4.7$Graphing$Rational$Functions$$ A"Practice!Understanding+Task!
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1 35 4.7$Graphing$Rational$Functions$$ A"PracticeUnderstanding+Task PartI:SeeingStructure Foreachfunction,determineintercepts,domain,asymptotes,andcompleteasignline.Usethis informationtosketchthegraph. 1. = ) 2. = ) ) 3. = )) ) 4. = ) 5. = 6. = 7.WhatobservationsdoyounoticeaboutthevariousgraphsfromPartI? PartII:SeeingMoreStructure 8.Determinethefeaturesandthensketchthegraphsofthefunctions. a. = b. = )) ) c. = )) )) d. = ) )) e. = ) ) f. = ) MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
2 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense. Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine Equation: Roots: yqintercept: Vert.Asymp.: Endbehavior: SignLine 36
3 4.7GraphingRationalFunctions TeacherNotes A"PracticeUnderstanding+Task Purpose:Thepurposeofthistaskisforstudentstobuildfluencyinfindingfeaturesofrational functionsandto curvesketch graphs. Standards.Focus:.. F.IF.4.Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesof graphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbal descriptionoftherelationship.keyfeaturesinclude:intercepts,intervalswherethefunctionis increasing/decreasing,positiveornegative;relativemaximumsandminimums;symmetries;end behavior;andperiodicity.*. F.IF.7d.Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin simplecasesandusingtechnologyformorecomplicatedcases.* d.graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and showingendbehavior. Related:.F.IF.5,.F.IF.9,.A.CED.1. LaunchPartIWholeClass):SLaunchthistaskbylettingstudentsknowthattheywillbe practicinggraphingrationalfunctionsusingthefeaturesofthefunction.foreachproblem,they shouldlabelthefeaturesusingtheirnotebookortheincludedworksheet)andsketchthegraph. ExplorePartIsmallgroups):Asyoumonitor,redirectstudentsifyounoticeerrors.Atthis time,theyshouldbepracticingsobedirectinguidingtheirworkiftheyarestuckormakingerrors. DiscussPartIWholeClass):Oncemoststudentshavecompletedthefirstset,selectstudents toshare.determineifyouwanttohaveallproblemsfrompartidiscussed.ifyourclassisdoing well,youmightchoosetoonlysharethosethatneeddiscussion.atthistime,furthersolidify studentunderstandingofendbehaviorasymptotesallthreesituations)anddiscussobservations question7).observationsmaybeaboutendbehavior,multiplicity,etc. ExplorePartIISmallGroup):Havestudentsmoveontolabelandgraphtheremaining problems.youmaywanttohavestudentsworkonacoupleofproblemsindividuallyfirstbefore workinginsmallgroups.monitorstudentworkforpartii,correctingerrorsastheycomeup.each probleminquestion8isunique,soplantogoovertheseasawholegroup. DiscussPartIIWholeClass):Selectstudentstosharetheirworkchooseonestudentfor eachproblem).letstudentsknowthattheyshouldexplaineachfeatureofthefunctionasthey present,butshouldfocusonwhatmakestheirproblemunusual. AlignedReady,Set,GoHomework:RationalFunctions4.7 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
4 NameRationalExpressionsandFunctions 4.7$ Ready,$Set,$Go$ Ready$ Topic:Findingtheasymptotes Solvingrationalequations,besuretocheckyousolutions = = = = = = Set$ Topic:Keyfeaturesofrationalfunctions Foreachofthegivenfunctionslistthekeyfeaturesofthefunctionincludingthedomain,range, increasinganddecreasingintervals,theintercepts,asymptotes,endbehavior = 2 = 2 MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
5 NameRationalExpressionsandFunctions 4.7$ = + 3 = 1 2) + 2) Sketchagraphandcomparethegivenfunctions.Explainwhatyouthinkaccountsforthe similaritiesanddifferencesbetweenthetwofunctions. 11.a. b. Similarities: Differences: Why? 12.a. Similarities: Differences: Why? = = 1 1 2) + 2) b. = = 1 2) + 2) MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
6 NameRationalExpressionsandFunctions 4.7$ Go$ Topic:Attributesofrationalfunctions 13.Howdoyouknowifafunctionisevenoroddorneither? 14.Howdoyoudeterminetheendbehaviorofafunction? 15.Istheendbehaviorforarationalfunctionalwaysthesame? 16.Whatisthedifferencebetweenaproperandanimproperrationalfunction? 17.Whatattributedoallproperrationalfunctionshave? MathematicsVisionProject MV P LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
4.5$Watch$Your$ Behavior $ A"Develop!Understanding+Task!
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x(t)*=*2t"+*1* y(t)*=*3t* *4**
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Quadratic and Polynomial Inequalities in one variable have look like the example below.
Section 8 4: Polynomial Inequalities in One Variable Quadratic and Polynomial Inequalities in one variable have look like the example below. x 2 5x 6 0 (x 2) (x + 4) > 0 x 2 (x 3) > 0 (x 2) 2 (x + 4) 0
Chapter 2 Equations and Inequalities. 2. z = -6 then z + 11 = z + 11 = 5 3. m = -5 then 6 - m = 6 -(-5) 6 - m = 11
Chapter Equations and Inequalities Section. Definitions and Solving Equations Practice... 7 - y = - 7 - (8) - - = - 8 is a solution. - n = (n-) - (6) (6 - ) - () - 6 is not a solution. (x - ) = x - (-
first later later still successful collision ( reaction ) low conc. both high conc. blue high conc. both low conc. red
Collision theory Basic idea (basic premise) http://www.chem.iastate.edu/group/greenbowe/sections/projectfolder/animations/no+o3singlerxn.html - before molecules can react, they must collide. H 2 + I 2
Absolute Value. That was easy
Absolute Value The absolute value of a number is its distance from 0 on a number line. Absolute value is always nonnegative since distance is always nonnegative. Let s write an example using mathematical