite%each%of%the%rational%expressions%in%its%simplest%form.%14.% 15.% 16.%

!! + 8! + 12!! + 3! − 18%

%%%%%%%

!! − 3! − 40!! − 11! + 24%

!! + 8! + 12!! + 3! − 18 +

!! − 3! − 40!! − 11! + 24%

17.% 18.% 19.%!! + 5! − 36(! − 4) ! ∙ !3(! + 2)! − 9 % ! + 3

!! − 4 +!! + 4! + 6

! − 2 % !! − 2! − 3! + 1 !÷ ! − 35 %

%%!!

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