Algebra 2, Spring Semester Review 2013

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1 Class: Date: Algebra, Spring Semester Review (1 point) Find the annual percent increase or decrease that y = 0.5(.) x models. 0% increase 0% decrease 10% increase d. 5% decrease. (1 point) An initial population of 80 quail increases at an annual rate of %. Write an exponential function to model the quail population. What will the approximate population be after years? f(x) = 80(1.) x ; 15 f(x) = ( 80 0.) x ;,708,494 f(x) = 80() x ; 9,97,940 d. f(x) = 80(0.) x ; 15. (1 point) The half-life of a certain radioactive material is days. An initial amount of the material has a mass of 1 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 5 days. Round your answer to the nearest thousandth. y = 1 1 ËÁ Ê ˆ x 1 Ê 1 ˆ x ; 0 kg y = ; kg ËÁ 1 Ê y = 1 1 ˆ x ;.945 kg d. y = 1 ËÁ 1 1 Ê 1 ˆ x ; kg ËÁ 1 4. (1 point) Use the graph of y = e x to evaluate e 1.7 to four decimal places d (1 point) Use the table feature on a graphing calculator to evaluate e 1.8 to four decimal places d (1 point) Suppose you invest $100 at an annual interest rate of 4.% compounded continuously. How much will you have in the account after 4 years? $800. $,701.8 $10,18.07 d. $1,9. 7. (1 point) How much money invested at 5% compounded continuously for years will yield $80? $95.70 $ $ d. $ Write the equation in logarithmic form. 8. (1 point) 5 = log = 5 log = 5 log = 5 d. log 5 = 1

2 Evaluate the logarithm. 9. (1 point) log d (1 point) log d. 11. (1 point) log d (1 point) The ph of a liquid is a measure of how acidic or basic it is. The concentration of hydrogen ions in a liquid is labeled H + + Use the formula ph = log H to find the ph level, to the nearest tenth, of a. liquid with [H + ] about d (1 point) Use the Change of Base Formula to evaluate log d (1 point) Use the Change of Base Formula to evaluate log d (1 point) A construction explosion has an intensity I of W/m. Find the loudness of the sound in decibels if L = 10 log I I 0 and I o = 10 1 W/m. Round to the nearest tenth decibels 10.9 decibels decibels d decibels 1. (1 point) A company with loud machinery needs to cut its sound intensity to 7% of its original level. By how many decibels would the loudness be reduced? Use the formula L = 10 log I. Round to the nearest I o hundredth..01 decibels 1.7 decibels.1 decibels d. 4. decibels

3 Solve the exponential equation. 17. (1 point) 1 1 = 4 4x (1 point) 4 4x = (1 point) 15 9x = d (1 point) Solve 15 x =. Round to the nearest ten-thousandth d (1 point) The generation time G for a particular bacteria is the time it takes for the population to double. The t bacteria increase in population is shown by the formula G =, where t is the time period of the. log a P d. d. population increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the end of the time period. If the generation time for the bacteria is hours, how long will it take 8 of these bacteria to multiply into a colony of 781 bacteria? Round to the nearest hour. 177 hours 7 hours 4 hours d. 85 hours Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary.. (1 point) log x = d (1 point) Solve log(4x + 10) = d. 990 Use natural logarithms to solve the equation. Round to the nearest thousandth. 4. (1 point) e 4x = d (1 point) e x + 1 = d

4 . (1 point) e x = d (1 point) e x = d (1 point) The sales of lawn mowers t years after a particular model is introduced is given by the function y = 5500 ln(9t + 4), where y is the number of mowers sold. How many mowers will be sold 4 years after a model is introduced? Round the answer to the nearest whole number. 0,89 mowers 8,811 mowers 41,709 mowers d. 19,71 mowers Generate the first five terms in the sequence using the explicit formul 9. (1 point) y n = 5n 5 0, 5, 0, 15, 10 0, 5, 0, 15, 10 10, 15, 0, 5, 0 d. 10, 15, 0, 5, 0 0. (1 point) c n = 1n (1 point) 49, 7, 5, 1, 1 1, 1, 5, 7, 49 1, 1, 5, 7, 49 d. 49, 7, 5, 1, 1 What is the 15 th term in the sequence using the given formula? c n = n d. 44. (1 point) Write a recursive formula for the sequence 7, 1, 19, 5, 1,... Then find the next term. a n = a n 1 +, where a 1 = 7; 7 a n = a n 1 +, where a 1 = 7; 7 a n = a n 1, where a 1 = ; d. a n = a n 1, where a 1 = 7; 7 4

