4.1 The Gift - Utah Education Network · Interpret$expressions$that$represent$a$quantity$in$terms$of$its$context.$ a.$Interpret$parts$of$an$expression,$such$as$terms,$factors,$and$coefficients.$

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

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

(

4.2$All$in$the$Family$

A"Develop!Understanding+Task!!

We!have!studied!several!families!of!functions!over!the!past!few!years!including!linear,!exponential,!quadratic,!logarithmic,!square!root,!and!polynomials!in!general.!In!this!task,!we!will!examine!features!of!families!of!functions!from!our!previous!work!and!also!look!at!the!features!of!the!functions!we!call!rational!functions.!

Part!I:!Finding!features.!Use!the!table!to!describe!the!process!you!would!use!to!find!a!given!feature!based!on!the!function!and!then!write!how!this!feature!can!be!found!for!any!function.!

1.! The!process!I!use!to!f ind!roots!for!the!following!functions:!Linear! Logarithmic! Polynomial!(in!factored!form)!!!!!!

! !

In!general,!you!find!the!roots!of!a!function!by…!!!!

2.! The!process!I!use!to!determine!end!behavior!for!the!following!functions:!Quadratic! Exponential! Polynomial!!!!!!

! !

In!general,!you!determine!the!end!behavior!of!a!function!by…!!!!

3.! Asymptotes!occur!when…!!Logarithmic! Exponential! !!!!!

! !

In!general,!asymptotes!of!a!function!occur!when…!!!And!you!can!determine!asymptotes!by…!!!

www.flickr.com/photos/pagedooley/8207781361$

7

!Mathematics!Vision!Project!|!MVP!

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

(

(

!4.! The!Domain!of!a!function!is…!!Square!root! Logarithmic! Polynomial!!!!!

! !

In!general,!the!domain!of!a!function!is…!!!!!

Part!II:!Characteristics!of!Rational!Functions!

In!Birthday!Gift!we!saw!a!rational!function!used!to!model!the!situation!with!Chile.!Rational!

Functions!are!any!function!!(!)!such!that!! ! = !(!)!(!)!where!P!and!Q!are!polynomials!in!x!and!Q!is!

not!the!zero!polynomial.!In!other!words!a!rational!function!is!a!ratio!between!two!polynomials.!

Below!are!examples!of!rational!functions.!Like!other!functions!we!have!studied,!rational!functions!come!in!different!forms,!with!each!form!highlighting!different!aspects!of!the!function.!!

! ! = 1! !!!!!!!!!!! ! = !

! + 3 + !5

! − 2 !!!!!!!!!!!!! ! = !! + 5! − 1!! + 3!! − 6 !!!!!!!!!!!!! ! = ! − 3 ! + 1

!(! − 1)(! − 4)!

Based!on!other!functions!we!have!studied,!make!a!conjecture!as!to!how!you!would!find!the!following!features!of!a!rational!function.!

5.!!

Conjecture!as!to!how!to!determine!each!feature!of!a!rational!function:!

Find!the!features!of!this!function:!

! ! = ! − 3 ! + 1!(! − 1)(! − 4)!

To!find!roots…!!!

Roots:!

To!determine!end!behavior…!!!

End!Behavior:!

To!find!asymptotes…!!!

Asymptotes:!

!

!

!

8

!Mathematics!Vision!Project!|!MVP!

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

(

(

Part!III:!Parent!Functions!and!transformations.!

The!linear,!quadratic,!square!root,!and!rational!parent!function!are!below.!Sketch!a!graph!of!the!parent!function!and!then!sketch!the!graphs!of!“parents!transformed”.!Use!a!table!of!values!to!assist!you.!

5.!!! ! = !! 6.!!! ! = !!!!

7.!!ℎ ! = !!!

8.!!! ! = !!!

!!

! ! !

! ! = ! ! + 3! ! ! = ! ! + 3! ! ! = ℎ ! + 3! ! ! = ! ! + 3!

!!

! ! !

! ! = ! ! − 2 ! ! ! = ! ! − 2 ! ! ! = ! ! − 2 ! ! ! = ! ! − 2 !

!!

! ! !

