Algebra 1 Semester 2 Final Eam Part 2 Don t forget to study the first portion of the review and your recent warm-ups. 1. Michael s teacher gave him an assignment: Use an initial term of 5 and a generator of + 7 to create an arithmetic sequence, she told the class. Michael does not think this is a good assignment because there are too possible sequences that work! he complains to his mom after school. Do you agree with Michael? Eplain completely. 2. A sequence can be represented by the formula t(n) 5n 2. What would be a recursive equation for this same sequence?. A sequence can be represented by the formula What would be an eplicit equation for this same sequence? 4. What is the difference between an eplicit equation to represent a sequence and a recursive equation? Be clear and complete. 5. Write both an eplicit equation and a recursive equation for the sequence 7, 10, 1, 16, 6. For each sequence defined recursively below, write the first five terms, and then write an eplicit formula for the sequence. a. b. c. 7. Antonio is president of Maths-R-Us and has asked his financial advisor for a report about the company s epected profits. The advisor told him that the profits were epected to follow the equation y 150210(0.82) over the net 12 months. Tell Antonio everything you can tell about the company from the equation. 8. For each sequence defined recursively below, write the first five terms, and then write an eplicit formula for the sequence. a. b. a 1 9 and a n 1 a n 2.5 c. a 1 16 and a n 1 a n
9. Each table below represents an eponential function of the form y ab. Copy and complete each table on your paper and find the corresponding rule. a. b. y 0 2 1 10 2 4 y 0 80 1 40 2 4 10. Cleo noticed that two of the equations below correctly fit the table. 4 2 1 0 1 2 y 11 9 7 5 1 1 1) y 1 2) y (2 ) ) y 2 4) y 7 a. Which two fit? Eplain how you made your decision. b. Eplain why it makes sense that a table can have two rules? 11. Consider the sequence 7,, 1, 5, 9, a. What kind of sequence is it? Eplain how you know. b. Make a table and graph the first si terms of the sequence. c. Write a rule for each term in the sequence, where n is the term number. 12. An arithmetic sequence has a first term of 0 and a 9 th term of 12. If 27 is an output of the sequence, which term number is it? Show and eplain how you know you have the correct term number. 1. Choose the correct eplicit equation for an arithmetic sequence in which t(4) = 8 and t(10) = 2. Eplain your reasoning or show the work that helped you decide on your answer. A. t(n) 8 2n B. t(n) 2 8n C. t(n) 4n 8 D. t(n) 4n 8
14. Consider the sequence, 6, 12 a. Write the net four terms of the sequence. b. What is the generator? c. What kind of sequence is it? How can you tell? d. Write a rule for this sequence. 15. Rewrite each of the epressions below with no parentheses and no fractions. Negative eponents are acceptable in your answer. a. (2 5 y ) (4y 4 ) 2 8 7 y 12 b. m 2 n 2m (mn) 2 16. Find the missing dimensions (length and width) or area of each part and write the area of the rectangle as a product and a sum. a. b. 1-20 +1 5y 5y 4 17. Solve the following equations for the indicated variable. Show all of your work. a. Solve for : 2( 1) 12 b. Solve for y: 8 2y 4 c. Solve for m: 4 p 4 2(m p) d. Solve for : y 4 18. Tinesse is a new student in your study group. She does not understand how to fill in this generic rectangle. Eplain what to do in several complete sentences. 1 24 12y 0.5y 19. Write an algebraic equation for each figure below to epress the relationship, Area as a product equals area as a sum. a. b. +1 2 2 6
