Section 5.1 - Different Forms of Linear Equations Lambrick ...€¦ · 192 ♦ Chapter 5 - Linear ... Graphing a Line Using the Slope and y-Intercept Step 1: Write the equation in

Section 5.1 - Different Forms of Linear Equations ♦ 191

Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.

Standard Form of a Linear Equation

IfA,BandCarerealnumbers,theequationAx By C+ = iscalledthestandard formoftheequationofaline. Wheneverpossible,itisbesttowritetheequationwithA,BandCasintegers,andA 0$ .

Forexample: x y3 4- + = canbeexpressedas x y3 4- =- ←multiply each term by ( − 1)

x y32 2 3+ = canbeexpressedas x y2 6 9+ = ←multiply each term by 3

Slope - Intercept Form of a Linear Equation

Theequationy mx b= + istheslope-intercept formoftheequationofaline.They-interceptofthelineis (0,b),andtheslopeofthelineism.

Thestandardformofanequationofalinecanbere-writteninslope-interceptformasfollows:

Ax By C By Ax C yBA x

BC

" "+ = =- + =- +

TheslopeofAx By C+ = isBA-

They-interceptofAx By C+ = isBC ,andthepointatwhichthegraphcrossesthey-axisis ,

BC0` j.

Forexample,considerthelinearequation x y2 3 12- = .Theslopeinterceptformofthelinecanbefoundin twoways:

x y2 3- =12 mBA

32

32=- =-

-=

y3- = x2 12- + or y-intercept=BC

312 4=-=-

y = x32 4- y x

32 4= -

Theslopeofthelineis32 ,andthey-interceptis(0,−4).

runrise

x

y

5.1

Lambrick Park Secondary

192 ♦ Chapter 5 - Linear Equations


Graphing a Line Using the Slope and y-Intercept

Step1: Writetheequationinslope-interceptformbysolvingfory. Step2: Identifythey-intercept(0,b)andgraphthispoint. Step3: Graphanotherpointusingtheslope,countingfromthey-intercept. Step4: Drawthelineconnectingthetwopointstoobtainthegraph.

Graph x y3 2 12+ = byusingtheslopeandy-intercept.

►Solution: Step1: x y3 2+ =12 y2 = x3 12- +

y = x23 6- +

Step2: They-interceptis(0,6):markthispoint.

Step3: Theslopeis: runrisem

23= =- .

From(0,6),gototheright2units, andgodown3units,toobtainthe point(3,2).

Step4: Drawthelinethroughthepoints (0,6)and(3,2).

Graphing a Line Using the Slope and a Point

Step1: Locateandgraphthegivenpoint. Step2: Graphanotherpointusingtheslope,countingfromthefirstpoint. Step3: Drawalineconnectingthetwopointstoobtainthegraph.

Graphthelinethrough(−2,−4)withslope3.

►Solution: Theslopeis3,therefore,fromthepoint(−2,−4),goup3units,andtotheright1unitto obtainthepoint(−1,−1).

Example 1

Example 2

(0,6)

(3,2)–3

+2

x

y

+3

+1 x

y




Writing an Equation of a Line Using a Slope and a Point

Bysubstitutinggivenvaluesforaslopeandpointofalineintoy mx b= + ,theline’sequationcanbefound.

Writetheequationofthelinewithslope2thatrunsthrough(−4,1)inslopeintercept-form.

►Solution: Thepoint(−4,1)givesax-valueof−4anday-valueof1.

y mx b= + " b1 2 4= - +^ h b1 8=- +

b 9=

Therefore,theequationofthelineisy x2 9= + .

Point - Slope Form of a Linear Equation

Theequationy y m x x1 1- = -^ histhepoint-slopeequationofaline.Thegivenpointis ,x y1 1^ handthe slopeofthelineism.Thisformulacomesfromre-arrangingthedefinitionofslope,m

x xy y

1

1=-- .

Writetheequationofalinewithslope2thatpassesthrough(−4,1)inslopeinterceptform.

►Solution: Substitutingthegivenpointandslopeintothepoint-slopeequationgives:

y y m x x1 1- = -^ h " y x1 2 4- = - -^^ hh y x1 2 4- = +^ h y x1 2 8- = +

y x2 9= +

Writetheequationofalinewithslope54 thatpassesthrough(3,−2)instandardform.

►Solution: Substitutingthegivenpointandslopeintothepoint-slopeequationgives:

y y m x x1 1- = -^ h " y x254 3- - = -^ ^h h

y x254 3+ = -^ h

y x5 2 4 3+ = -^ ^h h

y x5 10 4 12+ = -

x y4 5 22- =

Example 3

Example 4

Example 5




5.1 Exercise Set

1. Completeeachstatement.

a) Theformulaforthepoint-slopeformofalineis.

b) Intheequationy mx b= + ,(0,b)iscalledthe.

c) Theequationy mx b= + iscalledtheformoftheequationofaline.

d) Thestandardformoftheequationofalineis.

e) TheslopeofAx By C+ = is.

f) They-interceptofAx By C+ = is.

