Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4...
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NAME:___________________________ Math ________ , Period ____________ Mr. Rogove Date:__________ Study Guide: Slope and Linear Equations 1 Study Guide: Slope and Linear Equations Rates and Unit Rates A rate is a proportional relationship between two quantities. Unit rate is a rate where the second quantity is 1. Example: Pauline can mow 35 square feet of lawn is 2.5 minutes. (this is a rate). She can mow 14 square feet per minute (this is a unit rate). We set up proportions to find unit rates and write linear equations in 2 variables. We solve for y. Example: = 35 2.5 Set up proportion 2.5 = 35 Multiply the numerators by the denominator of the other fraction. = 14 Solve for y. Rates are directly proportional if their graph lies on a straight line AND passes through the origin. Slope Slope is a measure of steepness or slant of a line. It is also the measure of the rate of the change of a line. Slope can be expressed as a unit rate: How many vertical units does a line move for every one horizontal unit? The slope formula states that: = − − = = − − Where the coordinates of point P on a line are represented by = ( ! , ! ) and the coordinates of point R on the line are represented by = ( ! , ! ).
Transcript of Study Guide: Slope and Linear Equationsmrrogove2016.weebly.com/uploads/4/3/0/0/43009773/g... · 4...
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations1
Study Guide: Slope and Linear Equations
Rates and Unit Rates Arateisaproportionalrelationshipbetweentwoquantities.Unitrateisaratewherethesecondquantityis1.Example:Paulinecanmow35squarefeetoflawnis2.5minutes.(thisisarate).Shecanmow14squarefeetperminute(thisisaunitrate).Wesetupproportionstofindunitratesandwritelinearequationsin2variables.Wesolvefory.Example:
𝑦𝑥 =
352.5
Setupproportion
2.5𝑦 = 35𝑥
Multiplythenumeratorsbythedenominatoroftheotherfraction.
𝑦 = 14𝑥 Solvefory.RatesaredirectlyproportionaliftheirgraphliesonastraightlineANDpassesthroughtheorigin.
Slope Slopeisameasureofsteepnessorslantofaline.Itisalsothemeasureoftherateofthechangeofaline.Slopecanbeexpressedasaunitrate:Howmanyverticalunitsdoesalinemoveforeveryonehorizontalunit?Theslopeformulastatesthat:
𝒎 =𝒑𝟐 − 𝒓𝟐𝒑𝟏 − 𝒓𝟏
= 𝒓𝒊𝒔𝒆𝒓𝒖𝒏 =
𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒚− 𝒗𝒂𝒍𝒖𝒆𝒔𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒙− 𝒗𝒂𝒍𝒖𝒆𝒔
WherethecoordinatesofpointPonalinearerepresentedby𝑃 = (𝑝!,𝑝!)andthecoordinatesofpointRonthelinearerepresentedby𝑅 = (𝑟!, 𝑟!).
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations2
Slope Triangles Wecanusetheconceptofslopetrianglestoprovetheslopeofalineisthesamebetweenanytwopointsontheline:
• ShowAAsimilarityoftwotriangles.• Knowthatcorrespondingsidesofsimilartrianglesare
proportional…andthismeansthattheslopeofalineisthesamefortheentiredistanceoftheline.
Ifyoucreateseveraldifferentrighttrianglesusingthepointsabove,youcanseethatseethatallofthetrianglesaresimilartrianglesusingtheconceptofAAsimilarity.Example:∆𝑃𝐿𝐾 ~∆𝑅𝑀𝐾becausetheyshareacommonangleANDtheyeachhaverightangles…hence,theymeettheAAcriterionforsimilarity.Thismeanstheproportionsofcorrespondingsidesofeachtriangleareequal.Example:Theproportionsofthecorrespondingsidesofthetrianglesmentionedaboveareequal…
𝑅𝑀𝑀𝐾
=𝑃𝐿𝐿𝐾
126 =
105
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NAME:___________________________ Math________,Period____________
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StudyGuide:SlopeandLinearEquations3
Slope (cont’d)
PositiveSlopeLeft-to-rightinclining
lines
NegativeslopeLeft-to-rightdeclining
lines
ZeroslopeHorizontalLines
Thegreatertheabsolutevalueoftheslope,thesteepertheline!!Horizontallineshavezeroslope,andarewrittenas𝑦 = 𝑐.Verticallineshaveanundefinedslopeandarewrittenas𝑥 = 𝑐.
