Algebra 1 Unit 4 Practice

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1 Lesson The size of a tet file is kilobytes. The size of a video file is 1 kilobytes. How many times greater is the size of the video file than the size of the tet file? A. 4 B. 7 Algebra 1 Unit 4 Practice Lesson Assume that fi 0. For what value of y will y always be equal to? Eplain your answer. C. 17 D. 60. Arrange the epressions in order from least to greatest a b 8. Simplify and write the epression 7 a b without negative powers. a. 4? 4 b. c. 7 d.? 7 9. For what value of n is 4m n m 4?. The formula for density is D M, where D is V density, M is mass, and V is volume. The density of an object is 4 kilograms per cubic meter. Its mass is 7 kilograms. What is the volume of the object? 4. Simplify the epression? 9 1. A. B. 1 C. 1 D. 10. For what value of a is b? b a 1? Justify your answer.. Write an epression containing multiplication and division that simplifies to y Reason abstractly. Determine whether the statement below is always, sometimes, or never true. Eplain your reasoning. If is a positive integer, then the value of a is negative. 6. Critique the reasoning of others. Nestor says that the value of 6 8 6? is 6 1. Is he correct? If so, 6 6 eplain why. If not, identify Nestor s error and give the correct value. 014 College Board. All rights reserved. 1 SpringBoard Algebra 1, Unit 4 Practice

2 Lesson Simplify and write each epression without negative powers. 18 a. 1 b. 6 y 18 c. (a b c ) 4 (abc 4 )(ab) 1. Which epression is not equal to 4 1? b. Does Brooke s method always work? Eplain why or why not. 16. Model with mathematics. The area of a rectangle is given by the formula A lw, where l is the length and w is the width. A rectangular patio has an area of (ab) square feet and a length of ab feet. Write a simplified epression that represents the width of the patio. A. B. 1 C. ( ) D. 14. Write an epression involving at least one negative eponent and a power of a product that simplifies to mn. Lesson Kurt is cutting diagonal crossbars to stabilize a rectangular wooden frame. If the frame has dimensions of feet by feet, what is the length of one crossbar? Give the eact answer using simplified radicals. 1. When a quotient is raised to a negative power, Brooke claims that you can invert the quotient and write it with a positive eponent. For eample, when asked to simplify 4 a b, Brooke begins by writing b 4 a. a. Simplify 4 a b by using Brooke s method. Then simplify without using Brooke s method. How do your answers compare? 18. For each radical epression, write an equivalent epression with a fractional eponent. a. 7 b College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

3 1 19. a. What is the value of 7? b. Make use of structure. How can you use your answer to part a to help you find the value of n for which 7 n 9? Find the value of n and eplain your reasoning. Lesson 0-. The perimeter of a rectangle is 8 8 feet and the width is 4 feet. How many feet longer is the length of the rectangle than its width? 0. Which of the following epressions is not equivalent to ( 16y) 4? 4. Write in simplest radical form. State whether the result is rational or irrational. 4 A. 16y B. 8 4 y C y D. 16 y 4 1. a. What is 1? What is 1? Eplain your answers.. Find the value of a for which a. Eplain how you found your answer. b. Let n be a positive integer. What is the value of 1 n 1? Eplain your answer. 6. Which is the sum of 0 and 8? A. 1 B. 1. A cube-shaped bo has a volume of 1 cubic inches. Celia has. square feet of wrapping paper. Does she have enough paper to cover the entire surface of the bo? Eplain your reasoning. C. 1 D College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

4 7. Critique the reasoning of others. Identify and correct the error in each addition or subtraction problem. a a. Write in simplest form. Is the result rational or irrational? b c Ted is fencing in an area composed of a rectangle and a right triangle as shown below. b. What can you conclude from your answer to part a about whether the irrational numbers are closed under multiplication? Eplain Lorraine solved the equation? 41 6 and found that 4. Verify that Lorraine s solution is correct. He still needs to buy fencing for the side labeled. How much fencing does Ted need to buy for this side? Epress the answer in simplest radical form. Lesson 0-9. Which of the following is in simplest radical form? 7 A. B. 1 C D Critique the reasoning of others. Deanna says that 1 is in simplified form. Is she correct? If so, eplain why. If not, correct her mistake. 0. Jed has a rope that is 8 18 meters long. He cuts it into smaller pieces that are each meters long. How many smaller pieces of rope does Jed now have? College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

