MPM 2D Final Exam Prep 2 June 2017 1. Express each equation in standard form and factored form: ~ ~ +et's 'leu t W (.. ".>tak( a) y = (x + 5)2 + 1 on ::t~'t.{1'" ~heeh v 1' K 1 C'. T.) '. (J. lr lov J l ("' + ' " b) Y = 2(x + 1)218 ~..: 2. (xl 1:'}")( t J') ' ( B vi::: 2 ~ 1'+ 4 1<. t:2 ( 6! '.'..."...' u " r'. SF: 1' 21 if _ v!. X + )< :.:? '.' ; _~.. i v ) " '}.. " (X + tt:7k T (;24 ) l (X t 4 ) (X +(; ).. 2. Express each equation in factored form and vertex form: a) y = 4x 2 + 20x + 25 ' " :'1 x + S' ;: J ~ ('. "". J. p{:' b) y=2x 2 +24x64 ~J~). ( X J.. ;).J<. + 3.1) V):: 2. ( 7 +.) (X 6" 0' : Lf ( AJ.~.S'X t.&2) _c..2s)tl) ~ '. :~U (X). +.S'1'+0.2.5"l.;2 fr.2.r Vi : Lf (X. +2. S'J?. tyvl' S '.. 4 " (L.ve1 6 '2 c). Vj:=2.(+1 X ~ (0. " l j ' Y' "' )..' V ) ().') 0' r V'f :! ):: J.
3. For each quadratic write in factored form and vertex form and state all the key features (direction of opening zeros vertex axis of symmetry yintercept) a) y = x 2 8x + 15 Gil ::: fx: t:).) ( X.3'). '. ~_. b) y = 3x 2 +12x + 36 V ::: 3 (<.;) 4;x J ') J i 1 rr:'. ltj"::_ ~ ( [. & ) ( X. +~ j 'F '. V C '. v) :;:...?:> ( " ~ Lf X J + 6 ~. V :;: v (:~:~_ ~ ( A 1:...l{ X + 4') t' d' to :31. ').. '.. '.... ")::. ' '"A ( X '2.) ").r if 8 J.. v ". S ~'AAA. vo So f.o {(~ 2 4. Factor each of the following: a) x 2 6x 27 = (Xq') ('lt3).'. c) 25x 2 49 :: (S.~ 1) ( [; K T~') " d) 6x 2 X 2 ~.3X 2 )( :).X t. ) "..c..'... e) 2x 2 7x 15 :;(2K+ 3)( X 5') '. f) 2x 2 2x 24 ~.2 ( 1 "0 ).) ".'.: ) ( X 4) (1< t~} '.. "
9) 49x 2 42x + 9 ( '1 :.('h?') :!'..'. h) 4x 2 49 ~ 6 K ~ 'x2 K t?~ ~. i) 2x 2 + 3x + 1 =(2x:+) '. (XT) j) 8x 2 2x 21 :!y '..1) (2y tj') ".~ '.. k) 10x 2 28x + 16 ::2 (Sx 1!y X + 9 ) 'r 11(V_) = "J. () >( r.x. r... ) 7x 2 28 = ~1( x. ").Lf) ;: t (X ' 2) ( ')(+ 2. ) "'.._. ~.. 5. Determine the equation of the parabola that has zeros 4 and 8 1 V(2 1 9) and yintercept (08) (note: lots of information here l so there are different ways to do this one) ta S ;y"w1 Uv"V ~ f r W1"vj V ~v4 < v' j=' )0 ')..0. V) :: ~ ( X ~."") to ll ~ ").. f).:: 0 (02) +0 0 ~. N J.. v1 ';:.:L ( 'f. 2) t L:).. D j. 'v LL ' Lf"". j l 6. Determine the equation of the quadratic with vertex (31 2) and point (11 24) V) ::{t ( X tj")j. t tl...1 )... d4 ::; Ct (t 3) + 2. :ld::: ' Y 4t Lf Ct>S"S
