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ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (6.1) What is an equation for the translation of y y 1 5 x3 y 1 3 x5 y 1 5 x3 y 1 3 x 5 1 that has asymptotes at x 3 and y 5 ? x (6.1) What is the equation of the vertical asymptote of y 2 ? x6 x6 x0 x2 x 6 (6.1) Sketch the asymptotes and graph of y 1 2 . Identify the domain and range. x 3 5 7 cm was cut into two pieces. If one piece is cm, express the x2 x2 length of the other board as a rational expression. (6.3) A board of length 2 x 24 ( x 2)( x 2) 2 x 24 ( x 2) 2 6 x 24 ( x 2) 2 Revised January 2015 Page 1 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ 6 x 24 ( x 2)( x 2) (6.3, 6.4)Use the following expressions to answer the questions. A x5 x 4x 3 2 B 2 x x 2 C x3 2x 2 D 7 2x (a) Explain how to add two rational expressions. Simplify A B as an example. Be sure to give reasons for each step. Simplify completely. (b) Explain how subtracting (such as simplifying A B ) would be different from adding. (c) Explain how to multiply rational expressions, simplifying B C as an example. (d) Show how to divide B by C . (e) Explain how you would solve a rational equation like C D . Then describe, in detail, the strategy for solving an equation like B C D . Do not completely solve the equations. Instead, concentrate on the first two or three steps in solving; show what to do and explain why. (6.6) The rate of heat loss from a metal object is proportional to the ratio of its surface area to its volume. (a) What is the ratio of a steel sphere’s surface area to volume? (b) Compare the rate of heat loss for two steel spheres of radius 2 meters and 3 meters, respectively. (6.3) Multiply. Simplify your answer. 8 x 4 y 2 9 xy 2 z 6 3z 3 4 y4 6x5 y 8 z 9 6x5 z3 6x 4 yz 2 3 3 2 x y z 2 Revised January 2015 Page 2 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ 8 x6 z 4 4 x 4 z 2 (6.2) Which expression represents the quotient? 4 x2 z 2x4 z3 x2 z 2x3 z 4 x2 z 2 4 x 4 z 3 3x 2 z 4x3 z 4 3x2 z 2 (6.6) Last week, Wendy jogged for a total of 10 miles and biked for a total of 10 miles. She biked at a rate that was twice as fast as her jogging rate. (a) Suppose Wendy jogs at a rate of r miles per hour. Write an expression that represents the amount of time she jogged last week and an expression that represents the amount to time she biked last week. (hint: distance=rate time ) (b) Write and simplify an expression for the total amount of time Wendy jogged and biked last week. (c) Wendy jogged at a rate of 5 miles per hour. What was the total amount of time Wendy jogged and biked last week? 1 (6.2) Which expression is equivalent to y2 4 8x 3 1 5 x3 y 2 for all x, y 0 ? 6 1 5 2x 6 y 2 3 5 4x 3 y 2 3y2 4 4x 9 y3 48 x 4 Revised January 2015 Page 3 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (6.2) Which set contains all the real numbers that are not part of the domain of f ( x) x4 ? x 4 x 32 2 {8} {-4} {-4, 8} {-8, 4} (6.4) Solve the equation x 7 1 . x2 49 x 7 x 7 -8 7 8 No solution (6.6) A sight-seeing boat travels at an average speed of 20 miles per hour in the calm water of a large lake. The same boat is also used for sight-seeing in a nearby river. In the river, the boat travels 2.9 miles downstream (with the current) in the same amount of time it takes to travel 1.8 miles upstream (against the current). Find the current of the river. (6.6) A baseball player’s batting average is found by dividing the number of hits the player has by the number of at-bats the player has. Suppose a baseball player has 45 hits and 130 at-bats. Write and solve an equation to model the number of consecutive hits the player needs in order to raise his batting average to 0.400. Explain now you found your answer. (6.1) Which intervals correctly define the domain of f ( x) 1 2 x4 (, 4) and (4, ) (, 4) and (4, ) (, 4) and (4, ) (, 4) and (2, ) Revised January 2015 Page 4 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ 1 (6.1) Which statement is true for the function f ( x) ? x4 4 is not in the range of the function. 4 is not in the domain of the function. -4 is not in the range of the function. -4 is not in the domain of the function. 8 (7.1) What is the value of (15 4n) ? n 3 -42 -17 88 363 (7.2) Given the sequence 1, 2, 4, 8, …. Find the sum of the infinite series. 