2. Findtheslopeandy-intercept.

a) x y3 2 6- = slope b) x y4 3 12+ = slope

y-intercept y-intercept

c) x y2 5 7- =- slope d) x y5 2 0+ = slope


e) x y4 4- =- slope f) x y6 3- =- slope


3. Rewritethestandardformequationinslope-interceptform.

a) x y2 6+ = b) x y3 4- =

c) x y4 3 12+ = d) x y2 3 6- =

e) x y5 4 3+ = f) x y6 3 4- =




4. Rewritetheslope-interceptequationinstandardform.

a) y x2 1=- + b) y x3 1= -

c) y x3= d) y x32 1=- +

e) y x43 5= + f) y x

52

21=- +

5. Rewritethepoint-slopeequationinslope-interceptform.

a) y x2 3 1- = +^ h b) y x4 2 1+ =- -^ h

c) y x131 2- = +^ h d) y x4

52 3+ =- -^ h

e) y x32

41 8- = -^ h f) y x

41

21

32- = +` j

6. Rewritethepoint-slopeequationinstandardform.

a) y x2 3 1- = +^ h b) y x4 2 1+ =- -^ h

c) y x131 2- = +^ h d) y x4

52 3+ =- -^ h

e) y x32

41 8- = -^ h f) y x

41

21

32- = +` j




7. Matcheachdescriptionwithanequation.

a) Slope=−3,passingthrough(−1,2) i) y x3=

b) Slope=3,y-intercept(0,−6) ii) y x31=-

c) Passingthrough(0,0)and(3,−1) iii) y x3=-

d) Passingthrough(0,0)and(−1,3) iv) x y3 6- =

e) Passingthrough(2,0)and(0,−6) v) x y3 6- =

vi) y x2 3 1- =- +^ h

vii)y x2 3 1+ =- -^ h

8. Matcheachequationwiththegraphitmostcloselyresembles.

a) y x 2= - i) ii)

b) y x 2=- -

c) y x 2=- +

d) y x 2= +

iii) iv)

9. Writetheequationofeachlineinslope-interceptform.

a) (0,2),m=2 b) (0,−3),m=21

c) (0,3),m=0 d) (0,−2),m=32-

e) ,021-` j,m=

43- f) (0,2.3),m=0.4

x

y

x

y

x

y

x

y




10. Graphthelinearequation.

a) x y4 3 12- = b) y x32 4=- +

c) y x321 4- = +^ h d) x y2 3 10+ =

e) y x232 5+ =- +^ h f) x y5 2 0- =

g) y x25

21

23- =- +` j h) y x

35

27= -

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y




11. Writetheequationinstandardform,slope-interceptform,andpoint-slopeform.

a) b)

standardform standardform

slope-interceptform slope-interceptform

point-slopeform point-slopeform

c) d)




e) f)




x

y

x

y

x

y

x

y

x

y

x

y


Section 5.2 - Special Cases of Linear Equations ♦ 199


Horizontal Lines

Ahorizontallinecanbethoughtofasallthepointsonagraphwhereyhasthesamevalue.Fromsection5.1, itwasshownthattheslopeofahorizontallineis0.

Usingaslopeof0intheslope-interceptequationofaline,y mx b y x b y b0" "$= + = + =

Equation of a Horizontal Line with y-Intercept k

y k=

Forexample:y 3=

Vertical Lines

Averticallinecanbethoughtofasallthepointsonagraphwherexhasthesamevalue.Fromsection5.1it wasshownthattheslopeofaverticallineisundefined.

Theequationofaverticallineisx=kbydefinition,sincetheslopeisundefined.

Equation of a Vertical Line with x-Intercept k

x k=

Forexample:x 3=

x

y

x

y

Special Cases of Linear Equations5.2




Writing the Equation of a Line Through Two Points

Withourknowledgefromsection6.1itispossibletowritetheequationofalinewhenthecoordinatesoftwo pointsonthelineareknown.

WritetheequationofthelinepassingthroughA(5,2)andB(1,−4)inslope-interceptform.

►Solution: First,findtheslopeoftheline.

mx xy y

5 1

2 4

46

23

2 1

2 1=

--

=-

- -= =

^ h

Pickeitherpoint,andsubstituteitintothepoint-slopeformequation.Forthisexample,A(5,2) isused.

y y m x x1 1- = -^ h → y x223 5- = -^ h

y x223

215- = -

y x23

215 2= - +

y x23

211= -

Parallel and Perpendicular Lines

Inchapter5itwasshownthatparallellineshavethesameslopebutdifferenty-interceptsandperpendicular lineshaveslopesthatarenegativereciprocalsofeachother.Wecannowusetheseconceptstodetermineif equationsareparallel,perpendicularorneither.

