8Bindi is 18 years old and

Transcription

1 Expoetil fuctios 8Bidi is 8 ers old d ivestig $0 000 i fixed term deposit pig 6% p.. compoud iterest. Whe Bidi hs $0 000 she iteds to put deposit o home. How log will it tke for Bidi s $0 000 to grow to $0 000? The ivestmet tht Bidi hs mde is exmple of expoetil fuctio. I this chpter ou will ler how to use expoetil fuctio to model the growth of such ivestmet d other similr prolems.

2 8 Mths Quest 0 for Victori Idex lws I Mths Quest 9 we looked t idices d the idex lws. These lws re the sis for expoetil fuctios, which we will cover lter i the chpter. Let s first revise this work. A umer i idex form hs two prts, the se d the idex d is writte s: se x idex The first two idex lws relte to multiplictio d divisio of idex expressios. First Idex Lw: Whe umers with the sme se re multiplied, the idices re dded. m m + Secod Idex Lw: Whe umers with the sme se re divided, the idices re sutrcted. m m Exmple Simplif ech of the followig. m p m p c Write the expressio. m p m p x - 0x Multipl the umers with the sme se ddig the idices. Note tht p p. m + + p + m 6 8 p Write the expressio. Simplif multiplig the coefficiets, the multipl the umers with the sme se ddig the idices c Write the expressio. c x 0x Simplif dividig oth coefficiets the sme fctor, the divide umers with the sme se sutrctig the idices. x x -

3 Chpter 8 Expoetil fuctios 9 The Third Idex Lw is used i clcultios whe zero idex is ivolved. Third Idex Lw: A term (excludig 0) with idex of 0, is equl to. 0 Exmple Simplif ech of the followig. ( ) 0 ( ) 0 Write the expressio. ( ) 0 Appl the Third Idex Lw which sttes tht term (excludig 0) with idex of 0, is equl to. Write the expressio. ( ) 0 The term iside the rckets hs idex of zero so the rcket is equl to. Simplif. The Fourth, Fifth d Sixth Idex Lws ivolve removig rckets from idex expressio. Fourth Idex Lw: To remove rckets, multipl the idices iside the rckets the idex outside the rckets. Where o idex is show, ssume tht it is. ( m ) m Fifth Idex Lw: To remove rckets cotiig product, rise ever prt of the product to the idex outside the rckets. () m m m Sixth Idex Lw: To remove rckets cotiig frctio, multipl the idices of oth umertor d deomitor the idex outside the rckets. Exmple Simplif ech of the followig. ( ) ( 7 ) c x m - m m Write the expressio. ( ) Appl the Fourth Idex Lw multiplig the idices iside the rckets the idex outside the rckets ( ). 8 Cotiued over pge

4 60 Mths Quest 0 for Victori Write the expressio. ( 7 ) Appl the Fifth Idex Lw multiplig the idices iside the rckets the idex outside the rckets. Where o idex is show, ssume tht it is. Simplif c Write the expressio. c Appl the Sixth Idex Lw multiplig the idices of oth umertor d deomitor the idex outside the rckets. Simplif. x - - x 6x - 6 rememer rememer To simplif expressios with umers d/or proumerls i idex form, the followig idex lws re used.. m m +. m m. 0. ( m ) m. () m m m 6. m - m m EXCEL Spredsheet 8A Idex lws Mthcd Idex lws Idex lws Exmple, Simplif ech of the followig. c d e m 6 m 7 f c c g mp m p h i j m m 6m k x x 6x l x x x

5 Chpter 8 Expoetil fuctios 6 Exmple c Exmple Simplif ech of the followig. 7 c 6 d 7 e 8m - f 8 7 m g m 7 x - h - m x i 6x 7 8x j 7 c c k 0m p x l z - 6m p 8x z Simplif ech of the followig. 0 () 0 c (m ) 0 d x 0 e 0 f () 0 g 0 h 0 i x 0 (x ) 0 Exmple Simplif ech of the followig. ( ) ( ) c m - d e ( ) f ( ) g (m ) m h i j m - k 7x - l multiple choice m 0 is the simplified form of: A m m B 6m 0 - C (m ) D (m ) E m - The vlue of () 0 is: A B 9 C D E 6 Evlute ech of the followig. c ( ) d 6 e ( ) f g 6 h ( ) 0 i ( ) 0 7 Simplif ech of the followig. ( m + ) p c x m p m q d (x ) z e (p q ) 0 f m (m) 0

6 6 Mths Quest 0 for Victori Negtive idices So fr we hve delt ol with idices tht re positive whole umers or zero. To exted this we eed to cosider the meig of idex tht is egtive whole umer. Cosider the expressio. Usig the Secod Idex Lw, Writig terms i the expded ottio we hve: - - B equtig the results of simplifictio, usig the two methods, we hve:. I geerl terms, 0 ( 0 ) 0 (usig the Secod Idex Lw) Seveth Idex Lw: The covetio is tht expressio should e writte usig positive idices d we use the Seveth Idex Lw to do this. Exmple Express ech of the followig with positive idices: x m c - Write the expressio. x Appl the Seveth Idex Lw. x Write the expressio. m Appl the Seveth Idex Lw to write the expressio with positive idices. m c Cop the expressio d rewrite the frctio, usig divisio sig. Appl the Seveth Idex Lw to write the expressio with positive idices. c To divide the frctio use d tip.

