Math Algebra 2 Midterm Exam with Answers

Algebra 2 Honors Name________________________ Hour___________ Midterm Review Quadratics **This ENTIRE Chapter Should be done with a 4-function calculator only.** Factor the following completely. 1. +2 15 4. 3 7. 4 2. 81 +8 5 5. 5 10. 2( + ) + 5( + ) 8. 3 +4 8 1 +1 16 11. 2 5 3. 3 6. 16 9. 2 8 +5 + 40 + +3 Solve by Factoring Only. Find All solutions, both real and imaginary. 12. ( + 2)( 14. 5 4) = 0 +4=0 13. ( 1)( 15. (3 4)( + 3 + 2) = 0 2) = 1 + 25 21 16. 27 18. ( 17. (2 +5=4 + 1) 7( + 1) = 10 19. 5 + 13√ Solve the following Inequalities. 20. +9≤6 22. 21. 9 <9 23. Simplify the following expressions. 24. √ 27 √ 12 + 2√75 25. 27. 5) = 12 28. 4 (3 ) 3 +7 2 6=0 >0 + 10 > 0 26. (3 + 2 )(2 3) 29. √ 27 √ 12 2√75 30. 31. 32. Solve by Completing the Square. 33. 12 = 4 34. 35. 3 36. 2 + 12 + 10 = 0 4 +2=0 3 8=0 Write in Vertex Form by _______________________________. Then, determine if the function has a maximum or a minimum and give the value. 37. ( ) = +8 +6 38. ( ) = 3 6 2 Identify all key pieces of information and graph the following quadratics. You should know how to find each piece of information if the function is given in STANDARD Form and if given in VERTEX Form. 39. ( ) = +4 +1 40. ( ) = 6 6 AoS:___________ AoS:___________ Vertex:_________ Vertex:_________ Max/Min Value:__ Max/Min Value:__ x-int:___________ x-int:___________ y-int:___________ y-int:___________ Domain:________ Domain:________ Range:_________ Range:_________ 41. ( )=4 +4 5 2= 42. 2( + 1) AoS:___________ AoS:___________ Vertex:_________ Vertex:_________ Max/Min Value:__ Max/Min Value:__ x-int:___________ x-int:___________ y-int:___________ y-int:___________ Domain:________ Domain:________ Range:_________ Range:_________ Graph the equation = . Identify the transformations of the additional equations and graph them as well. This is to be done without a calculator. List the Domain and Range of each. 43. = 2 +3 44. =( 3) Trans:_____________ Trans:_____________ ___________________ ___________________ Domain:_____________ Domain:_____________ Range:_____________ Range:_____________ 4 45. Find the domain, the range, and the zeros of the function (without a calculator): ( ) = 2( D = ____________ Solve using the Quadratic Formula. 46. 2 ( + 2) = 5 R = ____________ 47. 1)( + 8) Zeros = _________ 5 2=0 Give the value of the discriminant and determine the nature of the roots of each equation. If the equation is factorable, say so. 48. 4 + 3 + 12 = 3 3 49. 7 2 =5 + ( + 2) + 2 = 0 has a double root. 50. Find the value(s) for k for which: 51. If + = 0, > 0, > 0, which statement is false? A. Zero cannot be a root. B. The roots must be imaginary. C. The sum of the roots is negative. D. The product of the roots is negative. E. There are no values of b and c for which the roots are equal. 52. Find a quadratic function ( ) = and x-intercepts of -3 and 1. + + for the parabola with the minimum value -8, 53. A rectangular field with an area of 5000 m2 is enclosed by 300 m of fencing. Find the dimensions of the field. 