7. Solve each equation a) (2x 5)(3x + 8) b) x 2 + 12x + 32 = 0 ~:::C;. ). 1'1. ; 2J c ' l~+ Lt.') ( v + q :: 0 r la" t. f'.. v.) c) 3x 2 lox 8 = 0 ;' ( ~ '.'~x+2 ) y 4):: l' ". d) 2x 2 + 5x 1 = 0 ( TU(" (...::.S t J (C)) '. _ 2jtJ _...._;(._;~.. X. ::_ L. ~ r.. ~:. S~ ~33 ~...'~ 4' 1 ::'..5 +.33 '~'. L{ 'f ::: ~ S.~~...:r" " X '; 0 q ~_ :; ) b'1 e) 3x 2 4x 10 = 0 _._ 'Y:: 4 i Jc L1._......_.._. ~"'~_ :2 (0) X;: Lf'! J ~&' ;i=~q L.3 ) C(0)..~_.._..._~._._.~._._~. _. ( =: f' Jljb _._. "''. ''~ l.t? f) 2(x 3)2 10 = 0 1 _X?J) ~ (0 ~~ '.. d' d ) y~) ~5' _ r' '1. ~ ~'!J5 y=~~ 5 X.::3JS' x::.j.b 8. Without solving determine the number of solutions for each of the following: a) 2x 2 5x + 1 = 0 ~:t4ac:;(s) 1 4{J.)( j. ~.. b) y= 1.8(x 3)2 + 2 cr ). clv'ajf ''. 2 S.)Lvh~0 ~ i:2. +1 f. '.X. ~.. ' ~ '..~
9. Determine the vertex by completing the square of each of the following: a) y = 3x 2 + 12x 2 ) ~ yj. _Lt. v" _.1.' ~_. l.' ' 1 _ ~. b) y = 0.2x 2 OAx + 1.);:.0.2. ( X )2x.)t "r j. lj::' 0.2 (X 2 K + (. t ( '. ' V) ~..? ex:' Lf X + Lf + ;JJ ::. 3 ( Y. Jj~+ l0 l) ~:J::...0.1 ( X '} 2. X 1:.) ~0 2 1 Vl_:: C! ( '1" + r B i. J6" { L.. '.. 1 _'. V ( 2 (VO ) v (... 10. The profit P of a band depends on the ticket price and can be modelled by the equation P = 15x 2 + 600x + 50 where x represents the ticket price in dollars. Determine the maximum profit and the ticket price that should be charged to achieve the maximum profit. ~0 ;: (PO 0 ~ 2 (!S') 11 r( "2 ( ~ ::.~ ~20) + b D D 2:0 ) + ~?0..' 11. A basketball player makes a long pass and the path of the ball can be modelled by h = 0.2x 2 + 2Ax + 2 where x is the horizontal distance from the player and h is the height of the ball above the court both in metres. Determine the maximum height of the ba. ).. rx::~).lf.. ~~_._ ;2{Q. 7) _. (!:. k? q j V. V
12. The height h in metres of a water balloon that is launched across a field can be modelled by h = 0.1x 2 + 2.4x + 8.1 where x is the horizontal distance from the launching position in metres. How far has the balloon travelled when it is 10 m above the ground? 1(;:: O""X?td~X. +'8.1 o (C"1' 'Y:: 2. Lf + 150 'D '::. 0. (. 1+ d. l{ X!1 _..._.. m '._._.. _' ) 2 l.... 1.1' ::. ~. '... q ~~ '10.. d V.) _. ~ _. e.. ~~. ~'~:."" '.....~..._"._ ;2. (_. O. ')...._. 0 J: '. ( '7 (j r}. rzrl ". L 3j 1;8 0 B 2 22 30se h i 1{.). c; JL!..._ i 1_.+_._._.~ 3 d. o = O~J 42 ~ (03..._".. _._.... ".. _.. _... m_' ' _ ' ().:3 0.3 t' ~. 1 Lf ==( c ~ )......" _'. 1 fa k.c:j :;"lf S {::j (Ci! '( cl :: 3 1:. 1 Lf' r ~.. 0' r. ~ ~... f'..j.. j' ".