Inthesystemofequationsx yx y2 32 6

- + =+ =' 1,determineifthelinesareparallel,perpendicular,

orneither.

►Solution: Theslopeofthestandardformoftheequationofaline,Ax By C+ = ,isBA- .

x y2 6+ = hasslope21-

x y2 3- + = hasslope2

Theslopesarenegativereciprocalsofeachother,thereforethelinesareperpendicular.

Example 1

Example 2




Inthesystemofequationsx yx y6 2 123 5- + =

- =' 1,determineifthelinesareparallel,perpendicular, orneither.

►Solution: Thisproblemcanbesolvedbychangingbothequationstoslope-interceptform.

x y3 5- = x y6 2 12- + =

y x3 5- =- + y x2 6 12= +

,y x m3 5 3= - = ,y x m3 6 3= + =

Theslopesareequal,thereforethelinesareparallel.

Inthesystemofequationsx yx y2 44 3 7- =+ =' 1,determineifthelinesareparallel,perpendicular,

orneither.

►Solution: Leavingthesystemofequationsinstandardform:

x y4 3 7+ = hasslopemBA

34=- =-

x y2 4- = hasslopemBA

12 2=- =--=

Changingthesystemofequationstoslope-interceptform:

x y4 3 7+ = x y2 4- =

y x3 4 7=- + y x2 4- =- +

,y x m34

37

34=- + =- ,y x m2 4 2= - =

Bothmethodsproducethesameresult:theslopesareneitherthesame,nornegativereciprocals, thereforethelinesareneitherparallelnorperpendicular.

Example 3

Example 4




5.2 Exercise Set

1. Matchthegraphy mx b= + withitsclosestdescription.

a) ,m b0 01 1 i) ii) iii)

b) ,m b0 02 1

c) ,m b0 01 2

d) ,m b0 02 2

e) m 0= iv) v) vi)

f) b 0=

2. Matchthegraphwiththelinearrelation.

a) x y4 8- =- i) ii)

b) x y4 8+ =

c) x y4 2- =-

d) x y4 2+ =

iii) iv)

3. Determinetheequationofthegraph.

a) b)

c) d)

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y




4. Determinetheequationofalinethroughthegivenpairofpoints.

a) (−4,1)and(6,1) b) (1,−4)and(1,6)

c) (−2,0)and(5,0) d) (0,−2)and(0,5)

e) (a,b)and(c,b) f) (b,a)and(b,c)

5. Writetheequationofthelinewiththegivencharacteristics.

a) vertical,passesthrough(3,6) b) vertical,passesthrough(−2,−4)

c) horizontal,passesthrough(3,6) d) horizontal,passesthrough(−2,−4)

6. Foreachpairofequations,determinewhetherthelinesareparallel,perpendicular,orneitherparallelnor perpendicular.

a) x y2 5 7+ = b) x y4 3 7- + =

x y4 10 2+ = x y8 6 0- + =

c) x y4 3 6- = d) x y3 5 4- = x y4 6 3+ =- x y5 3 4- =

e) x y4 3 5- = f) x y2 5 3- =-

x y3 4 2+ = x y10 4 1+ =

g) x y4 3- = h) x y5 2 7- =

x y4 2- =- x y2 5 7+ =




7. Writetheequationofalinepassingthroughthegivensetofpointsinslope-interceptform.

a) (3,5)and(2,4) b) (5,−2)and(−3,1)

c) (−4,1)and(−2,−3) d) (−1,−2)and(−6,−4)

e) (6,−2)and(−3,2) f) (0,0)and(−3,2)

g) (0,−6)and(−4,0) h) (5.2,−6.8)and(−1.6,−3.8)

i) (2,5)and(−2,5) j) (3,7)and(3,−1)




8. Reasoning.

a) Ifalineishorizontal,whatistheslopeofany b) Ifthegraphofalinearequationhasonepointthat linethatisperpendiculartoit? isboththex-interceptandy-intercept,whatpoint thatbe?

c) Whatistheequationofthex-axis? d) Findthex-interceptof x y3 2 8- = .

e) Findthevalueofcsothatthegraphof f) Findthey-interceptof x y4 3 2=- + . x c y4 3+ = hasanx-interceptof(−2,0).

g) Findthevalueofcsothatthegraphof h) IfAisnotzero,whatwillthegraphofAx C D+ =

x c y3 2- = hasay-interceptof(0,5). looklike?

i) IfBisnotzero,whatwillthegraphof j) Whatistheequationofalinewithxandycoordinates By E F+ = looklike? thatareequal,andpassesthroughtheorigin?

k) Whatistheequationofalinewithxandy l) Whatistheequationofalinepassingthroughthe coordinatesthatareoppositeinvalueand point ,a b^ hwithslope0? passesthroughtheorigin?

m) Whatistheequationofalinepassingthrough n) Whatisthey-interceptofax by c+ = ? thepoint ,a b^ hwithanundefinedslope?

o) Whatisthex-interceptofax by c+ = ? p) Whatistheslopeofthelineax by c+ = ?




