MCR UI EXAM REVIEW Hour Eam
Unit : Algebraic Tools for Operating with s: Rational Epressions. Simplif. State an restrictions on the variables. a) ( - 7-7) - (8 - - 9) b) ( - ) - ( + )( + ) - c) -6 d) - - - + - + - 8 - - - - e) f) + 6 + 8 + - 6 + - - + 6 + - + g) + - h) - - - + - i) - j) + + + - - + + - - l) "#+ -"#- 0"$ +"." &+" $ #&+" k) "# " $ %& + "$ %& " $ %"%& )"#* &"#* Unit : Radical Mathematics and Quadratic s. Simplif. a) 0 b) c) 6 e) 0 f) ( ) g) 8-0 i) 8-7 + j) 6 ( + 8) k) ( - )( + ) l) m) n) 7-6 + NOTE: Simplif (l) (m) and (n) b rationalizing the denominator.. Solve b factoring. a) - 7 = b) = 6-7. Solve using the quadratic formula. a) - = b) = - + 7 d) h) 7 6 8. Complete the square and Partiall Factor each of the following. State the maimum or minimum value of each function and the value of when it occurs. a) = - 7 + b) = - - 8 + c) = - + + d) = - + 6 6. Quadratic Stor Questions. A. The function h t = t + 0t + gives the approimate height h metres of a thrown football as a function of the time t seconds since it was thrown. The ball hit the ground before a receiver could get near it. a) How long was the ball in the air to the nearest tenth of second? b) For how man seconds was the height of the ball at least 7 m? c) What is the maimum height of the ball?
B. The height of an object h(t) in metres can be modelled b the equation h t = t t where t is the time in seconds after the object is released. Can the object ever reach a height of 0 m? Eplain. C. The profit function for a compan is modelled b P = + 7 + 9 where is the number of items produced in thousands. Determine the break-even point(s). 7. Calculate the value of k such that k + k = 0 has: (a) one root. (b) two roots. 8. Does the linear function g = + intersect the quadratic function f =? How can ou tell? If it does intersect determine the point(s) of intersection. 9. Determine the equation of the parabola with roots + and and passing through the point ( 0) Unit : Transformations of s 0. For each of the following state the domain range and whether or not it is a function. a) { ( ) () (79) ( ) (-7) } (- 6) (0-6) ( -6) (-6) c) d) - b) { } - - - -. If f ( ) - = find: æ ö a) f () b) f ç - è ø. Describe the transformations of the following functions from the graph of f (). a) = f ( - ) - b) = - f ( + ) - c) = f (-) + d) = - f (( + )) + 6. Describe the transformations on f ( ) = required to graph. Find the inverse of each function. Is the inverse a function? Eplain. a) = - b) = - 7 c) = ( + ) d) = - é ù = - ( 8) ê - ë ú. û
. i) Use transformations to sketch the graphs of each of the following pairs of functions on the same set of aes. The first function is the Parent/Base. a) = and = - b) = and = - ( + ) - c) = and = - d) = and = + ii) State the domain and range of each function. 6. The graph of = is stretched verticall b a factor of translated units to the left and translated units upward. Write the equation of the transformed function and state its domain and range. 7. Given f ( ) = + 6 a) Write equations for - f () and f (- ). b) Sketch the three graphs on the same set of aes. c) Determine an points that are invariant for each reflection. 8. Cop and complete the chart below. Relation Rough Sketch Domain Range? Yes or No a) = b) = ( -) - c) = - + d) e) = = f) + = Unit : Eponential s 9. Simplif. Epress each answer with positive eponents. - - a) b) ( - - ) c) - - - d) (6 )(- ) e) 0 6 - æ f) ç - è 8 ö ø -