14. The profit per week for sales of Canada 150 flags is modelled by P = 250 + 5n 2n 2 where n is the number of flags sold and P is profit in dollars. s it possible to make a profit of $500 in a week?. rl() 1 C (k1l '{ VV i 15. Describe the transformations applied to the graph of y = x 2 to obtain the graph of each quadratic given a) y = 2(x 4)2 + 16 b) y = ~ (x + 6)2 12 c) y = 5(x 1? + 4 "~ Vf2.( ( "'r y tj...f. S o V. c flj.. V. bh~e(ra J2; S o V. ~'ty t ha! ~ 2. V'x~t Lf '... 'lay) 10 let.1 G. clowyl (2 o f"7~t ( (~...". (.~plf' 16. Write the equation of the quadratic given the following information: a) shift up 4 reflection in the x axis shift right 2 b) vertical compression by a factor of 15 shift left 3 shift up 7 ).. c) shift left 2 vertical stretch by a factor of 3 reflection in the x axis shift down 9 d) shift down 5 reflection in the x axis vertical stretch by a factor of 4 shift right 8 J. l. o
17. n.6pqr<r= 90 and p = 12 cm. a) Determine r when <Q = 53 (round to 1 decimal place) b) Determine <Pwhen q = 16.5 cm (round to the nearest degree) C c."j....tfj ) :..J' V ~. ' ' 1d. V.' _.. { :J53 ~._'. ~.. 18. A plane is flying at an altitude of 1200 m toward a disabled ship. The pilot notes that the angle of depression to the ship is 12. How much further does the airplane have to fly to end up directly above the ship? Round to the nearest metre. ; "'Q"..J f ' ' 11.0] [;)00 _..... '_.". 'J:: t;:).oo 19. The angle of elevation from the top of a 16 m building to the top of a second building is 48. The buildings are 30 m apart. What is the height of the taller building? Round to one decimal place. r CVV' L q ~..'J:. l l.. _.~ 3 _ v 'X ::~JC>~Lf g ~. :.. '3 '3.3 ~'Y _' tojk2v bla~(d~ J ("~ LG 3 Vy
20. A swimmer observes that from point A the angle of elevation to the top of a cliff is 30. She swims toward the cliff to point B and estimates that the angle of elevation to the top of the cliff is now 42. f the height of the cliff is 70 m how far did the swimmer swim? (round to the nearest tenth of a metre) 1CVV 30:;. _~Q. c iff A :::.X :J.v.. e_.. '"' ~ L. l ) 1c! "''~' "tv ~1?O f'~42..'!'t). L. A R. 1' A lv' {" la ~ ~ X... X ~~. J. w ' LA. 21. n i1abc <B = 31 b = 22 em and c = 12 em. degree. 1ft: f"'.. ')2 cw'. (7{... " (j "..."'" L' =3...... _~ c. a : f3 S vyl.~ ~..r4~t2vv' C~fu. ~.Ar l 'yylvn'v v ~ Svt.lZ W' Determine L ~'vlc _' <S ln 3 L J _..". ~' ;)..2. <Crounding to the nearest.. "") J~ 22. Solve i1abc if <A = 75 <B = 50 and the side between these angles is 8 em. Round to one decimal place..: f' CvY' ~. SO. _... ~ t> Cl _'. C: CJS' {3 _0~ '":'. "';0 S1"rlf5; S 1( :5S {(::: S b 1Y1i'f'...' u)'n )~ L2. ::. ~ 5) 'n W cs J h ';"5 k) ~.~ csl~s~()_ S 1'v1 5'S b ~ 7. 5 c~
23. Allison is flying a kite. She has released the entire 150 m ball of kite string. She notices that the string forms a 70 angle with the ground. Mark is on the other side of the kite and sights the kite at an angle of elevation of 30. How far apart are Mark and Allison? Round to the nearest metre. 'X _._._ ' _~ ' Q t'" ''.1"f'o'.J"... _~_._"~.. _c_... "" 00 ~_. ~ <t"v ".".""..".... _ ft 24. Solve llabc if <A = 58 b = = 10 cm and c 14 cm. Round to the nearest tenth. 01;. q}':.r (o~2(! Lf'Y:D)(t{;0 'J"e J( S " ;Vj 1. r; : ~) c:;g n_._'. _ 10 3:: ( "" tf3 : C. :: 25. Find <B in llabc if a 1 S' (f. "" '! l.(' = 7 m b V'. ~ V ' p.j1 ~. 16 :4. o W. = 14 m and c '''.... = ''...._..'._... _ 15 m. Round to one decimal place.. S) ( 1C"...........' L. 1 vv'. 26. Points P and Q lie 240 m apart on opposite sides of a communications tower. The angles of elevation to the top of the tower from P and Q are 50 and 45 respectively. Calculate the height of the tower to one decimal place. d.l{.o V?~. _"....J.'.~ f)'n ')0 si Lt. r:::. V 10'1. fc 6'))18)' ~. S S~ ~' ;:: t. ~ Lf. ~'t= 0 S}1 'P j;).. 0 s)n ~S5'YJ L{ ~ l 30.S' v~