Algebra 2 "Algebra II 2024" - Public Release

Shell Measurements
n	Student A	Student B
1	8.1 cm	7.6 cm
2	4.2 cm	3.7 cm
3	2.1 cm	1.8 cm

Which number is equivalent to $open parenthesis, 5 plus i, close parenthesis, squared$ , where $i$ is the imaginary unit?

Select one answer.

i2 i1 i3 i4

What are the solutions of the equation $\frac{x}{x + 6} = \frac{2 x - 3}{x + 12} ?$

Select all that apply.

KEY i7

KEY i1

Value that makes the right side of the equation undefined i2

Value that makes the left side of the equation undefined i4

Value that makes the left side of the equation equal to $0$ i5

Value that makes the right side of the equation equal to $0$ i6

Student solves and factors to $0 = (x - 6) (x + 3)$ but uses $3$ instead of $- 3$ as solution

The recursive rules for two sequences are shown in the table, where $n$ represents a positive integer.

For each of the two sequences, identify the equation that represents the explicit rule.

Drag and drop an equation into each box.

Recursive Rule	Explicit Rule
f of 1 equals 5, f of n equals 3 plus f of open parenthesis n minus 1 close parenthesis, for n greater than 1
f of 1 equals 5, f of n equals 3 times f of open parenthesis n minus 1, for n greater than 1

The binomial $open parenthesis, x minus 2, close parenthesis$ is a factor of the polynomial function $P (x) = x^{3} - x^{2} + a x - 6,$ where $a$ is a constant.

What is the value of $a ?$

Enter your answer in the space provided.

$a =$

Function $f$ is defined as follows.

$f (x) = {(0.5)}^{2 x}$

Describe how the value of $f (x)$ is changing.

Select from the drop-down menus to correctly complete the statement.

For every increase of 1 in the value of the value of

f (x)

by .

An expression is shown.

${(\frac{64}{27})}^{- \frac{1}{3}}$

Use the properties of exponents to rewrite the expression.

Enter your answer as a fraction in the spaces provided.

fraction bar

The function $f$ is defined by $f (x) = 2 x + 5$ .

If $f (a) = c$ , which of the following equations gives the value of $a$ in terms of $c$ ?

Select one answer.

The expression x cubed minus 3 x squared minus 4 x plus 12 can be written as open parenthesis x squared minus a close parenthesis open parenthesis x minus b close parenthesis, where a and b are constants.

Determine the values of $a$ and $b$ and indicate them on the number line shown.

Select the places on the number line to plot the points.

Which value of $x$ makes the equation $5 {(e)}^{4 x} = 25$ true?

Correcti1

This option is the result of dividing 5 by 4 within the natural logarithm instead of dividing $\ln 5$ by 4.i2

This option is the result of thinking that 2 should be used within the natural logarithm since $5^{2} = 25 .$ i4

This option is the result of taking the natural logarithm of both sides of the equation $5 {(e)}^{4 x} = 25$ and thinking the result is $5 \cdot 4 x \ln e = \ln 25,$ or $20 x = \ln 25 .$

The graph of a polynomial function is shown in the following xy‑plane.

Which equation could represent the graph of the polynomial function?

Select one answer.

Right triangle A B C is shown in the xy-coordinate plane. Point B has coordinates 8, 6.

What are the values of sine of theta and sine of pi plus theta?

Enter your answers in the spaces provided.

sine of theta equals

sine of pi plus theta equals

Order the quantities from least to greatest.

Drag and drop the quantities into the correct order.

Least		Greatest

The graph of the polynomial function $f$ is shown in the xy-coordinate plane.

Which of the equations shown could define $f ?$

Correct i2

Thinks the function is the negative of the correct function i3

Thinks the function has zeros that are the negatives of the correct zeros i4

Thinks the function has zeros that are the negatives of the correct zeros and has the negative of the function

The table shows the percent of US households with computers for selected years between $1984$ and $2003 .$

Year	Percent
Percent of Households with Computers
$1984$	$8.2$
$1989$	$15.0$
$1993$	$22.9$
$1997$	$36.6$
$2000$	$51.0$
$2001$	$56.3$
$2003$	$61.8$

Which of the equations shown best models the data in the table, where $y$ is the percent of households with computers and $x$ is the number of years after $1980?$

The functions $f$ and $g$ are defined as shown.

$f (x) = x^{3} + x^{2} - 2 x$

$g (x) = 0.5 x^{2} + 1$

The values of $x$ for which $f (x) = g (x)$ lie in which of the intervals shown?

Select all that apply.

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 $y = 1000 {(1.005)}^{4 x}$ 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 $y equals 1500$ are shown in the xy-coordinate plane.

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

What is revealed by the coordinates of the point P?

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

Claim: There is no quadratic equation with real coefficients for which x equals negative 5 I is a solution.