5 . (1 point) Write a recursive formula for the sequence 7, 4, 1,, 5,... Then find the next term. a n = a n 1, where a 1 = 8; 7 a n = a n 1, where a 1 = 7; 8 a n = a n 1 +, where a 1 = ; d. a n = a n 1 +, where a 1 = 7; 8 4. (1 point) Write an explicit formula for the sequence 8,, 4,, 0,... Then find a 14. a n = n + 10; 1 a n = n + 8; 0 a n = n + 8; 18 d. a n = n + 10; (1 point) Suppose you drop a tennis ball from a height of 8 feet. After the ball hits the floor, it rebounds to 80% of its previous height. How high will the ball rebound after its third bounce? Round to the nearest tenth.. feet 4.1 feet 5.1 feet d. 1 feet. (1 point) The table shows the predicted growth of a particular bacteria after various numbers of hours. Write an explicit formula for the sequence of the number of bacteri Hours (n) Number of Bacteria a n = 1n a n = n + 1 a n = 1 1 n d. a n = 1n + 1 Is the sequence arithmetic? If so, identify the common difference. 7. (1 point) 1, 0, 7, 4,... yes; 7 yes; 7 yes; 1 d. no 8. (1 point) 14, 1, 4, 77,... yes; 7 yes; 7 yes; 14 d. no 9. (1 point).4, 9.8,, 4.,... yes; 1 yes; 1. yes; 1. d. no 40. (1 point) Find the 110 term of the sequence 7,, 1,, d (1 point) Find the missing term of the arithmetic sequence,, 4, d. 40 5

6 4. (1 point) Viola makes gift baskets for Valentine s Day. She has 1 baskets left over from last year, and she plans to make 1 more each day. If there are 15 work days until the day she begins to sell the baskets, how many baskets will she have to sell? 19 baskets 05 baskets 15 baskets d. 181 baskets Is the sequence geometric? If so, identify the common ratio. 4. (1 point), 1, 4, 48,... yes; yes; yes; 4 d. no 44. (1 point), 4, 1,,... yes; yes; yes; d. no 45. (1 point) 1, 9, 4 7, 8 81, 1 4,... yes; yes; 1 9 yes; 1 d. not geometric What is the fifth term of the geometric sequence? 4. (1 point) 5, 15, 45, d. 45 Write the explicit formula for the geometric sequence. Then find the fifth term in the sequence. 47. (1 point) a 1 = 4, a = 8, a = 1 a n = 4 () n ; 4 a n = 4 ( ) n ; 18 a n = 4 ( ) n 1 ; 4 d. a n = ( 4) n 1 ; (1 point) A rope is swinging in such a way that the length of the arc is decreasing geometrically. If the first arc is 18 feet long and the third arc is 8 feet long, what is the length of the second arc? 1 feet 10 feet 5 feet d. 7 feet

7 49. (1 point) What is a possible value for the missing term of the geometric sequence? 9,, 975, d What is the sum of the finite arithmetic series? 50. (1 point) d (1 point) ( 0.) + ( 1.5) d (1 point) d (1 point) ( 5) d (1 point) Justine earned $,000 during the first year of her job at city hall. After each year she received a % raise. Find her total earnings during the first five years on the jo $18,07.5 $1,004,704.0 $4,00.51 d. $108, (1 point) A rubber ball dropped on a hard surface takes a sequence of bounces, each one 5 one. If this ball is dropped from a height of 10 feet, what is the total vertical distance it has traveled after it hits the surface the 5th time? 7 15 feet feet feet d feet 5. (1 point) Find the sine of 180º. Round your answers to the nearest hundredth if necessary d (1 point) Find the cosine 15º. Round your answers to the nearest hundredth if necessary d as high as the preceding 7

8 58. (1 point) Find the exact value of sin 10º. sin = sin = 59. (1 point) Find the exact value of cos 00º. cos = 1 cos = 1 0. (1 point) Find the exact values of cos 150º and sin 150º. cos = 1, sin = cos = 1, sin = 1. (1 point) Find the radian measure of an angle of 80º. 9 14π 14π 9. (1 point) sin = 1 d. sin = 1 cos = d. cos = cos = d. cos = Find the degree measure of an angle of π 5 radians. 9π 14, sin = 1, sin = º 108πº 108º d.. (1 point) Find the radian measure of an angle of 110º π 18π (1 point) Find the degree measure of an angle of 4π radians π d. d. 14 9π π 00 º 18π 4.19º 40πº 40º d. π 15 º 5. (1 point) Ê Find the exact values of cos π ËÁ 4 radians ˆ Ê and sin π ËÁ 4 radians ˆ., 1,, d. 11, 1 8