54. There are currently 20 members in a school’s Ski Club and the dues are $8 per member. To encourage the recruiting of new members, the club treasurer suggests that for each new member recruited, the dues for all members be reduced by 10 cents. How many new members are needed for the club’s total membership money to be a maximum? 55. A ball is thrown vertically upward with an initial speed of 80 ft/s. Its height after t seconds is given by = 80 16 . a. How high does the ball go? b. When does the ball hit the ground? 56. One diagonal of a rhombus is 6cm longer than the other. If the area of the rhombus is 56cm2, find the length of the longer diagonal. (HINT: the area of a rhombus equals half the product of the lengths of the diagonals.) 57. A rectangle is 15 cm wide and 18 cm long. If both dimensions are decreased by the same amount, the area of the new rectangle is 116cm2 less than the area of the original. Find the dimensions of the new rectangle. 58. A signal flare is fired upward from ground level with an initial speed of 294 m/s. A balloonist cruising at a height of 2,450 m sees it pass on the way up. How long will it be before the flare passes the balloonist again on the way down? ( = 4.9 ) 59. A rectangular pasture is to be enclosed with 100 m of fencing. One side of the pasture lies along a river and need not be fenced. What is the maximum area that may be enclosed? Algebra 2 Honors Midterm Review Polynomials Name________________________ Hour___________ **This ENTIRE Chapter Should be done with a 4-function calculator only.** Simplify the expression. ) 1. ( 2. 4. 5. ( ( 7. 10. ( + ) 8. ) ) ( 13. 3( ) ( ) ( ) 3. ) ( + ) (2 ) 9. 11. 12. 14. 15. (3 ( + + ) ÷ (2 + 1) 17. (4 2 + 6) ÷ (2 ( 2 + 4)( ) 1) ) Perform the operation using Long Division. Your final answer should be in the notation of: 16. (6 ) ( + ) 6. ( ) (3 3) ( )+ ( ) ( ) Perform the operation using Synthetic Division. Your final answer should be in the ( ) notation of: ( ) + ( ) 18. ( +3 5) ÷ ( 19. (2 1) 20. Find k such that when +3 4 +4 27 is divided by Given one factor of the polynomial, find the others. 21. +3 6 8; 2 22. 4 List the possible rational roots for the polynomial. 23. ( ) = 2 5 +8 +3 5 24. 3 + 6) ÷ (2 + 5) 3, the remainder is zero. 12 ( )= + 3; 7 +3 3 + 20 Use Descartes’ Rule of Signs. State the possible number of positive real zeros, negative real zeros, and imaginary zeros of each function. 25. ( ) = 4 2 + +3 26. ( ) = 2 6 + 6 + 24 40 Find ALL roots of the polynomial. You may not use a calculator. 27. ( ) = 3 4 + 12 28. ( ) = 3 29. ( )=3 +5 7 3 +2 Write an equation with the given roots 31. 3 , 2 30. 32. 4 ( )= 9 6 10 +6 8 + 24 40 , For Questions 33-34, graph the following polynomials using their zeros, y-intercept, and what you know about multiplicity and end behavior. ( )= ( 33. 3) ( + 4) 34. ( ) = ( + 1) ( 2) Use your graphing calculator to find the Zero(s), Local Max(s), and Local Min(s) of the following polynomial function. Be sure to include the domain and range and end behavior sentences. Round answers to the nearest tenth. 