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FINAL EXAM REVIEW MCF3M Functions ______________________________________________________________________________ Exponent Laws 1. Evaluate. (No decimals) 2 30 40 125 3 5 2 33 1 3 a) 4 3 b) c) 27 3 d) 16 4 e) f) 21 8 42 41 1 1 1 1 2 12 0 g) 8 3 27 3 64 3 h) ( 4 2 2 3 ) 1 2. a) If a 3m and b n , simplify the expression and then express in terms of m and n: 2 3 2 5a 3b 2 2ab b) Solve the simplified expression if m 1 and n 3 . 3. Simplify: 24 x 9 y 7 15x 3 y 11 1 1 a) (2 x 3 y 5 z 2 ) 3 (3x 5 y 9 z ) 2 8 x 4 y 18xy 3 b) c) (49 x 10 y 12 ) 2 ( 27 x 9 y 15 ) 3 ( 4 x 3 y 2 ) 2 (3x 5 y 4 ) 3 ( x 3m y n 2 ) 7 d) e) (6 x 7 y 2 ) 2 x 2m1 y n 30a 4b 2 (15a 2b5 ) 4. Simplify, and then evaluate for a = -1, b = 2: (5ab 4 )2 45a3b6 2 485 2 489 5. Simplify: 2 485 2 488 6. Solve the following exponential equations: 1 a) 2 x 128 b) 4 64 x 1 c) 32 n 7 27n 2 81n d) 62 x 1 36 e) 42 22 x 3 16x 2 g) 5x 5x 3 15750 h) 5 x 17 Expanding and Factoring Polynomials 1. Simplify. a) ( 2 x 1) 2 b) (9 x 1)(9 x 1) c) ( x 1) 2 ( x 5)( 2 x 3) d) (3x 2) 2 ( x 7) 2 e) 3x 2 4x 1 x 2 2x 2 f) 4y 1 2y 1 y y 2 2y 1 3 g) 3z 1 2z2 3z 4 h) 2 a 2b 3 a b 2a b 2 i) x 2 5 x 3 j) x 2 x 1 x 4 x 4 2 2 2 2 2 2. Factor the following: a) 68 x 3 y 9 z 13 52 x17 y 7 z 11 12 x 4 y 56 z 7 b) x 2 x 72 c) x 2 11x 24 d) x 2 9 x 22 e) x 2 20 x 64 f) x 2 23x 50 g) 5 x 2 19 x 4 h) 4 x 2 81 i) 6 x 2 17 xy 12 y 2 j) 6 x 3 26x 2 20x k) 128 x 2 y 50 y 3 l) 5 x 2 8 x 3 m) a 2 2ab 15b 2 n) 6 x 2 13 xy 6 y 2 o) (2 x 1) 2 4 36 x 2 121 y 2 p) q) 49 42 x 9 x 2 r) 625 x 8 y 4 16 z 8 s) 1 6 x 9 x 2 4 y 2 49 169 Solving Quadratic Equations 1. 1. Solve each of the following quadratic equations using the best method: a) 3 y 2 7 y 4 0 b) x 2 4 x 11 0 c) 3x 2 27 0 1 2 d) 2x 2 3 x 3 e) y 4y 2 f) 6 x 2 5 x 4 2 g) 3 x 2 4 x 7 h) 3 x 2 48 0 i) 5 x 2 4 x 20 0 900 900 j) 2x 2 5 x 3 0 k) 25 x x6 2. Determine the optimal value in each case, whether it represents a maximum or a minimum, and at what value of x it occurs: 1 2 a) f ( x) x 2 16 x 24 b) f ( x) 4 x 2 24 x 17 c) f ( x) x 11x 5 2 1 d) f ( x) 2 x 2 7 x e) f ( x) x 2 8 x 9 3 3. A basketball player takes a shot. The height of the ball, h, in metres, can be modeled by the function h 4t 2 8t 3 , where t is time in seconds. a) What is the height of the ball after 2 seconds? b) What is the maximum height of the basketball? c) At what time does the basketball reach its maximum height? 4. A rocket firework follows a parabolic path. Its height, h, in metres, is given by h 5t 2 30t 12 , where t is the time in seconds, after the firework has been fired. a) At what height was the firework fired? b) For how many seconds was the rocket in the air? c) Determine the maximum height of the rocket. 5. A rectangle is 5 cm longer than it is wide, and has a diagonal 20 cm long. Find the dimensions of the rectangle. Functions and Transformations 1. For each of the following: i) State if each represents a function, and explain your reasoning in each case. ii) State the domain and range. 1 a) b) c) y ( x 4) 2 d) y 4 x7 2. If f ( x) 2 x 2 x 3 and g ( x) 3x 5 , determine: a) f (5) b) g ( f ( x)) c) f ( g ( x)) d) x when g ( x) 11 3. Describe the transformations on f ( x) x 2 that give the following functions: 1 a) f ( x) 2( x 3) 2 4 b) f ( x) ( x 4) 2 6 5 4. Write the transformed equation if f ( x) x 2 undergoes the following transformation: a vertical stretch of factor 3, shifted down 6 units, reflected in the x-axis and shifted to the left 2 units. 5. Use mapping notation to graph each of the following parabolas, and state the domain and range for each: 1 1 a) f ( x) 3 x 2 2 b) f ( x) ( x 2) 2 c) f ( x) 3( x 4) 2 1 d) f ( x) ( x 4) 2 3 4 2 Exponential Functions 1. Graph each of the following using a table of values. State the domain and range for each: x 1 a) f ( x) 3 x b) f ( x) 4 2. Sketch each of the following graphs on the same grid: x 3 f ( x) 0.5 , f ( x) 7 , f ( x) , f ( x) 11x x x 4 3. The population of a strain of bacteria doubles every 30 minutes. If there are initially 100 bacteria: a) Write a formula that represents the number of bacteria after x hours. b) Determine the amount of bacteria after 10 hours. c) After how many hours will there be 75000 bacteria? 