NJ DOE Unit 2_Grade 3



Name ________________________ Period ______ Date ____________

Algebra II Unit 3 Model Curriculum Assessment

1. For the functions f and g defined below, sketch a graph without the use of a calculator and use the graph to identify the solution set to [pic]

[pic]

[pic]

Solution set: _______________________

2. [pic]

For the functions defined above, fill in the tables of values. Then give the solution set to [pic] Explain your answer.

|x |y1 |y2 |

|2 | | |

|3 | | |

|4 | | |

|5 | | |

|6 | | |

|7 | | |

|8 | | |

|9 | | |

|10 | | |

Solution set: ________________

3. Based on the graph below, which of the following is the solution to the equation [pic]

[pic]

a. [pic]

b. [pic] or [pic]

c. [pic] or [pic]

d. [pic] or [pic]

4. A ball is dropped, and for each bounce after the first bounce the ball reaches a height that is a constant percent of the preceding height. After the first bounce it reaches a height of 10 feet, and after the third bounce it reaches a height of 4.9 feet.

Part A: The height the ball reaches after the nth bounce is represented by [pic] below. Write the value for each [pic] below.

[pic]

Part B: Write an explicit rule for the height after the nth bounce, [pic], where n represents the bounce number.

Explicit rule: ____________________

5. A job pays a salary of $8.50 an hour for the first year and $8.85 an hour for the second year. The hourly salary for year n follows an arithmetic sequence.

Part A: Write a recursive rule for the hourly salary.

Part B: Write an explicit rule for the hourly salary.

6.

[pic]

For the pattern above, the total number of boxes used in Figure n can be described by a geometric sequence. Write a recursive formula to find the number of boxes in Figure n.

Recursive formula: ____________________

7. [pic]

A recursive formula for a sequence is given above. Write an explicit formula for the sequence.

Explicit formula: ____________________

8. Write an expression for the inverse of [pic]

[pic]____________________

9. Which of the following is the inverse function for [pic]

where [pic]

a. [pic]

b. [pic]

c. [pic]

d. [pic]

10. If [pic] find [pic] where [pic] Show your work.

[pic]____________________

11. Describe each of the following key features of the graph of [pic]

Part A: Where does the graph intersect the x-axis?

Part B: Where does the graph intersect the y-axis?

Part C: Is the value of f positive or negative for [pic]

Part D: Is the graph increasing or decreasing when [pic]

12.

[pic]

Use the graph of the function f above to answer each of the following questions.

Part A: What is the value of f when [pic]

Part B: For what values of x is [pic]

Part C: For what values of x is [pic]

Part D: For what values of x is f increasing?

13. The area of a rectangular garden is expressed by the function [pic] where the length of the garden is x feet and the width of the garden is [pic] feet.

Part A: What values of x make sense in the context of the problem?

Part B: Graph the function [pic] in the coordinate plane below for the x-values you identified in Part A.

[pic]

Part C: What are the dimensions of the garden that will result in the maximum possible area?

14.

A sensor is placed on the rim of a tire, as shown in the figure above. The height of the center of the sensor above the center of the tire changes as the tire rotates and is represented by a function [pic] The sensor starts level with the center of the tire and begins rotating counterclockwise at time [pic] seconds. If [pic] reaches a maximum of 1 foot and the sensor makes 3 revolutions per second, which of the following graphs could represent [pic]

a. b.

[pic] [pic]

c. d.

[pic] [pic]

15. Robert is collecting books to donate to the library. The number of books he collects, n, is defined by [pic] where d is the number of days he spends collecting books.

Part A: What does 14 represent in the context of Robert’s book collecting?

Part B: What does 21 represent in the context of Robert’s book collecting?

16. A sample of carbon-14 is placed into a jar. The formula [pic] is used to compute the amount of the substance that will remain in the jar after a certain number of years. Which of the following statements explains what the t in the formula represents?

a. It is the number of years that will have passed after the carbon-14 was placed in the jar.

b. It is the number of years that it takes for half of the initial quantity of carbon-14 to decay.

c. It is the amount of carbon-14 that will have decayed 5,730 years after it was put in the jar.

d. It is the amount of carbon-14 that will remain t years after it was placed in the jar.

17. The population, in thousands, of a certain city can be modeled by the function [pic] where t is the number of years since 2000.

Part A: What was the population of the city in the year 2000 ?

Part B: Describe the rate of change of the city’s population.

18. In the table below, fill in each empty cell with the corresponding exact measure in radians or degrees.

|Angle measure in radians |Angle measure in degrees |

| |45° |

| |90° |

| |200° |

| |450° |

|[pic] | |

|[pic] | |

|[pic] | |

|[pic] | |

19. The top of a door is to be decorated with stained glass panes that are arranged in a semicircular shape as shown below. The radius of the semicircular shape is 1 meter and its outside edge is trimmed with metal cord. The red and blue sectors are trimmed with gold cord and the yellow and green sectors are trimmed with silver cord, as shown in the diagram below.

[pic]

If the angle [pic] in the red and blue sectors measures 0.5 radians, what length of silver cord is needed?

20. A unicyclist rides his unicycle across a stage. The wheel has a diameter of 2 feet, and the distance he rides across the stage is 40 feet. What is the angle, in radians, that the wheel turned in rolling that distance?

21. If [pic] and [pic] then in which quadrant does the terminal side of [pic] lie when it is placed in standard position? What are the values of [pic] and [pic] Explain your reasoning and show your work.

22. If the angle [pic] is placed in standard position, its terminal side lies in quadrant II and [pic] What is the value of [pic]

a. [pic]

b. [pic]

c. 0.75

d. 0.8

23. List or give a rule that finds all values of [pic] in radians, for which [pic]

24. Musical notes can be modeled as trigonometric functions. If the note

A above middle C has a frequency of 440 Hz (1 Hz is 1 cycle per second), which of the following functions could be used to model this note, where t is measured in seconds and a represents the amplitude of the sound wave ?

a. [pic]

b. [pic]

c. [pic]

d. [pic]

25.

[pic]

A mass is attached to a spring, as shown in the figure above. If the mass is pulled down and released, the mass will move up and down for a period of time. The height of the mass above the floor, in inches, can be modeled by the function [pic] t seconds after the mass is set in motion.

The mass is 4 feet above the floor before it is pulled down. It is pulled 3 inches below the starting point and makes one full oscillation in 0.2 second. If the spring is at its lowest point at [pic] which of the following functions models h ?

a. [pic]

b. [pic]

c. [pic]

d. [pic]

26. In Mercerville, New Jersey, the amount of daylight, in hours per day, can be approximated by the function [pic] where t is the number of days since the most recent January 1 (including January 1). Using this approximation, what are the maximum and minimum amounts of daylight throughout the year?

Maximum: _________

Minimum: _________

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