0. Use eponent laws to evaluate the following. NO DECIMALS a) - 0 b) 6 c) (- ) d) 0 + - g) (- ) 0 e) æ 8 ö h) ç ø è6. Epress using eponents. Simplif where necessar. a) - b) f) - æ ö ç ø è 7 æ 7 ö - i) ç è ø c) ( )( ). An insect colon with an initial population of 0 triples ever da. (a) Which function models this eponential growth: n A: p( n) = 0 B: p( n) = 0 n C: p( n) = 0 (b) For the correct model eplain what each part of the equation means.. Shlo is ver ecited about her brand new car Although she paid $0000 for the car its resale value will depreciate (decrease) b 0% of its current value ever ear. The equation relating the car s depreciated value v in dollars to the time t in ears since her purchase is v( t) = 0000( 0. 7) t. (a) Eplain the significance of each part of this equation. (b) How much will Shlo s car be worth in (i) ear? (ii) ears? (c) How long will it take for Shlo s car to depreciate to 0% of its original price?. (a) Is an eponential function either alwas increasing or alwas decreasing? Eplain. (b) Is it possible for an eponential function of the form = ab to have an -intercept? If es give an eample. If no eplain wh not.. Match each transformation with the corresponding equation using the function = 0 as the base. Give reasons for our answers. Not all transformations will match an equation. Transformation Equation (a) horizontal stretch b a factor of A = 0 + (b) shift units up + (c) shift units left B = 0 (d) vertical compression b a factor of C = -0 (e) vertical stretch b a factor of (f) shift units right (g) reflect in the -ais D E F G = 0 = 0 = 0 n - - æ ö = ç 0 è ø
6. (a) Describe the transformations that must be applied to the graph of graph of : i) = ( ) - = to obtain the - æ ö ii) = -ç +. è ø (b) Graph each function from part a). (c) Identif the following properties of the transformed function. (i) domain (ii) range (iii) equation of the asmptote (iv) intercept(s) if the eist Unit : Trigonometr 7. Determine the value of c to one decimal place. a) b) C c) A 0 cm 0 cm 0o 0 C 8 cm B A B A 0 cm C cm B 8. Solve each triangle. Round each side length and angle to the nearest tenth. A a) b) In D KLM 9.7 cm B 7. o C ÐK = 90 m =. cm and l = 8. 8 cm. 9. The Toronto Stock Echange is housed in the Echange Tower. From the top of the building the angle of depression to a point on the ground 00 m from the foot of the building is.6. Determine the height of the building to the nearest metre. 0. The point (0 ) is on the terminal arm of an angle q in standard position. Find sin q and cos q.. Find Ð A to the nearest tenth of a degree if 0 A 80. a) sin A = 0. 67 b) cos A = 0. 76 c) cos A = -0. 8988. Solve each triangle. Round each side length and angle to the nearest tenth. a) In D ABC ÐA =. ÐB = 7. and b = 6. 6 cm b) In D RST r =. 6 m s =. mand t =. m c) In D EFG ÐF = 67.8 f =. 6 mand e = 9. 8 m. An isosceles triangle has two. cm sides and two. angles. Find: a) the perimeter of the triangle to the nearest tenth of a centimetre. b) the area of the triangle to the nearest tenth of a square centimetre.
. Airport X is 0 km east of airport Y. An aircraft is 0 km from airport Y and north of due west from airport Y. How far is the aircraft from airport X to the nearest kilometer?. Two ships left Port Hope on Lake Ontario at the same time. One travelled at km/h on a course of. The other travelled at km/h on a course of 0. How far apart were the ships after four hours to the nearest kilometer? 6. Determine the number of triangles that could be drawn with the given measures. Then find the measures of the other angles and the other side in each possible triangle. a) In D GHI ÐG = 0 g = cm and h = cm b) In D XYZ ÐX = = mand = m c) In D ABC ÐB = 0. c =. mand b =. 9 m Unit 6: Trigonometric s 7. The coordinates of a point P on the terminal arm of an angleq in standard position where o 0 q 60. Determine the eact values of sin q cos q and tan q. a) P( ) b) P(7 -) 8. Find the eact value of each trigonometric ratio: o a) tan b) cos 0 9. If 0 q 60 find the possible measures of Ð A : a) cos A = b) tan A = - 0. Sketch one ccle of the graph of each of the following. State the domain range amplitude period vertical translation (when necessar) and phase shift (when necessar). a) = sin b) = - sin + o c) = sin( + ) d) = cos e) cos o = f) = cos ( -80 ) +. Prove each identit. - sin a) = cos cos b) + tan = c) cos cos - sin = sin tan d) - tan = cos - sin + tan e) ( - cos )( + tan ) = tan f) (sin - cos ) = - sin cos g) ( + cot )tan = sec h) sin sec = tan i) tan ( + cot ) = + tan
