PBA_acc (PBA-_acc) Name: Date: 1

Transcription

1 Name: Date: 1

2 1. The city s swimming pool is being filled with water. The dimensions of the pool are in feet, and the time to fill the pool is measured in minutes. The shape of the swimming pool is shown in the diagram. Elliott says that the rate at which the pool is filled can be expressed in square feet per minute. Explain why you agree or disagree with Elliott. Water is pumped into the pool at a constant rate. The graph shows the height, in feet ( y ), of the water in the pool over a period of 70 minutes ( x ). The height is measured from the bottom of the deep end of the pool. Elliot cannot understand why the graph is a curve even though the water is pumped into the pool at a constant rate. Describe the graph in terms of the rate of change as a function of time. Use your description to explain why the graph is not linear. 2

3 2. The value of Roger s investment account, V dollars, is given by the function where t is the number of years since Roger opened the account. No additional deposits or withdrawals have been made. Determine the values of the account at the end of years 1, 2, 3, and 4. Use the appropriate level of accuracy for reporting money values. Show your work or explain your answers. Roger s account can also be found using the recursive function shown below. Use the recursive function to write the sequence,,,,, writing each term in the sequence with the appropriate level of accuracy for money values. Show your work. Explain why the sequence may not exactly match the values based on the exponential function given at the beginning of the problem. 3

4 PBA_acc (PBA-_acc) 3. A vendor sells cotton candy and sodas at baseball games. From past experience, he knows that he must buy at least twice as many sodas as cotton candy. The vendor plans to spend no more than $250 buying both items for the next game. He will buy x packages of cotton candy at $1.00 per package. He will buy y sodas at $0.50 per soda. This graph models the situation. The two lines divide the quadrant into four regions, labeled A, B, C, and D. Write a system of inequalities that represents the graph. Which region represents the solution of the system of inequalities? Show your work or explain your answer. The vendor makes a profit of $1.50 on each package of cotton candy sold and $0.50 on each soda. If he sells all the sodas and cotton candy he bought, what is the greatest profit he could make? Show your work or explain your answer. 4

5 4. In the formula shown below, A is the amount of money in an investment account when P dollars are invested at an annual interest rate of r for 2 years. What is the amount of money in the account if $2,000 is invested and the annual interest rate is 2.5%? Show your work or explain your answer. What is the minimum amount that must be invested at an annual rate of 3% for the amount A to equal at least $2,000? Show your work or explain your answer. Solve the given equation for r. Show your work. 5

6 5. In the given functions shown, a, b, c, and d all represent different real numbers. What type of functions are f and g? Explain your answer. If the given functions are graphed on the same coordinate plane, would the functions intersect one another? Explain your answer. What is the solution to the equation Show your work. Part D Nathan is deciding between two different phone plans. Plan A offers 900 minutes for $59.99 plus $0.40 for each additional minute. Plan B offers unlimited minutes for $ Using the functions given, assign values for a, b, c, and d to represent the two different phone plans for when 6

7 6. The function C shows the relationship between the circumference of a circle and the length of its radius x. The function P shows the relationship between the perimeter of a square and the length of its side, Write the equation of function C and the equation of function P. Graph both of the functions on the same coordinate grid. Name two properties that the graphs of the two functions have in common. To the nearest tenth, for what value of x are both of the functions equal? Show your work or explain your answer. 7

8 7. The final score, f, a player receives during each level of a game is given by the equation where p is the number of points earned, a is the average expected score, and d is the difficulty of the level. Two friends are playing this game at the same level of difficulty. Ryan earns 140 points and receives a final score of 14. Tony earns 125 points and receives a final score of 8. What is the difficulty of the game at this level? Explain your answer. Rene decides to play the same game at a difficulty of 3.8. If she earns 124 points and a final score of 5, what is the average expected score of the game? Show your work. 8

9 8. The possible dimensions of Joe s rectangular garden are shown in the coordinate grid. Use inequalities to represent all of the possible dimensions, w and l, of Joe s garden. Use equations and/or inequalities to represent the constraints of the graph. Explain what these constraints represent. What is the perimeter of Joe s garden? Explain your answer. 9

10 9. Three functions are given. Use a table to find the two solutions for Explain how you used the table to find the solution. Graph and label the three functions on the coordinate plane. What is the solution for What is the solution for all three functions? Explain your answer. 10

11 10. The functions and are graphed on a coordinate grid. Write an equation to represent the linear function. Which values of x satisfy the equation Explain your answer. Create a different function, that shares one or more solutions with the function of How many solution(s) will the functions share? 11

12 11. The graph shows the distance a car traveled as a function of time. Calculate the rate of change and interpret its meaning in terms of the context. Show your work and explain your answer. Use function notation to express four and one-half hours of distance traveled. The function can be used to model the car s gas mileage, where represents the total distance in miles and x represents the number of gallons of gas used. What is the value of x when miles? Show your work. 12

13 12. A teacher wrote a sequence of positive integers on the board. She wrote this equation to describe the sequence. Write the first 5 terms of the sequence. Show your work. Express this sequence as a function of x, and include the appropriate domain. Then evaluate your function for Show your work. The teacher wrote another sequence on the board. Write a function of n that produces this sequence, where n is a positive integer. Evaluate your function for Explain one way to check your answer. 13

14 13. The table shows the input and output of a function. PBA_acc (PBA-_acc) Explain what makes this set of data a function. What could cause this set of data not to be a function? Write an equation, in function notation, that would fit the inputs and outputs in the table. Show work that proves your equation is true for the given data. 14