9 . (1 point) Find the exact value of sin π ËÁ Ê radians ˆ. 1 d (1 point) Ê Find the exact value of cos 7π ËÁ 4 ˆ radians. 1 d. 1 Use the graph of y = sin θ to find the value of sin θ for each value of θ. 8. (1 point) 70º d. 1 9

10 9. (1 point) Find the period of the graph shown below. π π 1 π d. 4π 70. (1 point) A particular sound wave can be graphed using the function y = sin x. Find the period of the function. period = π period = period = d. period = (1 point) Find the amplitude of the sine curve shown below. π 8 d (1 point) A particular sound wave can be graphed using the function y = sin 5x. Find the amplitude of the function. amplitude = 5 π amplitude = amplitude = π d. amplitude = 5 10

11 What is the graph of one cycle of a sine curve with the given characteristics? Using the form y = a sin bθ, what is an equation for the sine curve? 7. (1 point) amplitude =, period = π, and a > 0 y = sin θ d. y = sin θ y = sin θ y = sin θ 11

12 74. (1 point) amplitude = 4, period = 1 π, and a < 0 y = 4 sin 8θ d. y = 4 sin 4θ y = 4 sin 8θ y = 4 sin 1 4 θ 1

13 75. (1 point) y = sin θ d. 1

14 Sketch one cycle of the cosine function. 7. (1 point) y = cos θ d. 14

15 77. (1 point) y = cos θ d. 15

16 78. (1 point) What is the graph of one cycle of a cosine curve with amplitude, period π, and a < 0? d. Use the unit circle to find the inverse function value in degrees. 79. (1 point) Ê ˆ sin 1 ËÁ d

17 80. (1 point) Ê ˆ cos 1 ËÁ d (1 point) Find the height of the triangle. 8. (1 point) d d.. 17

18 8. (1 point) The line of sight from a small boat to the light at the top of a 5-foot lighthouse built on a cliff 5 feet above the water makes a 5 angle with the water. To the nearest foot, how far is the boat from the cliff? 84. (1 point) 141 feet 18 feet 7 feet d. 75 feet In ΔABC, C is a right angle, what is the measure of x? 85. (1 point) d d

19 8. (1 point) 87. (1 point) What are the focus and directrix of the parabola with equation y = 1 1 x? focus: Ê Ë,0 focus: Ê ËÁ 0, ˆ ; directrix: y = focus: Ê Ë,0 d. focus: Ê ËÁ 0, ˆ ; directrix: y = Identify the focus and the directrix of the graph of x = 1 0 y. focus (0, 5), directrix x = 5 focus (0, 5), directrix x = 5 focus ( 5, 0), directrix x = 5 d. focus ( 5, 0), directrix x = (1 point) A mirror with a parabolic cross section is used to collect sunlight on a pipe located at the focus of the mirror. The pipe is located inches from the vertex of the mirror. Write an equation of the parabola that models the cross section of the mirror. Assume that the parabola opens upward. y = 1 4 x y = 1 8 x y = 1 8 x d. y = 1 4 x Write an equation of a circle with the given center and radius. 89. (1 point) center (, 4) and radius 5 ( x ) + Ê ËÁ y + 4 ˆ = 5 ( x + ) + Ê ËÁ y 4 ˆ ( x + ) + Ê ËÁ y 4 ˆ = 5 d. ( x ) + Ê ËÁ y + 4 ˆ 90. (1 point) center ( 7, ) and radius ( x + 7) + Ê ËÁ y + ˆ = 4 ( x + 7) + Ê ËÁ y + ˆ ( x 7) + Ê ËÁ y ˆ = d. ( x 7) + Ê ËÁ y ˆ = 5 = (1 point) Write an equation for the translation of x + y = 49 by units left and 4 units up. ( x + ) + Ê ËÁ y + 4 ˆ = 49 ( x ) + Ê ËÁ y 4 ˆ = 49 ( x + ) + Ê ËÁ y 4 ˆ = 49 d. ( x ) + Ê ËÁ y + 4 ˆ = 49 What is the center and radius of the circle with the given equation? 9. (1 point) (x 1) + (y + 1) = 4 center ( 1, 1); radius 4 center ( 1, 1); radius center (1, 1); radius 4 d. center (1, 1); radius = = 4 19