7 Chpter 8 Expoetil fuctios 6 Worked exmple c shows the method used to demostrte the coverse of the Seveth Idex Lw. Tr this s exercise for ourself! All lws discussed i the previous sectio re pplicle to the terms with egtive idices. Exmple Simplif ech of the followig, expressig the swers with positive idices. x c m - x Write the expressio. Appl the First Idex Lw. Multipl umers with the sme se ddig the idices. Express the swer with positive idices. + ( ) + Write the expressio. Appl the Secod Idex Lw. Divide umers with the sme se sutrctig the idices. x - x x x - Express the swer with positive idices. x - c Write the expressio. c m - Appl the Sixth Idex Lw. Multipl the idices of oth umertor d deomitor the idex outside the rckets. Rememer tht. Express ll umers d proumerls with positive idices. Simplif. m 6 - m 6 - m 6

8 6 Mths Quest 0 for Victori Numers i idex form c e esil evluted, if the re expressed with positive idices first. Cosider the followig exmple: Evlute 6 without usig clcultor. Write the multiplictio. 6 Express with positive idex. 6 6 Exmple 6 Multipl the umertor of the frctio the whole umer. Evlute the deomitor. Ccel dividig oth umertor d deomitor the sme umer A clcultor c e used to evlute umers with egtive idex. The se is lws etered efore the idex. Exmple Evlute usig clcultor. 7 Usig scietific clcultor: Press x +/. (Note tht the x utto could e or x o our clcultor.) Usig grphics clcultor: Press ^ ( ) ENTER. x /DISPLAY 0.0 rememer rememer. A umer with egtive idex c e expressed with positive idex usig the followig idetities. () m - () - m m m. All idex lws ppl to terms with egtive idices.. Alws express swers with positive idices uless required otherwise.. Numers d proumerls without idex re uderstood to hve idex of.

9 Chpter 8 Expoetil fuctios 6 Exmple Exmple Exmple 6 Exmple 7 8B Negtive idices Express ech of the followig with positive idices. x c 9 d e x f m g 6 c h i 6 j 6 7 k m l - Simplif ech of the followig, expressig the swers with positive idices. x x c m m d 7 e x 6 x f x 6x g 6m x - h 9 m - i - m 6 x 7 6m j ( m ) k (p 7 q ) l ( ) m p o 6 q Evlute ech of the followig without usig clcultor. 6 c d e f 6 g 6 h i j k 0 - l 8 0 Evlute ech of the followig, usig clcultor. 6 c 7 d e f (0.0) 8 7 Negtive idices EXCEL Idex lws Mthcd Spredsheet multiple choice m Which of the followig, whe simplified, gives -? A m - B m C D E m m m Whe simplified, 7 is equl to: A B - C - D - E 6 6 -

10 66 Mths Quest 0 for Victori GAMEtime Expoetil fuctios 00 c Whe (x 6 ) is simplified, it is equl to: x 8 A - B - C - D - E x 8 8 d If - x is equl to, the x d (i tht order) re: x 8 8 A d 6 B 6 d C d D d E d x 8 x 8-6 Simplif 8 c c. d Simplif e f. 6e f Simplif ( 9g 7 h). Simplif ij -. ijk Evlute 9l 0 ( m 6 ) 0. Simplif questios 8 0 expressig our swer with positive idices p q 6 pq r s r s t u 8t u 6 Evlute Express with positive idex. MATHS MATHS QUEST C H A L LL E N G G E E Oe millio c e writte s List other two ws i which oe millio c e writte s the product of two umers cotiig the digits d 0. Fid w of writig oe millio s the product of two umers, either of which cotis the digit 0. Imgie piece of pper tht ou ter i hlf, put the two pieces together, ter those i hlf, put the pieces of pper together, ter those i hlf, d so o. After how m ters would ou hve out millio pieces of pper? Without usig clcultor, which of these umers is the lrger: 0 or 0?

11 Chpter 8 Expoetil fuctios 67 Frctiol idices We re ow red to look t idices tht re frctiol. Numers (or terms) with frctiol idices c e writte s surds, usig the followig lws: m - m ( ) m To uderstd how these lws re formed, cosider the followig umericl exmples. Usig the First Idex Lw ( ) so If these two idetities re true the. Similrl: Usig the First Idex Lw ( 8) so If these two idetities re true the 8 8. This c e geerlised to. m - Now cosider: or m ( ) m m ( ) m The Eighth Lw of Idices is m ( ) m. As c e see from the ove idetities, the deomitor of frctio () idictes the power or tpe of root. Tht is, implies cue root, implies fourth root, etc. Note tht whe (squre root), it is the covetio ot to write t the squre root sig. Tht is, for, we write rther th. Exmple m Evlute ech of the followig without usig clcultor. 9 6 m - m - m Rewrite the umer usig the Eighth Lw of Idices. 9 9 Evlute. m - Rewrite the umer usig ( ). 6 ( 6) Evlute the squre root. Evlute the result. 6 8

12 68 Mths Quest 0 for Victori A umer with frctiol idex c e evluted o scietific clcultor directl - x eterig the frctiol idex usig the c fuctio or usig the fuctio. O grphics clcultor, the ^ utto is still used to eter the frctiol idex or the MATH meu to clculte surd. Exmple Usig clcultor, evlute ech of the followig. Give the swer correct to deciml plces. 7 Method Eter the se the the frctiol idex o clcultor d evlute. O grphics clcultor, eter rckets roud the frctiol idex. Roud to deciml plces. 9 /DISPLAY Method Express s surd. Evlute usig our clcultor. O grphics clcultor, eter the power of the root,, the press MATH, select : x, press d ENTER. Roud to deciml plces. Note: Both methods re show o the scree ove right. Method Eter the se the the frctiol idex o clcultor d evlute. O grphics clcultor, press ^ ( 7 ) ENTER. Roud to deciml plces. Method Express 7 s surd. 7 ( 7 ) Evlute usig our clcultor. O grphics clcultor, press (, eter the power of the.7 root, 7, press MATH, select : x d press. Close the rckets pressig ) d squre the result pressig ^ d ENTER. Roud to deciml plces. Note: Both methods re show o the scree ove right.