35. ( )= 2 +8 36. ( )= 2 +8 16 Zero(s):___________________ Zero(s):___________________ Local Max(s):______________ Local Max(s):______________ Local Min(s):_______________ Local Min(s):_______________ Domain:__________________ Domain:__________________ Range:___________________ Range:___________________ Algebra 2 Honors Midterm Review Radicals Name________________________ Hour___________ **This ENTIRE Chapter Should be done with a 4-function calculator only.** Express the following in simplest radical form. No decimal answers. 1. 5 2. 4. 4 5. 5 7. 22 8. √ 10. 13. √ √ √ 3. √36 6. 8 9. 11. 12. 14. 15. √16 √ √ 16. √ 125 17. 25. √6 + √ √ 18. √6 27 20. √8 ÷ √4 19. √8 √16 22. 3 + 2√6 4 3 5√6 23. 5√2 21. √10 3√2 26. √10 √2 √5 2√10 24. 27. √8 √13 √ √ √3 √13 + √3 √ √ + √ Solve. 28. +8+5=2 29. 3 = 7√ 30. √3 31. +1 = 4 32. 2( + 3) = 54 5 33. ( 1 5) 2 7 +9=2 9 = 72 For 34-36, Graph the Parent Function = √ . Then, graph each of the following functions using what you know about transformations. Identify the Domain and Range of the graph. 34. = 3√ 35. = √ +2 36. = 2√ 4 Domain:_______________ Domain:_______________ Domain:_______________ Range:________________ Range:________________ Range:________________ For 37-39, Graph the Parent Function = √ . Then, graph each of the following functions using what you know about transformations. Identify the Domain and Range of the graph. 37. = 2√ 38. = √ +3 1 39. = √ +4 Domain:_______________ Domain:_______________ Domain:_______________ Range:________________ Range:________________ Range:________________ Find the Domain of the Following Function. Utilize the Domain to create a basic Table of Values and Graph. Identify the Range. 40. = √4 x y x y Domain:______________ Range:______________ 41. =√ 4 Domain:______________ Range:______________ Algebra 2 Honors Midterm Review Logarithms & Exponentials Name________________________ Hour___________ **#1-31, and 46-49 should be done with a 4-function calculator only. #32, 34-37 should be able to be completed with a 4-function calculator AND a scientific calculator. #33, 38-45 should be done with a scientific calculator. Let ( ) = necessary. 1. ( 2) 4. + and ( ) = (2) 2. ( + 2) Find the equation for ( )= 7. . Perform the indicated operation and state the domain when 5. ( 8. ( ) = log 1) 3. ( ) 6. ( + 2) 9. ( )= ( ). 2 Without Graphing, determine whether or not the following functions have inverse functions. ( )= ( )=2 ( ) = 5 + 17 10. + 17 11. 18 + 12. Write in Log form. 13. 2 = 32 14. 10 Solve. 17. log 18. log =2 = 0.1 = Write in Exponential Form. 15. ln 3 = 16. log 19. log = 2.5 20. log = = 8 Condense the following logarithmic expressions into a single logarithm. 21. 2 log + log 22. log 6 + log 5 23. 4 2 log 24. 2 log log Solve the following logarithmic equations. 25. log = 2 log 3 + log 5 26. log ( + 1) + log 27. ln + ln( + 3) = ln 10 28. log log ( log 2 3 log 2 =1 1) = Solve the following equations. #29-31 must be exact. Round #32-34 to the nearest thousandth. 29. 25 = 125 30. 4 = 31. √3 32. 2 33. 7 34. 12 + 3 =5 =5 =9 = 20 Solve the following exponential equations. Round your answers to the nearest thousandth. 35. = 12 36. 