15 18 30 (7.2) During a flu outbreak, a hospital recorded 12 cases the first week, 54 cases the second week, and 243 cases the third week. a) Write a geometric sequence to model the flu outbreak. b) How many cases will occur in the sixth week if the hospital cannot stop the outbreak? Revised January 2015 Page 5 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.2) Given the geometric sequence with common ratio r , write a rule for the nth term of the sequence 4, -28, 196, -1372… an 7(4)n1 an 4(7)n1 an 7(4)n1 an 4(7)n1 (7.1, 7.2) In a classic math problem a king wants to reward a knight who has rescued him from an attack. The king gives the knight a chessboard and plans to place money on each square. He gives the knight two options. Potion 1 is to place a thousand dollars on the first square, two thousand on the second square, three thousand on the third square, and so on. Option 2 is to place one penny on the first square, two pennies on the second, four on the third, and so on. Think about which offer sounds better and then answer these questions. a) List the first five terms in the sequences formed by the given options. Identify each sequence as arithmetic, geometric, or neither. Option 1 Option 2 b) For each option, write a rule that tells how much money is placed on the nth square of the chessboard and a rule that tells the total amount of money placed on squares one through n . Option 1 Option 2 c) Find the amount of money placed on the 20th square of the chessboard and the total amount placed on squares 1 through 20 for each option. Option 1 Option 2 d) There are 64 squares on a chessboard. Find the total amount of money placed on the chessboard for each option. Option 1 Option 2 e) Which gives the better reward, Option 1 or Option 2? Explain why. Revised January 2015 Page 6 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.4) If f ( x) e x , then which of the following is f 1 (7) ? e7 7 log 7 ln 7 (7.4) Which is the inverse of f ( x) 2log3 x ? f 1 ( x) 1.5 x f 1 ( x) 0.5(3) x f 1 ( x) 30.5 x f 1 ( x) 2(3) x (7.5) Find the value of log 2 32 . 5 1024 16 4 (7.5) Consider the function f ( x) log x . a) b) c) d) Identify the transformation applied to f ( x) to create g ( x) log x 1 . Identify the transformation applied to f ( x) to create h( x) log(10 x) . Compare the graphs of g ( x ) and h( x) . What do you notice? Use the properties of logarithms to explain your answer to part c. Revised January 2015 Page 7 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ x (7.6) Which is the same function as f ( x) ln ? 3 g ( x) ln x ln 3 g ( x) ln x ln 3 g ( x) ln 3 ln x g ( x) ln x ln 3 (7.6) Rewrite log9 92 x3 y in exponential form. 9 y 9(2 x 3) 9 y 92 x3 y 2x 3 9 y 18 x 27 (7.6) Psychologists try to predict the activation of memory when a person is tested on a list of words they learned. The following model is used to make this prediction: A ln(n) 0.5ln(T ) 0.5ln( L) where A is the number of words learned, n is the number of exercises, T is the amount of time between learning and testing and L is the length of the list that was tested. a) Write the formula as the ln of a single expression. b) Discuss the influence on A (going up or down) when increasing n, T, and L, according to the formula. Do these results make sense? c) If you want A to be bigger than 0, what conditions must be placed on L, T, and n? (7.8) If log 4{log 2[log 3(3 x)]} 1 , then what is x ? 2 81 48 27 9 Revised January 2015 Page 8 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.8) Which equation has the same solution as log 4 ( x 7) 5 ? 4x7 5 5x7 5 54 x 7 45 x 7 (7.8) A biologist studying the relationship between the brain weight and body weight in mammals uses the formula: ln( wbody ) ln( wbrain ) 669 Where wbody =body weight in grams and wbrain =brain weight in grams. What is the formula for the body weight? wbody ( wbrain )(e669 ) wbody ( wbrain ) (e669 ) wbody e( wbrain )( e 669 ) wbody 669( wbrain ) (7.9) Choose the function that describes the graph below: f ( x) log x 2 f ( x) log( x 2) f ( x) log x 2 f ( x) log( x 2) Revised January 2015 Page 9 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.9) What function is represented by the following graph? f ( x) 3x f ( x) 3x 2 f ( x) 2 3x 3 f ( x) ( ) x 2 (7.9) The graph of the equation y log(2 x 3) is translated right 3 units and down 3.5 units to form a new graph. Which equation best represents the new graph? y log(2 x 9) 3.5 y log(2 x 9) 3.5 y log(2 x 3) 3.5 y log(2 x 3) 3.5 (7.9) John graphs the equation y 5 x . Lana graphs the equation y 5 x 2 . How does Lana’s graph compare to John’s graph? Lana’s graph shifts 2 units downward Lana’s graph shifts 2 units upward Lana’s graph shifts 2 units to the left Lana’s graph shifts 2 units to the left Revised January 2015 Page 10 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.10) In 1950, the city of San Jose had a population of 95,000. Since then, on average, it grows 4% per year. What is the best formula to model San Jose’s growth? 