# Explore Graphs of Linear Relations. 1. a) Use a method of your choice to determine how much water will be needed each day of a seven-day cruise.

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1 . Graphing Linear Relaions Focus on Aer his lesson, ou will be able o graph linear relaions mach equaions o linear relaions wih graphs solve problems b graphing a linear relaion and analsing he graph Tina is in charge o ordering waer supplies or a cruise ship. She knows he amoun o waer required per da or each passenger and crew member as well as he amoun o waer reserves ha he ship carries. She decides o use her knowledge o linear relaions o draw a graph represening he relaionship beween he amoun o waer needed and he lengh o a cruise. I Tina were o develop an equaion, how could she deermine i he graph and he equaion represen he same relaionship? Maerials grid paper ruler Wha values will ou plo along he horizonal ais? along he verical ais? Eplore Graphs o Linear Relaions On a cruise, he average person requires a minimum o L o waer per da. The cruise ship has capaci or passengers and crew. The ship also carries a reserve o L o waer in case o emergenc.. a) Use a mehod o our choice o deermine how much waer will be needed each da o a seven-da cruise. b) On grid paper, plo he daa and label our graph. Compare our graph wih ha o a classmae.. a) Predic how much waer is needed or a en-da cruise. b) Wha linear equaion represens he lires o waer needed per da? c) How could ou veri our answer or par a)? Tr ou our sraeg. Relec and Check. Do our graph and he equaion represen he same relaionship? Eplain.. Discuss wih a parner i i would be appropriae o inerpolae or erapolae values using a racion o a da. Eplain wh or wh no.. a) I he cruise ship used L o waer, approimael how long did he rip las? Compare he mehod ou used wih a classmae s. b) Is here more han one wa o answer par a)? Eplain. Which mehod seems more eicien?. Graphing Linear Relaions MHR

2 Link he Ideas Eample : Graph a Linear Equaion The world s larges cruise ship, Freedom o he Seas, uses uel a a rae o kg/h. The uel consumpion,, in kilograms, can be modelled using he equaion =, where is he number o hours ravelled. a) Creae a graph o represen he linear relaion or he irs h. b) Approimael how much uel is used in h? Veri our soluion. c) How long can he ship ravel i i has approimael kg o uel? Veri our soluion. Describe he connecion beween he equaion and he graph. Soluion Mehod : Use Paper and Pencil a) Creae a able o values. Graph he coordinae pairs. Time, (h) Fuel Consumpion, (kg) Fuel Consumpion (kg) Cruise Ship Fuel Consumpion Time (h) Wha dieren mehods migh ou use o represen and hen solve he problem? b) Draw a sraigh line o connec he daa poins. Eend he line pas he las daa poin. Approimael kg o uel are used in h. Fuel Consumpion (kg) Cruise Ship Fuel Consumpion MHR Chaper Time (h)

3 Check: Subsiue he value = ino he equaion =. = () = The approimae soluion is correc. c) The uel will las approimael. h. Check: Subsiue = ino he equaion and solve or. = =. The approimae soluion is correc. Mehod : Use a Spreadshee a) In he spreadshee, cell A has been labelled Time,. Cell B has been labelled Fuel Consumpion,. Ener he irs eigh values or in cells A o A. Then, ener he ormula or he equaion ino cell B. Use an = sign in he ormula and * or muliplicaion. The value or comes rom cell A. Use he cursor o selec cells B down o B. Then, use he Fill Down command o ener he ormula in hese cells. The appropriae cell or will auomaicall be insered. For eample, =*A will be insered ino cell B. Use he spreadshee s graphing command o graph he able o values. Noe ha dieren spreadshees have dieren graphing commands. Use our spreadshee s insrucions o ind he correc command. Tech Link You could use a graphing calculaor o graph his linear relaion. To learn abou how o do his go o and ollow he links. A B C D E F G H I Fuel Consumpion (kg) Cruise Ship Fuel Consumpion Time (h). Graphing Linear Relaions MHR