4. The population of a city grows at a rate of 15% per year. If the population in 1978 is 250000, determine the projected population of the city in 2015. 5. A photocopier is purchased for $5200. Its value depreciates by 20% each year. a) Write a formula that represents the value of the photocopier after x years. b) Use your formula to determine the value of the photocopier after 12 years. c) The owner of the machine decides to replace the photocopier when its value hits $250. Determine when the owner will replace the machine. 6. A radioactive isotope has a half life of 1 week. Suppose you have 100 g of the isotope today. Determine the mass of the isotope after 10 weeks. 7. There are 50 bacteria present initially in a culture. In 3min., the count is 204800. What is the doubling period? 8. Peter has a new job. He is given the choice of two methods of payment: Method 1: A starting wage of $12/h with a raise of $0.50/h per year Method 2: A starting wage of $12/h with a raise of 5%/h per year. Peter expects to stay at the job for 6 years. a) Which type of growth does each method represent? Explain. b) Which method should Peter choose? Justify your answer. Finance 1. In order to have $7250 for tuition in a year and a half, you plan to deposit money into a savings account every three months. If that account pays 5.2% compounded quarterly, a) what will your regular deposits be? b) how much interest would you have earned? 2. Leo’s parents want him to be able to withdraw an allowance of $2700 every 6 months. How much should they invest into an account today that pays 8.5% compounded semi-annually so that Leo can withdraw his allowance for the next 5 years? 3. Determine: a) the amount $8500 will grow to if invested at 8% compounded quarterly for ten years. b) the principal that must be invested now at 6% compounded annually to be worth $10 000 in 5 years. c) The total accumulated amount of $500 invested every month at 7% compounded monthly for 8 years. 4. How long will it take, to the nearest year, for $800 to grow to $1000 in an account that pays 4.5% annually compounded semi-annually? 5. If a savings account compounds daily, what would the annual interest rate need to be for $500 to grow to $650 in 3 years? 6. Danika deposits $500 every 6 months in an account that pays 3.4% interest compounded semi annually. After 11 years she stops depositing and lets her money accumulate interest for another 12 years. How much money does Danika end up with in her account? Trigonometry 1. Use trigonometry to solve each of the following triangles: a) MNO, N 90, O 20, m 15cm b) PQR, P 62, Q 90, r 8cm c) JKL, K 90, J 32, j 7cm d) ABC, a 32cm, b 44cm, B 32.8 e) ABC, A 32.3, B 74.2, c 21.5cm f) ABC, a 39cm, b 46cm, c 53cm g) ABC, a 35.4cm, B 121.8, c 41.8cm 2. A tree casts a shadow 40 m long when the sun’s rays are at an angle of 36 to the ground. How tall is the tree? 3. Two office towers are 30 m apart. From the 15th floor (4o metres up) of the shorter tower, the angle of elevation to the top of the other tower is 70 . Find the height of the taller tower. 4. A hockey net is 1.83 m wide. A player shoots from a point where the puck is 12 m from one goal post and 10.8 m from the other. Within what angle must he shoot? 5. From the top of a lighthouse, Patti sees a whale in difficulty. The angle of depression to the whale is 34 . From another observation point 15 metres directly below, Savanga sees the same whale with an angle of depression of 17. a) Draw and label a diagram that illustrates the situation described above. b) How far is the whale from the foot of the observation tower? 6. Graph the following trigonometric functions where 360 x 360 , and state the domain and range of each: 1 1 a) f ( x) 2sin 3( x 90) 1 b) f ( x) cos x 2 2 1 c) f ( x) sin ( x 60) 2 d) f ( x) 3cos( x 90) 4 3