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1 TOPIC Non-lnear relatonshps. Overvew.. Introducton Most people use socal meda. If we were to graph the number of hours after a message or rumour had begun aganst the number of people who had seen or read t, what would t look lke? Have ou ever heated a cup of coffee n the mcrowave? You can t drnk t when t comes out of the mcrowave, or ou ll burn ourself. It doesn t take long for a cup of coffee to cool off enough so ou can drnk t, but t wll stll sta warm for a long tme. And t doesn t get cooler than room temperature. What would the graph of the cup of coffee coolng stuaton look lke? These eamples are functons or graphs, but the are not lnear functons or graphs, because the don t follow a straght lne. DISCUSSION How long do ou thnk t takes a rumour on socal meda to reach people? Does t matter who posts the rumour on socal meda? Does t matter what form of socal meda t s posted on? LEARNING SEQUENCE. Overvew. Eponental functons. Quadratc functons. Recprocal functons. Revew Full worked solutons are avalable for ths topc n the Resources secton of our ebookplus. CURRICULUM CONTENT Students: use an eponental model to solve problems AAM construct and analse a quadratc model to solve practcal problems nvolvng quadratc functons or epressons of the form = a + b + c, for eample brakng dstance aganst speed AAM recognse that recprocal functons of the form = k, where k s a constant, represent nverse varaton, dentf the shape of these graphs and ther mportant features AAM Number of vews Tme TOPIC Non-lnear relatonshps

2 . Eponental functons.. Identfng eponental functons An eponent s another name for a power or nde. The term a has an eponent of. An eponental functon s of the form: = a or = a where a > and a. Features of an eponental functon nclude: t s ether alwas ncreasng or alwas decreasng as ncreases t never equals zero the -ntercept s the pont (, ) consecutve -values ncrease b a common multple; for eample, the double, trple, or halve. At one end, the graph approaches the -as but never touches t. Ths lne that the graph approaches s called an asmptote. The equaton of ths lne s =. horzontal asmptote.. Graphs of the form = a, where a > The graph of = a where a > s a contnuous curve that s: alwas postve, meanng t wll be above the -as ncreasng -values as the -values ncrease. The -ntercept s the pont (, ). Methods for graphng functons nclude: completng a table of values and plottng the ponts recognsng the general shape and features usng technolog. GeoGebra, Desmos and Mcrosoft Mathematcs are eamples of free graphng programs. WORKED EXAMPLE Plot the graph of = b: a. completng a table of values for =,,,,,, b. usng technolog. THINK WRITE a.. Draw up a table of values. Jacaranda Maths Quest Mathematcs Standard E for NSW

3 cnonlnearrelatonshps_prnt // : page #. Complete the table of values. When =, = =. = =. = =. = =. When =, = =. When =, = =. When =, = =. When =, When =, When =, E Plot the ponts and jon them to form a curve. = G. FS Fnd the -values b substtutng n the values of. Hnt: Use our scentfc calculator to fnd the -value as a fracton or decmal where necessar. O. PR O (, ) PA EC TE D ) (, ) (, ) (, Enter the equaton n the technolog of our choce. Note: Some programs wll requre ou to enter the full equaton, =, whereas others wll onl requre ou to enter the rght-hand sde,.. The graph of = wll be drawn. R O U N C (, ) (, ) R b.. (, ) Weblnk: Desmos graphcal calculator TOPIC Non-lnear relatonshps

4 .. Graphs of the form = a, where a > The graph of = a where a > s a contnuous curve that s: alwas postve, whch means t wll be above the -as decreasng n -values as the -value ncreases. The -ntercept s the pont (, ). WORKED EXAMPLE Plot the graph of = b: a. completng a table of values for =,,,,,, b. usng technolog. THINK WRITE a.. Draw up a table of values.. Fnd the -values b substtutng n the values of. Hnt: Use our scentfc calculator to fnd the -value as a fracton or decmal where necessar.. Complete the table of values.. Plot the ponts and jon them to form a curve. When =, = ( ) =. When =, = ( ) =. When =, = ( ) =. When =, = =. When =, = =. When =, = =. When =, = =. = (, ) (, ) (, ) (, ) (, ) (, ) (, ) Jacaranda Maths Quest Mathematcs Standard E for NSW

5 cnonlnearrelatonshps_prnt // : page # Enter the equaton = n the technolog of our choce.. The graph of = wll be drawn. b.. = O PR O (, ) FS E Snce the graphs n Worked eamples and both have powers of, the graphs are smpl a reflecton of each other over the -as... Eponental models PA G Weblnk: Mcrosoft Mathematcs R R EC TE D Eponental functons grow b a common factor over equal ntervals. As such, eponental functons are used to model a wde range of real-lfe stuatons such as populaton growth, radoactve deca, compound nterest and deprecaton. Eponental growth s when a quantt grows b a constant factor or percentage n each fed perod of tme, for eample the growth of the bactera causng swne flu. Eponental deca s when a quantt decreases b a constant factor or percentage n each fed perod of tme, for eample the value of the faml car. When eponental functons are models of practcal growth and deca problems, an ntal quantt that s ether growng or decang needs to be gven. Ths gves the general forms: U N C O Eponental growth: = k a, where a > and k s the ntal quantt that s growng. Eponental deca: = k a, where a > and k s the ntal quantt that s decang. Interactvt: Eponental growth and deca (nt-) TOPIC Non-lnear relatonshps

6 cnonlnearrelatonshps_prnt // : page # WORKED EXAMPLE. c. d.... =. C U N PR O E The radus s ntall. m long. R =. (.) The radus wll be. metres after ears. R (m) R =. (.) (,.) 7 n (ears) R (m) R =. (.) (, ) O. =. G b.. R =. (.) EC TE D. R =. (.)n R. Wrte the equaton. The ntal length of the radus s the value of k n = ka. It s the length of the radus when n = ; that s, R =. (.). State the ntal length. After ears, n =. Substtute n = nto the equaton R =. (.)n and smplf. Wrte the answer. Enter the equaton =. (.) nto the technolog of our choce. The graph wll be drawn. Remember: Label the -as: n (ears) Label the -as: R (metres) Start from (,.), the ntal length of the radus. Hnt: You ma need to alter the settngs to change the aes to a sutable scale. Double the ntal radus, R =. Locate the pont on the curve wth a -value of : (, ). The -value s the number of ears. Answer the queston R a.. WRITE PA THINK O FS The radus of a tree ncreases ever ear accordng to the eponental functon R =. (.)n where R s the radus n metres after n ears. a. Fnd the ntal length of the radus. b. Calculate the length of the radus after ears. c. Use technolog to sketch the graph of the functon R aganst n. d. Use the graph to estmate the number of ears for the length of the radus to double. 7 n (ears) It takes ears for the radus of the tree to double. Weblnk: GeoGebra Jacaranda Maths Quest Mathematcs Standard E for NSW