13 Chpter 8 Expoetil fuctios 69 All idex lws discussed so fr re vlid for frctiol idices. The followig exmple illustrtes this. Exmple Simplif ech of the followig: m m ( ) 6 c 0 x - Write the expressio. Appl the First Idex Lw to multipl umers with the sme se ddig the idices. m m m Write the expressio. ( ) 6 6 Use the Fourth Idex Lw to multipl ech idex iside the rckets the idex outside the rckets. Simplif the frctios. 6 c Write the expressio. c Use the Sixth Idex Lw multiplig the idex i oth the umertor d deomitor the idex outside the rckets. x x 8 rememer rememer. Frctiol idices re those which re expressed s frctios.. Numers with frctiol idices c e writte s surds, usig the followig idetities: m - m ( ) m. All idex lws re pplicle to frctiol idices.

14 70 Mths Quest 0 for Victori 8C Frctiol idices EXCEL Mthcd Frctiol idices Spredsheet Idex lws Exmple 8 Exmple 9 Evlute ech of the followig without usig clcultor. 6 c d 8 e 6 f g 6 h i 7 j 00 k 6 l Usig clcultor, evlute ech of the followig. Give the swer correct to deciml plces. c d 89 e 8 f ( 0.6) g h i SkillSHEET 8. Exmple 0 Simplif ech of the followig. 8 8 c d x x e m m f 7 7 g 9 h i x x Simplif ech of the followig. x 9 x c z z d 6m7 m e x f x6 8 c c SkillSHEET 8. Simplif ech of the followig. c 6 d 7 7 e x x f m m 9 x 7 g h - i x - 0

15 Chpter 8 Expoetil fuctios 7 6 Simplif ech of the followig. x x 9 c m d 0x x e - f q p8 7 p q Simplif ech of the followig. c 7 6 SkillSHEET 8. d - ( ) 0 e m9 8 f g p7 - m - h x p i m c Exmple 0, c 8 Simplif ech of the followig. c ( ) d c e x z f 7 x 8 m g h i 7 8 c9 x 7-9 multiple choice If m - is equl to, the m d could ot e: A d B d 6 C d 8 D d 9 E oth C d D - p m - Whe simplified, - is equl to: p m - m p A - B - C D - p E - m p - m mp - - m m m p m - p SkillSHEET 8. GAMEtime 0 Simplif ech of the followig. m c d 6x e 8 9 f 6x 8 g 7m 9 h p q 0 i Expoetil futios 00 WorkSHEET 8.

16 7 Mths Quest 0 for Victori Creer profile JIM ELLIOT Flood Egieer Qulifictios: Diplom of Civil Egieerig Bchelor of Egieerig (Hos) Mster of Egieerig Sciece Mster of Busiess Admiistrtio Memer Istitutio of Egieers (Austrli) Emploer: Bureu of Meteorolog Comp We site: Civil Egieerig ws recommeded s turl progressio from m iterest/ptitude i mthemtics d phsicl scieces. Hdrolog d wter resources were prt of the ltter sectio of m egieerig studies d I ws iterested eough i the ide of mthemticl modellig i tht re to pursue creer i it. As egieer workig i the re of floodig, it is importt to e le to estimte how much wter flows dow river durig flood evet, prticulrl t the pek. Flood flows re ormll preseted s the volume of wter per uit of time (for exmple, cuic metres per secod). These flows c e difficult to mesure directl, so it is commo to hve to mke estimte fter the evet. Deris d other mrkigs re used to idetif the mximum level reched the flood. From these mrkigs the slope of the river durig flood c e mesured. The formul for reltig the slope of the flood to the velocit of flow is clled Mig s S C R formul. This formul is: v - where C.0 for SI uits v velocit of flow (m/s) S slope R hdrulic rdius (m) ( re/wetted perimeter) roughess coefficiet A difficult prt of this process is the estimte of, the roughess coefficiet. There re vrious empiricl reltioships for doig this. For exmple, i grvel-ed rivers where S > 0.00, equtio tht c e used is: 0.S 0.8 R 0.6 where R is the hdrulic rdius (m), d S is slope. Mig s formul is used to estimte the velocit of flow of flood wter t the pek of the flood. Multiplig this velocit the verge cross-sectiol re provides estimte of the pek flow rte. A tpicl d s work for me could ivolve usig these estimtes of pek flow to get some ide of the severit of the floodig d the dmge cused. This would iclude uderstdig the reltioships etwee hev rifll d floodig. Flood volumes re essetil iput to this. I ws lws fscited the et, cocise w mthemtics could e used to descrie thigs. I lso ejo its strct ture. Questios. List uit for flood flow.. Descrie how Jim c idetif the mximum level reched flood.. Wht is the me of the formul used to clculte the velocit of flow?. Ivestigte wht egieerig courses re ville. How m differet tpes of egieerig re there?