9. Findthexandyinterceptsoftheline 10. Showthattheequationofalinewithx-intercept ax by ab+ = . (a,0)andy-intercept(0,b)canbewritteninthe

formaxby1+ = .

11. Showthattheequationofalinewithpoints 12. Determinetherelationshipbetweenthegraphsof ,x y1 1^ hand ,x y2 2^ hcanbewrittenintheform theequationsAx By C+ = andBx Ay C- = . y y x x y y x x1 2 1 2 1 1- - = - -^ ^ ^ ^h h h h.

13. Ifthetwopointsthatalinepassesthroughare 14. Thinkofdifferentpointsonthegraphofthe known,itsequationcanbefound.Explainhow horizontalliney 2= .Whatdothepointshave thisisdone. incommon?Howdotheydiffer?

15. Whatistheslopeofallorderedpairsoftheform 16. Giventhat0cCisthesametemperatureas32cF, ,x x3-^ h? and100cCisequivalentto212cF,determinethe equivalentof68cFincC.

17. Intheequation,ax by x y2 3 6+ = - + ,find 18. Intheequation,ax by x y2 3 6+ = - + ,find a andb ifthegraphisahorizontallinepassing a andb ifthegraphisaverticallinepassing through ,0 3^ h. through ,3 0^ h.


Section 5.3 - Equations of Parallel and Perpendicular Lines ♦ 207


Towritetheequationofaline,apointandslopeisneeded.However,insomeproblemsthisinformationisnot directlygiven,andfurtherstepsmustbetakentofindapointorslope.

Whendeterminingtheequationofalinethatisparalleltoagivenslope,theconcepttorememberisthat parallellineshaveequalslopes.Whendeterminingtheequationofalinethatisperpendiculartoagivenslope, theconcepttorememberisthatperpendicularlineshaveslopesthatarenegativereciprocalsofeachother.

Writetheequationofalineparallelto x y3 2 6- = ,andwhichgoesthroughthepointA(4,−2).

►Solution: x y3 2 6- = hasslope m=BA- =

23--

=23

Therefore,theslopeofalineparallelto x y3 2 6- = hasslopem=23

Substitutingthegivenpointandslopeintothepoint-slopeequationofalinegives:

y y m x x1 1- = -^ h → y x223 4- - = -^ ^h h

y x223 6+ = -

y x23 8= - (slope-intercept form)

x y3 2 16- = (standard form)

Writetheequationofalineperpendicularto x y4 2 7+ = goingthroughthepointB(−2,5).

►Solution: x y4 2 7+ = hasslopem=BA- =

24- =−2

Thereforetheslopeofalineperpendicularto x y4 2 7+ = hasslopem=21 .

Substitutingthegivenpointandslopeintothepoint-slopeequationofalinegives:

y y m x x1 1- = -^ h → y x521 2- = - -^^ hh

y x521 2- = +^ h

y x521 1- = +

y x21 6= + (slope-intercept form)

x y2 12- = (standard form)

Example 1

Example 2

Equations of Parallel and Perpendicular Lines5.3




5.3 Exercise Set

1. Findtheslopesoflinesparallelandperpendiculartotheequation.

a) y x3= m ; ; b) y x2=- m ; ;

m= m=

c) y x32 2=- + m ; ; d) y x

53 1= - m ; ;

m= m=

e) x y2 3 4- = m ; ; f) x y3 2+ = m ; ;

m= m=

g) x y5 0- = m ; ; h) x 2= m ; ;

m= m=

i) y 2=- m ; ; j) x y2 1= - m ; ;

m= m=

k) x y43

31

21= + m ; ; l) . . .x y0 2 2 3 1 4+ = m ; ;