o. Solve each equation for 0 60. - b) cos + = 0 c) sin - = 0 a) sin = d) tan = e) ( cos + )(sin -) = 0 f) cos + cos = - g) cos + = sin Unit 7: Sequences and Series h) cos - = sin i) sin + sin =. Find the formula for the nth term and find the indicated term for each arithmetic sequence. a) 7... ; t b) 0-0... ; t8. Find the number of terms in each arithmetic sequence. a) 9... 69 b) 9... - 9. The Women s World Cup of Soccer tournament was first held in 99. The net two tournaments were held in 99 and 999. a) Write a formula for finding the ear in which the nth tournament will be held. b) Predict the ear of the th tournament. 6. Find the formula for the nth term and find the indicated term for each geometric sequence. a) 7 9... ; t b) 6-9... ; t7 7. Find a r and tn for each geometric sequence. a) t = and t6 = 96 b) t = -6 and t = -6 8. Use the recursion formula to write the first terms of each sequence. a) t = ; t = ; t = + n tn- tn- b) f ( ) = 8; f ( n) = 0. f ( n -) 9. Identif whether the series is Arithmetic or Geometric. Then find n. a) + + +... + 0 b) - - + +... + c) 68 + 096 +... +
0. Find the indicated sum for each arithmetic series. a) S for - 0-8 -6 b) + + +... + 0. The side lengths in a quadrilateral from an arithmetic sequence. The perimeter is 8 cm and the shortest side measures cm. What are the other side lengths?. Find the indicated sum for each geometric series. S for - 8 + 6 - +... b) 6 - + 0 -... + a). A ball is kicked from the ground 6. m into the air. The ball falls rebounds to 60% of its previous height and falls again. If the ball continues to rebound and fall in this manner find the total distance the ball travels until it hits the ground for the fifth time (assume the ball bounces verticall with no curvature in its path). Unit 8: Compound Interest and Annuities. Find the amount of each investment: a) $00 for ears at % per annum compounded monthl. b) $600 for ears at 6.7% per annum compounded quarterl.. Jean Paul is saving for a car. He puts $00 in a Guaranteed Investment Certificate paing.% per annum compounded quarterl. How much mone will he have available to bu a car ears from now? 6. What is the present value for each amount? a) $9000 in ears invested at.6% per annum compounded semi-annuall b) $0000 in a ear invested at 8.7% per annum compounded quarterl. 7. Marianna deposited $00 into her bank account at the end of each month for 8 months. a) The account pas.9% per annum compounded monthl. How much is in her account at the end of the 8 months. b) If the amount deposited each month were doubled how much would be in the account at the end of the eight months. 8. Faris needs $000 for universit in ears. His parents plan to invest some mone in an account paing interest at a rate of 7.% per annum compounded quarterl. How much should the invest now to have $000 in ears? 9. a) Michael wants to make an investment so that he would receive $000 ever 6 months for ears with the first pament due in 6 months. How much mone should he invest now at 7% per annum compounded semi-annuall? b) How much interest would be earned over the life of the investment?
60. To provide an annual scholarship for ears a donation of $0000 is invested in an account for a scholarship that will start a ear after the investments is made. If the mone is invested at.% per annum compounded annuall how much is each scholarship? 6. Brooke won $00000 in a lotter. The prize will be paid in earl installments of $0000 each ear for 0 ears. What is the present value of her winnings if current interest rates are 6.% compounded annuall? 