4 =7 37. +5 6=0 For the following story problems, round all necessary answers to the nearest thousandth. It is strongly advised that you know how to use ALL formulas from this section. 38. Of an original 100 g of a radioactive substance, 70 g remained after 12 days. How much will be left after 20 days? 39. The population of Ubetchakan grew from 65.6 million in 1975 to 80 million in 1985. What is the rate of annual growth? 40. The value of a new $3500 sailboat decreases 10% per year. Find its value after 20 years. 41. You are saving for your senior year spring break trip. You want to have $3000 in your savings account after 3 years. Find the amount you should deposit for each of the situations described. a. The account pays 2.25% annual interest compounded quarterly. b. The account pays 3.5% annual interest compounded monthly. c. The account pays 4% annual interest compounded continuously. 42. A man invests $10,000 in an account that pays 6.5% interest per year, compounded quarterly beginning on the first of February of 2006. a. Find the amount after 6 years. b. During which month of what year will the amount triple? 43. During which month of what year will it take for an investment of $2,000 deposited on February 1st, 1985 to double in value if the interest rate is 10.5% per year, compounded continuously? 44. A culture contains 1,600 bacteria initially and doubles every 30 minutes. a. Find a function that models the number of bacteria at time . b. Find the number of bacteria after three hours. c. After how many minutes will there be 6,000 bacteria? 45. Radium-123 has a half-life of 1,400 years. Suppose a sample of this substance has a mass of 32 mg. a. Find a function that models the amount of the sample remaining at time . b. Find the mass remaining after 6,000 years. c. How long will it take for the sample to decay to a mass of 12 mg? Without using a calculator, graph the exponential equation ( ) = . Then, graph each of the transformed functions. Be sure to list all transformations in the order in which they must be graphed. Then, find the Domain and Range. ( )= 2 2 Domain: 46. Range: 47. ( )=2 3 Domain: Range: Without using a calculator, graph the logarithmic function ( ) = log . Then, graph each of the transformed functions. Be sure to list all transformations in the order in which they must be graphed. Then, find the Domain and Range. ( ) = log ( + 4) Domain: 48. Range: 49. ( ) = 2 log 4 Domain: Range: Midterm Review Rational Algebra 2 Honors Name________________________ Hour___________ **This ENTIRE Chapter Should be done WITHOUT a calculator** Simplify each expression. 1. 2. 3. 4. 5. 6. Simplify each expression. Determine the domain of the function and find the zeros. 7. ( )= 9. ( )=( 5 + 6 )(9 ) 8. ( )= 10. ( )= ( ) Simplify each expression. 11. 14. 12. ÷ ÷ 13. 15. 4 ÷ ÷ 16. 17. 3 18. 19. 20. 21. 22. Solve. Be sure to check for extraneous solutions. 23. 26. = 24. + = + = 25. 27. =1 + = Quadratics Answers 1. ( + 5)( 3) 4. 3)( (2 7. 9. ( 3 2. (5 1)( + 1) + 3 + 9) )(2 + 5 ) (2 + 7 )( 8. 3 ) ( + 2 )( 10. (2 + 2 12. = 2( 16. = , 20. =3 21. 3, 24. 10√3 + √3 25. 28. 