5 Lesson Tell whether each sequence below is arithmetic, geometric, or neither. If it is arithmetic, identify the common difference. If it is geometric, identify the common ratio. a. 4, 4,, 1 9, b. 1, 4, 9, 16,, c., 4.7, 6., 7.9, d., 6, 18, 4,. Model with mathematics. When school has been cancelled, a principal calls 4 teachers. These 4 teachers each call 4 other teachers who have not yet been called. Then those teachers each call 4 other teachers who have not yet been called, and so on. a. The principal represents Stage 1. Make a tree diagram and a table of values to represent this situation. b. Can this situation be represented by a geometric sequence? If so, identify the common ratio. If not, eplain why not. c. How many teachers will receive phone calls at Stage 4? 7. The terms in a geometric sequence alternate between positive and negative numbers. What must be true about this sequence? A. The first term is negative. B. The first term is greater than the second term. C. The common ratio is between 0 and 1. D. The common ratio is negative. Lesson 1-8. Write the first five terms of the geometric sequence represented by the recursive formula below. f (1) 1 f( n) f n ( 1) 9. Ernie scores 0 points in Level 1 of a video game. In each subsequent level, he scores twice as many points as he did in the previous level. a. Write a recursive formula that represents this situation. b. Write an eplicit formula that represents this situation. 6. Consider the sequence 1,,,. a. Find a value of for which the sequence is arithmetic. Eplain your answer. b. Find a value of for which the sequence is geometric. Eplain your answer. c. Use either the recursive formula or the eplicit formula to find the number of points that Eddie scores in Level 10. Why did you choose the formula you did? 014 College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

6 40. Write a geometric sequence in which every term is an odd integer. Write both the eplicit and the recursive formulas for your sequence. Then identify the 9th term. Lesson Rajiv bought a rare stamp for $1. A function that models the value of Rajiv s after t years is v(t) 1? (1.0) t. What is the value of Rajiv s stamp after 0 years? A. $11. B. $1.66 C. $,6.00 D. $, Attend to precision. The function f(t) 40,000? (1.) t can be used to find the value of Sally s house between 1970 and 010, where t is the number of decades since a. Identify the reasonable domain and range of the function. Eplain your answers. Use the geometric sequences below for Items 41 and 4. Sequence 1 Sequence a n? n1 a1 an a n 1 b. Sally wants to calculate the value of her house in 199. What number should Sally substitute for t in the function? Eplain. 41. Which statement is incorrect? A. The terms in Sequence increase more quickly than the terms in Sequence 1. B. Both sequences have the same second term. C. The eplicit formula for Sequence contains raised to a power. D. The common ratio for Sequence is equal to the first term of Sequence Persevere in solving problems. How many terms in Sequence 1 are less than 00? Eplain how you found your answer. c. Find the value of Sally s house in The function h(t),000? (.1) t models the value of Ms. Ruiz s house, where t represents the number of decades since 190. In what year did the value of Ms. Ruiz s house first eceed $,000? Eplain how you can use a table to find the answer College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

7 46. The function h(t) 1,000? (1.) t models the value of Sam s house, where t represents the number of decades since The value of Kendra s house has been doubling each decade since In 010, the value of Sam s house was greater than the value of Kendra s house. Is it possible that the two houses had equal values in 1980? Eplain. 0. Compare the graph of an eponential growth function to the graph of an eponential decay function. Describe the similarities and differences. Lesson Identify the constant factor for the eponential function y 1. How can you use the constant factor to tell whether the function represents eponential growth or eponential decay? 1. Model with mathematics. Troy bought a book with 1 pages. The net day he read half the book. On each subsequent day, he read half of the remaining pages. The eponential decay function y 1(0.) gives the number of remaining pages days after Troy bought the book. a. How many pages did Troy have left to read after 6 days? 48. Mia bought a new computer for $1,00. A function that models the value of Mia s computer after t years is v(t) 1,00? (0.68) t. How much is Mia s computer worth after. years? b. Blake says that the value of the eponential function can never be 0, so Troy will never finish reading the book. Do you agree with Blake? Eplain why or why not. 49. Jane bought a new car for $0,000. A function that models the value of Jane s car after t years is v(t) 0,000? (0.8) t. In how many years will the car be worth less than half of what Jane paid for it? A. B. Lesson -. Without graphing, tell which function increases more slowly. Justify your answer. f() 99 g() 9 C. 4 D College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