95,000(1.04)t 95,000(0.96)t -.04t + 95,000 .04t + 95,000 (7.10) Sarai bought $400 of Las Vegas Cellular stock in January 2005. The value of the stock is expected to increase by 6.5% per year. a) Write a model to describe Sarai’s investment. b) Use the graph to show when Sarai’s investment will reach $1100? (7.10) The loudness of sound is measured on a logarithmic scale according to the formula I L 10 log( ) , where L is the loudness of sound in decibels ( db ), I is the intensity of sound, and I0 I 0 is the intensity of the softest audible sound. a) Find the loudness in decibels of each sound listed in the table. b) The sound at a rock concert is found to have a loudness of 110 decibels. Where should this sound be placed in the table in order to keep the sound intensities in order from least to greatest? Sound Intensity Jet taking off 1015 I 0 Jackhammer 1012 I 0 Hairdryer 107 I 0 Whisper 103 I 0 Leaves rustling 102 I 0 Softest audible sound I0 c) A decibel is 1 of a bel. Is a jet plane louder than a sound that measures 20 bels? Explain. 10 Revised January 2015 Page 11 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.10) Aaron invested $4000 in an account that paid an interest rate r compounded continuously. After 10 years he has $5809.81. The compound interest formula is A Pert , where P is the principal (the initial investment), A is the total amount of money (principal plus interest), r is the annual interest rate, and t is the time in years. a) Divide both sides of the formula by P and then use logarithms to rewrite the formula without an exponent. Show your work. b) Using your answer for part (a) as a starting point, solve the compound interest formula for the interest rate r . c) Use your equation from part (a) to determine the interest rate. (7.10) Denise is reviewing the change in the value of an investment. Time (decades) Value ($1000s) 0 2 1 1 2 0.5 3 0.25 Which statement can Denise use to model the data? Why is this type of function a good model for the data? 1 v 2( )t ; an exponential function is a good model because the value of the investment 2 changes by a constant amount in each time period. 1 v 2( )t ; an exponential function is a good model because the value of the investment 2 changes by a constant factor in each time period. v t 2 ; a linear function is a good model because the value of the investment changes by a constant amount in each time period. v t 2 ; a linear function is a good model because the value of the investment changes by a constant factor in each time period. Revised January 2015 Page 12 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.10) Amy recorded the total number of ladybugs observed in a garden over a 7-day period. The scatterplot below represents the data she collected. Total Number of Ladybugs Ladybugs Seen in Garden Over Time Number of Days Which type of function do these data points best fit? Cubic Exponential Linear Quadratic (7.10) Public Service Utilities uses the equation y a b log x to determine the cost of electricity where x represents the time in hours and y represents the cost. The first hour of use costs $6.66 and three hours cost $18.11. a) Determine the value of a and b in the model. b) What is the x-intercept of the graph of the model? What is the real world meaning of the x-intercept ? c) Use the model to find the cost for 65 hours of electricity use. d) If a customer can afford $40 per month for electricity, how long can he or she have the electricity turned on? Revised January 2015 Page 13 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.10) On an x-y coordinate plane the earth is located at (2, -1) and an asteroid is traveling on the path of g ( x) e3 x 3 . a) Write an equation representing the distance from the earth to the asteroid. b) If the asteroid is currently located at (4, e12 3) , what is