15 14. Two water tanks are being drained. The tables show the number of gallons, g, of water left in each tank after h hours. For each tank, the drainage rate will remain the same. What type of function can be used to model the relationship between hours and gallons for Tank 1? Explain your answer. Write a recursive formula to express the number of gallons left in Tank 1 after h hours, in terms of the number of gallons left in Tank 1 the previous hour, What type of function can be used to model the relationship between hours and gallons for Tank 2? Explain your answer. Part D Write a recursive formula to express the number of gallons left in Tank 2 after h hours, in terms of the number of gallons left in Tank 2 the previous hour, 15

16 15. The value of a machine used in manufacturing is modeled by the function, where x is the age of the machine in years and y is the value of the machine in thousands of dollars. Tyler says the function is an exponential function. Is Tyler correct? Explain your answer. What does the number 16 in the equation represent in this situation? What does the fraction in the equation represent in this situation? Part D Sketch a graph of the function on the coordinate plane shown. Label at least two points on the graph with their coordinates. 16

17 16. In the table, the values of are real numbers. PBA_acc (PBA-_acc) What type of function is Construct the equation of Explain your answer. in terms of s and t. Show your work and explain your answer. 17

18 17. P is a function that assigns, to every positive integer x, the probability that x heads will appear when x fair coins are tossed once. Write the numerical sequence that corresponds to Explain your answer. What type of function models this sequence? Explain your answer. Write the equation for a function f that generates the term of the sequence. Show your work and explain your answer. 18. A train transports one ton of freight 2,400 miles on 4.8 gallons of diesel fuel from one location to another location. Harrison makes a graph to represent the miles per gallon the train uses. What would be an appropriate domain for Harrison to use to graph this situation? Explain your answer. What should the origin of Harrison s graph represent for this situation? What rate of change should be shown in Harrison s graph? Show your work. 18

19 19. For each situation given in and : PBA_acc (PBA-_acc) Discuss what additional factor(s) would need to be considered. Identify what unit of measure and degree of accuracy should be used. Explain the reason for your choice of unit measure and degree of accuracy. A construction worker measures the dimensions of each classroom in a school to determine the number of tiles needed to cover the floor for each classroom. For a science project, a student plants basil seeds in several different environments and graphs the heights of all the plants over a period of time to compare the different growth rates. 20. Following the year 1983, the area of the island of Hawaii (the Big Island ) increased due to lava flows from two active volcanoes. One model for the increase in new land area, in acres, is given by the function where x represents the number of years after Determine the value of and interpret its value in the context of the problem. According to this model, in what year did the number of new acres exceed 400? Explain your answer. Let and let n represent a whole number. Write a recursive formula that gives the number of new acres, in terms of the number of new acres from the previous year, 19

20 21. The graphs of and are shown on the coordinate plane. What are the domains of these two functions? Label your answers. What are the ranges of these two functions? Label your answers. What is the value of Part D What value(s) of x make obtained algebraically. Show work or explain your answer. true? Explain how the answer can be obtained using the graph and how it can be 20

21 22. Jordan is on a 100-mile bicycle race that started at 8 a.m. He wants to finish the race in 5 hours or less. The table shows what time it is when Jordan reaches each 10-mile interval for the first 60 miles. During which 10-mile interval does Jordan have the highest average speed? Show your work or explain your answer. What is Jordan s average speed, in miles per hour, for the first 60 miles of the race? Round your answer to the nearest tenth. Show your work or explain your answer. For the next 30 miles, Jordan rides at an average of 20 miles per hour. What is the minimum average speed, in miles per hour, Jordan needs for the last 10-mile interval in order to reach his goal of finishing the race in 5 hours or less? Round your answer to the nearest tenth. Show your work. 21

22 23. A printing company s costs for printing a book are $500 for labor and running the necessary machinery plus $3 per book printed. The total cost for printing a particular book must be less than $1,300. Write an inequality to represent this situation. Let n represent the number of books printed. Solve your inequality for n. What is the greatest number of books that can be printed? Show your work or explain your answer. What are the constraints on the variable n? Explain your answer. 24. Zach is saving money to purchase a car. He has already saved $800 in an account. He is going to deposit $75 in the account every Friday. He will not make any withdrawals or any other deposits. Write an equation that represents the cost of the car, C dollars, in terms of n, the number of Fridays he will have to save $75. What are the constraints on the variable n? Explain your answer. Zach decides to buy a car that costs $3,500. Solve your equation to find out how many Fridays he needs to deposit $75 in the account. Show your work or explain your answer. 22

23 PBA_acc (PBA-_acc) 25. James and Elena are driving separate cars to a destination using the same route. James leaves at 9:00 a.m. and averages 40 miles per hour due to heavy traffic. Elena leaves at 11:00 a.m. and averages 50 miles per hour. They both arrive at the destination at the same time. Write a system of equations to represent this situation. Remember to define your variables. Solve the system of equations from. Show your work. What was the distance from where James and Elena started to their destination? Explain your answer. Part D What time do James and Elena arrive at their destination? Explain your answer. 23

24 PBA_acc (PBA-_acc) 26. Dana owes $12,000 on her car loan and has $2,500 in her savings account. Each month, Dana plans to pay $500 on her car loan and deposit $100 into her savings account. Let m represent the number of months. Write two algebraic expressions for the situation described: an expression representing the balance on the car loan after m months an expression representing the amount in the savings account after m months Use the expressions from to write an inequality, in terms of m, to determine the number of months required for the amount in Dana s savings account to exceed the balance on her car loan. Solve the inequality you wrote in. After how many months will Dana s savings account exceed the balance on her car loan? Show your work and explain your answer. 24