8 . Use a graphing calculator to graph the function g() a. Identify the values of a and b (from f() = ab ), and describe their effects on the graph. c. To keep the club from becoming overcrowded, the maimum club membership is 00 people. Does this additional information change your recommendation from part b? Eplain why or why not. b. Graph f() 1 on the same screen as the graph of g(). Describe the similarities and differences between the graphs. 4. Which function increases the fastest? A. y 14 B. y 17 Lesson On the coordinate grid below, p represents the amount of money in Paola s savings account, and v represents the amount in Vincent s account. Whose account had a higher initial deposit, and how much was it? Use the graph to justify your answer. y C. y 10 D. y v. Make sense of problems. A health club with 100 members is trying to increase its membership. Judy has a plan that will increase membership by members per month, so that the number of members y after months is given by the function y Desmond has a plan that will increase membership by 10% each month, so that the number of members y after months is given by the function y 100? 1.1. a. Whose plan will increase club membership more quickly? Use a graph to support your answer p Four students deposit money into accounts with interest that is compounded annually. The amount of money in each account after t years is given by the functions below. Use these functions for Items 79. Felicity: f(t) 00 (1.0) t Raisa: r(t) 800 (1.01) t Sanjay: s(t) 1,000 (1.01) t Megan: m(t) 00 (1.0) t b. Whose plan would you recommend? Eplain College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

9 7. Identify the constant factor in Sanjay s function and eplain how it is related to his interest rate. 8. a. Write a function to represent the amount of money Felicity will have after m months if her interest were compounded monthly rather than annually. b. Will Felicity earn more money when % annual interest is compounded annually or monthly? Eplain. Lesson - The population of Arizona from 1970 to 000 is shown in the table below. Use the table for Items 616. Arizona Year Resident Population ,77, ,716, ,66,8 000,10, ,9,01 9. Which shows the students names in order from greatest initial deposit to least initial deposit? A. Megan, Felicity, Raisa, Sanjay 61. Use a graphing calculator to find a function that models Arizona s population growth. Write the function using the variable n to represent the number of decades since B. Felicity, Raisa, Megan, Sanjay C. Sanjay, Megan, Raisa, Felicity D. Sanjay, Raisa, Felicity, Megan 6. Use a graphing calculator to create a graph showing the data from the table and the function you wrote in Item 61. Make a sketch of the graph. Is the function a good fit for the data? Eplain why or why not. 60. Use appropriate tools. The function t() 00 (1.01) represents the amount of money in Tracy s savings account after years. The function j() 00 (1.0) represents the amount of money in Julio s savings account after years. Eplain how to use your graphing calculator to determine when the amount in Julio s account will become greater than the amount in Tracy s account. Round to the nearest whole year. 6. Before the 01 population count was final, the Census Bureau predicted that Arizona s population in 01 would be 6,,. a. Use the function from Item 61 to predict Arizona s population in 01. What number did you substitute into the function? Eplain. b. How does your prediction in part a compare to the prediction from the Census Bureau? College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

10 64. Which function is the best model for the data in the table? y A. y B. y.6 16 C. y D. y Write a polynomial in standard form that has an even number of terms and whose degree is Attend to precision. Which shows the polynomial a 1 6a 16 a written in standard form? A. a 1 6a 16 1 a B. a 1 6a 1 a 16 C. a 1 6a 16 1 a D. a 1 6a 1 a Critique the reasoning of others. The function y 10,94(1.17) n represents the population of Nate s hometown, where n is the number of decades since Nate wants to rewrite the function to show the growth per year. He rewrites the function as y 10,94(0.117) n where n is now the number of years since Did Nate write the new function correctly? If so, eplain why. If not, eplain why not and write the correct function. 69. a. Is the epression a polynomial? Eplain why or why not. b. Karina says that the epression is not a polynomial because 1 is not a whole number. Do you agree with Karina? Eplain why or why not. Lesson Copy and complete the table below. Polynomial Number of Terms Name Leading Coefficient Constant Term Degree Lesson Add. Write your answers in standard form. a. ( 1 1 4) 1 (6 1 4) b. ( 1 ) 1 (7 1 9) c. ( ) 1 (4 1 ) 1 d College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