6. Mrs. Behnke bought a new car. She financed $00 at.9% /a compounded monthl and chose to make monthl paments for ears. a) What amount does Mrs. Behnke pa per month? b) The dealership told Mrs. Behnke that her paments would be $.89 per month for 8 months. If Mrs. Behnke didn t correct them how much etra would she pa? ANSWERS UNIT a - - + b - -9 + 6 c g - 8 - h UNIT a g m - ¹ i ¹ d - - 0 j ( + )( + )( - ) ¹ - - + + ¹ - e - f + ¹ - - - - k + 6 + ( + )( - ) + ¹ - - ± b c 6 d e 0 f 0 h i j k - 0-7 l ( + ) - a - b min = 7 = 6C items 7a UNIT 0a D : R : Not a { 7} {-7-9} function n ( 6 ) - a ma = 9 = - k ± c 7b = - 6 ma = 8 = 0b b d k R < k < R : {-6 6} = - = min = - a ± 77 = 6.A a). s b). s c) 7 m 8 D=9 so points of intersection 9 D : {- 0 } 0c b 6B ( -)( - ) ( + )( - ) ¹ - - l + 0 7 7 - ± 9 = 6 Yes Discriminant =6 9 = + 8 D : {- R : {- Î R} Not a function Î R} 0d D : {0 Î R} a -7 b R : {- Î R} Not a function a Translated units right Translated units down b Reflected in -ais Translated units left Translated unit down c Reflected in -ais Vertical compression b factor of Horizontal compression b factor of / Translated units up d Reflected in -ais Vertical stretch b a factor of Horizontal compression b factor of Translated units left Translated 6 units up Reflected in -ais Vertical compression b factor of ¼ Horizontal stretch b factor of Translated 8 units right a - f ( ) = +
b - f ( ) = ± + 7 Not a function c - f ( ) = ± - Not a function d - f ( ) = + a Translated units down b Reflected in -ais Vertical compression b factor of / Translated unit left Translated units down c Reflected in -ais d Vertical stretch b factor of Translated units left 6 f ( ) = ( + ) + 7a D : { Î R} R : { ³ Î R} - f ( ) = - - 6 f (-) = - 6 7c 8c 8f - f ( ) : (0 0) (-6 0) 8a f (-) : (0 0) D : { ³ 0 Î R} 8d R : { Î R} D : {- Î R} R : {- Î R} Not a function D : { Î R} 8b R : { Î R} D : { ¹ 0 Î R} 8e R : { ¹ 0 Î R} D : { Î R} R : { ³ - Î R} D : { Î R} R : { > 0 Î R} UNIT 9a 9 0a 0 b 0g 6 0 h a C b a Yes b e ( 0 ) = : not listed 6cii Domain: { Î R } Range: { < Î R} Asmptote: = -intercept: @. 9c 6 b 0c 0i p n = 0 ( ) ( ) n 0: Initial population : rate of increase n: number of das 9d 0d 8 a 9 v t = 0000 0. 7 a ( ) ( ) t 0000: Initial value of the car 0.7: percent of value carried to the net ear t: number of ears No a = 0 : not listed f = 0 - : not listed UNIT 7a 6. cm 7b.9 cm 7c 6.6 cm 8a g - 6 9e 9f 0e 0f (- ) C 6ai Vertical stretch of factor Horizontal compressi on of factor / Shift down unit b c bi $ 000 bii $ 9800 c 6. ears b A c B d G A =.9 8b a = 9.0 cm b =. cm 6ci Domain: 6aii { Î R} Range: { > - ÎR} Asmptote: = - -intercept: @ 0.7 L =. 9 6 m M =.6 k =. cm 0 Reflection over -ais Horizontal compression of factor / Shift units right Shift units up
0 sinq = 9 0 cosq = 9 a A = 8.0 or A =.0 b A = 7.0 c A =.0 a C =. b a = 0. cm c = 0.9 cm S =.9 R = 60.9 T = 66. c G = 66. a 0. cm b. cm 8 km 98 km E = 6. g =. m 6a Triangles H = 8.8 I = 0. i =.7cm or H =. I = 8.8 i =.7cm 6b 0 Triangles 6c Triangle C = 0. A =. a =.m UNIT 6 7a 8b 0a 0d sin q = cosq = tanq = - D : {0 60 Î R} 0b R : {- Î R} Amplitude = Period = 60 Phase Shift = none D : {0 60 Î R} 0e R : {- Î R} Amplitude = Period = 60 Phase Shift = none 7b - 7 8a sinq = cosq = tanq = - 6 6 7 9a A = 9b A =0 00 D : {0 80 Î R} 0c R : {0 Î R} Amplitude = Period = 80 Phase Shift = none Up units D : {0 080 Î R} 0f R : {- Î R} Amplitude = Period = 080 Phase Shift = none D : {- Î R} R : {- Amplitude = Period = 60 Î R} Phase Shift = left D : {80 900 R : {- Î R} Amplitude = Period = 70 Phase Shift = right 80 Upunit Î R} a = 0 00 b = c = 0 0 d = 60 0 e = 90 f =0 80 0 g = 60 80 00 h = 0 80 60 i = 9. 60. 0.6 6. UNIT 7 a t n = n + b t n = 7n - a b a t n = n + 987 b 7 t0 = 6 t8 = - n n- n- 6a tn = 7( ) 6b t = (- ) 7a a = r = t ( ) n n = or 7b a = - r = 8a 6 9 8b 8 n - n - t7 = 79 a = - r = - tn = -(- ) t n = -( ) 0. t6 = 9 9a Geometric 9b Arithmetic 9c Geometric 0a 00 0b 67 8 n = n = 7 n = 8 cm a -60 b 7 9. m UNIT 8 a $996.7 b $668. $6. 6a $688.8 6b $970.89 7a 6.60 7b $7.0 8 $08. 9a $66. 9b $67.8 60 $77.7 6 $7.9 6a $0. 6b $88.6