32. – 36. = ± ), 2 √ ± √ 38. ( ) = 3( 1) 13. = 17. = 2 )( 1( 14. = ±1, ±2 15. = 1, 18. = ±1, ±2 19. = 23. ( 5, 2) ∪ (0, ∞) 3+5 26. 12 27. 6 + 3 29. 180√3 30. 1 33. = 6 ± 2√10 34. 37. ( ) = ( + 4) = 5 5 31. = 2 ± √2 10; = 10 @ = 35. = 2± √ 4 =1 40. AoS: = 3 Vertex: ( 3,15) Max @ 15 x-int: 41. AoS: = 42. AoS: = , 6) Min @ -6 x-int: y-int: (0, 1) Domain: ( ∞, ∞) Range: [ 3, ∞) 2 ± √3, 0 3 ± √15, 0 ±√ ,0 y-int: (0,6) Domain: ( ∞, ∞) Range: ( ∞, 15] y-int: (0, 5) Domain: ( ∞, ∞) Range: [ 6, ∞) 1 Vertex: ( 1,2) Max @ 2 x-int: (0,0) & ( 2,0) y-int: (0,0) Domain: ( ∞, ∞) Range: ( ∞, 2] 43. , . 3, 4 : ( ∞, ∞) 48. : 52. ( )=2 57. 11 (4 + 5 ) 2 + 1) 6. 11. ± √ 2 Vertex: ( 2, 3) Min @ -3 x-int: 55a. (2 + 1)(4 1) +4 ) ), 2, 1 = 45. 5. 5)( 1)( + + 3) 39. AoS: 44. (3 22. ( ∞, 0) ∪ (0,9) 5; Vertex: ( 3. 108; 2 +4 : 100 14 2, : ( ∞, ∞) : [ 4, ∞) : : , ∞) 49. 6 : ( ∞, ∞) 3 : 144; 2 = 1, : ( ∞, 3] 46. 8 = , 53. 50 100 54. 30 55b. : 5 56. 58. 40 ±√ 59. : 8 47. = 50. =2 : 14 : 1250 ±√ 51. Polynomials Answers 1. 2. 5. 6. 9. 3 5 +6 4 13. 17. 2 21. 3. + = ±2, 3 31. ( )= 15. 16. 3 33. : : 1; +2 +2+ = 3, ± : 2 +9 = : → 4 ( 1) : = : 36. = → 26. : 3 29. = 1, 32. ( )=5 34. : = 0, 24. ±{1, 2, 4, 5, 10, 20} 1; : 1; : 2 , 1 ± √2 42 0 = ±2, 3 ± 30. + 101 34 = 0 ( 1), 1 ( 4), 2 ( 3) → : (0, 0) ∞ ( )→∞ →∞ ( )→∞ ∞ : → +1+ 20. ) : ∞ ( )→∞ 2, 2; : : (0, 36) : (1.1, 5.0); 4, 0, 2; : √ ( , ∞ ( )→∞ →∞ ( )→ 35. ± 18 = 3 ( 2), + 19. 23. ± 1, 5, 0 = 4, 28. 2 4 14. 22. 27. 8. 12. = 2, 4, 1 0; + 7. 11. 18. : 2 4. 10. +3+ 25. 72 : ( 2.4, 16.9); →∞ ( )→ ∞ ( )→∞ : ( ∞, ∞) ∞ : ( 0.9, 21.1); ; : ( ∞, ∞) : ( ∞, ∞) : [ 21.1, ∞) →∞ ( )→∞ Radical Answers 1. 5√ 2. 5. √25 6. 9. √4 10. 13. √ 14. 17. 3| | 18. 21. 10 22. 25. 29. √ = ,4 √6 4. 2√2 4√2 7. 2√2 8. √ √ 11. 2 12. 3√ 15. 4√3 + 2√7 16. 19. √2 20. √2 3. √ √ 48 7√6 26. 2| |√5 30. = 21 5| |√2 23. 50 27. 31. √ 26√5 5 24. 17 + 7√6 28. = 32. =6 33. = 32, 22 36. : [4, ∞) 38. : ( ∞, ∞) 40. : [ 2,2] 35. : [0, ∞) : ( ∞, 2] : [0, ∞) 37. : ( ∞, ∞) : ( ∞, ∞) : ( ∞, ∞) 39. : ( ∞, ∞) : ( ∞, ∞) 41. : ( ∞, 2] ∪ [2, ∞) 34. : [0, ∞) : [0, ∞) : [0, 2] : [0, ∞) Logarithms and Exponentials Answers 1. ( 2) = 4. ( + 2) = 7. ( )= 12 +4 5. ( =3 1) = 28. =2 31. = 32. = 35. = 2023 42b. +1 ( ) ≈ 1991 = 57.207 45a. = 32 46. . , . 2 : ( ∞, ∞) 48. . , 44a. : ( ∞, ∞) 22. log 15 =2 29. = 30. =1 33. ≈ 37.266 1.322 0.162 36. ≈ 2.004% = ≈ 0.280 40. $425.52 42a. $14,723.58 = 1600(2) 44b. 102,400 45c. 3, 47. 49. . = 1981.052 3 : ( ∞, ∞) 2, 4 5. 7. 2. ( 3. ( 6. ) : = , 3 : ( ∞, 3) ∪ ( 3,0) ∪ (0,1) ∪ (1, ∞) ) : ( 3, ∞) : (0, ∞) Rational Answers 1. = 27 18. 45b. 1.641 : ( ∞, 0) 1 26. 41c. $2660.76 : ( 4, ∞) 14. log 0.1 = = 45 41b. $2701.39 . 5 25. 39. 44c. 4 ≈ + 18 10. No = 36 38. 55.186 43. 9 21. log √ =2 3 +7 +9 ( ) = ln 9. =4 27. 41a. $2804.71 ( + 2) = 17. 24. log =0 6. 6 16. 10 = 23. log 37. ( ) = 13. log 32 = 5 20. ≈ 0.379 3 3. 12. No 19. 2 √2 34. = + ( )=3 8. 11. Yes 15. (2) = 27 2. 4. : ( ∞, ∞) 8. : = 9. : = 2 : 10. = 13. 17. ( ) ( 21. 25. ) , 4 : ∪ ,∞ : ( ∞, 3) ∪ ( 3,0) ∪ (0,3) ∪ (3, ∞) 3 : ( ∞, ∞) 11. 12. 14. 15. 16. 19. 20. 18. 22. =1 ∞, 26. ( )( ( )( = ) ) 1 23. = 27. = 24. ±√ = 3, 2