1. A student has learned that test scores in math are determined by this quadratic function:
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1 SEMESTER EXAMS 1. A student has learned that test scores in math are determined by this quadratic function: s( t) ( t 6) 99 In the function, s is the score and t is the number of hours that a student spends on homework each week. a) How many hours must a student spend on homework to achieve maimum score? b) What is the maimum score? c) Based on the function, what will be the score if a student does no homework?. Show that ( i) is a root of Solve 5 0 over the set of comple numbers. 5i 5 5i 5 4. Which of the following quadratic equation has no real roots? Page 1 of 9 Revised November 01
2 SEMESTER EXAMS 5. According to the Fundamental Theorem of Algebra, how many roots does the following equation have? Function A and Function B are continuous quadratic functions. Function A Function B f ( ) 6 Which function has a greater positive -intercept? Function A Function B Page of 9 Revised November 01
3 SEMESTER EXAMS 7. What is the equation of the parabola shown? y 4 y y y Factor ( 11)( 11) ( 11)( 11) ( 11 i)( 11 i) ( 11 i)( 11 i) 9. Solve the equation by factoring. 7 or 7 or Page of 9 Revised November 01
4 SEMESTER EXAMS 10. Solve the quadratic equation by taking the square root i Solve the equation by using the quadratic formula. 5 0 =1 or = -6 = 1 or = - 1 or 1 1. Simplify(- 4i) + (5-6i). 8+10i -9-8i 8-10i 6-10i 1. Simplify(1- i) - (-+ 7i). 4-10i 4 + 4i i -+ 4i Page 4 of 9 Revised November 01
5 SEMESTER EXAMS 14. Find the verte of y = + - and state if it is a maimum or a minimum. (-1, -4); maimum (-1, -4); minimum (-4, -1); maimum (-4, -1); minimum 15. The height of Carl, the human cannonball, is given by and t is in seconds after the launch. h( t) 16t 56t 40 where h is in feet a) What was his height at the launch? b) What is his maimum height? c) How long before he lands in the safety net, 8 feet above the ground? 16. What is the solution set of y y y 14? y 7 y or y 7 7 y y Which of the following is a factor of ( a 1) b? a b 1 a b a 1 a b Consider the function f ( ) 48. a) Determine the roots of the function. Show your work. g is the point (, 0). Write the function rule for g in verte form. b) The verte of c) Eplain how f transformed to become g Page 5 of 9 Revised November 01
6 SEMESTER EXAMS 19. Several values of the quadratic function f( ) are given in the table. f ( ) The function gis ( ) given by g( ) ( ). Which function has the greater maimum for which value of? f ( ); for 0 f ( ); for 6 g( ); for g( ); for 0. Which statement best describes these two functions? f 4 g 7 The maimum of f( ) is less than the minimum of g. ( ) The minimum of f( ) is less than the maimum of g. ( ) The maimum of f( ) is greater than the minimum of g. ( ) The minimum of f( ) is greater than the maimum of g. ( ) 1. Given the general form of a quadratic equation b c 0, determine the effect of each condition on the solutions. a) b 0 b) c 0 c) c 0 d) What is needed for the solutions to have imaginary parts? Page 6 of 9 Revised November 01
7 SEMESTER EXAMS. The amount of fuel F (in billions of gallons) used by trucks from 1990 through 009 can be approimated by the function F f ( t) t where t 0 represents a) Describe the transformation of the common function f () t t. Then sketch the graph over the interval 0 t 19. f(19) f(0) b) Find and interpret c) Rewrite the function so that t 0represents 000. Eplain how you got your answer. d) Use the model from part (c) to predict the amount of fuel used by trucks in 015. Does your answer seem reasonable? Eplain.. Use the graph provided to choose the best description of what the graph represents. Height (ft) Time (s) A ball I dropped from a height of 4 feet and lands on the ground after seconds. A ball is dropped from a height of 4 feet and lands on the ground after 1.5 seconds. A ball is shot up in the air and reaches a height of 4 feet after 1 second. A ball is shot up in the air, reaches a height of 4 feet, and lands on the ground after 1.5 seconds Page 7 of 9 Revised November 01
8 SEMESTER EXAMS 4. The table lists all the real roots of a 5 th degree polynomial p ( ) and the multiplicity of each root. Multiplicity Which general factorization correctly represents p? ( ) a( )( 1) ( ) a( )( 1) ( ) a( )( 1)( 1) ( ) a( )( 1) ( ) 5. A 4 th degree polynomial with real coefficients is found to have eactly two distinct real roots. What must be true about the other two roots? One root is real and the other is imaginary. Both roots must be real. Both roots are imaginary roots that are comple conjugates. All the roots have been found Page 8 of 9 Revised November 01
9 SEMESTER EXAMS 6. Consider the graph of p ( ) below. Which general factorization correctly represents p. ( ) Which general factorization correctly represents p? ( ) a( )( )( 4) a( )( )( 4) a( )( )( 4) a( )( )( 4) 7. The graph of pis ( ) shown below. Which general factorization correctly represents p? ( ) 4( )( )( 4) 6( )( )( 4) 1 ( )( )( 4) 4 4( )( )( 4) Page 9 of 9 Revised November 01
