MCR3U FINAL EXAM REVIEW (JANUARY 2015)

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1 MCRU FINAL EXAM REVIEW (JANUARY 0) Iroducio: This review is composed of possible es quesios. The BEST wa o sud for mah is o do a wide selecio of quesios. This review should ake ou a oal of hours of work, provided ou ca refer o our oes easil for quesios ou have difficul wih. Oce ou are doe, ou will have a iveor of all possible pes of eam quesios.. Cosider he followig ses: (a) Sae he domai of f. ( Sae he rage of g. (e) Is i rue ha f = (g) Sae he value of ( ) g f = g = h = (i) True or false: he domai of {(, ), (, ), (,) } {(,),(, ), (, )} {(, ), (,),(,),(,) } (b) Which of he ses are fucios? (d) Sae he value of h()? Eplai. (f) h() =. Sae he value of h (h) Creae a mappig diagram for h. g is he same as he rage of f.. Sae he domai ad rage for he fucio whose defiig equaio is = ( ) ou kow. +. Eplai how. Deermie he iverse of he followig fucios. The sae he domai ad rage of boh he origial ad is iverse. (a) ( ) = + f (b) f ( ) = +. A relaio f is show o he graph below. - - (a) Is he relaio a fucio? Eplai. (b) Evaluae f ( 0) ( Deermie he value of such ha f() =. (d) Sae he domai ad rage of f (e) Graph he lie = ad he iverse, f same grid as he relaio f., o he - -. Suppose ha f ( ) = ad g ( ) = + (a) Deermie g ( ) (b) Deermie g ( ) ( Deermie f ()., ad full simplif our aswer.. Graph he followig base fucios o graph paper. Be sure o show a appropriae umber of ke pois. Graph a asmpoes wih a doed lie ad label wih is equaio. (a) = (b) = ( = (d) =. Describe i words, usig vocabular suiable for his course, how (a) he graph of = + + is rasformed from he graph of =. 0.+ is rasformed from he graph of ( ) (b) he graph of = ( ) =.. Sae he equaios of he asmpoes for he followig raioal ad epoeial fucios. (a) (d) = (b) = ( = + + = ( ) + (e) = ( ) (f) = ( ) +. Skech he followig fucios o graph paper. Draw a asmpoes wih a doed lie ad label wih is equaio. (a) ( ) = + (b) = + ( = ( + ) +

2 The graph of = si() is refleced i he -ais, vericall sreched b a facor of ad raslaed uis up ad uis righ. Wha is he equaio of he ew graph?. Complee he char for he followig siusoidal fucios. The graph = siθ, = si[/(θ 0 )], ad = -.si[(θ + 0 )] +. o he same grid over ccles. Fucio Ampliude eriod Ma Mi hase shif Verical Shif Domai Rage = siθ = si ( θ 0 ) =.si θ [ ( )] [ ( )] [ ( + 0 )] = cos θ + = si θ + = cos θ. The graph below depics he deph of he waer o a pical da a ocea por. a) Deermie he maimum ad miimum dephs of he waer. b) Sae he ampliude, period, ad verical displaceme for his relaioship. Assumig he graph is a rasformed cosie graph, deermie a possible equaio for he graph.. A Ferris Wheel akes secods from is embarkme poi o reach is maimum heigh of m above he groud. The Ferris Wheel s miimum heigh of m above he groud occurs. secods laer. Below is he siusoidal curve ha models his sceario. Heigh (m) 0 h a) Deermie a equaio ha models he above sceario for ime,, i secods ad heigh, h, i meers, i he form [ k( d )] c h = a cos +. 0 b) Use he equaio o deermie he heigh of he Ferris Wheel afer secods, o he eares eh of a meer. Time (se. Solve he followig riagles. Do forge o check for he AMBIGUOUS CASE, where ecessar. Roud agle measures o he eares degree ad side leghs o oe decimal place. a) ABC where B = 0, a = cm, ad c = cm. b) DEF, where d =.cm, f =.cm ad D =. GHI, where G =, H =, ad i =.cm. d) JKL, where j = cm, l = cm, ad J = 0.