the distance from the earth to the asteroid? c) Sketch a graph of g ( x ) . d) Find the point when the asteroid is closest to the earth. (7.10) Rashid is in Biology class and has gathered data on fruit flies. The table below shows the number of fruit flies in his sample at the end of each day for a week. Day # of flies 1 5 2 10 3 20 4 40 5 80 If the population continues to grow in this manner, which function will Rashid use to predict the population of fruit flies on any given day? p (t ) 5t 2 p(t ) 5( x 3) 20 p(t ) 2 ln t t 1 p(t ) 5 2 (7.10) Which function best fits the data shown in this scatter plot? f ( x) x 2 f ( x) x f ( x) 2 x 1 f ( x) x 3 Revised January 2015 Page 14 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (7.10) The graph below shows the change in temperature of a burning house over time. a) Describe the graph. b) This graph was found in an old math book and next to it was written: Rise of temperature = t0.25 Show that this function does not describe the graph correctly. c) Assume that the power function r At 0.25 is a good description of the graph. Find a reasonable value for A . Graph the new function. d) Compare the graph in part (c) to the original one. Do you think that a different power of t might result in a better model? Would a larger or smaller power produce a better fit? Explain. e) Use the original graph to find data. Carry out a power regression on the data to find a function that would produce a better fit. Use ABC for questions 47 and 48. (8.1) Which of the following is equal to cos38 ? sin 38 A 38 tan52 cos52 sin 52 B 22 C (8.1) Which expression represents the length of AB ? 22sin38 22cos38 22 tan 38 Revised January 2015 Page 15 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ 3 (8.1) If tan , what is sin ? 4 4 5 5 3 3 5 4 3 (8.1) Find sin cos2 when sin 1 and is in Quadrant I. 4 15 4 15 64 15 256 3 16 (8.2) Convert 60 to radians. 3 6 radians radians radians 3 3 radians Revised January 2015 Page 16 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.3) An analog watch had been running fast and needed to be set back. In resetting the watch, the 5 minute hand on the watch subtended an arc of radians. 3 Part A: Suppose the radius of the watch is 1 unit. What is the length of the arc on the outside of the watch that the angle subtends? Part B: If the watch was at 10:55 before being reset, what is the new time on the watch? 10 Part A: units 3 Part B: 9:05 Part A: 5 units 3 Part B: 10:05 Part A: 5 units 3 Part B: 9:20 Part A: 10 units 3 Part B: 8:40 (8.2) Convert 3 radians to degrees. 8 (8.2) Convert 16 radians to degrees. 3 (8.3) Suppose each paddle on the wall of a clothes dryer makes 80 revolutions per minute. Part A: What angle does one paddle subtend in 10 seconds? Give your answer in radians. Part B: Write an algebraic expression to determine the measure in radians of the subtended angle after x seconds. Show how the units simplify in your expression. Part C: You are interested in determining the total distance a point on the drum travels in a 20minute drying cycle. Can you use your expression from Part B? What other information, if any, is needed? Explain. Revised January 2015 Page 17 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ 5 (8.3) What is the exact value of tan ? 4 1 1 2 2 2 2 (8.3) Which expression has the same value as tan ? 3 tan 3 tan 3 cos 3 sin 3 (8.3) What is the reference angle corresponding to 7 ? 4 5 2 4 7 4 3 2 Revised January 2015 Page 18 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.3) A regular hexagon is inscribed in the unit circle. One vertex of the hexagon is at the point 3 1 , . A diameter of the circle starts from that vertex and ends on another vertex of the 2 2 hexagon. What are the coordinates of the other vertex? 3 1 , 2 2 1 3 , 2 2 3 1 , 2 2 1 3 , 2 2 3 (8.3) For what angles x in 0, 2 does the cos x have the same value as sin ? 4 3 5 cos and cos 4 4 3 7 cos and cos 4 4 7 cos and cos 4 4 5 cos and cos 4 4 (8.3) For which radian measures x will tan x be negative? 5 7 13 15 21 23 , , , , , ,... 4 4 4 4 4 4 3 7 11 15 19 23 , , , , , ,... 4 4 4 4 4 4 3 5 11 13 19 21 , , , , , ,... 4 4 4 4 4 4 5 9 13 17 21 , , , , , ,... 