11 71. Write the perimeter of the triangle as a polynomial in standard form. Lesson 4-7. Subtract. Write your answers in standard form. a. ( ) ( 1 8 1) b. ( ) ( 6 1 ) Devon is fencing in a square garden. The length of each side of the garden is 1 feet. a. Show how Devon can use addition to find an epression that represents the total number of feet of fencing he needs for all four sides of the garden. Write the sum in standard form. b. Compare the epression for the garden s side length, 1, with your answer to part a. What do you notice? Does this make sense? Eplain. c d. ( 1 4 7) (9 1 10) 76. The perimeter of a rectangle is inches and the length is inches. Clark and Rachel were asked to find an epression for the width of this rectangle. a. Clark began by writing (1 1 0) (4 1 8). Find this difference and eplain Clark s reasoning. b. What should Clark do net? Eplain. 7. Which sum is equal to ? A. ( ) 1 ( 8) B. (8 1 1) 1 ( 1 8) C. (8 1 1) 1 ( 1 1 8) D. (8 1) 1 ( 1 8) c. Rachel began by writing (4 1 8) 1 (4 1 8). Find this sum and eplain Rachel s reasoning. d. What should Rachel do net? Eplain. 74. Make use of structure. Write two polynomials whose sum is: a b. 7 c e. Eplain how to finish solving the problem to find an epression for the width of the rectangle College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

12 77. Donna is planning a rectangular flower garden. The total area of the garden will be square feet. A square area in the garden measuring square feet will contain flowers, and the rest of the garden will contain vegetables. Write an epression for the area of the garden that will contain vegetables. 81. Which epression represents the area of the rectangle? 1 cm cm A. 1 1 cm B cm 78. Which difference is equal to 1 6 6? A. (7 1 ) (4 8 1 ) B. (7 1 ) (4 8 ) C. (7 ) (4 8 1 ) D. (7 ) (4 8 ) 79. Reason abstractly. Is it possible for the difference of two polynomials to be 1? If so, give an eample of two polynomials whose difference is 1. If not, eplain why not. C cm D cm 8. Each product below contains an error. Eplain how you can tell that the products are incorrect without multiplying. Then identify and correct each error. a. ( 1 8)( 1 7) b. ( 1)( 1) 1 1 c. ( )( 1 ) Lesson Find each product. Write your answers in standard form. a. ( 1 )( 7) b. ( 1 )( 1 9) c. ( 1)( 1 1) d. ( )( 4) 8. Find the missing number in each product. Show that your answer is correct. a. ( 1 )( 1 ) b. ( 1 )( ) 18 c. ( 7)( 1 ) 14 d. ( 1)( ) Make use of structure. As part of his math homework, Huong must show that ( 1 4)( 80) ( 80)( 1 4) 0. Huong does not want to multiply the binomials because the numbers are large. Describe how Huong can show that the epression is equal to 0 without multiplying the binomials College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

13 Lesson - 8. Find each product. Write your answers in standard form. a. ( 1 )( ) Lesson Find each product. Write your answers in standard form. a. ( 1 1) b. (4 1 )(4 ) b. ( 1 )( ) c. ( 1 7) c. ( 1)( ) d. (6 ) d. 4 ( 1 8)( 8) 86. Which product is equal to 6 1 9? A. ( ) B. ( ) C. ( 1 )( ) D. ( 1 )( ) e. ( )( 1 )(7 4) 90. Which product is equal to? A. ( ) B. ( ) C. ( ) D. ( ) 91. The formula for the area of a triangle is A 1 bh. 87. Ginny says that the area of this quadrilateral is square units. What assumption is Ginny making? 1 7 Cole and Brenda are finding a polynomial that represents the area of the triangle below. Cole plans to multiply 1 by 1 1 and then multiply the result by 6. Brenda plans to multiply 1 by 6 and then multiply the result by 1 1. Eplain why Brenda s solution method might be better Critique the reasoning of others. Shirley says that the product ( 1)( 1 1) is not a difference of two squares because the product is not in the form (a 1 b)(a b). Eplain to Shirley why she is incorrect College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

14 9. Epress regularity in repeated reasoning. For parts a d, find the degree of each polynomial. Then find their product and the degree of the product. Organize your results in a table. a. Polynomial 1: 4 Polynomial : 1 b. Polynomial 1: 4 Polynomial : c. Polynomial 1: 6 Polynomial : 1 d. Polynomial 1: 4 1 Polynomial : 1 e. When two polynomials are multiplied, what is the relationship between the degree of each polynomial and the degree of their product? 96. Give an eample of a polynomial with at least three terms that cannot be factored by factoring out the GCF. 97. The length of the side of a square is represented by the epression 1 4. When Carlos is asked to write an epression for the perimeter of the square with the GCF factored out, he writes 4( 1 4). Is Carlos correct? If so, eplain why. If not, eplain Carlos s error and give the correct answer. Lesson For which polynomial is the GCF of the terms? A. 1 1 B C D Factor each polynomial. a. 0 Lesson Factor completely. a. 4 b c b d. 6 4 c d What factor would you need to multiply by ( 1) to get 1? A. B. C. 1 1 D Model with mathematics. Adam is planning a rectangular patio that will have an area of square feet. The length of the patio will be 1 feet. Write an epression to represent the width of the patio Sergio claims that 1 6 is a perfect square trinomial. Eplain how you can tell by eamining the polynomial that Sergio is incorrect College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