7 In real-lfe problems, the ncrease or decrease n quanttes s sometmes gven as a percentage for the gven tme perod, gvng the value for a n the eponental equaton = a. In practcal growth problems, the quantt ncreases b a fed percent over equal tme ntervals. If the percentage ncrease s r%, the quantt ncreases b a factor of ( + r). The growth factor s ( + r), where r s the percentage epressed as a decmal. The constant, k, represents the ntal value of the quantt. The amount,, after ears s gven b = k ( + r). In practcal deca problems, the quantt decreases b a fed percent over equal tme ntervals. If the percentage decrease s r%, the quantt decreases b a factor of ( r) %. The deca factor s ( r), where r s the percentage epressed as a decmal. The constant, k, represents the ntal value of the quantt. The amount,, after ears s gven b = k ( r). Note: Snce ( r) <, the eponental equaton used to model ths stuaton s = a. WORKED EXAMPLE The student enrolment,, of a hgh school was n and ncreased b % per ear untl 7. a. What s the equaton whch models the eponental growth of the student enrolment? b. How man students were enrolled n the hgh school n? THINK a.. Use the general equaton for eponental growth. Identf the ke components of the equaton.. Substtute these values nto the general equaton. WRITE = k ( + r) k =, r = % =. = k ( + r) = ( +.) = (.). Wrte the answer. The equaton whch models the growth of the student enrolment s = (.), where s the student enrolment after ears. b.. Determne the requred value of. = for the ear. Therefore, corresponds to =.. Substtute ths value nto the equaton found n part a. When = : = (.) =.. Wrte the answer. In the ear, students were enrolled n the hgh school. TOPIC Non-lnear relatonshps 7

8 WORKED EXAMPLE A new car costng $ s deprecatng b 7% each ear. a. What s the equaton that models the eponental deca of the value of the car? b. Calculate the value of the car, to the nearest $, after:. ear. ears. c. Use technolog to estmate how man ears t would take for the car to be valued at less than $. THINK a.. Use the general equaton for eponental deca when deca s gven as a percentage. Identt the ke components of the equaton.. Substtute these values nto the general equaton. WRITE = k ( r) k = r = 7% =.7 ( r) =. = (.). Answer the queston. The equaton that models the eponental deca for the value of the car,, after ears s b... Substtute = nto the equaton and calculate. = (.) = (.) = 7. Answer the queston. The value of the car after one ear s $ 7... Substtute = nto the equaton and calculate. = (.) = (.) = 9.7. Answer the queston. The value of the car after three ears s $ 9 (to the nearest $). c.. Graph = (.).. Locate the pont where =.. Plot the pont (.9, ). Value ($) (, ) = (.) (.9, ) 7 9. Answer the queston. The car wll be valued less than $ after ears. Year Jacaranda Maths Quest Mathematcs Standard E for NSW

9 Eercse. Eponental functons Understandng, fluenc and communcatng. WE Usng technolog of our choce, graph the followng functons on the same set of aes. Descrbe the smlartes and dfferences between these graphs. a. = b. = c. = d. =. WE Usng technolog of our choce, graph the followng functons on the same set of aes. Descrbe the smlartes and dfferences between these graphs. a. = b. = c. = d. =. WE A partcular form of mcroscopc sea lfe gans mass accordng to the eponental functon m = (.) t, where m s the mass n mllonths of a gram and t s tme measured n das. a. Fnd the ntal mass of the mcroscopc sea lfe. b. Calculate the mass after das. c. Usng technolog of our choce, sketch a graph of mass aganst tme for das. d. Use the graph to estmate the number of das t takes the mass to double.. A partcular form of radoactve materal loses mass accordng to the rule m = (.) t where m s the mass and t s tme, measured n das. a. Fnd the ntal mass. b. Calculate the mass of radoactve materal after ears. c. Usng a technolog of our choce, graph the functon m aganst t. d. Use the graph to estmate the number of ears t would take for the mass to halve.. It s thought that the core temperature, T C, of a bun t mnutes after beng taken out of the oven s gven b the eponental functon T = (.) t. a. What s the temperature mmedatel after the bun s taken out of the oven? b. What s the temperature after mnutes?. WE In Januar 99, there were about nternet hosts. Durng the net ears, the number of hosts ncreased b 9% per ear. a. Wrte an eponental equaton whch models the number (n mllons) of hosts t ears after 99. b. Eplan the meanng of the ntal amount. c. Graph the model usng technolog. d. Use the graph to estmate the ear when there were mllon hosts. 7. The populaton of a rare brd n a regon of outback Australa ncreases b % per ear. When frst observed, the populaton was brds. a. Wrte an eponental equaton that models the number of brds,, after observng them for ears. b. Calculate the number of brds n ths regon after 7 ears. Gve our answer to the nearest whole number.. WE Cars deprecate n value each ear. A new car costng $ s to be deprecated at a rate of % each ear. a. What s the equaton that models the eponental deca of the value of the car, where V s the value of the car after t ears? b. Calculate the value of the car, to the nearest $, after:. ear. ears TOPIC Non-lnear relatonshps 9

10 9. Durng a controlled det n preparaton for a major trathlon, an athlete decreases n weght b % per month. He weghed 7 kg at the start of the controlled det. a. Wrte an eponental equaton that models the weght, kg, of the athlete, months nto the controlled det perod. b. Calculate the weght of the athlete after months on the controlled det. Gve our answer correct to decmal place.. A substance dsntegrates accordng to the eponental functon: M = (.) t, t where M s the mass n grams after t ears. a. Fnd the ntal mass. b. Calculate, to the nearest nteger, the mass of the substance after:. ears. ears. ears v. ears c. Sketch the graph of the mass, M, aganst tme, t, b usng ether part b or technolog of our choce. d. Estmate the number of ears t would take for the mass to reduce to a quarter of ts orgnal mass. Problem solvng, reasonng and justfcaton. The number of frut fles, N, n a colon after t das of observaton s modelled b N = A (.) t where A s a constant. The graph of ths equaton s shown. N 7 (, ) t a. What s the value of A? b. What s the dal growth rate of the number of frut fles n the colon? Wrte our answer as a percentage. Comment on the reasonableness of our answer. c. How man of the nsects were present after das? Gve our answer to the nearest nteger. Justf our answer. d. How man das does t take for the number of frut fles n the colon to double? Justf our answer. Jacaranda Maths Quest Mathematcs Standard E for NSW

11 . The value, V, of a certan machne after t ears of V operatng s modelled b the eponental deca equaton V = A (.7) t, where A s a constant. The graph of the value of ths machne s gven below. (, ) a. What s the value of A? b. What s the earl deca rate of ths machne? Gve our answer as a percentage and justf. c. Calculate the value of the machne after:. ears. ears. 7 9 t d. How man ears appromatel would t take for the machne to halve n value?. Harr nvested $ n a fund pang an nterest rate of.% p.a. wth nterest compounded annuall. a. Fnd the equaton whch models the eponental growth of the value of Harr s nvestment. b. Calculate the value of hs nvestment, to the nearest dollar, after:. ear. ears. c. Usng technolog, estmate how long t would take for Harr s nvestment to reach $.. Megan nvested $ n a fund pang an nterest rate of.% p.a. wth nterest compounded monthl. a. State the nterest rate per month. b. Fnd the equaton whch models the eponental growth of the value of Megan s nvestment. c. Calculate the value of her nvestment, to the nearest dollar, after:. ear. ears. d. Usng technolog, estmate how long t would take for Megan s nvestment to reach $. e. Dscuss the dfferences between Megan s nvestment and Harr s nvestment from the prevous queston.. The ntal pulse rate (beats per mnute) of a dancer s 7. The pulse rate decreases b % per mnute. a. Wrte an eponental equaton whch models the pulse rate of the dancer. b. Calculate the pulse rate after mnutes. c. Graph the model usng technolog. d. What practcal range of values would be used to model the pulse rate of the dancer? Eplan.. A populaton of bactera doubles ever hour. At noon the number of bacteral cells s. a. Wrte an eponental functon to model the growth of the bactera. b. What s the bactera populaton at pm? c. What was the populaton at 9 am, three hours before the number of bacteral cells were counted? TOPIC Non-lnear relatonshps