25 PBA_acc (PBA-_acc) 27. At a large university, a class is considered a preferred class if the class begins between the hours of 10:00 a.m. and 4:00 p.m. A class is considered undesirable if the class begins before 10:00 a.m. or after 4:00 p.m. Let x represent the number of preferred courses offered and let y represent the number of undesirable courses offered. If the total number of courses offered is 120 or fewer, write an inequality to represent this situation in terms of x and y. Sketch the set of all possible solutions to the inequality you wrote in on the coordinate grid. The university administrators want to better utilize the classroom space to save money. They issue the following two guidelines to the team responsible for creating the class schedules. Preferred classes cannot exceed 40% of all classes offered. At least 25% of all classes offered must be in the undesirable time slots. Write two inequalities to represent the guidelines and graph the inequalities on the same coordinate grid as. Part D Explain what your solution represents within the context of the problem. 25

26 28. In the system of linear equations shown, j and k are non-zero real numbers. Write a system of linear equations, using integer coefficients, with the same solution as the given system. Explain how you know the system you wrote in has the same solutions as the given system. Solve the system of equations for x and y in terms of j and k. Show your work. Part D Explain what the solution to your system represents on a coordinate plane. 26

27 PBA_acc (PBA-_acc) 29. A random number generator produces real numbers from 0 to 10. After two numbers are generated, the point is graphed where x is the first number generated and y is the second number generated. Write a system of linear inequalities to represent where all possible points lie in a coordinate plane. Write a system of linear inequalities that expresses the following relationships. The sum of x and y is greater than or equal to 4. The sum of x and y is less than or equal to 6. On the coordinate plane, graph all of the inequalities you wrote in Parts A and B. Explain the meaning of the shaded region. Part D What percent of all possible points produced by the random number generator have coordinates whose sum is between 4 and 6? Show your work or explain your answer. 27

28 30. A company s profit on x widgets sold is represented by the equation shown. State an appropriate domain for this situation and explain your answer. Using the domain from, graph this function. Find and explain what this represents. Part D Does this situation have a realistic x -intercept? Explain your answer. 28

29 31. The graph shows the average use of electricity in kilowatts per hour (KWH), y, by a small town in the year 2010, where represents the month of January. Describe the behavior of the function within the domain of Let for January. Show your work. and estimate the rate of change for this interval. What month did the function reach its maximum? Explain what the maximum represents for this data. Write your answer from, in function notation. Describe what this data represents in the context of this situation. Part D Describe what an x -intercept would represent in this graph. Is it likely to occur for this context? Explain your answer. 29

30 32. The data shows the number of houses sold in the southern region of the United States. Estimate the rate of change from the last four years of the given data. Show your work. Using the rate of change you found in, write an equation, in function notation, for the last four years of the given data. Let represent the year Using the function you found in, predict the number of houses that will sell in the year Show your work. Part D Explain if your answer in is or is not a reasonable answer in context to this problem. 30

31 33. The function represents the value of Gale s investment, V dollars, at the end of t years. Write a recursive rule to represent this function for Explain your function. What are the values of the function when Show your work. Round your answer to the nearest cent. Graph the function for the domain Remember to title your graph and label your axes. 31

32 34. During the first month of a new business, the expenses were $180 and the revenue was $15. During each of the next several months, the expenses and the revenue increased as shown in the table. Write an equation that expresses the expenses, E, as a function of the number of months, m. Write an equation that expresses the revenue, R, as a function of the number of months, m. By examining only the equations you wrote in and, is it possible to determine whether the monthly revenue will ever exceed the monthly expenses? Explain how you know without giving any dollar amounts. Part D The profit of a business is determined by finding the difference between the revenue and the expenses. If both the expenses and the revenue continue to increase at a constant rate, what is the total profit for this business for the first 6 months? Show your work and explain your answer. Part E Write an equation that expresses the total profit, P, as a function of the month, m. 32

33 35. In the year 2000, biologists reported the population of an endangered animal was declining by 5% per year. They speculated that same rate of decline had been continuing for many years prior to 2000, and the decline would continue under the present conditions. The graph shows this situation, where m represents the number of endangered animals in the year In terms of m, what is an estimate of the population of these animals in the year 2005? Explain your answer. In terms of m, what is an estimate of the population of this endangered animal in the year 1995? Explain your answer. If a law that makes conditions more favorable for this animal is passed t years after 2000, the population is expected to increase by 5% per year. Write an equation for the function, after the law was enacted. Explain your answer. to represent the size of the population x years 33

34 36. A square piece of fabric is bordered with leather thread. The length of each side of the square is x inches. The length of the leather thread needed is twice the length of each side. The cost of the fabric is $0.40 per square inch and the leather thread is $0.75 per inch. Write an equation for the function C, in terms of x, which represents the total cost for the fabric and the leather thread needed to make one square with x- inch sides. Explain your answer. Jay says function C is linear. Rachel says function C is exponential. Is Jay or Rachel correct? Explain your answer. 34

35 37. Anthony invested money in two companies at the same time. He invested $127 in Company S and $64 in Company T. The growth with Company S was linear, and the growth with Company T was exponential. The graph shows the values of the two investments for the first few years. The point of intersection of the two graphs is specific in your answer. Interpret this point in terms of Anthony s two investments. Be Write an equation of a linear function, S, that expresses the amount of money Anthony has in Company S as a function of the number of years, x, that it has been invested. Show your work. What does the slope of the line from tell you about the investment? Part D The exponential function expresses the amount of money Anthony has in Company T as a function of the number of years, x, that it has been invested. What does the base of the exponent tell you about the investment? 35