10 SEMESTER EXAMS 8. Use the graph of pto ( ) answer questions. a) True or False: The leading term of p, ( ) when written in standard form, is positive. b) True or False: From the graph, p( ) 0. The multiplicity of the factor ( ) is even. Eplain your answer. 9. If 4 f ( ) 7 8 4, find the possible rational roots of f( ). 1, 4 1, 4 1, 1, 1, 1,, 4 0. Given polynomial q, ( ) q(4) 6. Which statement is correct? 4 is not a root 4 is a root ( 4) is a factor ( 4) is not a factor Page 10 of 9 Revised November 01
11 SEMESTER EXAMS 4 1. Consider p( ) a) Show that 5 and 5 are zeros of p. ( ) b) Completely factor pwhere ( ) all the coefficients are rational numbers. c) h ( ) is ptranslated ( ) 4 units right and units up. What is the equation of h ( )?. 5 4 p( ) a) Show that p( ) is a root. b) Factor pcompletely ( ) c) If f ( ) p( ), what are the real roots of f( )? 4. Given the polynomial p( ) 1 16 : a) Show that p( i) is a root. b) What other root must also be a root of p? ( ) Eplain. c) Factor pcompletely. ( ) 4 4. Consider p( ) a) Show that 6 are roots of p, ( ) then write pas ( ) the appropriate factorizations at this point. b) Factor p ( ) completely. c) Let q( ) p(4 ). List out the roots of q. ( ) d) Let f( ) be pvertically ( ) stretched by, translated units to the right and 4 units up. Write out the algebraic relationship between f( ) and p. ( ) 5. What is the 4 th term of the epanded binomial ( 1)? Page 11 of 9 Revised November 01
12 SEMESTER EXAMS 6. For what values of c will c 0 have eactly one distinct real root? Write a cubic function that passes through the following points: (-, 0) (, 0) (-1, 0) and (1, ). y 7 6 y y y How many possible rational zeros eist for the polynomial function 6 y 9 4 1? Suppose y and ( y) 1. What is y? Page 1 of 9 Revised November 01
13 SEMESTER EXAMS 40. Which graph represents 5 f ( ) 6 9? Page 1 of 9 Revised November 01
14 SEMESTER EXAMS 41. Divide 4 ( 7) ( 1) using long division This polynomial function has at least one rational root. 4 p( ) k 9 a) What are all the possible integer values of k? Show your work or eplain how you know. b) What are all the possible real roots of the function? Show your work or eplain how you know. 4. The volume V( ) and height ( h ) of the prism is given. Find a polynomial epression for the area of the base ( B ) in terms of. (Hint: V Bh ) h V( ) Page 14 of 9 Revised November 01
15 SEMESTER EXAMS 44. Consider the function f ( ) 9 9. a) Use the leading coefficient and degree of f( ) to describe the end behavior. b) Write the rule for the function g( ) f ( ), and describe the transformation. c) Describe the end behavior of g. ( ) How does the end behavior of grelate ( ) to the transformation of f( )? 45. Use the information in the table. Interval Value of f() (, ) Negative (,1) Positive (1, 4) Negative (4, ) Positive a) What are the three real zeros of the polynomial function f? b) What can be said about the behavior of the graph of f at 0? c) What is the least possible degree of f? Eplain. Can the degree of f ever be even? Eplain Page 15 of 9 Revised November 01
16 SEMESTER EXAMS 46. The town of Frostburg eperienced a bit of a heat wave during January of this year. The graph below shows the curve of best fit that represents the low temperature of every day in January. A newspaper journalist is writing a story on the weather and needs to report some information. He needs a bit of guidance with interpreting the graph. 1) Write a few sentences describing the key characteristics of the graphs as it relates to the contet of the problem. Be sure to include domain, range, intervals where the function increases and decreases, and y intercepts, and any other important information Page 16 of 9 Revised November 01
17 SEMESTER EXAMS 46. (cont. ) The graph below shows the curve of best fit that represents the low temperature of every day in February. ) Three different models have been proposed that could be used to determine the temperature for a particular date in February. The models are given below: Model 1: y a b c Model : y a( )( 9)( 0) Model : y a( )( 9)( 0) Which model would best describe the low temperatures for February? Eplain why you chose that model. The weather in July showed a related pattern to the weather in February. The curve of best fit for July is shown below: ) Eplain the relationship between the graph for February and the graph for July. Use that relationship to create an equation for the temperatures in July Page 17 of 9 Revised November 01
18 SEMESTER EXAMS 47. If f () = -1 and g() = -1, which epression represents f () g() for 1? Which value of makes this equation true? ( - 7) = Solve for : No real solution 50. Solve for and No real solutions Page 18 of 9 Revised November 01