12 . Quadratc functons.. Identfng quadratc functons Quadratcs derve ther name from quadratus, the Latn work for squared. In a quadratc epresson: the varable s rased to the power of two, or squared the hghest power, or nde, s, for eample or + 7. The coeffcent of a partcular varable s the number the varable s multpled b. For eample, n the epresson + 7: the coeffcent of s the coeffcent of s. The general form of a quadratc functon s: = a + b + c where: a s the coeffcent of b s the coeffcent of c s the constant term. The graphs of quadratc functons are called parabolas. Methods to graph a quadratc functon nclude: constructng a table of values usng technolog, such as GeoGebra or Desmos. When graphed, a parabola has the followng shape: WORKED EXAMPLE State the coeffcent of n each of the followng quadratc epressons. a. + b. + c. d. + 7 THINK WRITE a. Fnd the term:. The coeffcent of s. b. Fnd the term:. The coeffcent of s. c. Fnd the term:. The coeffcent of s. d. Fnd the term:. The coeffcent of s. Jacaranda Maths Quest Mathematcs Standard E for NSW

13 WORKED EXAMPLE 7 Identf whether the followng graphs could show a quadratc relatonshp. a. b. c. THINK 7 WRITE a. Not a parabolc shape Not a quadratc relatonshp b. Parabolc shape A quadratc relatonshp c. The graph looks lke two parabolas joned together. A quadratc relatonshp has onl one parabola. Not a quadratc relatonshp.. Graphng the quadratc functon (or parabola) Important features of a parabola are the: the turnng pont, where the curve changes drecton the as of smmetr, where one sde of the parabola s the mrror mage of the other sde the as ntercepts, where the curve crosses the - and -aes. the mamum or mnmum values. The graph of a parabola s ether uprght or nverted. Uprght parabola Turnng pont Inverted parabola Turnng pont If the coeffcent of s negatve, the parabola s nverted. The parabola has eactl one turnng pont, lng on the as of smmetr. The as of smmetr s a vertcal lne through the turnng pont. The turnng pont s called ether a mamum or a mnmum turnng pont, as t gves ether the greatest (mamum) or least (mnmum) value of the quadratc epresson or parabola. TOPIC Non-lnear relatonshps

14 Mamum turnng pont Mnmum turnng pont As of smmetr As of smmetr A parabola wll alwas ntersect the -as at one pont, called the -ntercept. The coordnates of the -ntercept are (, c) for the general quadratc = a + b + c. The ponts at whch a parabola ntersects the -as are called the -ntercepts. A parabola wll ether: ntersect the -as at two ponts ntersect the -as at one pont not ntersect the -as at all. c Two -ntercepts c Two -ntercepts c One -ntercept c One -ntercept c No -ntercepts No -ntercepts c Jacaranda Maths Quest Mathematcs Standard E for NSW

15 WORKED EXAMPLE Consder the graph shown at rght. a. Descrbe the orentaton of the parabola (uprght or nverted). b. Wrte the equaton of the as of smmetr. c. State the coordnates of the turnng pont. d. State the coordnates of an:. -ntercepts. -ntercepts. e. State whether the parabola has a mnmum or mamum turnng pont. THINK a. The arms of the parabola are pontng upwards. b. The as of smmetr s the vertcal lne passng through the turnng pont. c. The turnng pont s at the lowest pont of the parabola. d.. The -ntercepts are the ponts where the parabola ntersects the -as.. The -ntercept s the pont where the parabola ntersects the -as. e. The parabola s uprght, so the turnng pont s a mnmum. WORKED EXAMPLE 9 WRITE The parabola s uprght. The equaton of the as of smmetr s =. The turnng pont s (, ). The -ntercepts are at (, ) and (, ). The -ntercept s at (, ). The parabola has a mnmum turnng pont. a. Complete the table of values for the quadratc functon =. b. Plot the ponts on a graph and jon them wth a smooth curve. c. State the coordnates of:. the turnng pont. the -ntercept. the -ntercepts. d. State the equaton of the as of smmetr. e. State whether the parabola has a mamum or mnmum turnng pont. TOPIC Non-lnear relatonshps

16 THINK a.. Substtute each -value nto the epresson to fnd the -value. WRITE When = : = ( ) ( ) = + = When = : = ( ) ( ) = + = When = : = () () = When = : = () () = = When = : = () () = = When = : = () () = 9 = When = : = () () = = When = : = () () = = When = : = =. Complete the table of values. b.. Plot the ponts.. Jon them wth a smooth curve. = 7 7 Jacaranda Maths Quest Mathematcs Standard E for NSW

17 c.. Locate the turnng pont. The turnng pont s (, ).. Fnd where the curve crosses the -as.. Fnd where the curve crosses the -as. d. The as of smmetr s the vertcal lne through the turnng pont (, ). e. Look at the shape of the graph. It s an upward parabola. The -ntercept s (, ). The -ntercepts are (, ) and (, ). The equaton of the as of smmetr s =. The turnng pont s a mnmum turnng pont. When sketchng a quadratc functon, or parabola, alwas show the coordnates of: the - and -ntercepts the turnng pont. If the -values are restrcted, stop at the end ponts. Show the coordnates of these endponts. WORKED EXAMPLE a. B completng a table of values, sketch the graph of = + for. b. State and label the coordnates of:. the -ntercept. the -ntercepts. the turnng ponts v. the end ponts. THINK a.. Create a table of values for the gven -values.. Substtute the -values nto the epresson to fnd the -values. WRITE a. When =, = ( ) + ( ) = = When =, = ( ) + ( ) = = When =, = () + () = When =, = () + () = + = When =, = () + () = + = TOPIC Non-lnear relatonshps 7

18 . Complete the table of values.. Plot the ponts and jon them wth a smooth curve, startng where = and endng where =. b. From the graph: b.. The -ntercept s where the curve crosses the -as.. The -ntercept s where the curve crosses the -as.. The turnng pont s where the curve changes drecton. v. The endponts are where the curve starts and stops... Quadratc models (, ) (, ) = + (, ) (, ) (, ) The coordnates of the -ntercept are (, ). The coordnates of the -ntercept are (, ). The coordnates of the turnng pont are (, ). The endponts are (, ) and (, ). Quadratc functons are also used to model a wde range of real-lfe stuatons to solve practcal problems. Quadratc functons can model the moton of a fallng object, the flght of a projectle or the shape of objects such as brdges. The are also used n economc models of cost and revenue. The turnng pont s used to fnd a mamum, the greatest value, or mnmum, the least value, of the quadratc model. The -ntercepts show where the -value changes from postve to negatve or negatve to postve. In a busness venture, for eample, t can show when ou start makng a proft. The -ntercept refers to the startng value, or ntal value, of the model. When a parabola s sketched to represent a practcal model, we must consder the -values and -values to ensure t makes sense n a practcal contet. Jacaranda Maths Quest Mathematcs Standard E for NSW