36 38. Austin purchased a bed frame and mattress for $1, from a furniture store. He pays $ at the time of the purchase and finances the remaining balance with the furniture store. Austin has six months to pay the remaining balance before he is charged a yearly interest rate of 15%. Austin plans on paying the same amount each month for the next six months. Write an equation that represents the remaining balance, y, after each month, x. Create a coordinate grid, like the one shown, to graph the equation you wrote in. Label the axes and scales. What is the domain of this situation? Explain your answer. Part D How would the equation and graph be affected if Austin does not pay off the remaining balance in six months? 36

37 39. Use the inequality to answer each part. PBA_acc (PBA-_acc) Solve the inequality for x. Show and explain each step used in solving the inequality for x. For which values of a and c would the inequality not have a solution? Explain your answer. Using the inequality from, let and What is the value of x? Show your work. 40. The cost to rent an ice cream truck is $1,400 per day, tax included. The cost to rent a food cart is $650 per day, tax included. Lisa needs to rent one ice cream truck and food carts for an event her company is holding over the two-day weekend. She has a budget of $10,000 to spend on the rentals. Write an inequality to represent the number of food carts, f, Lisa can rent. How many food carts is Lisa able to rent? Show your work. How many food carts can Lisa rent if the rental company lowers the rental costs of the food carts and the ice cream truck by 35%? Show your work. 37

38 41. Tara earns $38,000 during the first year at her job. Each year after the first year, she will receive a raise of 5%. Write the equation of a function S that gives Tara s salary, started her job. as a function of the number of years, x, after she Use the function from to determine Tara s salary after 4 years. Show your work or explain your answer. If no other conditions change, how many years will it take for Tara s salary to double from her initial salary of $38,000? Explain your answer. Part D At the end of the first year, Tara s employer decided to award her a $3,000 bonus in addition to the 5% yearly raise. The bonus will not be included in the salary used to compute Tara's 5% increase the next year. If the employer follows this practice for the next few years, write a recursive formula that describes her total pay for a given year, in terms of her total pay from the previous year. Include a definition of n in your formula. 38

39 42. Function f models Mr. Carson s weight n weeks after he started a diet. His initial weight was 195 pounds and is represented by The recursive rule for function is given. Beginning with Mr. Carson s starting weight of 195; write the numerical sequence of his weights for the first 6 weeks of the diet. Can the function f be represented by a single linear equation? Explain your answer. Give the value of and explain what it represents. Part D According to the given rule, when does equal 182? Explain your answer. 39

40 43. The expression is used to determine the volume of a pyramid, and the volume of a prism is calculated by using the expression Bh. The variables B and h are shown in each of the figures below. What does the B represent in each formula? What does the h represent? A prism and a pyramid have the same volume. The base of the prism has an area twice as large as the base of the pyramid. What is the ratio of the height of the pyramid to the height of the prism? Explain your answer. A cone has a circular base. If the height of the cone, h, and the radius of the base, r, are known, what is the formula for the volume of the cone using r and h? 40

41 44. Renee is a traveling salesperson. She pays for her own traveling expenses and receives a 22% commission of her total sales. Her average monthly traveling expenses are $1,540. Renee s goal is to earn at least $5,500 for the month after deducting her traveling expenses. Write an inequality that can be used to determine the total sales Renee needs to meet her goal. Be sure to define the variable you used in your inequality. How much will Renee need to sell to meet her goal? PBA_acc (PBA-_acc) After paying overhead and commission, Renee s employer has 15% of Renee s total sales amount left. The employer pays $1,280 per month for Renee s medical benefits from this 15%. Write an inequality that can be used to determine the minimum amount Renee s total sales need to be for the employer to be able to pay for Renee s benefits and still make a profit. Be sure to define the variable. How much must Renee sell in order for the employer to make a profit? Renee s employer offers three different compensation plans. The plans are detailed in the table shown below. Renee is considering changing from Plan C to either A or B. Her goal is to make the highest salary possible. Compare Plan A and Plan B to Plan C and determine the amount of sales that Renee would have to make in a month for Plan C to be more profitable than either A or B. 41

42 45. A sports team generates revenue from three major sources: ticket sales, parking, and concessions. In order to estimate the revenue from each game, the owners have created models for each revenue stream, based on the total attendance of each game. For all of the equations below, R represents revenue in dollars and n represents the number of people in attendance. For ticket sales revenue, the equation is For concession revenue, the equation is For parking revenue, the equation is Using the three models, create a single equation for total revenue equal to the sum of the ticket sales, concession, and parking revenue. Express the function in a simplified format. The function is a linear function. Identify the rate of change in this function. What does the rate of change represent in the real-life situation? Determine the minimum number of tickets the team must sell in order to break even in ticket sales. PBA_acc (PBA-_acc) The concession revenue is modeled by a quadratic equation. The coefficient of the term is negative. Find the vertex of this function and determine if it is an absolute maximum or minimum. What does the vertex represent in the real-life situation? 42

43 46. Craig purchased a cell phone for $100 and a telephone service that costs $29.95 per month. Write an equation showing the total cost, y, for Craig s telephone purchase and service where x represents the number of months he uses it. Use your equation to determine how many months Craig used his telephone and service if his total cost was $ Show your work. Craig creates a graph where y represents the total cost of his telephone purchase and service for the first 6 months. What ordered pair correctly represents the y -intercept of Craig s graph? Explain what the y -intercept means in this context. Ana purchased a different telephone and service at the same time as Craig. At the end of the first 5 months, Craig s total cost was $5 less than Ana s total cost. If the cost of Ana s cell phone was $125, what was the cost of her monthly service? Show your work. 43