4 b) and c) From he menu, selec Add Trendline o draw a sraigh line rom he irs daa poin o he las one. Eend he line pas he las daa poin. A B C D E F G H I J Fuel Consumpion (kg) Cruise Ship Fuel Consumpion Time (h) For par b), approimael kg o uel are used in h. For par c), he uel will las approimael. h. Did You Know? Fish inders operae using sonar, which uses sound waves o see objecs underwaer. The ish inder produces a sound wave and sends i hrough he waer. When he sound wave mees an objec wihin is range, i bounces back o he ish inder. The ish inder deermines he deph o he objec b measuring he ime beween when he sound wave was sen and when i reurns. The ish inder hen skeches he objec on he screen. Show You Know a) Graph he linear relaion = -. b) Use he graph o esimae he value o i =. c) Use he graph o esimae he value o i = -. Eample : Deermine a Linear Equaion From a Graph Grea Slave Lake, which is locaed in he Norhwes Terriories, is he deepes lake in Norh America. I has a maimum deph o m. Sam decided o check he deph using his ish inder. He colleced he ollowing daa up o a deph o m, which was he maimum deph ha his ish inder could read. Disance From Shore, d (m) Waer Deph, w (m) Lierac Link A deph, such as m, is epressed in dieren was. In a able and a graph, use he negaive value, -. In a senence, sa m below surace. MHR Chaper

5 Sam used a spreadshee o graph he daa. A B C D E F G H I J Waer Deph (m) Grea Slave Lake Deph Recordings Disance From Shore (m) a) Wha linear equaion does his graph represen? How do ou know he equaion maches he graph? b) I his paern coninues, how ar rom shore would Sam be when he waer is m deep? c) A wha rae is he deph o he waer decreasing? d) Is i appropriae o inerpolae or erapolae values on his graph? Eplain. Soluion a) Add a column o he able o help deermine he paern. Disance From Shore, d (m) Waer Deph, w (m) Paern: Mulipl d b The waer deph, w, decreases b. m or each -m increase in he disance rom shore, d. The equaion is w = -.d. Check b subsiuing a known coordinae pair, such as (, ), ino he equaion. Le Side = - Righ Side = -.() = - Le Side = Righ Side The equaion is correc. Wha is he connecion beween he graph and he equaion?. Graphing Linear Relaions MHR

6 Sraegies Solve an Equaion b) Subsiue w = ino he equaion and solve or d. - = -.d - -. = d d. How else could ou solve his problem? Sam would be approimael. m rom shore when he waer is m deep. c) The deph is decreasing a a rae o. m or each mere awa rom shore. The rae a which he waer deph is decreasing is he coeicien o d in he equaion. d) Yes, i is reasonable o inerpolae or erapolae values beween and beond he given daa poins since he values or disance and deph eis. However, i is unreasonable o erapolae values beond he maimum deph o m. Show You Know Ideni he linear equaion ha represens he graph. A B C D E F G H I Disance (m) d (, ) (, ) (, ) (, ) (, ) Time (s) MHR Chaper

7 Eample : Graph Horizonal and Verical lines For each able o values, answer he ollowing quesions: Table Table Time, (s) Disance, d (m) Disance, (m) Heigh, (m) a) Draw a graph o represen he able o values. b) Describe a siuaion ha he graph migh represen. c) Wrie he equaion. Eplain how ou know he graph represens he equaion. Soluion a) d Graph o Table. Graph o Table. Disance (m) Heigh (m) Time (s) Disance (m) b) Table : The graph could show he relaionship beween disance and ime when a pedesrian is waiing or a raic ligh o change. The disance rom he pedesrian o he opposie side o he road is consan. Table : The graph could show he relaionship beween he heigh o a ladder and is disance rom he wall where i is placed. The disance o he base o he ladder rom he wall is consan as he ladder is eended. Think o a dieren siuaion o represen each graph. c) Table : The disance, d, remains consan or each inerval o ime. The equaion is d =. For each value o in he able and he graph, he value o d is. Table : The disance,, remains consan or each inerval o heigh. The equaion is =.. For each value o in he able and he graph, he value o is... Graphing Linear Relaions MHR