19 WORKED EXAMPLE A ball s thrown vertcall nto the ar. The graph shows the heght h metres, of the ball after t seconds. a. What s the mamum heght the ball reaches above the h ground? (,.) b. The ball was thrown from a platform. How hgh s the platform above the ground? c. How man seconds does t take for the ball to reach ts greatest heght? d. When s the ball metres above the ground? (, ) e. Appromatel how man seconds does t take for the ball to reach the ground? (., ) f. Onl part of the graph of the parabola s drawn. Can ou t suggest wh? THINK WRITE a.. Locate the turnng pont. Turnng pont (,.). The greatest heght wll be the h-value. h-value =.. Answer the queston. The ball s mamum heght above the ground s. metres. b.. Locate the ntal pont, the h-ntercept. The h-ntercept s (, ).. The heght s the h-value. h-value =. Answer the queston. The platform s metres above the ground. c.. Locate the turnng pont. Turnng pont (,.). Tme s the t-coordnate. t-value =. Answer the queston. It takes seconds for the ball to reach ts greatest heght. d.. Locate ponts where h =. Ponts (, ) and (, ). Tme s the t-coordnate. t-value = or. Answer the queston. Intall the ball s metres above the ground. It s agan metres above the ground after seconds when t s fallng. e.. Locate the ponts where h =. Pont (., ). Tme s the t-coordnate. t-value =.. Answer the queston. It takes appromatel seconds for the ball to reach the ground. f.. The t-as s tme, so t, and the h-as s heght above ground, so h.. Answer the queston. Snce both tme and heght above the ground need to be greater than or equal to zero, the parabola would start at (, ) and stop at (., ). TOPIC Non-lnear relatonshps 9

20 WORKED EXAMPLE A landscaper has metres of edgng wth whch to construct a rectangular garden bed. He wshes to construct the garden bed that has the greatest area. a. If one sde of the garden bed s metres, calculate:. the length of the other sde. the area of the garden bed. b. If one sde of the garden bed s metres, fnd an epresson for:. the length of the other sde. the area, A, of the garden bed. c. State a sutable range of values for for ths practcal stuaton. d. B completng a table of values or usng technolog of our choce, sketch the quadratc functon whch models ths problem. e. Fnd the mamum area of the rectangular garden bed and ts dmensons. THINK a... State the permeter formula for a rectangle.. Substtute n the known values: metres for the permeter and metres for the length. WRITE w l l Permeter = l + w w = + w. Fnd the wdth, w. = + w w = w =. Answer the queston. The length of the other sde s metres. Note: Snce the permeter s metres, the length and wdth must add to metres, half the permeter... State the area formula for a rectangle and substtute n the known values. Area = length wdth = =. Answer the queston. The area of the garden bed s m. b... Name the other sde. Let the other sde be w metres.. The two sdes must add to metres. + w =. Answer the queston. The other sde s ( ) metres... State the area formula. Area = length wdth. Substtute for length and ( ) for wdth. A = ( ) A =. Answer the queston. The area s A =. Jacaranda Maths Quest Mathematcs Standard E for NSW

21 c. Fnd a sutable range of values for, rememberng that s a length of a garden bed and halfwa around the garden s metres. The length,, needs to be postve and less than. d.. Draw up a table of values from to. When =, A =. Substtute the -values nto the epresson to fnd the values of A. A = () () A = When =, A = () () A = A = 9 When =, A = () () A = A = When =, A = () () A = 9 A = When =, A = () () A = A = When =, A = () () A = A = When =, A = () () A = A = When = 7, A = (7) (7) A = 7 9 A = When =, A = () () A = A = When = 9, A = (9) (9) A = 9 A = 9 When =, A = () () A = A =. Complete the table of values TOPIC Non-lnear relatonshps

22 . Plot the ponts and jon them wth a smooth curve. Alternatvel, use A (, ) technolog of our choce to show the same graph of the quadratc model. = (, ) (, ) 7 9 e.. Locate the turnng pont. Mamum turnng pont (, ). Area s the -value. -value =. Answer the queston, rememberng that the dmensons are and ( ). WORKED EXAMPLE The mamum area of the garden bed s m, and ts dmensons are m b m. A bungee jumper dves from a platform from a brdge over a rver. The heght, h metres, above the ground after t seconds n the ar s gven b h = t +. a. Usng technolog, sketch the graph of the bungee jump. What restrctons need to be consdered? b. What wll be the bungee jumper s heght after second? c. From what heght dd the bungee jumper ntall jump? d. For how long was the bungee jumper n the ar? e. Is t realstc for the bungee jumper to spend. seconds on the jump? Eplan. THINK a.. Use technolog to sketch the graph of h = t +, where t represents the tme and h represents the heght.. For the practcal stuaton, the doman needs to be restrcted as the tme must be greater than. Tme cannot be a negatve value. The range also needs to be restrcted. If the heght s negatve, ths means that the bungee jumper has ht the surface of the rver and gone below water level. WRITE h t Jacaranda Maths Quest Mathematcs Standard E for NSW

23 b.. Use the data settngs or other functons of the technolog to fnd the value of h when t =.. Answer the queston. After second, the bungee jumper s m above ground. c.. The ntal heght can be calculated b fndng the h-ntercept. When t = : h = () + =. Answer the queston. The bungee jumper was ntall m above the ground. d.. Determne the tme when the bungee jumper s at ground level b fndng the t-ntercept. h (., ). Answer the queston. The bungee jumper was n the ar for. seconds. e.. Use the technolog to fnd the value of h when t =... Answer the queston. If the bungee jumper took. seconds, the bungee jumper would have gone metres below the rver surface. Ths does not seem realstc, as the rver s probabl not metres n depth and the bungee cord would not be ths long. The bungee jumpee would probabl stop just above the rver surface. Eercse. Quadratc functons Understandng, fluenc and communcatng. WE State the coeffcent of n each of the followng quadratc epressons. a. + b. + c. +. State the coeffcent of n each of the followng quadratc epressons. a. + b. + c. + t TOPIC Non-lnear relatonshps

24 . WE 7 Identf whch of the followng graphs s a quadratc relatonshp. a. b. c.. MC Whch of the followng graphs shows a quadratc relatonshp? 7 B C D d. 7 7 A. WE For each of the graphs shown:. descrbe the orentaton of the parabola. state the coordnates of the turnng pont. wrte the equaton of the as of smmetr v. state the coordnates of an -ntercepts and/or -ntercepts v. state whether the parabola has a mamum or mnmum turnng pont. Jacaranda Maths Quest Mathematcs Standard E for NSW