44 47. Brianna wants to buy some trees for her yard. A local garden store charges $25 to plant each tree plus a delivery fee of $50 for all trees purchased. A lawn service charges $35 to plant each tree with free delivery. Write the system of equations that could be graphed to represent this situation, where y is the total charge for delivery and planting x trees. Explain how the graph could be used to determine the solution of the system of equations you wrote. Determine the solution to the system of equations algebraically. Show your work, and explain what the solution means in this context. Brianna wants to plant 8 or more trees in her yard. Which service provider would be the most economical? Explain your answer. 44

45 48. A function is considered an even function if it is symmetric about the y -axis. Another way of demonstrating that a function is even is to show that The function is an even function because which is the same as Sketch the function Is the function even? Consider where Is this an even function? Justify your answer. Compare the graphs of the functions and where k is a real number. Describe how the graph of can be transformed into the graph of For which values of k is an even function? PBA_acc (PBA-_acc) Consider that two functions, and are even. Which function would always be even? Justify your answer. 45

46 49. The surface area of a cylinder can be found using the formula where r is the radius of the circular base and h is the height of the cylinder. A cylinder has a base with a radius of 2 units and a surface area of height? Show your work. square units. In units, what is the cylinder s A different cylinder has a surface area of cylinder? Show your work. square units and a height of 1 unit. What is the radius, in units, for this Suppose that the surface area and the radius of a cylinder are known. Rewrite the surface area formula to solve for the height of the cylinder. Part D Suppose that the surface area and the height of a cylinder are known. Rewrite the surface area formula to solve for the radius of the cylinder. 50. The function gives the revenue for the sales of n tickets to a concert. What is an appropriate domain for this function? Explain your answer. The cost of each ticket is $21. Write a function to represent this situation. The revenue for the concert was $15,792. How many tickets were sold? Show or explain your work. 46

47 51. The population of a bacteria colony doubles every 6 hours. For example, starting with a sample of 10 bacteria, after 6 hours there would be 20 bacteria and so on. After 12 hours there would be 40, and so on. Now consider a scientist collected a colony of 25 bacteria. Using the growth rate provided above, copy and complete the following table by providing all missing values. Explain how you got the values in your table. The expression can be used to describe the growth rate in. Interpret each term in the expression, including the meaning of the variable, x. How would you determine the approximate number of bacteria after 3 hours? Provide an estimate for the population at that time. Part D Let the variable d represent the number of days after the bacteria colony was started. Describe the net change in population after one full day. Using this result, create an exponential model that will predict the bacteria population for our original colony of 25 after d days of growth. 47

48 52. A company orders two types of gravel to use in a landscaping project. Black gravel costs $0.55 per pound, and red gravel costs $0.80 per pound. To complete the project they will need at least 5,000 pounds of gravel, but they can spend at most $3,500. Define variables to represent pounds of red gravel and pounds of black gravel. Write an inequality that can be used to determine possible combinations that will satisfy the requirement that the total weight must be over 5,000 pounds. Use the same variables to write another inequality to determine how much of each type of gravel the company can afford to buy. Graph the inequalities from and on a coordinate grid. Shade each half-plane to represent the solutions of the linear inequalities. Indicate which section of the graph satisfies both inequalities. Part D What is the maximum amount of red gravel that the company can purchase? Justify your answer by interpreting the graph. 48

49 53. When the point is rotated clockwise about the point the coordinates of the resulting point are If the point is rotated clockwise about the point what are the coordinates of the resulting point? Show your work. Prove that using the coordinates of points A, B, and B. Prove that using the coordinates of points A, B, and B. 49

50 54. Triangle PQR is shown on a coordinate plane. PBA_acc (PBA-_acc) Lester proves that PQR is a right triangle by showing that two of the sides are perpendicular. Andrea proves that PQR is a right triangle using the Pythagorean Theorem. Show the steps and computations that Lester could have used for his proof. Show the steps and computations that Andrea could have used for her proof. Triangle PQR is rotated about the origin to produce triangle Show that the sides of the are parallel to the sides of its pre-image. 50

51 55. Quadrilateral EFGH has vertices at and What is the perimeter, in units, of quadrilateral EFGH? Show your work and round your answer to the nearest tenth. Prove that and are parallel to show that EFGH is a trapezoid. Point Q is the y -intercept of and point R is the y -intercept of Use the information you proved in to prove Part D Devise a plan to determine the area of EFGH. For this task, it will not be necessary to work through the steps and provide a numerical answer. Give a detailed description of what you would need to do. 51

52 56. Figure JKLM is graphed on a coordinate grid. PBA_acc (PBA-_acc) What is the domain of the figure when it is translated 4 units to the left? Describe the effect the translation used in has on the perimeter of the figure. Explain your answer. Give possible coordinates for answer. that would represent a horizontal stretch of the original figure. Explain your 52

53 57. Triangle ABC is depicted below. Prove that triangle ABC is an isosceles triangle and not equilateral. What is the area of triangle ABC? Show your work. Sketch a new triangle JKL that is created by performing a rotation about the origin. Label all of the points. Determine whether or not is congruent to Explain your answer. 53