m= m=




2. Findtheequationoftheline,instandardform,thatpassesthroughthegivenpointandisparalleltothegiven line.

a) P(0,0);y x2 5= - b) P(0,0);x y2 5= +

c) P(1,3); x y3 6- = d) P(−2,0); x y2 5 3+ =

e) P(−6,3);y x4 8+ =- f) P(5,−2); y x3 1 4+ =-

g) P(−4,−3);x y43 2= - h) P(0,−5);x y

32 1=- +

i) P(−5,2);x 3= j) P(−5,2);y 4=-

k) P(−4,1); x y32

43 12+ = l) ,P

21

32-` j; .x y

31 0 4 2- =




3. Findtheequationoftheline,instandardform,thatpassesthroughthegivenpointandisperpendiculartothe givenline.

a) P(0,0);y x2 5= - b) P(0,0);x y2 5= +

c) P(1,3); x y3 6- = d) P(−2,0); x y2 5 3+ =

e) P(−6,3);y x4 8+ =- f) P(5,−2); y x3 1 4+ =-

g) P(−4,−3);x y43 2= - h) P(0,−5);x y

32 1=- +

i) P(−5,2);x 3= j) P(−5,2);y 4=-

k) P(−4,1); x y32

43 12+ = l) ,P

21

32-` j; .x y

31 0 4 2- =




4. Determinetheequationofalineparalleltothegraphgoingthroughthegivenpoint,instandardform.

a) (4,−2) b) (−5,−4)

c) (2,1) d) (5,−3)

5. Determinetheequationofalineperpendiculartothegraphgoingthroughthegivenpoint,instandardform.

a) (4,−2) b) (−5,−4)

c) (2,1) d) (5,−3)

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y




6. Findtheequationofalineparallelto x y3 4 8+ = 7. Findtheequationofalineparalleltox y3 8- =

withthesamey-interceptas x y5 3 10- = . withthesamey-interceptas x y3 2 6+ = .

8. Findtheequationofalineparallelto x y2 7 10+ = 9. Findtheequationofalineperpendicularto withthesamex-interceptas x y3 4 5- = . x y2 3 7- = withthesamey-interceptas x y5 2 10- = .

10. Findtheequationofalineperpendicularto 11. Findtheequationofalineperpendicularto x y3 2 9+ = withthesamex-interceptas 1x y2

321= + withthesamex-interceptas

x y2 5 0- = . x y2 3 9+ = .

12. Acirclecentredattheoriginpassesthroughthepoint 13.Arhombushascoordinates ,0 0^ h, ,3 4^ h, ,8 4^ h, ,3 4-^ h.Whatistheequationofalineperpendicular and ,5 0^ h.Whataretheequationsofthediagonals totheradiusatthispoint? oftherhombus?Whatrelationshipistherebetween thediagonals?


Section 5.4 - Linear Applications and Modelling ♦ 213


Graphsareeffectivevisualtoolsbecausetheypresentinformationquicklyandeasily.Sometimes,datacanbe betterunderstoodwhenpresentedbyagraphthanbyatablebecausethegraphcanrevealatrendor comparison.

Waterfreezesat32°F,or0°C.Waterboilsat212°F,or100°C.Graphthelinearrelation between°Cand°F,andfindaformulathatconvertsCelsiustoFahrenheit.

►Solution: Thefreezingpointonthegraphis(0,32) Theboilingpointonthegraphis(100,212)

m100 0212 32

100180

59=

-- = =

Byslope-intercept,F C59 32= +

Itcostsapopcornvendor$490tomake150bagsofpopcornand$610tomake350bags.

a) Graphthelinearrelationbetweencostandnumberofbags. b) Findthecostequation. c) Findthefixedcost. d) Findthecostof250bagsofpopcorn. e) Howmanybagsofpopcorncanbeboughtfor$724?

►Solution: b) .m350 150610 490

200120 0 60=

-- = =

.C B490 0 60 150- = -^ h .C B490 0 60 90- = -

.C B0 60 400= +

c) Thefixedcostis$400

d) . $C 0 60 250 400 550= + =^ h

e) . B0 6724 0 400= +

. B0 6724 400 0- =

. B0 6 3240 =

B 540=

Example 1

Example 2

212

32

100°C

°F

Cost($)

BagsofPopcorn

490

610

150 350

Linear Applications and Modelling5.4




Afamilyhasamedicalplanthatpays70%ofallprescriptioncosts,lessa$200deductibleeach year.

a) Writeafunctionthatmodelsthefamily’sresponsibilityforprescriptioncosts. b) Determinetheamountthemedicalplanwillpayon$1250inprescriptioncosts. c) Determinetheamountspentonprescriptionpurchasesiftheamounttheplanpaidwas $1250. d) Graphthisfunctionandlabeltheanswersfromb)andc).

►Solution: a) LetRbetherefund,andCbetheprescriptioncost.

m=0.70,y-intercept=−200, .R C0 70 200= -

b) R = . C0 70 200-

= .0 70 1250 200-^ h = 675

Theplanwillpay$675on$1250inprescriptioncosts.

c) R = . C0 70 200-

1250 = . C0 70 200-

1450 = . C0 70

C = 2071.43

$2071.43isspentonprescriptionpurchases,togeta$1250refund.

d)

Example 3

PrescriptionCost($)

c)

b)

Refund($)

01250

675

2071.73–200

1250




5.4 Exercise Set

Assumelinearappreciationorlineardepreciationforallproblems.