!9.!!Each!function!type!has!characteristics!that!separate!it!from!other!families!of!functions,!yet!there!are!also!connections!to!be!made!across!families.!Summarize!this!task!by!explaining!how!you!see!rational!functions!as!different!from!other!functions!you!have!studied!as!well!as!by!how!they!are!similar!to!all!functions.!

!

!

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

9

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

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

(

!

All"in"the"Family–!Teacher'Notes!$A"Develop!Understanding+Task!!

Purpose:$$The$purpose$of$this$task$is$to$develop$student$thinking$about$how$families$of$functions$are$similar$and$also$to$highlight$how$they$are$different.$Specifically,$how$rational$functions$are$similar$and$different$to$other$functions.$In$Part$I,$students$solidify$their$understanding$by$generalizing$how$to$find$roots,$and$behavior,$asymptotes,$and$domain$of$any$given$function.$The$focus$of$this$task$should$mostly$be$on$Part$II,$where$students$use$their$background$knowledge$(and$information$from$Part$I)$to$further$develop$their$understanding$of$rational$functions$using$features$of$the$function.$Part$III$of$this$task$has$students$practice$graphing$transformations$of$functions,$while$at$the$same$time$becoming$more$comfortable$with$transformations$of$the$rational$parent$function,$! ! = ! !!.$$

$Standards!Focus:!!A.SSE.1!Interpret$expressions$that$represent$a$quantity$in$terms$of$its$context.$a.$Interpret$parts$of$an$expression,$such$as$terms,$factors,$and$coefficients.$!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.$$F.IF.8!Write$a$function$defined$by$an$expression$in$different$but$equivalent$forms$to$reveal$and$explain$different$properties$of$the$function.$!F.BF.3!Identify$the$effect$on$the$graph$of$replacing$f(x)!by!f(x)+k,!kf(x),!f(kx),!and!f(x+k)!for$specific$values$of$k$(both$positive$and$negative);$find$the$value$of$k$given$the$graphs.$Experiment$with$cases$and$illustrate$an$explanation$of$the$effects$on$the$graph$using$technology.$Include$recognizing$even$and$odd$functions$from$the$graphs$and$algebraic$expressions$for$them.$$!!Related:!A.CED.3,!F.IF.6!

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

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

(

Launch!(Whole!Class):!!Begin!by!telling!students!that!they!are!going!to!complete!this!task!in!three!parts.!Part!I!is!the!launch,!where!students!will!create!an!example!of!a!given!function!and!then!describe!how!they!would!find!the!given!feature!of!that!function!(how!they!would!find!the!domain,!for!example).!For!each!problem,!they!should!also!create!a!generalization!for!finding!each!feature.!When!they!have!completed!Part!I,!they!should!share!their!generalizations!of!how!they!would!find!each!of!the!features!for!any!given!function!and!then!move!on!to!complete!Part!II!with!their!small!group.!!Explore!(Small !Group):!!Have!students!work!independently!on!Part!I!and!then!have!them!discuss!their!generalizations!with!each!other!before!moving!on!to!Part!II.!Monitor!student!thinking!as!they!work,!listening!to!students!in!their!small!groups!justify!their!generalizations!about!how!they!find!features!of!functions.!!As!students!work!on!question!5,!note!which!groups!are!choosing!functions!b,!c,!and!d.!As!students!are!discussing!how!to!find!the!features!of!the!function!they!have!chosen,!ask!them!why!they!chose!the!function!they!did.![While!function!b!‘looks!easy’!to!begin!with,!finding!the!roots!requires!simplifying!the!function—do!not!discourage!students!to!change!their!function!if!this!is!the!one!they!chose!to!work!on,!however,!if!they!say!the!root!is!at!(e5,!0),!ask!them!to!show!their!evidence!that!this!is!true.!Look!for!a!group!of!students!who!complete!function!b!with!the!correct!answers!for!the!roots!and!the!asymptotes.!While!function!c!does!allow!us!to!find!the!end!behavior!easily,!most!students!will!probably!not!choose!this!one!to!work!on!(since!hopefully,!they!will!realize!that!d!is!already!factored!therefore!easier!to!find!the!roots).!After!students!have!spent!a!‘decent!amount!of!time’!on!question!5!(and!you!know!who!you!plan!to!have!share!out!during!the!large!group!discussion,!have!the!class!move!onto!Part!III!of!the!task.!Have!students!work!on!this!part!if!the!task!long!enough!that!the!whole!group!discussion!can!also!include!student!thinking!for!question!9.!!!Discuss!(Whole!Class): !You!may!wish!to!start!the!whole!group!discussion!by!selecting!a!student!who!correctly!worked!through!function!b!in!question!5!(the!correct!answers!of!the!roots!being!at!(3,0)!and!at!(e2,!0)!and!that!the!asymptotes!are!at!x!=!2!and!x!=!4).!Have!them!go!through!their!process!for!finding!the!features!(using!their!explanations!from!Part!I).!Next,!have!a!student!share!their!solutions!for!function!d.!After!this!student!shares!their!results!(and!explains!how!they!found!the!roots!and!asymptotes),!ask!the!class!why!function!d!is!in!an!easier!‘form’!than!function!c.!This!is!a!good!time!to!talk!about!the!forms!of!a!rational!function!and!to!highlight!that!function!d!is!the!best!form!to!graph!these!features.!Later!in!the!module,!students!will!spend!more!time!discussing!end!behavior!so!you!may!wish!to!focus!the!attention!of!this!task!on!how!it!is!best!to!graph!rational!functions!when!they!are!in!factored!form!and!to!make!sure!students!know!how!to!find!the!roots,!asymptotes,!and!domain!at!this!time.!To!conclude!the!discussion,!discuss!question!9.!Summarize!this!task!by!explaining!how!you!see!rational!functions!as!different!from!other!functions!you!have!studied!as!well!as!by!how!they!are!similar!to!all!functions.!