54 58. Golf balls have a diameter of approximately 4.25 centimeters (cm). Use 3.14 as an approximation of as needed for the following parts. Company M ships golf balls in cartons that have the least volume needed to hold 24 golf balls. Each carton is in the shape of a rectangular prism. The base of a carton is 17 cm long and 8.5 cm wide. The largest cross section of the carton is a rectangle. What is the area of the rectangle? Give your answer to the nearest square centimeter. Company N ships golf balls in cartons that hold twelve balls. Each carton measures 4.25 cm by cm by 17 cm. What percent of the total volume of the carton do the golf balls fill? Round your answer to the nearest percent. Company P wants to create a container that will take up less space. The supplier cuts a 4.5 cm by cm face from a carton and rotates it about the perpendicular bisector of its shorter side. The shape generated by the rotation is a cylinder. What is the volume of the cylinder? Round your answer to the nearest cubic centimeter. 54

55 59. Rectangle WXYZ is graphed in the coordinate plane. PBA_acc (PBA-_acc) Ally applies a rotation of 90 clockwise about the origin to rectangle WXYZ to form rectangle RSTV. What are the coordinates of the vertices of rectangle RSTV and compare the size and orientation of rectangles WXYZ and RSTV. Show your work or explain your answer. Kramer applies a dilation centered at the origin by a factor of to rectangle WXYZ to form rectangle MNPQ. What are the coordinates of the vertices of rectangle MNPQ and compare the size and orientation of rectangles WXYZ and MNPQ. Show your work or explain your answer. Consider the segments XW and YZ. Is it possible to map segment XW onto segment YZ using a translation? Explain your answer. 55

56 60. Rectangle WXYZ is graphed in the coordinate plane. PBA_acc (PBA-_acc) Prove algebraically that the diagonals of the rectangle are congruent. Show your work. Rectangle WXYZ is translated 3 units to the right and 2 units down to form Rectangle If represents a point on WXYZ, write the coordinate pair, in terms of x and y, that represents that point s image on Rectangle What are the ordered pairs representing each vertex in Rectangle Show your work using the representation that you defined for the image of each vertex. Rectangle is a dilation of Rectangle WXYZ centered at by a scale factor of 5. What are the ordered pairs representing each vertex of Rectangle Show your work. 56

57 61. Point is the center of circle Q. Circle Q is tangent to the x -axis at R and to the y -axis at T. Segment SV is a diameter of circle Q. Circle Q is shown in the diagram. Points P, S, Q, and V lie on line m. Point P is at the origin. What are the coordinates of points T and R? Use the information given about the location of circle Q to explain how you determined these coordinates. What is the intersection of and Explain your answer using the correct mathematical terms. What are the exact values of the coordinates of points V and S? Explain your reasoning. Part D Prove that opposite angles in the quadrilateral URTW are supplementary. 57

58 62. In the triangle MNP shown below, and Q is the midpoint of Using a series of statements, prove that Describe a transformation that would transform Be sure to state the reason for each statement. onto itself. Given that determine the distance from point M to point N. Show your work. 58

59 63. In the figure shown, The line containing each of these line segments is shown in the diagram. Use a series of statements to prove Parts A and B. Be sure to give a definition, theorem, or postulate for each of your statements. Prove that Prove that Given that and determine the value of x and the measure of Explain your reasoning for each step in your solution. Part D A translation of PQRS is defined as What are the coordinates of the vertices of the resulting image What is relationship between PQRS and Explain your reasoning. 59

60 64. In the figure, Determine so that Show your work and give appropriate definitions, theorems, and/or postulates to support your statements. If determine the value of x so that Show your work and give appropriate definitions, theorems, and/or postulates to support your statements. Draw on the figure. If and at R, what must be true about Show your work and give appropriate definitions, theorems, and/or postulates to support your statements. 60

61 PBA_acc (PBA-_acc) 65. An artist puts a 1-inch-wide frame around two of her paintings. The diagram shows the paintings with the frames. In Picture 1, determine whether quadrilateral is similar to quadrilateral JKLM. If they are similar, define the dilation required, including the center and the scale factor. If they are not similar, explain why. In Picture 2, determine whether quadrilateral is similar to quadrilateral PQRS. If they are similar, define the dilation required, including the center and the scale factor. If they are not similar, explain why. Suppose that the next picture the artist wants to frame is a rectangle with dimensions l and w. She will again use a 1-inch frame. What must be true about l and w if the outside edge of the frame is similar to the outside edge of the picture? Explain your answer. 61

62 66. Line segments RU and TV intersect at point S. The diagram shows the two triangles formed by drawing line segments RV and TU. The lengths of the sides of are a, b, and c units, and the length of is ka units. What must the lengths of and be for and to be similar? Explain your answer. Given: and are similar. Determine the similarity transformation(s) that will take to Explain why the resulting image of lies in the same position as Given: and are similar. A line segment congruent to is drawn in parallel to The end points P and Q create as shown in the diagram. Give an informal proof or explanation that shows 62

63 67. On the coordinate plane below, circle A is centered at Circle B is centered and is inscribed in The coordinates for one point on each circle are given. A new triangle, is drawn such that circle A is inscribed in and What scale factor transforms to Show your work and explain your answer. If the area of is 12.2 square units, what is the area of Show your work. Points D and H are on circle A. What is the measure of Explain your answer. Part D Point F is on and point G is on both circle B and The measure of What is the measure of Show your work and explain your answer. 63

64 68. Triangle RST is graphed in the coordinate plane. PBA_acc (PBA-_acc) The transformation produces triangle Describe this transformation in words. What are the vertex coordinates of triangle Triangle undergoes a transformation that produces triangle with coordinates and What transformation was applied to triangle to produce triangle Explain your answer. Apply the transformation to triangle Write the vertex coordinates and describe in words how this transformation changes the position of triangle Part D Of triangle RST, and which are congruent? Use the definition of congruence in terms of rigid motions to show that these triangles are congruent. 64