17 Chpter 8 Expoetil fuctios 7 Further use of idex lws So fr we hve cosidered situtios where oe prticulr idex lw ws used for simplifig expressios with idices. However, i most prcticl situtios more th oe lw is eeded to simplif the expressio. The followig exmples show simplifictio of expressios with idices, usig severl idex lws. Exmple Simplif ech of the followig. ( ) 7x m - c m 7 ( x ) 7m m Write the expressio. ( ) Appl the Fourth Idex Lw. Multipl ech idex iside the rckets the idex outside the rckets. Evlute the umer. 6 Multipl coefficiets d multipl proumerls. Appl the First Idex Lw to multipl proumerls with the sme se ddig the idices Write the expressio. Appl the Fourth Idex Lw i the deomitor. Multipl ech idex iside the rckets the idex outside the rckets. Appl the Secod Idex Lw. Divide umers with the sme se sutrctig the idices. m - m Use idices. to express the swer with positive 7x ( x ) 7x - 9x 6 7x 9 7-9x c Write the expressio. c Simplif ech umertor d deomitor multiplig coefficiets d the proumerls with the sme se. Appl the Secod Idex Lw. Divide umers with the sme se sutrctig the idices. m m 7 7m m 6m - 7m Simplif the umertor usig 0 6m m m 8-0 7

18 7 Mths Quest 0 for Victori Whe more th oe idex lw is used to simplif expressio the followig steps c e tke.. If expressio cotis rckets, expd them first.. If expressio is frctio, simplif ech umertor d deomitor the divide (simplif cross the dow).. Express the fil swer with positive idices. The followig exmple illustrtes the use of idex lws for multiplictio d divisio of frctios. Exmple Simplif ech of the followig. ( ) - 0 ( - 8m ) 7 ( 6m ) m 6m ( Write the expressio. ) ( ) 7 Remove the rckets i the umertor of the first frctio d i the deomitor of the secod frctio. Multipl umertors d multipl deomitors of the frctios. (Simplif cross.) Divide the umertor the deomitor. (Simplif dow.) Express the swer with positive idices. - Write the expressio. 8m ( 6m ) m 6m Remove the rckets. 8m - 6m m 6 6m 8m Chge the divisio sig to multiplictio - d tip the secod frctio ( d tip). 6m 6m 6 m 8m Multipl umertors d multipl deomitors. (Simplif cross.) 86m 6 Ccel commo fctors d divide proumerls with the sme se. (Simplif dow.) Simplif d express the swer with positive idices m 8m Note tht the whole umers could e ccelled i step.

19 rememer rememer Chpter 8 Expoetil fuctios 7. Simplifictio of expressios with idices ofte ivolves pplictio of more th oe idex lw.. If expressio cotis rckets, the should e removed first.. If the expressio cotis frctios, simplif cross the dow.. Whe dividig frctios, chge to d tip the secod frctio ( d tip).. Express the fil swer with positive idices. 8D Further use of idex lws Mthcd Exmple Simplif ech of the followig. ( ) ( ) 6 c m ( m ) 6 d ( pq ) ( p q ) e ( 7 ) ( ) f ( c ) ( c ) g i 6x q x h ( 6m ) q p p m Further use of idex lws Simplif ech of the followig. x 6 c ( ) ( x ) d x e - f x 7 ( 7 ) Exmple p 6 q g - h c i p q c ( m ) ( m ) 7 g h - g h x z x z Exmple Simplif ech of the followig. c m 6 m - c 6m 7 6 6x d x 6 ( 6x - e ) - f 9x x 6 9x x 7 g ( ) ( p h 6 q ) pq - i 6( ) p q ( pq ) 0m 6 m m m x x - 0x x 6x x x x

20 76 Mths Quest 0 for Victori Exmple Simplif ech of the followig. - 6 ( - 6 ) - 6 c m d - e f m 6m x - - m 0 x x9 0 p g 6 q p 6 q h - i q p 6 ( m ) - ( m 6 ) ( - m ) ( m) x - ( x ) x 6 x 7 x x - 9x x Exmple Simplif ech of the followig c d x 6 ( x 6 ) ( x ) 0x e x x 6-0 x ( x ) f m m m 6 m g m 6m - 8m h c - c 6c 6 Evlute ech of the followig. ( ) 0 ( 0 ) ( 6 ) ( ) ( 6 9 ) 0 6 ( ) 7 Evlute the followig for x 8. (Hit: Simplif first.) ( x) x ( - ) x 8 Simplif the followig frctio. 9 ( ) - ( ) ( ) Fid the vlue of if the frctio is equl to. WorkSHEET 8. 9 multiple choice Which of the followig is ot the sme s ( x)? A 8x B ( x) C 6x D ( x ) E ( ) x ( x )

21 Chpter 8 Expoetil fuctios 77 Expoetil fuctios d their grphs Reltioships of the form x re clled expoetil fuctios with se, where is rel umer ot equl to, d x is the idex or expoet. Expoet is other word for idex or power. (Note tht i this chpter we will tke ol positive vlues of.) The me expoetil is used, s the idepedet vrile x is the expoet. Exmple Complete the tle of vlues elow d use it to plot the grph of x. x 0 Sustitute ech vlue of x ito the fuctio x to oti the correspodig -vlue. x Plot ech poit geerted o set of xes. Joi with smooth curve. x x The grph i worked exmple hs severl importt fetures. The grph psses through (0, ). Tht is, the -itercept is. The grph of equtio i the form x will pss through this poit. The grph psses through the poit (, ). All grphs of the form x will pss through the poit (, ). > 0 for ll vlues of x. You will otice tht for egtive vlues of x, the grph gets ver close to ut will ever touch the x-xis. Whe this occurs, the lie which the grph pproches is clled smptote. The equtio of the smptote for x is 0.