Aligned!Ready, !Set , !Go!Homework:! !Rational!Functions!4.2!

!

Name%%%%%%%%%%%%%%%%%%%%%%Rational%Expressions%and%Functions% 4.2$!

!

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

!

!!

!

!!

Ready,$Set,$Go!$$!!!!!!!Ready$Topic:%Arithmetic%with%rational%numbers%%Simplify%each%of%the%following%expressions%1.% 2.%% 3.%%

12 +

13%%%%

45 −

13%

72 +

25%

4.% 5.% 6.%12 ! ∙ !

13%

%%%

16 ÷

25%

38 ! ∙ !

25%

%7.%Explain%the%standard%process%for%adding%and%subtracting%two%fractions?%%%%%%8.%Explain%the%standard%procedure%for%multiplying%two%fractions?%%%%%9.%Explain%you%procedure%for%dividing%two%fractions?%%%%

www.flickr.com/photos/pagedooley/8207781361%

10

Name%%%%%%%%%%%%%%%%%%%%%%Rational%Expressions%and%Functions% 4.2$!

!

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

!

!!

!

!!

$Set$Topic:%Features%and%families%of%functions%%Find%the%roots%%and%domain%of%each%function.%10.%% 11.% 12.%! ! = ! + 5 ! − 2 ! − 7 % ! ! = !! + 7! + 6% ! ! = (! + 1)(! + 6)

(! + 2) %%%%%%%%%%%%%

13.% 14.% 15.%ℎ ! = −! + 2

(! + 1)(! + 5)%! ! = ! + 1 ! − 1 ! − 5 % ! ! = !! − 2! − 24%

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

%%%%

11

Name%%%%%%%%%%%%%%%%%%%%%%Rational%Expressions%and%Functions% 4.2$!

!

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

!

!!

!

!!

Go$!

Topic:!Solve!each!inequality.!(Hint:!It!might!be!helpful!to!sketch!the!graph!and/or!to!create!a!sign!line).!

10.%% 11.% 12.%! + 5 ! − 2 ! − 7 ≥ 0% !! + 7! + 6 < 0% 3! − 5 > 2%

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

13.% 14.% 15.%−! + 2

(! + 1)(! + 5) ≥ 0% ! + 1 ! − 1 ! − 5 < 0% !! − 2! − 24 > 0%%%%%%%%%%%%%%%%

!!!!!!

12