25 a. b.. a. Draw up a table of values from = to = for each of the quadratc functons below and graph them on the same number plane.. =. =. = v. = b. Descrbe the smlartes and dfferences between the curves graphed. 7. a. Draw up a table of values from = to = for each of the quadratc functons below and graph them on the same number plane.. =. = +. = + v. = + b. Descrbe the smlartes and dfferences between the curves graphed. c. Sketch = wthout plottng ponts. Check our answer usng technolog.. WE 9 a. Complete the table of values for the quadratc functon = +. b. Plot the ponts on a graph and jon them wth a smooth curve. c. State the coordnates of:. the turnng pont. the -ntercept. the -ntercepts. d. State the equaton of the as of smmetr. e. State whether the parabola has a mamum or mnmum turnng pont. 9. WE a. B completng a table of values, sketch the graph of =, for. b. State and label the coordnates of:. the -ntercept. the -ntercepts. the turnng ponts v. the end ponts. TOPIC Non-lnear relatonshps

26 cnonlnearrelatonshps_prnt // : page #. WE A mssle s fred vertcall nto the ar from the top of a clff. The graph shows the heght, h metres above the ground, of the mssle after t seconds. h FS (, ) (.9, ) t PR O O (, ) What s the mamum heght the mssle reaches above the ground? b. The mssle was fred from a launchng pad. How hgh s the launchng pad above the ground? c. How man seconds does t take for the mssle to reach ts greatest heght? d. When s the mssle metres above the ground? e. Appromatel how man seconds does t take for the mssle to reach the ground? f. Onl part of the graph of the parabola s drawn. Can ou suggest wh?. The quadratc functon below models the path of a football where h s the heght above the ground n metres and d s the horzontal dstance, n metres, from the plaer s foot. PA G E a. h EC TE D (7.7, 7.) (, ) R (.7, ) d What s the mamum heght reached b the football? Gve our answer to decmal place. b. How hgh above the ground was the football when t was kcked? c. How far wll the football have travelled horzontall when t touches the ground? Gve our answer to decmal place. U N C O R a. Jacaranda Maths Quest Mathematcs Standard E for NSW

27 . WE A farmer has metres of fencng wth whch to create a rectangular feld wth the greatest area. a. If one sde of the feld s metres, calculate:. the length of the other sde. the area of the feld. b. If one sde of the feld s metres, fnd an epresson for:. the length of the other sde. the area, A, of the feld. c. State a sutable range of values for for ths practcal stuaton. d. B completng a table of values or usng technolog of our choce, sketch the quadratc functon whch models ths problem. (Hnt: For a table of values, use =,,,,.) e. Fnd the mamum area of the rectangular feld and ts dmensons.. The sum of two postve numbers s. a. Let one of the numbers be. What s the other number? b. Show that the product of these two numbers can be modelled b the quadratc functon =. c. Usng a method of our choce, sketch a graph of ths quadratc functon. d. Fnd the mamum product of the two numbers.. WE When a to rocket s launched, ts path s n the form of the quadratc functon = +, where metres s the heght of the rocket above the ground and metres s the horzontal dstance from the launch ste. a. Usng a method of our choce, sketch the graph of the quadratc functon modellng the flght path of the to rocket. (Hnt: For a table of values, use =,,,...) b. Fnd the horzontal dstance travelled b the to rocket. c. Fnd the mamum heght reached b the rocket. Problem solvng, reasonng and justfcaton. A suspenson brdge has overhead cables supportng the roadwa. The cables are attached to plons and form parabolas shapes. The heght, metres, of the cables above the roadwa s gven b the quadratc functon = + 7, where s the, n metres, between the plons. The graph of the quadratc functon s shown. (, 7) = + 7 (9, ) (, 7) TOPIC Non-lnear relatonshps 7

28 a. How hgh above the roadwa are the cables when the are attached to the plons? b. What s the dstance between each plon? c. What s the mnmum dstance between the roadwa and the cable? d. Onl part of the parabola s drawn. Can ou suggest a reason for ths?. The monthl proft or loss, p (n thousands of dollars), for a new brand of hamburgers s modelled b the quadratc functon p =, where s the number of months after the ntroducton of the new hamburger nto the market. a. Usng a method of our choce, sketch the graph of the quadratc functon. b. Fnd the frst month for whch a proft was shown. c. In whch month s the proft $? Justf our answer. 7. The graph shows the heght, h metres, of a golf ball as a functon of the tme, t seconds, after t s ht. a. For how man seconds s the golf ball n the ar? b. What s the greatest heght reached b the golf ball? c. For how man seconds s the golf ball above a heght of metres? Justf our answer. h (.,.) (, ) (, ). When ou are drvng a car, the brakng dstance requred to stop and the speed at whch ou are travellng s an eample of a quadratc model. One of these quadratc functons s gven below: t d = s (f) where d s the breakng dstance n metres s s the speed n km/h f s the coeffcent of frcton, a constant that depends on the road surface. a. When the road surface s dr asphalt, the coeffcent of frcton s appromatel... Rewrte the quadratc functon n terms of d and s.. Calculate the breakng dstance requred when travellng on dr asphalt at a speed of km/h. b. When the road surface s c, the coeffcent of frcton s appromatel... Rewrte the quadratc functon n terms of d and s.. Calculate the breakng dstance requred when travellng on dr asphalt at a speed of km/h. c. Comment on the dfference n breakng dstances at km/hour. Jacaranda Maths Quest Mathematcs Standard E for NSW

29 9. The path taken b a netball thrown b an Australan plaer s gven b the quadratc functon = where s the heght of the netball and s the horzontal dstance from the plaer s upstretched hand. a. Complete a table of values for for ths functon. b. Plot the graph of = c. What values of are not reasonable for ths model and wh? d. How far above the floor was the ball when t was thrown? e. Estmate the mamum heght reached b the netball. f. Assumng that nothng hts the netball, estmate how far awa from the plaer the netball wll strke the ground. Justf our answer.. A car engne spark plug produces a spark of electrct. The sze of the spark depends on how far apart the termnals are. The percentage performance, Z%, of a certan brand s thought to be modelled b the quadratc functon Z = g g, where g s the dstance between the termnals. Below s a graph of the quadratc functon =. (, ). = a. From the graph, what values for and would be sutable to model the performance of the spark plug? b. When s the performance greatest? c. From the graph, estmate the dstances between the termnals for whch the percentage performance s greater than %. (, ). TOPIC Non-lnear relatonshps 9

30 . The path of a baseball from the ptcher s hand can be modelled b the equaton = , where s the heght of the baseball above the ground (,.9) and s ts dstance from the ptcher. Both dstances are n metres. To help the team coach, ou have sketched the graph for ths quadratc model. It s shown below. (,.) a. State the range of values for and for whch our graph makes sense n ths practcal contet. (.9, ) (.9, ) b. At what heght does the baseball leave the ptcher s hand? c. How far does the baseball land from the ptcher s hand? Gve our answer to decmal place. d. How hgh wll the baseball be above ground when t s metres from the ptcher s hand? Gve our answer to decmal place. e. The catcher s mtt s. metres above the ground. If the ball hts the catcher s mtt wthout the catcher movng t, estmate how far the catcher s from the ptcher.. Recprocal functons.. Identfng recprocal or hperbolc functons Recall that the recprocal of the number s, or for an number s. The basc recprocal functon s: The general recprocal functon s: Methods to graph the recprocal functon nclude: completng a table of values usng technolog, ncludng a spreadsheet recognsng the general shape of the functon. Consder the recprocal functon =,. =, = k,, where k s a constant. The table of values from = to = s gven below. 7 Jacaranda Maths Quest Mathematcs Standard E for NSW