22 78 Mths Quest 0 for Victori Grphics Clcultor tip! Drwig expoetil fuctios Use our grphics clcultor to plot the grph of x. Exmple. Press Y.. For Y eter ^X. Plot the grph of x for x. Stte the -itercept. c Write the equtio of the horizotl smptote.. Press WINDOW d djust the settigs.. Press GRAPH to see the grph of x.. Now o the sme xes grph x. Press Y d for Y eter ^X. 6. Press GRAPH to see oth x d x o the sme set of xes. Whe the expoetil fuctio is of the form k x (or k x ), the coefficiet k hs the effect of diltig the grph, without chgig the sic shpe or the loctio of the smptote. This c e see i the followig worked exmple. Prepre tle of vlues tkig x-vlues from to. Fill i the tle sustitutig ech vlue of x ito the give equtio. Drw set of xes o grph pper to plot the poits from the tle d joi them with smooth curve. x x 0 x

23 Chpter 8 Expoetil fuctios 79 Locte where the curve cuts the -xis. Altertivel, fid the -vlue for x 0 i the tle. c Fid imgir lie to which the curve gets closer d closer ut does ot cross. As it is horizotl smptote, the equtio will e of the form costt. The -itercept is. c The equtio of the smptote is 0. The worked exmple elow cosiders the effect of egtive i the expoet. Exmple Plot the grph of x for x, clerl showig the -itercept d the horizotl smptote. Drw up tle of vlues. Sustitute the vlues of x ito the equtio to fid the correspodig -vlues. Drw set of xes, plot the poits geerted from the tle d joi with smooth curve. x x x 9-7 rememer rememer. Reltioships of the form x, where re clled expoetil fuctios with se.. To oti the grph of expoetil fuctio, costruct tle of vlues first d the plot the poits from the tle d joi them with smooth curve. Altertivel use grphics clcultor or computer grphig pckge.. A smptote is lie which the grph pproches ut ever cuts.

24 80 Mths Quest 0 for Victori Mthcd SkillSHEET Expoetil grphs 8. Exmple 8E Expoetil fuctios d their grphs You m use grphics clcultor or computer grphig pckge to ssist ou i this exercise. Complete the tle elow d use the tle to plot the grph of 0 x. x 0 EXCEL GC Spredsheet progrm Grphs of the form + c Plot the grph of ech of the followig expoetil fuctios. x x c 6 x O the oe set of xes drw the grphs of x, x d x. Use our swer to questio to descrie the effect of icresig the vlue of o the grph of x. Plot the grph of x for x. Exmple Stte the -itercept. c Write the equtio of the horizotl smptote. Expoetil grphs 6 Complete the followig tle of vlues d the plot the grph of x, x, d x o the sme set of xes. x 0 x x x Exmple 7 Stud the grphs i questio 6 d stte the effect tht the vlue of k hs o grphs with equtio k x. 8 Plot the grph of x for x, clerl showig the -itercept d the horizotl smptote. 9 O the oe set of xes sketch the grphs of x d x. 0 Use our swer to questio 9 to descrie the effect of egtive idex o the grph of x. Complete the tle of vlues elow d use the poits geerted to sketch the grph of x. x 0 x B writig with egtive idex, show lgericll tht the fuctios d x re ideticl. Drw the grphs of (.) x, (.) x d (.8) x. Drw the grph of 0 (.) x. Stte the -itercept. c Write the equtio of the horizotl smptote.

25 Chpter 8 Expoetil fuctios 8 Mug-He ivests $000 t 0% p.. iterest compouded ull. This ivestmet c e represeted the fuctio A 000 (.), where A is the mout to which the ivestmet grows d is the umer of ers of the ivestmet. Prepre tle of vlues for 0 6. Sustitute iteger vlues of ito the equtio d use clcultor to fid correspodig vlues of A. Plot the poits geerted the tle, clerl lellig the xes. Joi the poits with smooth curve. c Use the tle of vlues or the grph to fid the vlue of the ivestmet fter ers. Kevi us cr for $ The cr deprecites t the rte of % p.. The vlue, $V, of the cr fter ers c e give the equtio V (0.8). Prepre tle of vlues for 0. Sustitute iteger vlues of ito the equtio d use clcultor to fid correspodig vlues of A. Roud swers to the erest whole umer s required. Plot the poits geerted the tle, clerl lellig the xes. Joi the poits with smooth curve. c Descrie wht is hppeig to the vlue of the cr s icreses. d Fid the vlue of the cr fter ers. Give the swer to the erest dollr. Further expoetil grphs I the previous exercise we sw the effect of chgig the vlue of i x d chgig the vlue of k i k x. Now we will cosider the effect of chgig some of the other fctors i expoetil equtio. Use our grphics clcultor to drw the grph of x. Now use the clcultor to drw the grphs of ech of the followig o the oe set of xes. x x + c x For the grphs drw i prt, stte the equtio of the horizotl smptote. Use our swers to prts d to stte the effect tht chgig the vlue of c hs o the grph of x + c. O the sme set of xes drw the grphs of: x x + c x. 6 Use our swer to prt to stte the effect tht chgig the vlue of hs o the grph of x. Now let s look t sketchig some expoetil grphs. Rememer tht sketch grph shows the sic shpe of the reltioship d its ke fetures. A sketch grph is ot drw plottig poits from tle of vlues. 7 Sketch grphs of ech of the followig o the oe set of xes, showig the -itercept d the horizotl smptote. x d x + x d x c x d x + d x d x Check our swers with grphics clcultor or computer grphig pckge.