20 9. (1 point) ( x 4) + Ê ËÁ y 8 ˆ = 9 center at ( 4, 8); radius 9 center at (4, 8); radius center at (4, 8); radius 9 d. center at ( 4, 8); radius 94. (1 point) x + y + 14x 1y = 9 center at ( 7, ); radius 1 center at (7, ); radius 1 center at (7, ); radius 4 d. center at ( 7, ); radius 4 Write an equation of an ellipse in standard form with the center at the origin and with the given characteristics. 95. (1 point) vertex at (, 0) and co-vertex at (0, ) x 9 + y = 1 4 x 4 + y 9 = 1 d. 9. (1 point) vertex at (, 0) and co-vertex at (0, 5) x 5 + y = 1 x 5 + y 9 = 1 d. x + y x + y = 1 = 1 x 9 + y 5 = 1 x + y (1 point) vertex at ( 5, 0) and co-vertex at (0, 4) x 5 + y 1 = 1 x 5 + y 4 x 1 + y 5 = 1 d. x 4 + y 5 = 1 = 1 = 1 0

21 What are the foci of the ellipse? Graph the ellipse. 98. (1 point) x 49 + y 4 = 1 foci (0, ± 15 ) foci (0, ± 11 ) foci (0, ± 15 ) d. foci (0, ± 11 ) 1

22 Use matrices A, B, and C. Find the sum or difference if you can. A = B = C = (1 point) B A 100. (1 point) not possible d (1 point) d. d

23 10. (1 point) d (1 point) The first table shows the teams with the four best records halfway through their season. The second table shows the full season records for the same four teams. Which team had the best record during the second half of the season? Record for the First half of the Season Team Wins Losses Team 1 14 Team 7 1 Team 5 15 Team Record for the Season Team Wins Losses Team Team 5 4 Team 5 4 Team Team Team Team 1 d. Team 4 Find the values of the variables (1 point) 8 + t = y t = 5, y = t =, y = 7 t = 1, y = 5 d. t =, y = 5

24 105. (1 point) 1 w 10. (1 point) f = k f = 7, k =, w = 9 or 9 f = 7, k =, w = 81 or 81 f = 7, k =, w = 9 or 9 d. f = 7, k =, w = x 0 5 x y + = 1 1 4y + 8 x = 4 and y = x = 4 and y = x = 5 and y = 1 d. x = 5 and y = 1 Find the product (1 point) (1 point) d. d

25 109. (1 point) The table summarizes the scoring of a football game between Team A and Team B. A touchdown (TD) is worth points, a field goal (FG) is worth points, a safety (S) is worth points, and a point after touchdown (PAT) is worth 1 point. Use matrix multiplication to find the final score. TD FG S PAT Team A 1 0 Team B Team A: 5 Team B: Team A: Team B: 9 Team A: Team B: 40 d. Team A: 17 Team B: 8 Determine whether the product is defined or undefined. If defined, give the dimensions of the product matrix (1 point) 111. (1 point) defined; defined; defined; 1 d. undefined defined; defined; 1 defined; 1 d. undefined Are matrices A and B inverses? 11. (1 point) A = 11. (1 point) 5 18 and B = yes no A = 9 and B = 5 9 no yes 5 5

26 Evaluate the determinant of the matrix (1 point) 115. (1 point) d d. 0 Does the given matrix, A, have an inverse? If it does, what is A 1? 11. (1 point) A = (1 point) A = d. does not exist d. does not exist

27 118. (1 point) Ï a + 5b 5c = Write the system Ì Ô 7a + 7b + 4c = as a matrix equation. Then identify the coefficient matrix, 7a 4b 9c = 1 ÓÔ the variable matrix, and the constant matrix. 7

28 5 5 a 5 5 a b = b = c c variable matrix: coefficient matrix: a a constant matrix: b variable matrix: b c c coefficient matrix: constant matrix: a a b = d b = c c coefficient matrix: variable matrix: a a constant matrix: b coefficient matrix: b c c variable matrix: constant matrix: 1 1 8