31 Plottng these ponts and jonng them wth a smooth curve gves the graph, as shown below. Notce that the recprocal of zero s undefned. 7 (, ) = (, ) (, ) (, ) (, ) (, ) (, ) 7 7 (, ) (, (, ) ) (, ) (, ) 7 The graph of the recprocal functon s called a rectangular hperbola. The rectangular hperbola has the followng specal features. The graph has two parts. These are called the branches of the functon. It s alwas undefned for =, as we cannot dvde b zero. As ncreases to nfnt, the graph gets closer to the -as but never touches t. Smlarl, as decreases to negatve nfnt, the graph gets closer to the -as. The -as s called the horzontal asmptote of the graph. The curve approaches the lne = (the -as) but t never touches t. As gets closer to, the graph gets closer to the -as but never touches t. The -as s called the vertcal asmptote of the graph. The curve approaches the lne = (the -as) but t never touches t. These specal features are shown on the graph of = below. Horzontal asmptote 7 Branch = 7 7 Branch Vertcal asmptote 7 TOPIC Non-lnear relatonshps 7

32 cnonlnearrelatonshps_prnt // :7 page 7 # elesson: Sketchng quadratcs n turnng pont form (eles-9) O the table of values for =. PR O a. Complete FS WORKED EXAMPLE the ponts to sketch the graph of the recprocal functon =. c. Comment on the features of the curve. Plot the ponts to sketch the graph of the functon and jon wth a smooth curve. EC TE D b. Complete the table of values. R Undefned R O 7 Comment on the features. C U N c. G a. WRITE PA THINK E b. Plot The curve conssts of two branches. The curve approaches the lnes = and =, but t never actuall touches them. The horzontal asmptote s = and the vertcal asmptote s =. Jacaranda Maths Quest Mathematcs Standard E for NSW

33 .. Inverse varaton Varaton eamnes the effect that changng one varable has on another varable. If two quanttes var nversel, then ncreasng one varable decreases the other. For the two varables and, where > and >, we sa vares nversel wth f decreases as ncreases or vce versa. If vares nversel to, t s wrtten as or = k, where k s the constant of varaton or the proportonalt constant. The smbol means vares wth. Inverse varaton gves the graph of a rectangular hperbola defned for >, the postve branch... Recprocal models Recprocal functons ma also be used to model a wde range of real-lfe stuatons to solve practcal problems. Recprocal functons model practcal relatonshps such as speed and tme, volume and pressure, or current and resstance n an electrcal applance. In practcal stuatons, when two quanttes are nversel proportonal: the are modelled b the rectangular hperbola, = k, > (the postve branch) the product of the two quanttes equals the constant of varaton, k. That s, k =. WORKED EXAMPLE For a gven mass of gas kept at a constant temperature, the volume, V cm, vares nversel as the pressure, P unts. The graph shows ths relatonshp for a partcular tpe of gas. a. Fnd:. the volume when the pressure s unts. the pressure when the volume s cm. b. Fnd the recprocal functon that models ths graph. c. Usng the recprocal functon found n part b, fnd:. the volume when the pressure s unts. the pressure when the volume s cm. THINK WRITE a... Locate on the horzontal as. P = V 9 7. Locate the pont on the curve. Pont on the curve: (, ) 7 9 P. Answer the queston The volume s cm when the pressure s unts... Locate on the vertcal as. V =. Locate the pont on the curve. Pont on the curve: (, ). Answer the queston. The pressure s unts when the volume s cm. TOPIC Non-lnear relatonshps 7

34 b.. Wrte an epresson for V vares nversel wth P. V P. Rewrte wth an equals sgn. V = k P. Choose a pont: Choose a pont: (, ). Substtute to fnd k, k = V P k = k =. Answer the queston. V = s the recprocal functon of P ths model. c... Substtute P = nto the model. V = P = =. Answer the queston. The volume s cm when the pressure s unts... Substtute V = nto the model. V = P = P. To fnd P: P = P P multpl both sdes b P P = smplf P dvde both sdes b = smplf. P =. Answer the queston. The pressure s unts when the volume s cm. WORKED EXAMPLE The table shows how the tme, T hours taken to complete a task vares nversel wth the number of workers, n. Number of workers, n Tme taken (hours), T 7 Jacaranda Maths Quest Mathematcs Standard E for NSW

35 a. Plot the ponts to represent ths data. b. Wrte a mathematcal statement connectng T, n and k, the constant of varaton. c. Calculate:. the tme that t would take 7 people to complete the task.. the number of people requred to complete the task n half an hour. THINK a. The amount of tme taken to complete the task depends on the number of workers; tme s the dependent varable so place t on the vertcal as. The smallest number of hours s and the hghest s so use a scale of on the vertcal as. The smallest number of workers s and the hghest s. Use a scale of on the horzontal as. b.. The graph looks lke a hperbola so assume that the varables are nversel proportonal. Wrte ths mathematcall.. The proporton sgn can be replaced wth = k. Wrte the equaton.. Choose one par of coordnates to help fnd k, sa (, ). Substtute the values T = and n =. Solve for k b multplng both sdes b.. Replace the k wth ts value of n the equaton. Wrte the equaton. c... To fnd the tme taken for 7 workers to complete the task, substtute the value n = 7 nto the equaton T = n and solve for T. WRITE Tme (hours) T T n T = k n = k = k = k T = n n Number of workers T = 7 T =.7 hours. State the result. It would take 7 people.7 hours to complete the task... To determne the number of workers requred to complete the task n half an hour, substtute T = nto the formula. T = n = n TOPIC Non-lnear relatonshps 7

36 . Solve for n b: multplng both sdes b n multplng both sdes b. n = n n n = n = n =. State the answer. people are requred to complete the task n half an hour. WORKED EXAMPLE 7 The table of values shows the fracton of pzza receved when sharng a pzza between an ncreasng number of people. Number of people (n) 7 Fracton of pzza (A) Plot ths nformaton on a number plane. Note: A spreadsheet has been used n ths worked eample as an llustraton of technolog. a. Fnd the nverse varaton rule connectng A and n n ths eample. b. What fracton of pzza would each person receve f the pzza s to be shared equall between people? THINK a.. Usng a spreadsheet, cop the data nto two columns. WRITE 7 A n B A / / / / / /7 / 7 Jacaranda Maths Quest Mathematcs Standard E for NSW

37 . Use the spreadsheet functons to draw a scatterplot of the data. Sharng one pzza. Use the spreadsheet functons to add a trend lne. As a hperbola s requred, select the opton Power or equvalent. b. Use the equaton on the chart to fnd the rule connectng A and n. Fracton of pzza Fracton of pzza Number of people Sharng one pzza = 7 Number of people The equaton gven on the chart s =, whch s the same as =. corresponds to the fracton of pzza (A) and corresponds to the number of people (n). Therefore, the rule connectng A and n s A = n. c.. Substtute n =. A =. Answer the queston. Each person would receve of the pzza. TOPIC Non-lnear relatonshps 77