26 8 Mths Quest 0 for Victori importt evets of Complete the tle of vlues for the two expoetil fuctios d plot their grphs. The vlues of the fuctios d the letters eside the x-vlues give the puzzle code. x : x A D E 0 G H I Y L ( ) x + : x N O P 0 R S T U V x

27 Chpter 8 Expoetil fuctios 8 Simplif ( ) ( ). 6e Simplif d. ( 8e 8 d ) ( j Simplif k ) j -. 6( j k ) 6k i 8hi h Simplif -. 7h i 7h i Evlute x 7 - whe x. 9x x ( ) x 6 Drw the grph of x. 7 Drw the grph of 0 x. 8 Drw the grph of x. 9 Drw the grph of x. 0 Drw the grph of x. MATHS MATHS QUEST C H A L LL E N G G E E A tourig us verges 80 km/h, ut i fogg coditios, the driver slows dow to verge of 0 fewer km/h. How m miutes lter th usul will the us rrive t its ext stop 60 km w if there is fog? A squre is cut ito two sectios stright lie pssig through the cetre of the squre. Show tht o mtter which lie is drw, the res of the two sectios re lws equl. Wht re the possile iteger vlues of d such tht? Wht re the possile iteger vlues of, d c such tht ( ) c c?

28 8 Mths Quest 0 for Victori Modellig expoetil growth d dec Expoetil fuctios c e used to model m rel situtios ivolvig turl growth d dec. Expoetil growth is whe qutit grows costt percetge i ech fixed period of time. Exmples of expoetil growth iclude growth of ivestmet t certi rte of compoud iterest d growth i the umer of cells i cteril colo. Expoetil dec is whe qutit decreses costt percetge i ech fixed period of time. Exmples of expoetil dec iclude erl loss of vlue of item (clled deprecitio) d rdioctive dec. Both expoetil growth d dec c e modelled expoetil fuctios of the tpe k x ( k x ). The differece is i the vlue of the se. Whe >, there is expoetil growth d whe 0 < < there is expoetil dec. The vlue of k correspods to the iitil qutit tht is growig or decig. For exmple, iitil umer of cteri i culture dish, the popultio of cit t the egiig of certi time itervl, or the origil vlue of ivestmet. Exmple The umer of cteri, N, i Petri dish fter x hours is give the equtio N 0 x. Fid the iitil umer of cteri i the Petri dish. Fid the umer of cteri i the Petri dish fter three hours. c Drw the grph of the fuctio of N gist x. d Use the grph to estimte the legth of time tht it will tke for the umer of cteri to triple. 6 Write the formul. N 0 x The iitil umer of cteri i the Petri dish is N 0 x whe x 0. So sustitute x 0 ito the give formul d evlute. (Notice tht this is the vlue of k for equtios of the form k x.) Write the swer i setece. Whe x 0, N The iitil umer of cteri i the Petri dish is 0.

29 Chpter 8 Expoetil fuctios 8 To fid the umer of cteri i the Petri dish fter hours sustitute x ito the formul d evlute. Whe x, N Write the swer i setece. After hours there re 00 cteri i the Petri dish. c Drw set of xes, lellig the horizotl xis x d the verticl xis N. c Plot the poits geerted the swers to prts d. Clculte the vlue of N whe x d x d plot the poits geerted. At x, N 0 At x, N Joi the poits plotted with smooth curve. N 00 N 0 x x d Triple 0 is 0, so drw horizotl lie from N 0 to the curve d from this poit drw verticl lie to the x-xis. d N N 0 x 00 0 x The poit o the x-xis will e the estimte of the time tke for the umer of cteri to triple. The time tke will e pproximtel.6 hours.

30 86 Mths Quest 0 for Victori Exmple A ew computer costs $000. It is estimted tht ech er it will e losig % of the previous er s vlue. Fid the vlue, $V, of the computer fter the first er. Fid the vlue of the computer fter the secod er. c Fid the equtio which reltes the vlue of the computer to the umer of ers,, it hs ee used. d Use our equtio to fid the vlue of the computer i 0 ers time. Stte the origil vlue of the computer. V Sice % of the vlue is eig lost ech er, the vlue of the computer will e 88% or (00 )% of the previous er s vlue. Therefore, the vlue fter the first er (V ) is 88% of the origil cost. 7 Write the swer i setece. V 88% of The vlue of the computer fter er is $60. The vlue of the computer fter the secod er, V, is 88% of the vlue fter the first er. Write the swer i setece. V 88% of The vlue of the computer fter the secod er is $.0. c The origil vlue is V 0. c V The vlue fter the first er, V, is V otied multiplig the origil vlue The vlue fter the secod er, V, is otied multiplig V 0.88, or multiplig the origil vlue, V 0, (0.88). The vlue fter the third er, V, is otied multiplig V 0.88, or V 0 (0.88). B oservig the ptter we c geerlise s follows: the vlue fter the th er, V, c e otied multiplig the origil vlue, V 0, 0.88 times; tht is, (0.88). V ( ) (0.88) V 000 (0.88) (0.88) V 000 (0.88) d Sustitute 0 ito the equtio otied i prt to fid the vlue of the computer fter 0 ers. Write the swer i setece. d Whe 0, V (0.88) The vlue of the computer fter 0 ers is $8.0.

31 Chpter 8 Expoetil fuctios 87 Sometimes the reltioship etwee the two vriles closel resemles expoetil ptter, ut cot e descried exctl expoetil fuctio. I such cses, prt of the dt re used to model the reltioship with expoetil growth or the dec fuctio. Exmple 8 The popultio of certi cit is show i the tle elow. Yer Popultio ( 000) Assume tht the reltioship etwee the popultio, P, d the er, x, c e modelled the fuctio P k x, where x is the umer of ers fter 97. The vlue of P must e multiplied 000 i order to fid the ctul popultio. Stte the vlue of k, which is the popultio, i thousds, t the strt of the period. Use middle poit i the dt set to fid the vlue of, correct to deciml plces. Hece, write the formul, coectig the popultio, P, with the umer of ers, x, sice 97. c For the ers give, fid the size of the popultio usig the formul otied i prt. Compre it with the ctul size of the popultio i those ers. d Predict the popultio of the cit i the ers 00 d 00. From the give tle, stte the vlue of k tht correspods to the popultio of the cit i the er 97. k 8 Write the give formul for the popultio of the cit. P k x Replce the vlue of k with the vlue foud i. P 8 x Usig middle poit of the dt, replce x with the umer of ers sice 97 d P with the correspodig vlue. Middle poit is (98, ). Whe x 0, P, so 8 0. Cotiued over pge