1. Aninsurancecompanypurchasedcomputersforits 2. Inherfirstyearofpractice,apsychologisthas160 office.Thevalueofthecomputersaftertwoyears patients.Bythethirdyear,thenumberofpatients was$80000,and$56000afterfouryears. grewto246.Ifthistrendcontinues,howmany Determinethepurchasepriceofthecomputers. patientswouldshehaveinthefourthyear?

3. Thepercentof18-25yearoldswhosmoke 4. Ataxicabispurchasedfor$36000.Attheendof worldwidehaschangedfrom46.8%in1987to 10yearsitissoldforscrapfor$1800.Findthe 37.2%in2000.Predictthepercentageof18-25 depreciationequation. yearoldsthatwillsmokein2012.

5. Ahomewaspurchasedfor$410000.Theowner 6. Aprintercosts$960newandisexpectedtobe expectsthehometodoubleinvalueinthenext worth$140aftersixyears.Whatwillitbeworth 10years.Findtheappreciationequation. afterfouryears?




7. Apaintingisexpectedtoappreciate$75eachyear. 8. Agrandfatherclockisexpectedtobeworth$2700 Ifthepaintingwillbeworth$620intwoyears, inthreeyearsand$3200infiveyears.Whatwill whatwillitbeworthin14years? itbeworthineightyears?

9. Atimesharecottagepurchasedfouryearsagois 10. Aprintingcompanychargesafixedratetosetup nowworth$36200.Ifthecottagehasappreciated theprintingpress,plusacostof$3.50forevery $2150peryear,finditsoriginalpurchaseprice. 100copies.If800copiescost$64.00,howmuch willitcosttoprint1500copies?

11. Acitywithapopulationof62000had480police 12. Anelectricalsubstationisworth$246000when investigationsinayear.Whenthepopulationof itisinstallednew,butisworthnothingafterits thecityroseto74000,thenumberofinvestigations 15yearlifecycle.Findthedepreciationequation. was640inayear.Ifthistrendcontinues,howmany investigationswillthecityhavewhenitspopulation reaches100000?




13. Thetotalcostofacomputeristhesumoftheselling 14. Itcosts$1200tostartupabusinesssellinghot price,plusasalestaxof12%,plusa$20disposalfee. dogsonthebeach.Eachhotdogcosts40¢to produce.

a) Expressthetotalcostofthecomputerasalinear a) Findthecostequation. functionofthesellingprice.

b) Whatisthetotalcostofacomputerthatsells b) Howmanyhotdogsareproducedifthetotal for$1540? costis$1560?

c) Whatisthesellingpriceofacomputerwhose c) Howmanyhotdogsmustbesoldattwo totalcostwas$1061.60? for$1.00tobreakeven?

15. Itcostsacompany$2140toproduce500widgets 16. ToshipapackagefromVancouvertoWinnipeg and$3660toproduce900widgets. overnightcosts$27.30foraonepoundpackage, and$38.80forathreepoundpackage.

a) Whatisthefixedcostforproducingwidgets? a) Findthecostequation.

b) Findanequationrelatingthecostofproducing widgets. b) Findthecostofshippinga6.5poundpackage.

c) Whatisthetotalcostofproducing200widgets? c) Ifapackagecost$45.70toship,howmuch doesitweigh?

d) Howmanywidgetscanbeproducedfor$7308?




Thenotation f x^ hisanotherwayofwritingyasafunction.Forexample,thefunctiony x2 4= - maybe writtenas f x x2 4= -^ h ,where f x^ hisread“fofx”.

Withoutfunctionnotation,aproblemcouldbestated:Giveny x2 4= - ,findywhenx=5.Usingfunction notation,thesameproblemwouldbestated:Given f x x2 4= -^ h ,find f 5^ h.The notation f 5^ h implies the value of y when x is 5.Thestatement f 5 6=^ h saysthevalueofyis6whenxis5.Thisisthepoint(5,6).

Given f x x3 5= +^ h ,determinethecoordinatesofonepointonthelinefor f 2^ h.

►Solution: f 2 3 2 5 11= + =^ ^h h

Thereforethepointis(2,11).

Given f x x3 5= +^ h ,determinethecoordinatesofthepointwhere f x 7=-^ h .

►Solution: f x x3 5= +^ h x7 3 5- = +

x7 5 3- - =

x12 3- =

x4- =

Thereforethepointis(−4,−7).

Completethetablefor f x x3 5= +^ h .

x x3 5+ f x^ h ,x y^ h3

►Solution:

Example 1

Example 2

Example 3

x x3 5+ f x^ h ,x y^ h3 14 f(3) (3,14)

Function Notation5.5


Section 5.5 - Function Notation ♦ 219


Determinetheslope-interceptfunction f x mx b= +^ h if f 1 4=^ h and f 3 2=-^ h .