4 4 4 4 4 4 Revised January 2015 Page 19 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.3) The diameter of a bicycle tire is 20 in. A point on the outer edge of the tire is marked with a white dot. The tire is positioned so that the white dot is on the ground, then the bike is rolled so that the dot rotates clockwise through an angle of 16.75 radians. Part A: To the nearest tenth of an inch, how high off the ground is the dot when the wheel stops? Show your work. Part B: What distance was the bicycle pushed? Round your answer to the nearest foot. Part C: Would changing the size of the tire (value of r) change either of the answers found in Parts A or B? Explain your reasoning. (8.3) A ribbon is tied around a bicycle tire at the standard position 0 . The diameter of the wheel is 26 inches. The bike is then pushed forward 20 feet from the starting point. In what quadrant is the ribbon? Explain how you obtained your answer. (8.3) Find tan when sin cos and is in Quadrant IV. 2 2 2 2 1 1 (8.3) Two friends counted 24 evenly spaced seats on a Ferris wheel. As they boarded one of the seats, they noticed the edge of the wheel was 1 meter off the ground. They learned from the operator that the diameter of the wheel was 28 meters. After they got seated and started moving, in a counter-clockwise direction, they counted 13 chairs pass the operator, and then the Ferris wheel was stopped on the fourteenth chair to load another passenger. Part A: Design a representation of the Ferris wheel and locate where the friends were when the wheel stopped to load the next passenger. Part B: How many radians had they rotated through in the time before they stopped? Part C: To the nearest tenth of a meter, how far above the ground were they? Show your work. Revised January 2015 Page 20 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.4) If 0 and cos 1 sin , then what is the value of ? 3 3 2 2 5 6 6 (8.5) In RST , r 11.2 , s 9.8 , mT 38 , and mS 60 . Which expression can be used to find the length of side t? 9.8sin 82 sin 60 11.2 sin 82 sin 60 11.2 sin 60 sin 38 9.8sin 38 sin 60 (8.5) Solve ABC , given that A 47, B 52 , and b 78 . (8.5) Given ABC with a 10, b 13 , and A 19 , find c. Round your answer to two decimal places. (8.5) Solve ABC with A 110 , a 5 , and b 7.3 . (8.5) A 50 foot ramp makes an angle of 4.9 with the horizontal. To meet new accessibility guidelines, a new ramp must be built so it makes an angle of 2.7 with the horizontal. What will be the length of the new ramp? Revised January 2015 Page 21 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.6) Which equation would you use to find mS ? R 2 2 2 7 4 9 2 4 9 cos S 4 7 42 72 92 2 7 9 cos S S T 9 (8.6) Which expression can be used to find mR ? 48 cos 1 29 R 81 cos 1 4 6 4 S 29 cos 48 9 T 29 cos 1 48 (8.7) Give an expression for the height h of DEF , and use the expression to write a formula for 1 the height of the triangle in terms of the variables shown by replacing h in the formula A bh . 2 h f sin D, A h 1 ef sin D 2 1 1 1 f sin D, A e f sin D 2 2 2 h d sin D, A 1 ed sin D 2 h e sin D, A 1 2 e sin D 2 Revised January 2015 Page 22 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.7) RST is an isosceles right triangle. Part A: Determine the exact value of t. Use radical notation if necessary, and do not approximate. Show your work. Part B: Use RST to determine the exact value of sin 45 . Use radical notation if necessary, and do not approximate. Show your work. Refer to DEF to answer Parts C and D. Part C: Use your answer to Part B to determine the exact value for the area of DEF . Part D: Using a calculator, determine the area of DEF to the nearest tenth of a cm2. (8.7) Which expression represents the area of the triangle in square feet? 48 6 1319 72 30 37 5 42 ft 35 ft 67 ft 144 42 35 67 144 102 109 77 Revised January 2015 Page 23 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Which is the equation of the graph shown below? f x 2sin x f x 2sin 2 x 1 f x sin x 2 1 f x sin x 2 2 (8.9) The graph of which function has a period of and an amplitude of ? 1 y sin 2 x y sin 2 x 1 1 y sin x 2 1 y sin x 2 (8.9) Which function has an amplitude of 2 and a period of ? f x 2cos 2 x f x 2cos x 1 cos 2 x 2 1 f x cos x 2 f x (8.9) Which function has an amplitude of and a period of f x f x 2 2 ? cos 2 x 2 cos f x cos 1 x 2 x f x cos 2 x Revised January 2015 Page 24 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Which of the following is a vertical asymptote of the graph of f x tan x ? 