3 . A surveor is o oe side of a river. O he oher side is a cliff of ukow heigh. To deermie is heigh, he surveor las ou a baselie AB of legh 0m. From poi A, she selecs poi C a he base of he cliff ad measures CAB o be. She selecs poi D o he op of he cliff direcl above poi C ad measures a agle of elevaio of. She moves o poi B ad measures CBA as. Fid he heigh of he cliff, o he eares eh of a mere. D h C A 0m B. Skech each agle i sadard posiio ad sae he coordiaes of (, ) o he ui circle, o hree decimal places. Deermie hree oher agles ha are coermial wih i. Epress a leas oe as a egaive agle. Wrie he formula o deermie a coermial agle. a) θ = b) θ = - θ = d) θ = e) θ =. For each of he followig agles: a) 0 b) 0 ad i) Skech each agle i sadard posiios. ii) Sae he coordiaes of poi (, ) o he ermial arm of he ui circle usig eac values. iii) Sae he eac value of he rigoomeric fucio for each of he rig fucios. Do o use a calculaor. Use our special riagles.. Fid he values o hree decimal places of he rig raios for agle θ =.. Deermie each value of θ for 0 θ 0 o he eares degree. a) siθ = -0. b) cosθ = 0. aθ = -. d) cscθ =.0 e) secθ = -. f) coθ = rove he followig rigoomeric ideiies. Sae our reasoig where ecessar (i.e. TI, QTI, ad RTI). a) csc θ = csc θ cos θ b) a θ + coθ = secθ cscθ coθ + cscθ = csc θ d) secθ = aθ + cosθ siθ coθ. Simplif he followig raioal epressios ad sae he resricios o he variables. a) ab b ab b) d) m m 0 m m m + + m m. Solve he followig quadraic equaios usig he mos efficie mehod. Simplif radical aswers. No decimals! a) = + b) = + = d) =. Deermie he defiig equaio of he quadraic fucio ha has zeros passes hrough he poi (, ). NOTE: Epress our fial aswer i sadard form.. Deermie he soluio(s) o he followig liear-quadraic ssems. a) = ad = + b) = + ad = ad, ad. Wha kid of lie is he liear fucio i quesio #b? Eplai our aswer.. Wrie he followig powers as roos firs. The evaluae. Wrie aswers as fracios i simples form. No decimals! a) b)

4 . Simplif he followig powers o a sigle power (oe base, oe epoe). a) a a b) ( + )( ) ( ). The populaio of a colo of baceria doubles ever 0 hours. There are 00 baceria a he begiig of he eperime. a) Deermie a equaio o model he relaioship where A() is he populaio of baceria ad is he ime from he begiig of he eperime, i hours. b) Wha is he populaio das afer he sar of he eperime? Roud o he eares whole umber.. Complee he followig able. Sequece,, -, -, Ne erms Tpe of Sequece (A, G, N) Geeral Term (i simplified form) 0. Sae he firs erms for each of he followig sequeces. a) =, =, =, > b) f ( ) =. Draw ascal s Triagle up o ad icludig he h row.. Use ascal s Triagle o epad he followig biomial. Make sure our fial aswer is full simplified. a) ( b) m + a + b) ( ). Calculae he followig a) he amou of a $00 ivesme afer ears wih a aual ieres rae of.%, compouded semi-auall. b) he aual ieres rae required for a ivesme of $000 o double i ears if i is compouded mohl. Roud o he eares eh of a perce. he umber of ears required for a $00 ivesme o grow o $000 if ieres is.% p.a. compouded quarerl. Roud aswer o he eares eh of a ear. d) he amou ou eed o ives oda i order o have $000 i ears if ieres is.% p.a. compouded semi-mohl (wice a moh).. Jack s pares ives $00 a he ed of each moh for he e ears for his educaio. The ivesme pas.% p.a. compouded mohl. a) How much will Jack have i his educaio fud a he ed of he ears? b) How much ieres was eared o he ivesme?. Deermie he prese value of a loa wih pames of $00 ever mohs for ears if ieres is.% p.a. compouded semi-auall.. Sherri would like o have $ 000 afer ad a half ears. She will make a regular deposi ever mohs io a ivesme accou ha pas.% p.a. compouded quarerl. Calculae he amou she will eed o deposi ever mohs.. Joh has a ivesme of $ 000 ha pas.% p.a. compouded mohl. How much ca he wihdraw each moh for he e ears?