Equations of Lines StandardFormofanequation𝒂𝒙 + 𝒃𝒚 = 𝒄where𝑎, 𝑏, and 𝑐areallintegersand𝑎 > 0Wecangraphalineknowingthestandardformofalinearequationbyidentifyingthex-interceptandy-intercept.SlopeInterceptformofanequation𝒚 = 𝒎𝒙 + 𝒃wheremistheslopeandbisthey-intercept.Wecangraphalineknowingtheslope-interceptformofalinearequationbyidentifyingthey-interceptandusingtheslopetographanotherpoint.Minimumrequirementsforfindingtheequationofaline:1.Slopeandonepointontheline(oftentheonepointwillbethey-intercept)2.Twopointsontheline.(thiscanbethex-interceptandy-intercept)Convertfromslope-interceptformtostandardform:
• Gatheraxandbytermsonleftsideoftheequation.• Makesurethata,b,andcareallintegersandaispositive.
Convertfromstandardformtoslope-interceptform:
• Solvefory.
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations4
Problem set Completetheproblemsbelowbeforetakingourtest.Ifyouhaveanyquestions,besuretoaskeitheratareviewsession.Aphonecompanyhastwoplansfordatathatareasfollows:PlanAincludesanupfrontservicechargeof$45permonth,andcharges$2.50foreachGBofdataused.PlanBincludesnoservicecharge,andcharges$7.50foreachGBofdata.1.Writeequationsthatrepresenteachcellphoneplan,whereyisthecostofthedata,andxistheamountofdata(inGB).Identifytheslopeandexplainwhatitmeans.2.Graphtheequationsonthecoordinateplane.3.WhenisitbetterforyoutogowithplanA?WhendoesitmakesensetogowithplanB?SophiaandColbyweregiventhesetwopoints(2,−5)and(−3, 5),andtoldtowritetheequationoftheline.Sophiathoughttheequationwas𝑦 = −2𝑥 − 1.Colbythoughtshewaswrongandthattheequationwas𝑦 = − !
!𝑥 − 4.Who’sright?Why?
Whatdidtheotherpersondowrong?
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations5
AgallonofmilkatWholeFoodscost$6.ThegraphbelowcomparesthecostofmilkatWholeFoodsandSafeway.Estimatethecostofahalf-gallonofmilkatSafeway.Justifyyourreasoning.12candybarsatTargetcost$15.00.ThegraphbelowcomparesthecostofcandybarsatTargetandCostco.Estimatethecostof1candybaratCostco.Justifyyourreasoning.Inadditiontothesefourproblems,thefollowingexerciseshavebeenassignedinKhanAcademy.Pleasechecktheboxtoindicatethatyouhaveeithercompleted5inarowor8outof10.
¨ Graphfromslope-interceptequation¨ Slope-interceptequationfromgraph¨ ConvertingBetweenSlope-InterceptandStandardForm
Cost($)
Milk(gallons)
WholeFoods
Safeway
Costco
Target
Candybars
Cost($)
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations6
Usethegraphbelowtocompletethequestions.Onpage2,weidentifiedtwosimilartriangles(∆𝑃𝐿𝐾 𝑎𝑛𝑑 ∆𝑅𝑀𝐾).Identify3othersimilartriangles,andshowthattheyaresimilarbylookingattheproportionsofthecorrespondingsides.3triangles:Proportionsofcorrespondingsides(seepage2foranexample):Whatistheproportionofthecorrespondingsidesalsocalled?(ifyouconsiderthecorrespondingverticalsidesinthenumeratorandthecorrespondinghorizontalsidesinthedenominator).
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