4 x x x 4 4 2 3 x 4 x (8.9) What is the equation for the graph shown? y 4sin x 8 3 y 6sin 3 y 4 cos y 6 cos y 2sin 3 3 3 ,12 2 9 ,4 2 x4 x 8 x4 x 12 3 (8.9) Which function has an amplitude of 3 and a period of 1 ? 2 y 3sin 4 x y 3cos 4 x y 3sin x 2 1 y cos 3 x 2 Revised January 2015 Page 25 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Which function is represented by the graph shown? y 3 tan 2 x 1 y 3 tan x 2 y 3 tan x y 3 tan 2 x (8.9) Write an equation of the form y a sin bx , where a 0 and b 0 , with amplitude 2 and 3 period 12. (8.9) Write a function for the sinusoid. ,8 2 , 2 2 (8.10) A sound wave models a sinusoidal function. 3 Part A: If the wave reaches its maximum at ,12 and its minimum at , 0 , what are the 2 2 shift, amplitude, and period of the function? Part B: Write the function that models this sound wave. Part C: Graph the function. Revised January 2015 Page 26 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Is the function y 2sin 2 x 3 in the form y a sin b x h k ? Why or why not? 2 How does the amplitude and period of the function compare to the amplitude and period of y sin x ? How does the graph of the function compare to the graph of y 2sin 2 x ? (8.9) Graph y 2sin x . Revised January 2015 Page 27 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Using f x cos x as a guide, graph g x cos x . Identify the x-intercepts and 2 phase shift. x-intercepts: x n where n is an integer; phase shift: units to the right 2 2 x-intercepts: x n where n is an integer; phase shift: units to the left 2 2 x-intercepts: x n where n is an integer; phase shift: units up 2 2 x-intercepts: x n where n is an integer; phase shift: units down 2 2 Revised January 2015 Page 28 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Find the amplitude and period of the graph of y 3cos x . (8.9) Find the amplitude and period of the graph of y 2 cos 6 x . (8.9) Graph one cycle of the graph of the function f x 4 cos x (8.9) Graph one cycle of the graph of the function f x 6sin x 2 (8.9) Graph y tan 3 . x . Include vertical asymptotes in your sketch. 2 (8.9) The graph of a sine function has amplitude 5, period 72 , and a vertical translation 4 units down. Write an equation for the function. (8.9) The graph of a cosine function has amplitude 4, period 90 , and a vertical translation 3 units down. Write an equation for the function. Then sketch the graph without using graphing technology. Revised January 2015 Page 29 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.9) Sketch the graphs of y cos x, y cos 3 x, and y 3cos x . Tell how the graphs are alike and how they are different. (8.9) Consider the related equations y sin x, y 2sin x, and y sin 2 x . Explain the effect that the coefficient 2 has on the graphs of y 2sin x and y sin 2 x when compared to the graph of y sin x . (8.11) HONORS The unit circle centered at the origin has a radius of 1, and the coordinates x, y locate any point on the circle. Part A: Prove that cos2 sin 2 1 for representing the central angle of the arc intercepted by the point x, y and the x-axis. Part B: Does this formula work for all values for ? Explain. 3 Part C: Without using a calculator, evaluate cos if tan . 4 5 (8.11) HONORS For an angle , sin and . 13 2 Part A: Use the Pythagorean identity to find cos . 5 3 2 , does cos change? Explain. Part B: If sin and 13 2 (8.10) Tides can be modeled by periodic functions. Suppose high tide at the city dock occurs at 2:22 AM at a depth of 35 meters and low tide occurs at 9:16 AM at a depth of 9 meters. Write an equation that models the depth of the water as a function of time after midnight. When will the next high and low tides occur? Revised January 2015 Page 30 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.10) A Ferris wheel with a radius of 25 feet is rotating at a rate of 3 revolutions per minute. When t 0 , a chair starts at the lowest point on the wheel, which is 5 feet above ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). (8.10) Storm surge from a hurricane causes a large sinusoidal wave pattern to develop near the shore. The highest wave reached the top of a wall 20 feet above sea level. The low point immediately behind this wave was 6 feet below sea level, and was 20 feet behind the peak. What is the amplitude of the sinusoid? What is the vertical shift of the sinusoid from a wave at ground level? (8.10) The graph below shows how the reproductive rate of rodents varies depending on the season. On