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This page shows questions in the Alg II Modeling public release module at MSDE. Algebra 2
"Alg II Modeling"

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Expand All Official Practice Test

This is a multiple choice question that allows you to select only one option.

An account that pays 0.5% interest was opened with an initial deposit, and no additional deposits to or withdrawals from the account were made. The equation y=1000(1.005)4x models the amount, in dollars, in the account x years after it was opened.

The graph of the model and the graph of the equation y=1500 are shown in the xy-coordinate plane.

The intersection of the two graphs is marked with the point P.

Graph. The figure shows the graph of a curve and a line, in the x y plane. The numbers 0, through 40, in increments of 5, are indicated on the x axis. The numbers 0, through 2,000, in increments of 500, are indicated on the y axis. The curve begins on the y axis at 1,000, and moves upward, and to the right, through the plane, and passes through the point with coordinates approximately 35, comma, 2,000. The line begins on the y axis at 1,500 and moves horizontally, to the right, through the plane. The curve and the line intersect at point P, with the coordinates 20 comma, 1,500. End figure description.

What is revealed by the coordinates of the point P?

  1. A.
    the yearly interest rate for the account
  2. B.
    the amount of the initial deposit in the account
  3. C.
    the number of years it takes for the account value to reach $1500
  4. D.
    the number of years it takes for the initial amount in the account to double
i3

Correct i1

Thinks interest rate is revealed by the point i2

Thinks initial deposit is revealed by the point i4

Thinks amount of time to double is revealed by the point

This is a fill in the blank question that allows you to enter only numbers.

For the two pendulums shown, a weight swings back and forth from left to right. Both pendulums start at the left position at time t=0 seconds.

Diagram. The figure shows a diagram of two pendulums, pendulum Q and pendulum R. Each pendulum consists of a weight, suspended from a horizontal surface. The distance between the weight and the surface it is attached to, is greater for pendulum Q than for pendulum R. End figure description.

For each pendulum, the horizontal distance, in inches, of the weight from the center position at time t seconds is given by functions Q and R, respectively. A negative distance represents the weight being to the left of the center position, and a positive distance represents the weight being to the right of the center position.Equation 1: Q of t, equals negative 6 cosine of, open parenthesis, pi t close parenthesis. Equation 2: R of t, equals negative 4 cosine of, open parenthesis, four thirds pi t, close parenthesis.

The maximum horizontal distance of pendulum Q to the right of its starting position is how much greater, in inches, than the maximum horizontal distance of pendulum R to the right of its starting position?

Enter your answer in the space provided.

inches

This is a test question that allows you to enter extended text in your response.

The number of members, m, that a new fitness center expects to have t months after the opening of the center is shown in the scatter plot.

Fitness Center Members

The figure shows a scatter plot of 10 points in a coordinate plane. The horizontal t axis is labeled Months After Opening, and the numbers 0 through 10, in increments of 2, are indicated. The vertical m axis is labeled Number of Members, and the numbers 0 through 300, in increments of 50, are indicated. The coordinates of the points are as follows. Note that all values are approximate.1 month, 75 members.2 months, 140 members.3 months, 180 members.4 months, 210 members.5 months, 240 members.6 months, 250 members.7 months, 260 members.8 months, 270 members.9 months, 280 members.10 months, 290 members.

The fitness center constructed the model m of t, equals 300 minus 280, open parenthesis, 0.76, close parenthesis, raised to the power of t to project the number of members that can be accommodated t months after opening.

  • What is the average rate of change, in members per month, in the number of members the fitness center expects to have from the 4th to the 8th month after opening? Show your work.
  • What is the significance of the number 300 in the model? Explain your answer.

Enter your answers and your work and explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (37 files were searched.)

Holistic Rubric for 4-Point Modeling Constructed Response Items

Sample Top Score Response

An estimate for the number of members in the 4 t h month is 211 . An estimate for the number of members in the 8 t h month is 273 .

Average rate of change: 273 − 211 8 − 4 = 62 4 = 15 1 2 members per month

The expression 280 ( 0.76 ) t approaches zero as t increases. So, 300 − 280 ( 0.76 ) t approaches 300 as t increases. Therefore, 300 is the maximum number of members.

Scoring

Points

Sample Characteristics

4 Points

A four-point response provides full and complete evidence of the modeling process used to solve a real-world problem.

The response may:

  • identify the problem that needs to be solved.
  • determine information that is needed to solve the problem.
  • communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and essentially precise representations.
  • contain minor flaws that do not detract from correct modeling or demonstration of a thorough understanding.
  • evaluate or validate a partial or complete solution and show how to improve or refine the solution.

3 Points

A three-point response provides evidence of the modeling process used to solve a real-world problem.

The response may:

  • identify most of the problem that needs to be solved.
  • determine most of the information that is needed to solve the problem.
  • communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and precise representations with minor flaws.
  • evaluate or validate a partial or complete solution and show how to improve or refine the solution, but the improvement or refinement may include minor flaws.

2 Points

A two-point response provides partial evidence of the modeling process used to solve a real-world problem.

The response may:

  • partially identify the problem that needs to be solved.
  • determine some of the information that is needed to solve the problem.
  • include a partial solution path that may be incomplete.
  • contain some errors in identifying the mathematics that is needed to solve the problem.
  • evaluate or validate a partial or complete solution and attempt to improve or refine the solution.

1 Point

A one-point response provides limited evidence of the modeling process used to solve a real-world problem.