►Solution: f 1 4=^ h meansthepoint(1,4)

f 3 2=-^ h meansthepoint(3,−2)

m1 3

4 2

26 3=

-

- -=-=-

^ h

f x mx b= +^ h f b1 3 1=- +^ ^h h b4 3=- +

b 7=

Therefore f x x3 7=- +^ h

If f x x2 1= +^ h ,

a)Whatis f x3^ h?b)Whatis f x 3+^ h?

►Solution: a) f x3^ h= x2 3 1+^ h = x6 1+

b) f x 3+^ h= x2 3 1+ +^ h = x2 7+

If f x x2 1= +^ h ,determineh

f x h f x+ -^ ^h h ,h 0! .

►Solution: f x x2 1= +^ h , f x h x h2 1+ = + +^ ^h h

Thereforeh

f x h f x+ -^ ^h h =h

x h x2 1 2 1+ + - +^ h6 6@ @

=h

x h x2 2 1 2 1+ + - -

=hh2

= 2

Example 4

Example 5

Example 6




5.5 Exercise Set

1. Completethetableforthelinearfunctiondefinedbyg x x2 3=- +^ h .

x x2 3- + g x^ h ,x y^ h2−42c

c−2−c+1

2. For f x x3 2= -^ h ,find:

a) f 3^ h b) f 4-^ h

c) f k^ h d) f x2 1-^ h

e) f x h+^ h f) f x f h+^ ^h h

3. For f x x4 5= +^ h ,find:

a) f 3^ h b) f 4-^ h

c) f k^ h d) f x2 1-^ h

e) f x h+^ h f) f x f h+^ ^h h

4. For f x x5 2=- +^ h ,find:

a) f x 3=-^ h b) f x 7=^ h

c) f x 12=-^ h d) f x 5=-^ h

e) f x a=^ h f) f x a5 7=- +^ h




5. Grapheachfunctionovertherealnumbers.

a) f x x2 1= +^ h b) f x x21 3=- +^ h

c) f x x43 2= -^ h d) f x x

32 4=- -^ h

e) f x 3=^ h f) f x x41 42=- +^ h

f(x)

x

f(x)

x

f(x)

x

f(x)

x

f(x)

x

f(x)

x




6. Grapheachfunctionifthedomainis{−3,−2,−1,0,1,2}

a) f x x2 1= +^ h b) f x x21 3=- +^ h

c) f x x43 2= -^ h d) f x x

32 4=- -^ h

e) f x 3=-^ h f) f x x41 3=^ h

f(x)

x

f(x)

x

f(x)

x

f(x)

x

f(x)

x

f(x)

x




7. Forg r r2r=^ h ,find:

a) .g 0 5^ h b) g38` j

c) g h^ h d) g h 2+^ h

8. Forg r rh2r=^ h ,find:

a) .g 0 5^ h b) g38` j

c) g h^ h d) g h 2+^ h

9. Forg r r2r=^ h ,find:

a) g21` j b) g

38` j

c) g h^ h d) g h 2+^ h

10. Forg r r h2r=^ h ,find:

a) g21` j b) g

38` j

c) g h^ h d) g h 2+^ h




11. Usethegraphofeachfunctiontostatethedomain,statetherange,determine f 2^ h,andsolve f x 2=^ h forx.

a) domain

range

f 2^ h

f x 2=^ h

b) domain

range

f 2^ h

f x 2=^ h

c) domain

range

f 2^ h

f x 2=^ h

d) domain

range

f 2^ h

f x 2=^ h

f(x)

x

f(x)

x

f(x)

x

f(x)

x




12. Completethetable,andgraphthefunction.Also,givethedomainandrangeofthefunction.

a) x xg 2 1=- -^ h domain

range

b) g x x32

= -^ h domain

range

c) x xg 4= -^ h domain

range

d) x xg21= -^ h domain

range

x g x^ h−4−204

x g x^ h1−1−3−5

x g x^ h10−1−2

x g x^ h01−1−2

g(x)

x

g(x)

x

g(x)

x

g(x)

x




13. Determine f x mx b= +^ h .

a) f 0 3=-^ h b) f 2 4=^ h f 2 5- =^ h f 1 4- =-^ h

c) f 2 5=^ h d) f 3 6- =^ h f 3 3- =^ h f 1 2=-^ h

e) f 3 2=^ h f) f21

32=-` j

f 3 2- =^ h f25

38- =` j

14. Determineh

f x h f x+ -^ ^h h ,h 0! .

a) f x x3=^ h b) f x x3 4= -^ h

c) f x x5 2= -^ h d) f x x2=^ h




15. Thefunction 32f c c59= +^ h determinesthe 16. Aballisdroppedfromahighrisebuilding.The

FahrenheitequivalentofdegreesCelsius.Findthe heightoftheballinmetres,tsecondsafteritis Fahrenheitequivalentof: dropped,isgivenbythefunction .h t t9 8 1002=- +^ h .

a) 30°C a) Findh 0^ h.

b) 0°C b) Findtheheightoftheballafter2seconds.

c) −40°C c) Findthetimeittakesfortheballtohitthe ground.