8 Show You Know a) Wrie he linear equaion ha represens he graph. b) Eplain how ou know he graph maches he equaion. Disance (m) d Time (s) Ke Ideas You can graph a linear relaion represened b an equaion. Use he equaion o make a able o values. Graph using he coordinae pairs in he able. The graph o a linear relaion orms a sraigh line. k = j k - j k The graph o a linear relaion can orm a horizonal or a verical line. You can use graphs o solve problems b inerpolaing or erapolaing values. j Check Your Undersanding Communicae he Ideas. You are given a linear equaion. Describe he process ou would ollow o represen he equaion on a graph. Use an eample o suppor our answer.. Use eamples and diagrams o help eplain how horizonal and verical lines and heir equaions are similar and how he are dieren. MHR Chaper

9 . a) Describe a real-lie siuaion o represen he daa on his graph. A B C D E F G H I (,.) (,.) (,.) (,.) (, ) b) Eplain how ou would deermine he equaion ha represens he graph. Give our eplanaion o a classmae. c) Can ou inerpolae or erapolae values on his graph? Eplain our hinking. Web Link For pracice maching graphs and linear equaions, go o www. mahlinks.ca and ollow he links. Pracise For help wih # o #, reer o Eample on pages.. Ian works par-ime a a movie heare. He earns \$./h. The relaionship beween his pa, p, and he ime he works,, can be modelled wih he equaion p =.. a) Show he relaionship on a graph. b) Eplain how ou know he graph represens he equaion. c) Ian works h in one week. Use wo mehods o deermine his pa. A B. Andrea is ravelling b bus a an average speed o km/h. The equaion relaing disance, d, and ime,, is d =. a) Show he relaionship on a graph. b) How long does i ake Andrea o ravel km? C. Choose he leer represening he graph ha maches each linear equaion. a) = b) = + c) = +. Graphing Linear Relaions MHR

10 . Creae a able o values and a graph or each linear equaion. a) = b) r = s +. c) m = k + For help wih # o #, reer o Eample on pages.. The graph shows he relaionship beween he cos, C, in dollars and he mass, m, in kilograms o pears. C Cos o Pears Cos (\$) Mass (kg) a) Wha is he linear equaion? b) How much could ou bu or \$? c) Is i appropriae o inerpolae or erapolae values on his graph? Eplain.. The graph represens he relaionship beween he heigh o waer in a child s pool, h, and he ime,, in hours as he pool ills. Heigh (cm) A B C D E F G Waer Heigh in Pool Time (h) a) Deermine he linear equaion. b) Wha is he heigh o he waer aer h? c) Is i appropriae o inerpolae or erapolae values on his graph? Eplain. MHR Chaper m. Deermine he linear equaion ha models each graph. a) A B C D E F G b) A B C D E F. Wha linear equaion does each graph represen? a) b) A B C D E F A B C D E F

11 . Creae a graph and a linear equaion o represen each able o values. a) b) r Appl c) z d) h n The graph represens he aliude o a ho-air balloon he irs min aer i was released. Aliude (m) A B C D E F G Balloon Aliude Over Time (, ) (, ) (, ) (, ) Time (min) (, ) (, ) (, ) a) Wha was he approimae aliude o he balloon aer min? b) Esimae how long i ook or he balloon o rise o an aliude o km. c) Wha linear equaion models he graph? d) How as is he balloon rising?. Sanja conduced an eperimen o deermine how long i akes o hea waer rom C o is boiling poin a C. He ploed his daa on a graph. Temperaure (ºC) w Waer Temperaure Over Time Time (min) a) Approimael how long did i ake or he waer o reach boiling poin? Eplain our reasoning. b) Wha was he emperaure o he waer aer min? c) A wha rae did he waer emperaure increase? Eplain our reasoning.. Paul drives rom Time, Disance, Edmonon o Calgar. (h) d (km) He uses a able o record.. he daa... a) Graph he linear.. relaion... b) How ar did Paul.. drive in he irs h?.. c) How long did i ake Paul o drive km? d) Wrie he equaion ha relaes ime and disance. e) Wha was Paul s average driving speed? Wha assumpions did ou make? Web Link To learn abou using a graphing calculaor o ener daa on a able and plo he daa on a graph, go o and ollow he links.. Graphing Linear Relaions MHR