5 MCRU EXAM REVIEW ANSWER KEY ) a) D: {,, } b) boh f ad h because here is ol oe -value for each -value R: {,, } d) h() = e) es because each (, ) pair i f is swiched i g. f) = g) h - () = h) i) false ) D: { R}, R: {, R} because i is a quadraic fucio where he parabola has a vere a (, ) ad is opeig dow, sice a is egaive. ) a) ( ) f =, Origial (f): D: { R}, R: { R} Iverse (f - ): D: { R}, R: { R} b) f - () = ( + ), Origial (f): D: { -, R}, R: { -, R}, Iverse (f - ) D: { -, R}, R: { -, R} ) a) No a fucio because i does o pass he Verical Lie Tes b) f(0) = - = D:{ R, 0 }, R:{ R, - } d) = f - f ) a) g(-) = b) g( ) = + f() = ) a) b) = = 0

6 = 0 d) = / = =0 = 0 ) a) verical srech b a facor of, reflecio i he -ais, or horizoal reflecio, horizoal compressio b a facor of ½, ad raslaio of uis righ ad uis up. b) reflecio i -ais, or verical reflecio, verical compressio b a facor of /, horizoal srech b a facor of, ad raslaio of uis lef ad uis dow. ) a) = 0 ad = 0 b) = - ad = - = ad = d) = 0 e) = - f) = ) a) b) = = ( + ) = + = ( + ) = + 0) = si[-( )] + 0 ) Fucio Amp eriod Ma 0 Mi hase shif Verical Shif Domai Rage = siθ {θ R} { R, - } = si ( θ 0 ) =.si θ {θ R} { R, - } [ ( )] [ ( )] [ ( + 0 )] = cos θ {θ R} { R, } 0 {θ R} { R, } = si θ + -0 {θ R} { R, } = cos θ {θ R} { R, - -}

7 ) a) Maimum: m ad miimum: m b) amp:., period:., v. displaceme:. For a cosie curve, he firs ma represes oe possible phase shif, so d =, 0. a possible equaio is =.cos ( ) +. ) a) Ma: m, mi: m, a = ad c =, p = 0 (disace b/w maimums), k = 0 / = =. ad sice his is a cosie fucio ad he firs ma occurs a h, he d = a possible equaio for his fucio is: h = cos[.( ) ] + b).m ) a) b.cm, A, C 0 b) #: F, E, e.cm, ad I =, g.0cm, h.cm #: F, E, e.cm d) #: L, K, k.cm, ad #: L 0, K, k.0cm ) h.m ) a) (0., 0.), b) (0.0, -0.), Coermial agles: θ =,, 0,... Coermial agles: θ =,, 0,... or -, -0, -0, or -, -, -0, A coermial agle: θ = + 0 A coermial agle: θ = θ = θ = - (-0., 0.), d) (-0., -0.0), Coermial agles: θ =,,,... Coermial agles: θ =,,,... or -0, -, -, or -, -0, -, A coermial agle: θ = + 0 A coermial agle: θ = + 0 θ = θ = e) (0., -0.0) θ = Coermial agles: θ =, 00,, or -, -, -, A coermial agle: θ = + 0 ) a) θ = 0 θ = 0 (ii), (iii) si 0 =, cos 0 =, a 0 = csc0 =, sec0 =, co0 =

8 b) θ = 0 θ = 0 (ii), (iii) si0 = csc0 =, cos0 =, sec0 =, a0 =, co0 = θ = θ = (ii), (iii) si =, cos =, a = csc =, sec =, co = ) si 0., cos 0., a., csc., sec., co 0.00 ) a) θ, b) θ, 0 θ, 0 d) θ, e) θ, f) θ, 0) Aswers ma var. See eacher o check our proofs. ) a) ( a b), a 0, b 0 a ( ),,0,, ) a) =, b) = ± =, ) = ) a) (, ) ad (, ) b) (, ) b) d) m( m + ) ( m ), m ±,, ( )( ) d) =,0, ± ) I is a age lie because here is ol oe poi of iersecio bewee a age lie ad a quadraic. (i.e. There is oe soluio o he liear-quadraic ssem.) ) a) = ) a) a b) ) a) ( )0 b) ( ) = = + A ( ) = 00 b) A() 0 ) Sequece Ne erms Tpe of Sequece (A, G, N) Geeral Term (i simplified form),, -, -, -, -, -, -, Arihmeic = + ( )( ) 0 Geomeric Neiher Numeraor: Geomeric ad Deomiaor: Arihmeic = + = ( ) = + = ( ) ( )( )

9 0) a),, -, -, b),,,, ) ) a) a + a b + a b + a b + 0a b + a b + a b + ab + b b) m + m 0 + 0m + 0m + 0m + m + ) a) $0.0 b).%. ears d) $0. ) a) $0 0. b) $ 0. ) $. ) $. ) $.