17. Thefunction 1P d d32= +^ h givesthepressurein 18. ThetemperaturebelowthesurfaceoftheEarthis

atmospheresatadepthofdfeetintheocean. givenbyT d d10 20= +^ h ,whereTisincelsius anddisinkilometres.

a) Findthepressureat160feet. a) Findthetemperature12kmbelowthesurface oftheearth.

b) Atwhatdepthisthepressure9.6atmospheres? b) Whatdepthhasatemperatureof166°C?




Section 5.1

1. Findtheslopeandy-intercept.

a) x y2 5 7- = slope b) x y5 2+ =- slope


2. Writethestandardformequationinslope-interceptform.

a) x y6 3- = b) x y2 5 7+ =

3. Writetheslope-interceptequationinstandardform.

a) y x32 4=- + b) y x3

52=- +

4. Writethepoint-slopeequationinslope-interceptform.

a) y x132 4+ =- -^ h b) y x

32 4

21- =- +` j

5. Writethepoint-slopeequationinstandardform.

a) y x132 4+ =- -^ h b) y x

32 4

21- =- +` j

6. Writetheequationofeachlineinstandardform.

a) (0,−3),m=−4 b) (2,0),m=31-

Chapter Review5.6


Section 5.6 - Chapter Review ♦ 229


7. Determinetheequationin:standardform,slope-interceptformandpoint-slopeform.

a) b)




Section 5.2

8. Determinetheequationofthegraph.

a) b)

9. Writetheequationofthelinewiththegivencharacteristics.

a) vertical,passesthrough(−2,5) b) horizontal,passesthrough(−2,5)

c) vertical,passesthrough ,a b^ h d) horizontal,passesthrough ,a b^ h

x

y

x

y

x

y

x

y




10. Foreachpairofequations,determinewhetherthelinesareparallel,perpendicularorneitherparallelnor perpendicular.

a) x y3 2 7+ = b) x y5 2 4- =

x y4 6 2+ = x y4 10 3+ =

c) y x2 3= - d) x y3 2- =

x y2 3+ =- x y6 2 2- =

11. Writetheequationofthelinepassingthroughthegivensetofpointsinstandardform.

a) (−3,1)and(−4,−6) b) (−2,−3)and(−5,−1)

Section 5.3

12. Findtheslopesoflinesparallelandperpendiculartothefollowingequations.

a) x y3 4 6- =- m ; ; b) x y3 2= + m ; ;

m= m=

13. Findtheequationofthelinethatpassesthroughthegivenpointandisparalleltothegivenline.

a) P(−2,4); x y2 3 5- = b) P(4,−1); x y4 7 2+ =-

14. Findtheequationofthelinethatpassesthroughthegivenpointandisperpendiculartothegivenline.

a) P(−2,4); x y2 3 5- = b) P(4,−1); x y4 7 2+ =-


Section 5.6 - Chapter Review ♦ 231


15. Determinetheequationofaline,instandardformwhichisparalleltothelineandwhichgoesthroughthe givenpoint.

a) (5,2) b) (−3,4)

16. Determinetheequationofaline,instandardformwhichisperpendiculartothelineandwhichgoesthrough thegivenpoint.

a) (5,2) b) (−3,4)

Section 5.4

17. Thecosttoprint1200booksis$11140,andthecosttoprint2000booksis$17940.Assumingthereisalinear relationbetweenthecostsandthenumberofbooksprinted.

a) Findthecostequation. b) Findthe“setup”costofprintingthebooks.

c) Findthecostof3000books. d) Howmanybookscanbepurchasedfor$24740.

x

y

x

y

x

y

x

y




Section 5.5

18. For f x x3 2=- -^ h ,find:

a) f 3^ h b) f 4-^ h

c) f x 3=^ h d) f x 4=-^ h

e) f a^ h f) f x a=^ h

g) f x h+^ h h) f x f h+^ ^h h

19. Determine f x mx b= +^ h .

a) f 3 4=^ h b) f 1 4- =-^ h f 2 6- =^ h f 3 7=^ h

c) f 4 2- =-^ h d) f 4 2=^ h f 1 5=^ h f 2 4=^ h

e) f a a2=^ h f) f a b=^ h f b b2=^ h f b a=^ h


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Section 5.1 - Different Forms of Linear Equations Lambrick ...€¦ · 192 ♦ Chapter 5 - Linear ... Graphing a Line Using the Slope and y-Intercept Step 1: Write the equation in