29 What is the solution of the system? Solve using matrices (1 point) Ï x + y = 9 Ì Ô ÓÔ x + y = d. no solution 10. (1 point) Ï x + y = 10 Ì Ô ÓÔ 4x + y = d. no solution 11. (1 point) Ï x + y = 1 Ì Ô ÓÔ x y = d. no solution 1. (1 point) Ï 7x + y = Ì Ô ÓÔ 11x y = d. no solution Solve the system. 1. (1 point) Ï x + y 4z = 0 Ì Ô x + y + z = 8 5x + 5y + z = ÓÔ ( 0, 8, ) ( 4,, 4) (4,, 4) d. ( 4,, 4) 9

30 14. (1 point) Ï 5x y = 1 Ì Ô x 4y + 5z = 8 ÓÔ x z = 1 ( 4, 1, 1) (4, 1, 0) (1, 8, 1) d. ( 4, 1, 0) 0

31 Algebra, Spring Semester Review 01 Answer Section 1. ANS: B PTS: 1. ANS: A PTS: 1. ANS: B PTS: 1 4. ANS: A PTS: 1 5. ANS: A PTS: 1. ANS: D PTS: 1 7. ANS: D PTS: 1 8. ANS: B PTS: 1 9. ANS: C PTS: ANS: A PTS: ANS: B PTS: 1 1. ANS: C PTS: 1 1. ANS: D PTS: ANS: A PTS: ANS: C PTS: 1 1. ANS: D PTS: ANS: C PTS: ANS: C PTS: ANS: C PTS: 1 0. ANS: A PTS: 1 1. ANS: D PTS: 1. ANS: A PTS: 1. ANS: B PTS: 1 4. ANS: D PTS: 1 5. ANS: A PTS: 1. ANS: A PTS: 1 7. ANS: C PTS: 1 8. ANS: A PTS: 1 9. ANS: C PTS: 1 0. ANS: C PTS: 1 1. ANS: C PTS: 1. ANS: A PTS: 1. ANS: B PTS: 1 4. ANS: D PTS: 1 5. ANS: B PTS: 1. ANS: A PTS: 1 7. ANS: A PTS: 1 8. ANS: D PTS: 1 9. ANS: B PTS: ANS: A PTS: ANS: C PTS: 1 4. ANS: A PTS: 1 1

32 4. ANS: A PTS: ANS: D PTS: ANS: A PTS: 1 4. ANS: C PTS: ANS: B PTS: ANS: A PTS: ANS: B PTS: ANS: B PTS: ANS: B PTS: 1 5. ANS: C PTS: 1 5. ANS: C PTS: ANS: A PTS: ANS: B PTS: 1 5. ANS: C PTS: ANS: B PTS: ANS: B PTS: ANS: B PTS: 1 0. ANS: C PTS: 1 1. ANS: B PTS: 1. ANS: C PTS: 1. ANS: B PTS: 1 4. ANS: C PTS: 1 5. ANS: C PTS: 1. ANS: B PTS: 1 7. ANS: A PTS: 1 8. ANS: D PTS: 1 9. ANS: B PTS: ANS: A PTS: ANS: D PTS: 1 7. ANS: C PTS: 1 7. ANS: C PTS: ANS: C PTS: ANS: D PTS: 1 7. ANS: D PTS: ANS: C PTS: ANS: C PTS: ANS: B PTS: ANS: B PTS: ANS: A PTS: 1 8. ANS: A PTS: 1 8. ANS: B PTS: ANS: B PTS: ANS: B PTS: 1 8. ANS: C PTS: ANS: D PTS: ANS: C PTS: 1

33 89. ANS: D PTS: ANS: A PTS: ANS: B PTS: 1 9. ANS: D PTS: 1 9. ANS: C PTS: ANS: D PTS: ANS: A PTS: 1 9. ANS: C PTS: ANS: A PTS: ANS: A PTS: ANS: B PTS: ANS: C PTS: ANS: B PTS: ANS: C PTS: ANS: B PTS: ANS: D PTS: ANS: B PTS: ANS: C PTS: ANS: A PTS: ANS: B PTS: ANS: C PTS: ANS: B PTS: ANS: D PTS: ANS: A PTS: ANS: A PTS: ANS: C PTS: ANS: A PTS: ANS: B PTS: ANS: D PTS: ANS: C PTS: ANS: A PTS: ANS: A PTS: ANS: A PTS: 1 1. ANS: B PTS: 1 1. ANS: C PTS: ANS: D PTS: 1