The equation to any straight line can be expressed in the form:

Transcription

1 Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he parameric equaion for he curve creaed by he successive inersecion poins. Deermining Equaions Sar a new documen and inser a Graph applicaion. Use he [Menu] o adjus he window seings: Window/Zoom > Quadran 1. A series of sraigh line graphs will be consruced o form a sring paern similar o ha on he Chords bridge. The firs sraigh line graph passes hrough he poins: (0, ) & (1, 0) The resul is shown opposie. Use he quesions o help deermine he equaion for his line and all subsequen lines. The equaion o any sraigh line can be expressed in he form: m = gradien = c = y-axis inercep Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all

2 Quesion: 1. Deermine he equaion of his firs line, passing hrough he poins: (0, ) & (1, 0) a) Wrie down he y-axis inercep of he firs line. (0, ) b) Calculae he gradien of he firs line. m = - c) Wrie down he equaion of he firs line and graph i on he calculaor. y x Once he firs line is compleed, ry he second line. The second sraigh line graph passes hrough he poins: (0, ) & (, 0) As more graphs are added i may be desirable o remove he equaion labels. Seings > Auomaically hide plo labels Quesion:. Deermine he equaion of he line, passing hrough he poins: (0, ) & (, 0). Quesion:. y x Deermine he gradien and y inercep for he remaining sraigh lines in his family of lines. Record your resuls using exac values. Graph all equaions on he same se of axis. Eqn. No. Poin 1 Poin Gradien y-inercep Equaion 1 (0, ) (1, 0) y x (0, ) (, 0) (0, ) (, 0) (0, ) (, 0) (0, ) (, 0) (0, ) (, 0) (0, ) (, 0) Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all

3 (0, ) (, 0) (0, ) (, 0) (0, 1) (, 0) A single equaion can be deermined o graph all equaions by using a parameer () for he equaion number. Sudy each of your equaions above and compare wih he equaion number. Quesion:. The general equaion is of he form: a y c where a, b and c are expressions in erms of. b a) Deermine an expression for a in erms of. a (Refer also o par d) b) Deermine an expression for b in erms of. b (Refer also o par d) c) Deermine an expression for c in erms of. c d) Wrie down he general equaion for he family of sraigh lines. Verify your equaion by subsiuing a range of values for and comparing wih he corresponding original equaion. y Sudens may also have y however hey should be careful a answering as: y as he quesion required he form: y c b Inser a Calculaor applicaion and define as he se of inegers: {1,,,,,,,,, } Define your general equaion in erms of he variable x and parameer. Reurn o he Graph applicaion and graph your funcion: f ( x, ) Quesion:. Describe he shape of he curve formed by he family of sraigh lines. Iniially he curve may appear like a hyperbola; however he curve is a parabolic. There are several ways his can be illusraed. Par of his invesigaion (separae aciviy) shows ha his curve is a parabola roaed o. Graph he following exended family of sraigh lines: f ( x, ), f ( x, ) and f ( x, ). To see he full effec, zoom ou using he zoom ou ool in he Window / Zoom menu and place he magnifying glass close o he cenre of he screen. Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all

4 Quesion:. Describe he shape of he curve formed by he exended family of sraigh lines. The curve is parabolic. (As per Quesion ) Finding a Locus The poins of inersecion beween successive equaions can be used o produce he curve where infiniely many sraigh lines are generaed. 1 Quesion:. Show ha he firs wo lines passing hrough (0, ) & (1, 0) and (0, ) & (, 0) inersec when: and 0 y. Eqn 1: y x Eqn : y x x x x Quesion:. By subsiuion: y 0 y Use simulaneous equaions o deermine he nex poin of inersecion, beween equaions and. Eqn 1: y x Eqn : y x x x x By subsiuion: y y 1 The original curve or envelope would be angen o he sraigh line equaions, as he number of lines over he inerval is increased successive poins of inersecion would come closer and closer o he curve. Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all

5 Quesion:. Use CAS o deermine he poin of inersecion beween f( x,) and f( x,). 1 Quesion:. Complee he able below for he poins of inersecion beween successive lines. Quesion:. Use he difference able o help idenify he naure of he paern in he x coordinaes. Based on he resuls deermine an equaion in erms of he equaion number. Noe: When = 1 his will be he poin of inersecion beween equaions 1 and. When =, his will be he poin of inersecion beween equaions and. = Consan, herefore quadraic. Eqn. Nos. Poin of Inersecion x-coordinae 1 1 & 0 & & 1 1 & 0 0 & & 0 1 & 1 1 & 1 & 0 0 Rule: ( 1) Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all

6 Quesion: 1. Explain he CAS insrucion : solve f ( x, ) f ( x, 1), x This insrucion finds he x-coordinae for he poin of inersecion beween consecuive equaions. ( 1) Quesion: 1. Deermine he equaion for he y coordinae of he successive poins of inersecion. By subsiuion: Quesion: 1. 1 y On he Graph applicaion, change he graph ype o parameric and use he equaions from Quesion 1 for he x coordinae and Quesion 1 for he y coordinae. Change he sep size o 0.1 and he domain for : - 0. Window seings include x-min = -, x-max = 0, y-min = - and y-max =. Exension - Polynomials Quesion: 1. A polynomial is an expression conaining he summaion of one or more variables wih ineger powers and corresponding coefficiens. a. Use he parameric equaions o wrie a polynomial involving x and y for he curve produced by he poins of inersecion. 1 y y ( 1) y y x y by subsiuion ( 1) ( y)( 1 y) xy y y 0 xy y y To es his command he lis mus be deleed. The delee variable command is in he Acions menu: DelVar Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all

7 b. Use a selecion of appropriae poins o show ha your equaion is an accurae represenaion of he curve creaed by he poins of inersecion of consecuive lines. Answers will vary. Example: 0 1 y xy y y Noe ha he quesion saes o use a selecion of poins o show ha he equaion is an accurae represenaion of he curve. As he curve was derived from he successive poins of inersecion he poins should come from hese poins. If sudens use he solve command for values of x or y hey may ge wo answers for he corresponding x or y value and should herefore explain why. c. Show ha he derived coninuous equaion i no an accurae represenaion of he limiing case where infiniely many lines would form a smooh curve. There are many ways o show ha he coninuous equaion is no an accurae represenaion. One approach is o consider he x coordinaes of poins of inersecion. Given he naure of he sraigh line equaions i is reasonable o expec ha successive poins of inersecion would conain x coordinaes ha are greaer han zero. Le 0 xy y y 0 (0) (0) y (0) y y 0 y y 0 ( y)( y1) 0 y or y 1 so wha if y =? When y =, -.0 or.0 The same approach can be used o show ha curve crosses he y axis also. So he curve is a good approximae model, however o obain a model for he limiing case where poins ge closer and closer ogeher would require a differen approach. Teacher Noes: This is aciviy is par 1 of. In his invesigaion he iniial linear equaions are simplisic bu he curve is somewha more complicaed han for Par. In Par he iniial equaions are slighly harder bu he final curve is much easier. This aciviy can also be used in Year 1 Specialis Mahemaics as he resuling polynomial in he exension quesions can be used o compare he gradien using: Implici differeniaion Chain rule (from he parameric equaions) Sudens can also invesigae beer models by exploring soluions o: solve f ( x, ) f ( x, ), x as 0. The resulan equaion f ( x, y) xy y y 11 provides a much beer model for he curve. Texas Insrumens 01. You may copy, communicae and modify his maerial for non-commercial educaional purposes provided all