38 WORKED EXAMPLE Speed and tme for a partcular journe are nversel proportonal. a. If the speed s km/h and the tme s hours, wrte a mathematcal statement connectng these quanttes. b. Rewrte the statement usng the constant of varaton, k. c. It takes hours at an average speed of km/h to complete a certan journe. Fnd the value of k. d. How long would t take to complete the same journe at an average speed of km/h? THINK a. Speed and tme are nversel proportonal. b. Replace the varaton sgn wth an equals sgn and nclude k, the constant of varaton. WRITE c. Substtute = and = to fnd k. = k = k where k s the constant of varaton. = k = k d.. Rewrte the recprocal functon wth = the value of k.. Substtute =. = =. k =. Answer the queston. It would take. hours (or hours and mnutes) to complete the journe. Eercse. Recprocal functons Understandng, fluenc and communcatng. a. WE Complete the table of values for =. b. Plot the ponts to sketch the graph of the recprocal functon =. c. Comment on the features of ths curve.. a. Complete the table of values for =. 7 Jacaranda Maths Quest Mathematcs Standard E for NSW

39 b. Plot the ponts to sketch the graph of the recprocal functon =. c. Comment on the features of ths curve.. WE For a gven mass of gas, the volume, V cm, vares nversel as the pressure, P unts. The graph shows ths relatonshp for a partcular tpe of gas. Volume (cm ) Pressure (unts) a. Fnd:. the volume when the pressure s unts. the pressure when the volume s cm. b. Fnd the recprocal functon that models ths graph. c. Usng the recprocal functon found n part b, fnd:. the volume when the pressure s unts. the pressure when the volume s cm.. The tme, t hours, requred to complete a journe s nversel proportonal to the speed, s km/h. The graph shows ths relatonshp for a partcular journe. a. Fnd:. the tme requred to complete the journe f the average speed s km/h t (h). the average speed f the tme taken s hours. b. Fnd the recprocal functon that models ths graph. c. Usng the recprocal functon found n part b, fnd:. the tme requred to complete the journe f the average speed s km/h. the average speed f the tme taken s hours. s (km/h) TOPIC Non-lnear relatonshps 79

40 . WE The table below shows how the tme, T hours, taken to complete a task vares nversel wth the number of workers, n. Number of workers, n Tme taken, T hours a. Plot the ponts to represent ths data. b. Wrte a mathematcal statement connectng T and n to model ths problem. c. Calculate:. the tme that t would take people to complete the task. the number of people requred to complete the task n seven and a half hours.. A bag contans sweets. Snce the sweets are to be dvded equall between frends, the number of sweets, s, that each frend receves vares nversel wth the number of frends, n. a. Cop and complete the table to show the number of sweets each frend receves for varous numbers of frends. Number of f rends, n Number of sweets, s 7 b. Plot ths nformaton on a number plane. c. Wrte a mathematcal statement connectng s and n to model ths problem. d. Calculate:. the number of sweets each frend would receve f the sweets were dvded equall between 9 frends. the number of frends sharng the bag of sweets f the each receved sweets. 7. WE 7 The table below shows the fracton of a cake receved b each person when a cake s shared equall between an ncreasng number of people. Number of people, n 7 9 Fracton of cake, A a. Cop and complete the table to show the fracton of the cake that each person receves. b. Usng a method of our choce, fnd the nverse varaton rule connectng A and n. c. If the cake s to be shared equall between people, what fracton of the cake would each receve?. The pressure n a bccle tre pump, P, s nversel proportonal to the volume of ar, V. The table below shows measurements of the pressure for dfferent volumes of ar. Volume of ar, V Pressure, P a. Usng a method of our choce, sketch the graph to represent ths nformaton. b. Wrte a mathematcal statement connectng P and V to model ths problem. Jacaranda Maths Quest Mathematcs Standard E for NSW

41 cnonlnearrelatonshps_prnt // :7 page # Usng the recprocal functon found n part b, calculate:. the pressure when the volume s unts. the volume when the pressure s unts. 9. WE The tme requred for a partcular journe s nversel proportonal to the speed. a. Usng the smbols t for tme n hours and s for speed n km/h, together wth the varaton sgn, wrte a mathematcal statement that connects tme and speed. b. Rewrte the statement usng the constant of varaton, k. c. It takes hours at an average speed of 7 km/h to complete a certan journe. Fnd the value of k. d. How long would t take to complete the same journe at an average speed of 9 km/h? Gve our answer to the nearest hour.. For a partcular dstance, tme and speed are nversel proportonal. It takes hours at an average speed of km/h to complete a certan journe. How long would t take f the average speed was km/h? Gve our answer correct to the nearest half hour.. The cost per person to rent a mountan cabn s nversel proportonal to the number of people who share the rent. If the cost s $ per person when people share, what s the cost per person when people share? Problem solvng, reasonng and justfcaton G E PR O O FS c. The ptch of a muscal note vares nversel as ts wavelength. A tone wth a ptch of vbratons per second has a wavelength of.7 m. a. Fnd the ptch of a tone that has a wavelength of. m. b. Fnd the ptch of a tone that has a wavelength of cm.. The current, I amperes, that flows n an electrcal applance vares nversel as the resstance of the applance, R ohms. If the current s amperes when the resstance s ohms, fnd: a. the resstance when the current s amperes b. the current when the resstance s ohms.. A compan makng earplugs fnds that the number sold depends on the prce. If the prce s hgher, fewer are sold. The market research gves the followng epected sales results: Prce of the earplugs $ $ $ $ $ $ $ R EC TE D PA. Number sold (thousands) R Draw a graph of the number sold, n (n thousands), versus the prce, $P. O a. Wrte a mathematcal statement connectng n and P, gven that the number of earplugs sold vares nversel wth the prce of the earplug. c. Usng the recprocal functon found n part b:. predct the number of earplugs sold f the prce was $ each. calculate the prce f earplugs were sold. U N C b. TOPIC Non-lnear relatonshps

42 . A process to sterlse surgcal nstruments s tested and ts success s found to depend on the temperature used. The results are shown n the table. Temperature of sterlser ( C) 9 Mcrobes remanng alve (%) a. Usng a method of our choce, draw a graph of the percentage of mcrobes remanng alve, M, versus the temperature of the sterlser, T C. b. Wrte a mathematcal statement connectng M and T, gven that the percentage of mcrobes remanng alve vares nversel wth the temperature of the sterlser. c. Usng the recprocal functon found n part b:. predct the percentage of mcrobes remanng, to decmal place, f a temperature of 7 C was used. Justf our answer.. calculate the temperature requred to ensure that no more than % of the mcrobes reman alve.. a. Cop and complete the table shown. b. These ponts represent an nverse varaton problem. Fnd:. the constant of varaton. the equaton of the recprocal functon to represent these ponts. 7. The varables and var nversel for the followng graphs. Wrte an equaton relatng and for each graph. a. (, ) b. 7. There are 9 was to arrange square blocks to form a rectangle. Here are two was. Heght = Base = Heght = a. Fnd the other seven was and record our results n the table. b. Plot these 9 ponts on a number plane of heght, h versus the base, b. (, ) 7 Base = 9 Heght, h Base, b Area, A 9 Jacaranda Maths Quest Mathematcs Standard E for NSW