32 88 Mths Quest 0 for Victori 6 Solve the equtio for Roud the swer to two deciml plces. Rewrite the formul with this vlue of..06 So P 8 (.06) x c Drw tle of vlues d eter the give ers, the umer of ers sice 97, x, d the popultio for ech er, P. Roud vlues of P to the erest whole umer. Commet o the closeess of the fit. c Yer x P The umers for the popultio otied usig the formul closel resemle the ctul dt. d Fid the vlue of x, the umer of ers fter 97. Sustitute this vlue of x ito the formul d evlute. Roud to the erest whole umer. d For the er 00, x 0. P 8 (.06) 0 P P 7 Aswer the questio i setece. The predicted popultio for 00 is Repet for the er 00. For the er 00, x. P 8 (.06) P P 98 The predicted popultio for 00 is rememer rememer I the fuctio k x :. k represets the iitil mout or qutit.. is the se. If >, the fuctio represets expoetil growth. If 0 < <, it represets expoetil dec.. To fid the vlue of : (i) i the cse of expoetil growth dd the % icrese to 00% d chge the resultig percetge ito deciml (ii) i the cse of expoetil dec sutrct the % decrese from 00% d chge the resultig percetge ito deciml.

33 Chpter 8 Expoetil fuctios 89 8F Modellig expoetil growth d dec Exmple 6 The umer of micro-orgisms, N, i culture dish fter x hours is give the equtio N 000 x. Grphs of Fid the iitil umer of micro-orgisms i the dish. the form Fid the umer of micro-orgisms i dish fter hours. x + c c Drw the grph of N gist x. d Use the grph to estimte the umer of hours eeded for the iitil umer of micro-orgisms to qudruple. The vlue of ivestmet (i dollrs) fter ers is give A 000 (.07). Fid the size of the iitil ivestmet. Fid the vlue of the ivestmet (to the erest dollr) fter 6 ers. c Drw the grph of A gist. d Use the grph to estimte the umer of ers eeded for the iitil ivestmet to doule. EXCEL Spredsheet Exmple 7 multiple choice The fuctio P 00 (0.89) represets : A expoetil growth with the iitil mout of 00 B expoetil growth with the iitil mout of 0.89 C expoetil dec with the iitil mout of 00 D expoetil dec with the iitil mout of 0.89 E expoetil dec with the iitil mout of The reltioship etwee two vriles, A d t, is descried the fuctio A (.09) t, where t is the time, i moths, d A is the mout, i dollrs. This fuctio idictes: A mothl growth of $ B mothl growth of 9. cets C mothl growth of.09% D mothl growth of 9.% E erl growth of 9.% A ew wshig mchie cost $90. It is estimted tht ech er it will e losig 7 per cet of the previous er s vlue. Fid the vlue of the mchie fter the first er. Fid the vlue of the mchie fter the secod er. c Fid the equtio tht reltes the vlue of the mchie, $V, to the umer of ers,, tht it hs ee used. d Use our equtio to fid the vlue of the mchie i ers time. A certi rdioctive elemet decs i such w tht ever 0 ers the mout preset decreses %. I 900, 0 mg of the elemet ws preset. Fid the mout preset i 90. Fid the mout preset i the er 000. c Fid the rule tht coects the mout of the elemet preset, A, with the umer of 0-er itervls, t, sice 900. d Fid the mout preset i the er 00. Roud our swer to deciml plces. e Grph the fuctio of A gist t. f Use the grph to estimte the hlf-life of this elemet (tht is, the umer of ers eeded for hlf the iitil mout to dec). SkillSHEET SkillSHEET

34 90 Mths Quest 0 for Victori Exmple 8 6 Whe T-shirt mde of certi fric is wshed, it loses % of its colour. Fid the percetge of colour tht remis fter: i two wshigs ii five wshigs. Write fuctio for the percetge of colour, C, remiig fter w wshigs. c Drw the grph of C gist w. d Use the grph to estimte the umer of wshigs fter which there is ol 8% of the origil colour left. 7 The popultio of certi coutr is show i the tle elow. Yer Popultio (i millios) Assume tht the reltioship etwee the popultio, P, d the er,, c e modelled the formul P k, where is the umer of ers sice 960. Stte the vlue of k. Use the middle poit of the dt set to fid the vlue of rouded to two deciml plces. Hece, write the formul tht coects the two vriles, P d. c For the ers give i the tle, fid the size of the popultio, usig our formul. Compre the umers otied with the ctul size of the popultio. d Predict the popultio of the coutr i the er The temperture i the room, recorded t 0-miute itervls fter the ircoditioer ws tured o, is show i the tle elow. Time (mi) Temperture WorkSHEET 8. Assume tht the reltioship etwee the temperture, T, d the time, t, c e modelled the formul T c t, where t is the time, i miutes, sice the ircoditioer ws tured o. Stte the vlue of c. Use the middle poit i the dt set to fid the vlue of to deciml plces. c Write the rule coectig T d t. d Usig our rule, fid the temperture i the room 0, 0, 0 d 0 miutes fter the ircoditioer ws tured o d compre our umers with the recorded temperture. Commet o our fidigs. (Give swers correct to deciml plce.)