the x-axis, the months are grouped by season and on the y-axis, the reproductive rate is represented on a scale from poor to good. a) When is the reproduction of the rodents at the lowest? When is it at the highest? b) Put 0.2 and 2 as the minimum and maximum values on the y-axis. Design a formula that describes the graph. Explain how you determined your formula. c) Suppose the reproductive rate were put on a scale from 0, for extremely poor, to 10 for extremely good. In this case, use 0 and 10 as the minimum and maximum values on the y-axis. Design a formula that describes the corresponding graph. What changes did you have to make to your formula form part (b)? Revised January 2015 Page 31 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.10) An oscilloscope is a machine that measures the magnitude of fluctuating voltages by displaying a graph of the voltage over time. The figure below shows the shape of the fluctuating voltage. The horizontal axis displays time, t, and the vertical axis shows voltage. The person who is working with scope can see from the buttons that in this case, one step on the horizontal-axis scale is 0.5 sec and one step on the vertical-axis scale is 0.2 V. Design the formula for this fluctuating voltage. Explain how you determined your formula. (8.10) One modern application of the addition and subtraction of functions appears in the field of audio engineering. To help understand the idea behind this use, consider the following simplified example. Suppose that the function f(x) = sin(x) represents the sound of music to which you are listening. Unfortunately, there is background noise. Let g(x) = 0.75sin(1.5x) represent that noise. a) Sketch a graph of f and the sum f + g, on a single set of axes. Your graph represents both what you want to hear and what you actually do hear. They clearly are not the same. b) Now apply some mathematics to engineer away the noise. You can add a microphone to your headphones. In turn, the microphone picks up the background noise, g ( x ) , then plays back through our headphones an altered version, h( x) . Thus what you now hear in the headphones is the sum of three functions: f (what you want to hear), g (the noise you don’t want), and h (the correction for the noise). Suppose the compensating function is h( x) 0.75sin(1.5 x ) . Graph the new sum, f g h , to see what you hear now. Explain the idea behind the adjustment. c) Another engineer suggests adding h( x) 0.75sin(1.5 x) instead of the h defined in part (b). Discuss the merits of that solution. Revised January 2015 Page 32 ALGEBRA II Honors/Algebra II SEMESTER EXAM PRACTICE MATERIALS SEMESTER 2 2014–2015 ___________________________________________________________________________ (8.10) The price of oranges fluctuates, depending on the season. The average price during a complete year is $2.25 per kilogram (about 2.2 pounds). The lowest price will be paid in midFebruary ($1.60 per kilo); the highest price is paid in mid-August. Assume the price fluctuations are sinusoidal in nature. a) Design a formula using the cosine to describe the price of oranges by months during a year. Let t 0 represent January 1, t 1, February, and so forth. Explain how you determined your answer. b) How would your formula change if you used the sine function instead of cosine? Explain how you arrived at your answer. c) Sketch the graph of your model from part (a). Then read from your graph the times of the year when the price of the oranges will be below $2.45. (8.10) A satellite is circling west-to-east around the earth. Below are the projections of three orbits of the satellite, labeled as curves 1, 2, and 3. The projections are given in terms of longitude and latitude readings (both are in degrees). Curve 1, can be expressed as a sinusoidal function of the form y A sin( B( x C )) D . a) Determine values for A, B, C, and D to produce a sinusoidal model that you think best describes Curve 1. b) What constants in your sinusoidal model for (a) will you need to modify in order to describe Curves 2 and 3? What are your models describing these curves? (8.14) HONORS Suppose cos sin 1 . Part A: Solve the equation for in the interval 0, 2 . Part B: Why might you have to check solutions in Part A for extraneous roots? Part C: Suppose is between 5.25 and 6 and cos sin 1 . Find cos and explain your reasoning. Revised January 2015 Page 33