19 SEMESTER EXAMS 51. Solve for No real solutions 5. Identify the and y intercepts of the function f ( ) 8. (8,0) and (0,-) (,0) and (0,) (8,0) and (0,8) (-,0) and (0,8) 5. Which is the domain of the function f ( ) 5 4? { 4} { } { 0} { } 54. Compare the graph of y 6 with the graph of its parent function f ( ). Shifts 6 units down Reflects across the -ais and shifts 6 units down Reflects across the -ais and shifts 6 units up Reflects across the y-ais and shifts 6 units up Page 19 of 9 Revised November 01
20 SEMESTER EXAMS 55. If 1 8 4, what is the value of ? 56. 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(1.04) t 95(0.96) t -.04t t A biologist studying the relationship between the brain weight and body weight in mammals uses the formula: ln( w ) ln( w ) 669 body brain Where w body =body weight in grams and w brain =brain weight in grams. What is the formula for the body weight? 669 w ( w )( e ) body brain 669 w ( w ) ( e ) body brain wbody e ( w )( e 669 brain ) w 669( w ) body brain 58. Find the value of log Page 0 of 9 Revised November 01
21 SEMESTER EXAMS 59. Given the sequence 1,, 4, 8,. Find the sum of the infinite series During a flu outbreak, a hospital recorded 1 cases the first week, 54 cases the second week, and 4 cases the third week. a) Write a geometric sequence to model the flu outbreak. b) How many cases will occur in the sith week if the hospital cannot stop the outbreak? 61. Which is the same function as ( ) ln f? g( ) ln ln ln g ( ) ln g( ) ln ln g( ) ln ln 6. Rewrite log 9 y in eponential form. 9 9y 9( ) y 9 9 y 9y Page 1 of 9 Revised November 01
22 SEMESTER EXAMS 6. Given the geometric sequence with common ratio r, write a rule for the nth term of the sequence 4, -8, 196, -17 a n a n a n a n 7( 4) n 4( 7) n 7(4) n 4(7) n Choose the function that describes the graph below: f ( ) log f ( ) log( ) f ( ) log f ( ) log( ) 65. Sarai bought $400 of Las Vegas Cellular stock in January 005. The value of the stock is epected 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? Page of 9 Revised November 01
23 SEMESTER EXAMS 66. Consider the function f ( ) log. a) Identify the transformation applied to f( ) to create g( ) log 1. b) Identify the transformation applied to f( ) to create h( ) log(10 ). c) Compare the graphs of gand ( ) f( ). What do you notice? d) Use the properties of logarithms to eplain your answer to part c. 67. What is the value of (15 4 n)? n 68. What function is represented by the following graph? f( ) f( ) f( ) f( ) ( ) Page of 9 Revised November 01
24 SEMESTER EXAMS 69. The graph of the equation y log( ) is translated right units and down.5 units to form a new graph. Which equation best represents the new graph? y log( 9).5 y log( 9).5 y log( ).5 y log( ) John graphs the equation y 5. Lana graphs the equation y 5. How does Lana s graph compare to John s graph? Lana s graph shifts units downward Lana s graph shifts units upward Lana s graph shifts units to the left Lana s graph shifts units to the left Page 4 of 9 Revised November 01
25 SEMESTER EXAMS 71. 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 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 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 c) Find the amount of money placed on the 0 th square of the chessboard and the total amount placed on squares 1 through 0 for each option. Option 1 Option d) There are 64 squares on a chessboard. Find the total amount of money placed on the chessboard for each option. Option 1 Option e) Which gives the better reward, Option 1 or Option? Eplain why Page 5 of 9 Revised November 01
26 SEMESTER EXAMS 7. The loudness of sound is measured on a logarithmic scale according to the formula 10log( I L I ), where L is the loudness of sound in decibels ( db ), I is the intensity of sound, and I0 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? 0 Sound Intensity Jet taking off 10 Jackhammer 10 Hairdryer 10 Whisper 10 Leaves rustling 10 Softest audible sound 15 1 I 7 0 I I I I I c) A decibel is 1 of a bel. Is a jet plane louder than a sound that measures 0 bels? Eplain If log 4{log [log ( )]}, then what is? Which equation has the same solution aslog 4( 7) 5? Page 6 of 9 Revised November 01
27 SEMESTER EXAMS 75. Aaron invested $4000 in an account that paid an interest rate r compounded continuously. After rt 10 years he has $ The compound interest formula is A Pe, 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 eponent. 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. 76. What is the inverse of f ( ) 9? f f f f 1 ( ) ( ) ( ) 1 ( ) If f ( ) e, then which of the following is f 1 (7)? 7 e 7 log7 ln Page 7 of 9 Revised November 01
28 SEMESTER EXAMS 78. If 1 4 f ( ) 8, what is f( )? f ( ) ( 8) 4 f ( ) f ( ) 6 4 f ( ) ( 8) 79. Which is the inverse of f ( ) ( 1) 4? a( ) b ( ) 1 a ( ) a( ) Which is the inverse of f ( ) log? f f f f 1 ( ) ( ) 0.5() ( ) 1 ( ) () Page 8 of 9 Revised November 01
29 SEMESTER EXAMS 81. Which statement must be true if f and g are inverses of one another? ( f g)( ) ( g f )( ) ( f g)( ) f ( ) g( ) ( g f )( ) g( ) f ( ) ( f g)( ) f ( g( )) ( g f )( ) g( f ( )) 1 ( f g)( ) ( g f )( ) Page 9 of 9 Revised November 01
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