15 101. Make sense of problems. Alison has a square carpet whose area is square feet. Karl has a square carpet whose side length is 1 6 feet. Find a value of for which the areas of the carpets are equal. What is the area of each carpet for this value of? Eplain how you found your answers. 10. Reason abstractly. Jackie says that if the factored form of a trinomial is ( 1 1)( 1 c) for a positive number c, then c is the constant term of the trinomial and c must be a prime number because its only factors are 1 and c. Is Jackie correct? If so, eplain why. If not, give a countereample to disprove Jackie s claim. Lesson Factor each trinomial. Write your answer as a product of two binomials. a. 0 b c Lesson Factor each trinomial completely. a b. 8 c d d Which trinomial cannot be factored? A. 1 4 B C. 1 4 D An architect represents the area of a rectangular window with the epression Factor this trinomial to find possible epressions for the length and the width of the window For the trinomial 1 b 8, give all values of b for which the trinomial can be factored. Eplain how you know that you have found all possible answers The trinomial 6 1 b 1 1 can be factored. Which statement is true? A. The value of b could be an even number. B. The value of b cannot be greater than 7. C. The value of b must be positive. D. The value of b must be a multiple of,, or College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

16 109. Give an eample of a trinomial for which one of the factors is 7. Eplain how you found the trinomial. 11. Taina correctly simplified the rational epression shown below Which term appears in Taina s simplified epression? A. 8 B. 6 C. D Critique the reasoning of others. Gordon says that when is factored completely, there are three factors. Holly says there are two factors. Who is correct? Eplain. What error might the other student have made? 11. Kevin says that for the rational epression 11, cannot equal 1,, or. Is 116 Kevin correct? If so, eplain why. If not, describe Kevin s error a. Make use of structure. Write a rational epression that simplifies to 1. Eplain how you found your answer. Lesson Simplify each epression a. 1 6 b b. How many possible correct answers are there to part a? Eplain. 11. A catering service charges $16 for each guest s meal plus a flat fee of $00. Write a rational epression for the cost per guest for an event with g guests. c College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

17 Lesson Simplify by using long division a. 4 1 b. ( ) c Identify and correct the error in each division a. 1 ) b. ) Model with mathematics. The area of a rectangular flower bed is square feet. The length of the flower bed is feet. a. Write an epression for the width of the flower bed. Lesson Multiply or divide. a. 1? 1 6 b. What are the length, width, and area of the flower bed when? b. 10?( 4) ( 1 ) 118. Andy was asked to simplify each epression below using long division. For which epression should he have a remainder? A B C D. 4 c. d The figure shows a rectangular prism. The area of the rectangular face ABCD is The length AB is 1 and the length of BF is 1. Is it possible that BCFG is a square? Eplain. D A E C B F H G College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

18 1. Which product is equal to 1? A. B. C. D ? ? ? ? Construct viable arguments. In the quotient a ( 1 7) 4, a represents a real number. ( 1 7) Tony says that if the quotient is negative, then a must be negative. Is he correct? If so, eplain why. If not, eplain why not and correct Tony s error Roy subtracted as shown below. Identify and correct Roy s error Add or subtract. Simplify your answers if possible. a. 1 b c d Becky and Shannon each added as shown below. Whose solution is correct? Eplain. Lesson Which pair of epressions has a least common multiple that is the product of the epressions? A. 1 4 and 4 B. 4 and C. 1 4 and 16 D. 4 and ( 4) 6 6 Becky 1 6? Shannon ? ( ) College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice

19 18. Make sense of problems. On a hike, Rowena ran for a total of miles and walked for a total of miles. She ran at a rate that was twice as fast as her walking rate of r miles per hour. Write and simplify an epression for the total amount of time that Rowena walked and ran on her hike. What was the total time of her hike if she walked at a rate of miles per hour? College Board. All rights reserved. SpringBoard Algebra 1, Unit 4 Practice