43 cnonlnearrelatonshps_prnt // :7 page # c. Descrbe the relatonshp between the heght and the base.? e. Dscuss wh t s not sutable to use the graph of the hperbola for all values of > to model ths nverse relatonshp. d. Wh do the ponts all le on the rectangular hperbola = FS. Revew.. Summar EC TE D PA G E PR O O In ths topc ou have learnt: to graph and recognse an eponental functon n the form = a or = a (a > ) usng technolog to nterpret the meanng of the ntercepts of an eponental graph n a varet of contets to construct and analse an eponental model to solve a practcal problem to construct and analse a quadratc model to solve practcal problems nvolvng quadratc functons or epressons of the form = a + b + c to recognse the shape of a parabola and that t alwas has a turnng pont and an as of smmetr to nterpret the turnng pont and ntercepts of a parabola n a practcal contet to consder the ranges of values for and for whch the quadratc model makes sense n a practcal contet k to recognse that recprocal functons of the form =, where k s a constant, represent nverse varaton, and to dentf the shape of these graphs and ther mportant features to use a recprocal model to solve practcal nverse varaton problems algebracall and graphcall to graph an eponental, quadratc or nverse functon usng technolog. Dgtal doc: Topc summar a comprehensve summar of ke learnng ponts (doc-7) Eercse. Revew R Understandng, fluenc and communcatng MC Durng a growng spurt, our frend ncreases n heght b % per month. If her orgnal heght was cm, her heght, to the nearest centmetre, after months s: A. cm B. cm C. cm D. cm. MC The populaton of bats n a Northern European countr s decreasng at the rate of % per annum. If the orgnal populaton was bats, the appromate populaton after ears s: A. B. 7 C. D. 7. A collectable football card s bought for $. Each ear the value of the card ncreases b %. Fnd the value of the card after ears.. The value of a machne s deprecatng b % per ear. If the orgnal prce of the machne was $, fnd the value of the machne after ears. U N C O R. TOPIC Non-lnear relatonshps

44 . MC Whch of the followng equatons could the graph represent? A. = B. = C. = D. =. MC Whch of the followng graphs best represents the equaton =? A. B. (, ) (, ) C. (, ) 7. MC Whch statement s correct? A. The turnng pont s (, ). B. The as of smmetr s =. C. The -ntercept s (, ). D. The turnng pont s (, ). D. (, ) (, ) (, ) (, ) (, ) Jacaranda Maths Quest Mathematcs Standard E for NSW

45 . MC Whch statement s ncorrect? A. The mamum turnng pont s (, ). B. The equaton of the as of smmetr s =. C. The -ntercept s (, ). D. The -ntercepts are (, ) and (, ). 9. The populaton of rabbts n a certan area s thought to be modelled b the eponental functon = (.7) where represents the rabbt populaton after ears. a. What s the ntal populaton? b. What s the epected rabbt populaton after ears? Gve our answer to the nearest rabbts.. MC Whch statement about the graph s not true? A. It represents an eponental functon. B. It has two branches. C. The vertcal asmptote s =. D. The horzontal asmptote s =.. For each of the followng quadratc functons, state:. the turnng pont. the -ntercept. the -ntercepts v. the as of smmetr v. whether the functon has a mamum or a mnmum value. a. 9 7 (, ) (, ) (, ) 7 (, ) (, ) (, ) (, ) 9 7 (, ) (, ) b. TOPIC Non-lnear relatonshps

46 . The prce per person to hre a lmousne for a Year formal vares nversel as the number of passengers. It costs $9 each for students. How man students would there be f hrng the lmousne cost each student $.?. MC When an amount of gas s enclosed n a contaner, the pressure s nversel proportonal to the volume. Ths s notceable when usng a bccle pump. As the volume nsde the pump s reduced b pushng the plunger, the pressure ncreases. Usng P for pressure and V for volume, whch of the followng s not true? A. P B. As V ncreases, P decreases. V P C. If V s halved, P s doubled. D. V = k. MC If and = when =, then when =, equals: k A. B.. C. D.. For a rectangle wth a constant area, the wdth, w cm, s nversel proportonal to the length, l cm. a. Wrte a mathematcal statement connectng length, wdth and k, the constant of varaton. b. Fnd k when the length of the rectangle s cm and the wdth s cm. c. Fnd the length of another rectangle wth the same constant area when the wdth s 9 cm.. For a partcular dstance, tme and speed are nversel proportonal. It takes 7 hours at an average speed of km/h to complete a certan journe. How long would t take f ou ncreased our average speed b km/h? Gve our answer to the nearest hour. 7. The table below shows how the tme, T hours, taken to complete a task vares nversel wth the number of workers, n. Number of workers, n Tme taken, T hours a. Plot the ponts to represent ths data. b. Wrte a mathematcal statement connectng T and n to model ths problem. c. Calculate:. the tme t would take workers to complete the task. the number of workers requred to complete the task n hours.. The current, I amperes, that flows n an electrcal applance vares nversel as the resstance of the applance, R ohms. If the current s amperes when the resstance s ohms, fnd: a. the resstance when the current s. amperes b. the current when the resstance s ohms. 9. A partcular nvestment apprecates b % per ear. If an nvestment of $ was made now, how much would t be worth, to the nearest $ after: a. ear b. ears c. ears? Jacaranda Maths Quest Mathematcs Standard E for NSW

47 . The monthl proft or loss, $p (n thousands of dollars), for a new wdget s modelled b the quadratc functon p =, where s the number of months after the ntroducton of the new wdgets nto the market. a. Cop and complete the table of values for ths functon. 7 9 p b. Plot the ponts to graph the functon. c. Comment on wh the table of values started at =. d. Was the compan makng a proft or a loss ntall? e. Fnd the frst month for whch a proft was shown. f. Estmate from our graph n whch month the proft would reach $7. Problem solvng, reasonng and justfcaton. In 9 wnd turbnes n Europe generated about ggawatt-hours of energ. Over the net ears, the amount of energ that was generated ncreased b appromatel % per ear. a. How much wnd energ, to decmal place, was generated n:. 9 (Hnt: t = ) ? b. Usng a method of our choce, estmate n whch ear ggawatt-hours (GWh) of energ was generated.. The populaton of a partcular fsh, f, located n the Great Barrer Reef was observed to be n and was decreasng b % per ear. Gve our answers to the nearest nteger where necessar. a. Form an equaton to model the eponental deca of the fsh populaton, f, over t ears. b. How man of the fsh were estmated to be n the Great Barrer Reef n? c. Assumng the populaton of fsh contnues to decrease at the same rate, how man would ou estmate to be n the Great Barrer Reef n?. The graph shows the path of a hocke ball gven b = (m) ( ) where s the heght, n metres, of the ball above the ground and s the horzontal dstance, n metres, of the ball from the hocke stck. a. Determne where the hocke ball hts the ground. b. What s the mamum heght of the ball above the ground? Justf our answer. (, ) (, ) (m) TOPIC Non-lnear relatonshps 7