35 Chpter 8 Expoetil fuctios 9 Creer profile DONNA PULLEN Accoutig Mger Qulifictios: Bchelor of Commerce (Accoutig/ Commercil Lw) I studied ccoutig i Yer d rell ejoed it. I ws lws much etter with umers th words. Also, I fid tht lot of ccoutig work ivolves ivestigtig d solvig prolems which I ejo doig. Some of the ws I use mthemtics i m work re: Workig out lo repmets sed o vrious vriles; tht is, the mout of the lo, the iterest rte, the repmet period d so o. Txtio clcultios, icludig icome tx ple d frige eefit tx. These clcultios ivolve pplig differet tx rtes. Clcultig shiftworkers pelt rtes, overtime, holid d sickess rtes. Durig tpicl d I m e ivolved i slr clcultios, djustmets, queries, ccout recocilitio, testig of the ccoutig sstem, clcultig csh flows d preprig mgemet reports. I fid mthemtics chllegig d prcticl d I use it ever d. Questios. List three tsks Do m eed to work o i her jo.. Wht tertir courses re ville to ecome ccoutt?. Fid out wht sujects re prerequisites for studig ccoutig t tertir istitutio. A growig ivestmet At the strt of this chpter we met Bidi who ws ivestig $0 000 i fixed term deposit erig 6% p.. iterest. Whe Bidi hs $0 000 she iteds to put deposit o house. Write expoetil fuctio tht will model the growth of Bidi s ivestmet. Use our grphics clcultor to grph this fuctio. Fid the legth of time (correct to the erest er) tht it will tke for Bidi s ivestmet to grow to $ Suppose tht Bidi hd ee le to ivest t 8% p.. How much quicker would Bidi s ivestmet hve grow to the $0 000 she eeds? Clvi hs $ 000 to ivest. Fid the iterest rte t which Clvi must ivest his moe, if his ivestmet is to grow to $0 000 i less th 8 ers.

36 9 Mths Quest 0 for Victori summr Cop the seteces elow. Fill i the gps choosig the correct word or expressio from the word list tht follows. To multipl umers with the sme se, the idices re : m m +. To divide umers with the sme se, the idices re : m m. To remove rckets, the idices iside the rcket the idex outside the rcket: ( m ) m. A umer tht hs idex of is equl to : 0. To remove rcket cotiig product or frctio, multipl the idex of ever iside the rcket the idex outside the rcket: () m m m or. m m 6 A umer with egtive idex is equl to the of the sme umer with positive idex: -. 7 Numers with frctiol idices c e expressed s. 8 The deomitor of the frctiol idex idictes the or m tpe of root: ( ) m. 9 Fuctios of the form x re clled fuctios with se, where is rel umer, ot equl to. 0 For the grph of k x, the lrger the vlue of k, the the grph. If k < 0, the grph is i the x-xis. Whe >, the fuctio s x icreses. The the vlue of, the steeper the grph. Whe 0 < <, the fuctio s x icreses. The the vlue of, the less steep the grph. The grph of x hs horizotl with equtio 0. Expoetil growth or dec c e modelled with the fuctio k x. i The vlue k represets the or qutit. ii If the se >, the fuctio represets expoetil. iii If 0 < <, the fuctio represets expoetil. WORD LIST dded smller dec zero iitil mout multipl reciprocl icreses lrger expoetil power surds sutrcted reflected smptote steeper fctor growth decreses

37 Chpter 8 Expoetil fuctios 9 CHAPTER review Simplif ech of the followig. x x x c c 0m c d 6 - p7 q Evlute ech of the followig ( ) 0 ( ) 0 Simplif ech of the followig d express our swer with positive idices. 6 x 0x c ( m ) Evlute ech of the followig without usig clcultor. ( ) 9 c Simplif ech of the followig. c 6 Evlute ech of the followig, without usig clcultor. Show ll workig Simplif ( ) x 0 + 6x 6 8 Simplif ech of the followig. c 8 8A 8A 8B 8B ( ) 6 6 m - m - m x 9 6x x x x x 6 8C 8C 8C 8D

38 9 Mths Quest 0 for Victori 8D 8D 8E 9 Simplif ech of the followig d the evlute. ( 6 ) 0 If m, fid the vlue of: 6 m m ( ) m ( ) m ( 9 m ) ( 6 ) - 6 ( ) 0 For the expoetil fuctio x : complete the tle of vlues elow x 0 plot the grph. 8E 8E 8E 8E 8E 8F 8F test ourself CHAPTER 8 Drw the grph of 0 x for x. Drw the grph of 0 x for x. O the sme xes drw the grphs of (.) x d (.) x. Use our swer to prt to expli the effect of chgig the vlue of i the equtio of x. O the oe set of xes drw the grphs of x, x d x. Use our swer to prt to expli the effect of chgig the vlue of k i the equtio k x. 6 O the sme set of xes sketch the grphs of (.) x d (.) x. Use our swer to prt to expli the effect of egtive idex o the equtio x. 7 A rdioctive sustce decs so tht its mss, m, i grms fter t ds is give m. 0.t. Fid the iitil mss of the sustce. Fid the mss of the sustce fter ds. c Grph the fuctio. d Use the grph to estimte the umer of ds eeded for the mss of the sustce to reduce to of the origil mss A certi ivestmet ps 6.% p.. Costruct the formul which gives the totl vlue, A, of the ivestmet fter ers, if the origil mout ivested ws A 0. If $000 ws ivested, fid the totl vlue fter 7 ers. c Grph the fuctio. d Use our grph to fid the umer of ers eeded for the iitil ivestmet of $000 to doule i size.