12 . The relaionship beween degrees Celsius ( C) and degrees Fahrenhei ( F) is modelled b he equaion F = C +. a) Graph he relaionship or values beween - C and C. b) Waer boils a C. Wha is his emperaure in degrees Fahrenhei? c) Waer reezes a C. How did ou represen his on our graph? d) A wha emperaure are he values or C and F he same?. Scuba divers eperience an increase in pressure as he descend. The relaionship beween pressure and deph can be modelled wih he equaion P =.d +., where P is he pressure, in kilopascals, and d is he deph below he waer surace, in meres. a) Graph he relaionship or he irs m o diving deph. b) Wha is he approimae pressure a a deph o m? Veri our answer. c) The maimum pressure a scuba diver should eperience is abou kpa. A wha deph does his occur? Veri our answer. d) Wha does +. represen in he equaion? How is i represened on he graph? Did You Know? Aer deep or long dives, scuba divers need o undergo decompression. The do his b ascending o he surace slowl in order o avoid decompression sickness, also known as he bends. Eend. The graph shows he normal range o lengh or girls rom birh o age monhs. Lengh (cm) l Growh Char or Girls a Age (monhs) a) For wha age range does girls growh appear o represen a linear relaion? b) For wha age range, does girls growh appear o represen a non-linear relaion?. Janice le he school a noon riding her bike a km/h. Flora le school a : riding her bike a km/h. a) Draw a disance ime graph o plo he daa or boh ccliss during he irs our hours. Use a dieren colour or each cclis. b) How can ou ell rom he graph ha Flora has caugh up o Janice? c) Abou wha ime did Flora cach up o Janice? d) I Janice and Flora coninued o ride a heir respecive speeds, a wha ime would he again be apar b a disance o km?. An online music download sie oers wo monhl plans. Plan A oers \$ plus \$ per download and Plan B oers \$. per download. a) Graph boh linear relaions on he same grid. b) Eplain he condiions under which each deal is beer. MHR Chaper

13 . Simple ineres is paid according o he ormula I = p r, where p is he principal, r is he rae o ineres per ear, and is he ime in ears. The ineres is no added o he principal unil he end o he ime period. Canada Savings Bonds oer a simple ineres bond paable a.% per ear up o a maimum o en ears. a) Creae a able o values o show he ineres earned on a \$ bond or he en-ear period. b) Use a graph o show he ineres earned over en ears. c) How man ears would i ake o earn \$ ineres? \$ ineres? d) I ou could leave he principal beond he en-ear period, esimae he number o ears i would ake o earn \$ ineres. The world s ases submarines can reach speeds o km/h in s, saring rom res. I a submarine is alread moving, hen he ime o reach is op speed will dier. a) Choose our dieren saring speeds up o a maimum o km/h. For each speed, assume ha he acceleraion is he same. For each speed include: a able o values a linear equaion and a graph o represen he relaionship beween speed and ime Did You Know? A suden eam rom he Universi o Québec se a new world speed record or he ases one-person, non-propeller submarine. In, he submarine, OMER, reached a speed o. knos (. km/h) in he Inernaional Submarine Races. b) Describe each graph. Ideni an similariies and dierences ou observe beween he graphs and he equaions.. Graphing Linear Relaions MHR

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