The response may:

  • partially or incorrectly identify the problem that needs to be solved.
  • determine a minimal amount of the information that is needed to solve the problem.
  • include an incomplete or unorganized solution path.
  • contain errors in identifying the mathematics that is needed to solve the problem.
  • contain the correct solution, but work is limited or missing.
  • evaluate or validate a partial or complete solution but does not show how to improve or refine the solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a multiple choice question that allows you to select only one option.

The students in a class are analyzing several examples of spirals in nature.

The students measured the distances shown in the picture for different shells.

Diagram. The figure shows a diagram of a shell with three colinear segments that increase in length with each rotation. The students measured the shell at approximately one half rotation, one and one half rotations, and two and one half rotations. End figure description.

Each student measured the three distances shown in the picture by starting from the outside of the shell and moving toward the center.

One student’s measurements were 8.1 centimeters, 4.2 centimeters, and 2.1 centimeters. A second student’s measurements of a different shell were 7.6 centimeters, 3.7 centimeters, and 1.8 centimeters.

Which equation could be used to model the students’ data, where a represents the first measurement, in centimeters, and d represents the nth measurement, in centimeters, where n=1, 2, or 3?

  1. A.
    d equals a open parenthesis one-half close parenthesis to the power of n minus 1
  2. B.
    d equals a open parenthesis 2 to the power of n minus 1 close parenthesis
  3. C.
    d equals a open parenthesis one-half close parenthesis to the power of n
  4. D.
    d equals a open parenthesis 2 to the power of n close parenthesis
i1

Correcti2

Uses wrong basei3

Uses wrong exponenti4

Uses wrong base and exponent

This is a question with 2 parts, including a test question that allows you to enter extended text in your response.

The table shows the box office revenue, in millions of dollars, for the first 10 weeks after a movie opened in theaters.

Box Office Revenue for a Movie
Number of Weeks after Opening Box Office Revenue (in millions)
0 $85.63
1 $51.97
2 $37.21
3 $25.20
4 $15.22
5 $11.17
6 $7.02
7 $4.03
8 $4.42
9 $1.95

Part A

Create a function to model the data. Explain why you chose the type of function and explain the meaning of the parameters of the function.

Enter your answer and your explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Part B

The theaters will stop showing the movie when the weekly box office revenue is below $100,000. In what week does your model predict that the theaters will stop showing the movie?

Show your work or explain how you found your answer.

Enter your answer and your work or explanation in the space provided. You may also use the drawing tool to help explain or support your answer.

Drawing Box

There is no alt-text. (37 files were searched.)

Holistic Rubric for 4-Point Modeling Constructed Response Items

Sample Top Score Response

Part A

The function fx=81.670.67x models the data. An exponential function was chosen because the data seems to decrease rapidly at first, then level off a bit.

The constant 81.67 represents the box office revenue, in million dollars, predicted by the function 0 weeks after the movie opened, that is, during the movie’s opening week.

The constant 0.67 means that the revenue is decreasing on average by 1-0.67=0.33 or 33% each week.

Part B

100,000 is 0.1 million, so the time when the function has a value less than 0.1 should be determined. By graphing y=81.670.67x and y=0.1 on the same graph, it can be seen that the least number of weeks after the movie opened when the function value is less than 0.1 is 17.

Points

Sample Characteristics

4 Points

A four-point response provides full and complete evidence of the modeling process used to solve a real-world problem.

The response may:

  • identify the problem that needs to be solved.
  • determine information that is needed to solve the problem.
  • communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and essentially precise representations.
  • contain minor flaws that do not detract from correct modeling or demonstration of a thorough understanding.
  • evaluate or validate a partial or complete solution and show how to improve or refine the solution.

3 Points

A three-point response provides evidence of the modeling process used to solve a real-world problem.

The response may:

  • identify most of the problem that needs to be solved.
  • determine most of the information that is needed to solve the problem.
  • communicate an accurate, organized solution path that is aligned to the problem using appropriate, effective, and precise representations with minor flaws.
  • evaluate or validate a partial or complete solution and show how to improve or refine the solution, but the improvement or refinement may include minor flaws.

2 Points

A two-point response provides partial evidence of the modeling process used to solve a real-world problem.

The response may:

  • partially identify the problem that needs to be solved.
  • determine some of the information that is needed to solve the problem.
  • include a partial solution path that may be incomplete.
  • contain some errors in identifying the mathematics that is needed to solve the problem.
  • evaluate or validate a partial or complete solution and attempt to improve or refine the solution.

1 Point

A one-point response provides limited evidence of the modeling process used to solve a real-world problem.

The response may:

  • partially or incorrectly identify the problem that needs to be solved.
  • determine a minimal amount of the information that is needed to solve the problem.
  • include an incomplete or unorganized solution path.
  • contain errors in identifying the mathematics that is needed to solve the problem.
  • contain the correct solution, but work is limited or missing.
  • evaluate or validate a partial or complete solution but does not show how to improve or refine the solution.

0 Point

A zero-point response is completely incorrect, incoherent or irrelevant.

This is a test question that allows you to enter extended text in your response.

The world population in 1997 was 5.88 billion.

The world population in 2017 was 7.53 billion.

Assume that the ratio of the populations in any two consecutive years was constant from 1997 to 2017.

Write an equation of the form b equals a open parenthesis r to the power of n close parenthesis, where a, b, and n are constants, that can be used to find r, the rate of growth per year of the world population.

Enter your equation in the space provided. Enter only your equation.