20. The multiplication table below has factors along the top row and left column. Their product is where the row and column intersect. With your team, complete the table with all of the factors and products. Multiply ( ) ( 5) 2 2 9 5 2 2 8 6 2 2 6 8 21. On a recent online math quiz, Leonhard faced the question: True or false: (a + b) 2 = a 2 + b 2. Leonhard quickly typed in false, and the screen promptly showed Congratulations! You are correct! So if it doesn t equal a 2 + b 2, what does it equal? Now Leonhard was stumped. Help him out: what does it equal, and how do you know? Be clear so Leonhard can understand this question. 22. Solve for. ( 6)( 4) 20 ( 2) 4 24. If and y are positive then simplify: 4 6 4 y 25. Solve for : 8 8 4 26. Simplify the following radical epression: 75 27. Simplify: 21 + 7 28. Simplify: 7 + (-2 ) + 6 + 4 7 29. Multiply and simplify: ( 2 )(2 5 ) 0. Find using the Pythagorean Theorem. 1 12 1. Factor completely: a) 28m 5 5m 2 b) d 2 + 4d 5 c) 2 2-7 15 d) 6v 2 8v 8 e) 9b 2 64 f) 2w 2 + 5w g) 4m 2 + 25 h) 2 15
2. Find all solutions a) ( 5) = 0 b) 2 18 = 0 c) 2 2 2 = 0. Graph the equation: y = 2 + 2 + 4. Solve for : (c 2) 2 = 5 5. How many real roots does z 2 2z + 6 = 0 have? 6. Solve using quadratic formula: 2 4 1 = 0 7. If a quadratic equation has eactly one real root, then which of the following could be the graph of the related function? A. B. C. D. 8. Given the graph of the parabola below, find the equation of the ais of symmetry. 9. Identify the direction that the following parabola opens: y = 2 + 2 15 A] up B] down C] left D] right 40. Solve for using the quadratic formula: 2 + = 0
41. Match the graph with its function A) y = ² + 2 B) y = ( + )² C) y = ² 6 D) y = ( 5)² 42. The graph of y = ² + + 6 is shown. What are the solutions to y = -² + + 6? 4. Match the graph with its function A) y = ² + 4 B) y = ² + 4 + 5 C) y = ² 6 + 5 D) y = ² + 5 44. What are the roots of this function? f() = ² + 6 16 45. Solve the quadratic equation 2² + 5 = 12 46. What are the -intercepts of y = 2² + 15? 47. If the quadratic functions are graphed, which is the widest? A) y = 2² B) y = ² c) y = ² D) y = 5² 48. Solve : (Remember to CHECK for etraneous roots!!) 2 a) 2 5 b) 2 1 c) 7 2 5 d) 2 4 1
49. Simplify: a) 162 b) 24 c). 12 d). 8 6 2 f). 2 12 g). 2 15 e) 5 h) 22. 2 4 2 2 2 i) 12 4 j) 2 18 j) 28 7 k) 9 7 9 7 l) 7 8 12 50. The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers. Hint: There will be two sets of solutions. 52. The hypotenuse of a right triangle is 6 more than the shorter leg. The longer leg is three more than the shorter leg. Find the length of the shorter leg. 5. Using the quadratic equation 2 4 5 = 0, perform the following tasks: a) Solve by factoring. b) Solve by using the quadratic formula. 54. Students were told to graph the following: and a) Jolene claims that the graphs are the same. Alfred asserts that they are NOT. Both are somewhat correct. Clearly eplain why using correct vocabulary. b) Now graph them!! 55. A ball is kicked from ground level into the air. Its height h, in feet, after seconds can be represented by the equation. What is the total elapsed time, in seconds, from the time the ball is kicked until it reaches ground level again? 56. Suppose you throw a baseball straight up at a velocity of 64 feet per second. A function can be created by epressing distance above the ground, s, as a function of time, t. This function is s = 16t 2 + v 0 t + s 0 16 represents 1/2g, the gravitational pull due to gravity (measured in feet per second 2 ). v 0 is the initial velocity (how hard do you throw the object, measured in feet per second). s 0 is the initial distance above ground (in feet). If you are standing on the ground, then s 0 = 0. a) What is the function that describes this problem? b) The ball will be how high above the ground after 1 second? c) How long will it take to hit the ground? d) What is the maimum height of the ball? What time will the maimum height be attained?