65 69. The coordinates of the vertices of quadrilateral ABCD are and What type of quadrilateral is ABCD? Justify your answer. What are the lengths of the diagonals of ABCD? Show your work. What are the coordinates of the midpoint of each diagonal? Show your work. Part D Andre states that the diagonals of ABCD bisect each other. Using your answers to Parts B and C, explain whether Andre s statement is correct or incorrect. 65

66 70. Figure WXYZ is graphed on the coordinate plane. PBA_acc (PBA-_acc) Name a transformation consisting of one rotation about the origin followed by two reflections across the axes that carries figure WXYZ onto itself. Use coordinates to show this is true. Explain your answer. Determine whether the opposite sides of figure WXYZ are congruent. Show your work. Determine whether the opposite angles of figure WXYZ are congruent. Show your work. Part D Determine whether the diagonals of figure WXYZ are congruent. Show your work. Part E Based on Parts A D, what is the best classification of figure WXYZ? Explain your answer. 66

67 71. In the diagram, and intersect at point T, and is parallel to Provide a detailed explanation why the proportion is true in this situation. Explain why the equation given in can be written as Triangle RST is rotated about point T to create its image as shown in the diagram below. Explain why the length of is and why the length of is Part D Is parallel to Justify your answer. 67

68 72. The coordinate plane shows the graph of with vertices and When a dilation is performed on the resulting image is The center of the dilation is and the scale factor is What are the coordinates of points Explain your answer. Determine whether or not is congruent to Explain your answer. Is the line that contains parallel to the line that contains Explain your answer. Part D Suppose that the center of the dilation is and the scale factor remains What is the relationship between and Explain your answer. 68

69 73. The circle graphed below is tangent to the y -axis and to lines It is tangent to the line at point Copy the given circle on a coordinate grid and draw the diameter that intersects the circle at point R. Label the coordinates of the endpoints of the diameter. Explain how you determined the second endpoint of the diameter. Draw another diameter perpendicular to the diameter you drew for and label the endpoints with their coordinates. Explain how you determined the endpoints of this diameter. Draw an inscribed square in the circle using point R as one vertex. Part D Prove that the figure you inscribed in the circle is a square. 69

70 74. The diagram shows with line segment QU parallel to side RT. Use the diagram to complete each part. Determine the value of x in the diagram. Explain your work. Determine the measure of Round your answer to the nearest tenth of a degree. Explain your work. Is Part D Is Explain your answer using appropriate postulates, theorems, and definitions. Explain your answer using appropriate postulates, theorems, and definitions. 70

71 75. Triangle has vertices at Determine the coordinates of D, the midpoint of and E, the midpoint of Show the computations you used to determine these coordinates. Prove that the line segment joining the midpoints of two sides of the triangle is parallel to the third side of the triangle. Prove that divides and proportionally. 71

72 76. In the figure shown, N is the midpoint of and PBA_acc (PBA-_acc) Describe a rigid transformation that could carry onto Prove that referring to your answer in and congruence postulates for triangles. Prove Part D Assuming what is true about the angles of the triangles? Can this be used to prove that two triangles are congruent? Explain your answer. 72

73 77. Triangle ABC is graphed as shown with Triangle ABC is translated such that is the origin. Determine the coordinates of its image, then dilated by a scale factor of 2. The center of dilation Show your work. Prove without using the slope formula. Use the work from to verify that by Angle-Angle Similarity. Explain your answer. Part D Triangle is dilated by a scale factor of with the center of dilation at vertex C. The resulting image is What are the coordinates of the vertices of Explain your answer. 73

74 78. The scatter plot shows the relationship between the popcorn sales and the bottled water sales for each basketball game. Which statement can be concluded from the data? A. The data has a weak correlation and demonstrates causation. B. The data has a strong correlation and demonstrates causation. C. The data has a weak correlation and does not demonstrate causation. D. The data has a strong correlation and does not demonstrate causation. 79. A group of 7 senior citizens is traveling together on a bus tour. Their mean age is 81.4 years, and the standard deviation is 2.4 years. The bus driver is 25 years old. If the bus driver s age is included in the data set, which statement could be true? A. The new mean is 50% less. B. The new mean is 50% greater. C. The new standard deviation is 50% less. D. The new standard deviation is 50% greater. 74

75 80. The dot plot shows Kim s quiz scores for last semester. The maximum score on each quiz is 10. The table shows Kim s quiz scores for this semester. Create a dot plot for Kim s quiz scores for this semester. What is the difference between the means of the two data sets? Show your work. Kim takes two more quizzes this semester. The mean for this semester s quiz scores decreases by 1 from what is shown in the table. What could be Kim s scores on these two quizzes? Show your work or explain your answer. 75

76 81. The two-way frequency table shows the adoption data for an animal shelter for the first five months of a year. What is the difference between the interquartile ranges of the number of cats adopted and the number of dogs adopted during the first five months? Show your work or explain your answer. Create a conditional relative frequency table for this data. Explain the values you used in your table. 76

77 82. The table shows the amounts Anita earned for babysitting. A linear model for the data is in the context of the given data? What do the slope and y -intercept of the linear model represent Analyze the residuals for the data shown in the table. Show your work and explain your answer. 77

78 83. Kei is reading a book for a book report. He records the total number of pages he has read at the end of each day. The line of best fit for Kei s reading data is where y represents the total number of pages Kei has read in terms of x, the number of days Kei has read the book. What are the slope and y -intercept of the given function? Explain the meaning of the slope and y -intercept within the context of Kei s reading data. The residual for each x -value is shown in the table. PBA_acc (PBA-_acc) Using the residuals in the given table, make another table that shows the y- values that are paired with the x- values. Sketch a coordinate plane and create a scatter plot using the data you generated. Then graph the line of best fit on the same coordinate plane. Remember to title your graph and label the axes. 78

79 84. The table shows the number of points Tatiana scored per game and the total number of points scored by the whole team for each of those games. Write an equation for the line of best fit. Determine the correlation coefficient for the data, rounded to the nearest hundredth. Explain what the correlation coefficient implies about the relationship between the number of points Tatiana scores and the total number of points the team scores in a game. Explain what the slope and y -intercept of the line of best fit represent in this context. 79

80 85. Jeff collected information from 100 cities in the United States. He plotted the average monthly temperature for September ( x ) and the total number of inches of rainfall in September ( y ). Using his calculator, Jeff determined the equation for a line of best fit. The correlation coefficient for his data is What information does the correlation coefficient give about the strength of the linear relationship between temperature and rainfall? Explain your answer. How useful is Jeff s line of best fit for predicting rainfall from temperature? Explain your answer. Jeff switched the x and y values for all of the ordered pairs of his original data and plotted the results. How does the r of this new data set compare with his original r? Explain how you know. Part D After Jeff switched the x and y values for the original data and plotted the results, is the line of best fit of this new data set the same as his original line of best fit? Explain how you know. 80

81 86. A car dealer summarizes his business by recording the number of customers who purchased or leased cars. He also records the size of the car (small, midsize, and full-size). The table shows his summary for one week. Regardless of whether a customer purchased or leased a car, which size was most preferred? Justify your answer using the information in the table. Of the people who purchased a car, what trend (in terms of car size) appears to be true for that week? Explain your reasoning. What association do you see between the type of sale (purchased or leased) and the size of the car? Justify your answer using the information in the table. 81

82 87. Several students completed a paper-and-pencil maze multiple times in an experiment. Their times, in seconds ( y ), were plotted against the number of previous attempts they had made ( x ) in a scatter plot as shown. A linear model that describes the data is given by the equation Interpret the slope of the line in terms of the context. Determine the y -intercept of the line. Explain what the y -intercept represents in terms of the context. Determine the x -intercept of the line. What is the significance of the x -intercept in terms of the context? Explain your answer. 82

83 88. All of a city s schools participated in a food-collection drive to stock the community food pantry. Each school reported the number of pounds of food collected. The histogram shows the results of the community food drive. In which interval is the median number of pounds collected by the schools located? Explain your answer. The same schools participated in the food-collection drive the previous year. The box plot gives the results for the previous year. Estimate the difference, in pounds, between the medians of the two sets of data. Could the two medians be equal? Justify your answer. With the information you have from the box plot, create a histogram that could represent the data from the previous year. Use the same weight intervals that are used in the given histogram. Remember to title the new histogram and label your axes. Explain your strategy for determining the frequencies you used for the weight intervals in your histogram. 83

84 89. Jelena is considering buying a used car from two different dealerships. The list shows the prices of the cars Jelena is considering buying from Dealership A. The box plot displays the prices of the 7 cars Jelena is considering buying from Dealership B. Draw a box plot to represent the prices of the cars Jelena is considering buying from Dealership A. What is the difference in the interquartile ranges of the two data sets? Show your work. For which dealership is the mean of the prices of the cars that Jelena is considering buying greater? Explain your answer. Part D What is the median car price for each dealership? Compare the medians and means for each dealership. Which best represents the prices at each dealership, the mean or the median? Explain your answer. 84

85 90. The table shows the number of bacteria in a culture at different times during an experiment. What was the number of bacteria at the beginning of the experiment? Show your work or explain your answer. Write an exponential equation that fits the data for n, the number of bacteria, in terms of h, the number of hours since the experiment began. Use your equation from to find the number of bacteria 4 hours after the experiment began. Show your work. Part D Sketch a graph of the equation that represents the data for the first 4 hours of the experiment. Label the scales for both axes. Include an appropriate title for the graph and for each axis. 85

86 91. The scatter plot shows Grace s distance, in miles, from her destination at the end of each hour during her car trip. Write a linear function that fits the data shown in the scatter plot. Explain how you determined your function. Identify and interpret the slope, x -intercept, and y -intercept of the linear function in the context of the situation. Define the domain of the function that best fits this situation. Explain your answer. 86

87 92. Incoming freshmen at a community college can take a placement test to determine if they are eligible for a college Calculus class. A student needs to score 50 or more points to enroll in Calculus I. Students scoring less than 50 points must enroll in a Pre-Calculus course first. The box plots show the data collected from the scores of 24 females and 24 males who took a recent placement test. Is the interquartile range (IQR) for females greater, less, or equal to that of males? Explain your answer. Estimate the median score for females and the median score for males. Justify your estimates. Use the shape of the distribution to estimate the mean score for females. Explain your answer. Part D Use the shape of the distribution to estimate the mean score for males. Explain your answer. Part E One of the 48 students selected at random is eligible to enroll in Calculus I based on the student s score. What is the probability that the randomly-selected student is a male? Explain your reasoning. 87

88 93. A distribution of the weekly earnings, in dollars, of students in a statistics class is shown in the histogram. In which interval is the median of this distribution located? Explain your answer. If the data set is graphed as a box-and-whisker plot, describe how the upper and lower whiskers would compare with each other. One of the students made a dot plot of this data set. Explain what information a dot plot would provide that is not provided by the histogram. Part D One of the students was absent the day the data was collected, and his weekly salary of $200 is added to the data set. Explain what effect, if any, this new data point has on the median. 88