Part A

Show that the claim is not correct by providing an example of a quadratic equation for which x equals negative 5 I is a solution. Include any other solutions to your equation in your answer.

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

Is it possible for a quadratic equation with real coefficients to have x equals negative 5 I as its only solution? Justify your answer.

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

Drawing Box

Holistic Rubric for 4-Point Reasoning Constructed Response Items
Sample Top Score Response Part A A quadratic equation with real coefficients that has $x = â 5 i$ as a solution must also have $x = 5 i$ as a solution. One such equation is $(x â 5 i) (x + 5 i) = 0,$ which is equivalent to $x^{2} + 25 = 0 .$ Part B There is no quadratic equation with real coefficients that has $x = â 5 i$ as its only solution. If the only solution is $x = â 5 i,$ then the quadratic equation is a multiple of ${(x + 5 i)}^{2} = 0,$ which is equivalent to $x^{2} + 10 i x â 25 = 0,$ and that equation cannot be equivalent to one with real coefficients because $\frac{10 i}{â 25}$ is not a real number. Scoring
Points	Sample Characteristics
4 Points	A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning. The response may: be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary. demonstrate a thorough understanding of the mathematics. contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.
3 Points	A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning. The response may: be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary. contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.
2 Points	A two-point response for reasoning items provides evidence of partially correct mathematical reasoning. The response may: display an incomplete reasoning process. contain mathematical flaws.
1 Point	A one-point response for reasoning items provides limited evidence of correct mathematical reasoning. The response may: demonstrate the beginning of a valid chain of reasoning. reflect a lack of essential understanding of the underlying mathematical concepts. contain the correct solution, but work is limited or missing. contain errors in the fundamental mathematical procedures or reasoning. contain omissions or irregularities that lead to an inadequate solution.
0 Point	A zero-point response is completely incorrect, incoherent or irrelevant.

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

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

Consider the polynomial function $P,$ defined by P of x equals x cubed plus C X squared plus x plus 2, where $c$ is a constant, and consider the graph shown in the xy-coordinate plane.

If P of negative 2 equals 0, can the graph shown be the graph of y equals P of x?

Correct.i1

Incorrect conclusion and supporting factsi2

Incorrect conclusion and some incorrect supporting factsi3

Correct conclusion but incorrect supporting facts

When a customer pays with a debit card at a clothing store, the owner pays a processing fee. The function $P$ represents the processing fee, in dollars, for a debit card purchase of $x$ dollars.

$P (x) = 0.035 x + 0.15$

Complete the statement to describe how the processing fee is calculated based on the purchase amount.

Select from the drop-down menus to correctly complete the statement.

The processing fee consists of a fixed amount of plus of the purchase amount.

An equation is shown.

$\sqrt{x^{2} + 19} - 100 = 0$

What are the solutions of the equation?

A.

$x equals 9$ only
B.

$x = 3 \sqrt{1109}$ only
C.

$x = - 9$ and $x equals 9$
D.

$x = - 3 \sqrt{1109}$ and $x = 3 \sqrt{1109}$

The function $C$ approximates the number of calories recommended per day for a dog, based on the weight, in pounds, of the dog.

The table gives values of $C$ for selected values of $p,$ where $C (p)$ represents the recommended number of calories per day for a dog that weighs $p$ pounds.

$p$ (pounds)	$C (p)$
Recommended Number of Calories for Dogs
$11$	$250$
$22$	$450$
$33$	$750$
$44$	$1000$
$55$	$1250$
$66$	$1500$
$77$	$1700$
$88$	$1880$

Which of the equations shown most closely approximates $C ?$

Claim: $\sqrt[3]{x} \leq x,$ where $x$ is a real number.

Determine whether the claim is true for values of $x$ in each interval.

Select only one box per row.

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 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 $4 th$ to the $8 th$ 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

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: $\frac{273 â 211}{8 â 4} = \frac{62}{4} = 15 \frac{1}{2}$ members per month The expression $280 {(0.76)}^{t}$ approaches zero as $t$ increases. So, $300 â 280 {(0.76)}^{t}$ 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.

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.

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 $n th$ measurement, in centimeters, where $n equals 1,$ $2,$ or $3?$

Correcti2

Uses wrong basei3

Uses wrong exponenti4

Uses wrong base and exponent

Consider the functions $P$ and $Q,$ defined as shown.Equation 1: P of x, equals x squared plus 7 x, minus 14. Equation 2: Q of x, equals negative 3 x plus 10.

In the xy-coordinate plane, what are the coordinates of the points at which the graphs of the equations $y = P (x)$ and $y = Q (x)$ intersect?

Explain how you determined your answer.

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

Drawing Box

Holistic Rubric for 4-Point Reasoning Constructed Response Items
Sample Top Score Response The two graphs intersect at the points $(x, y) = (â 12, 46)$ and $(x, y) = (2, 4) .$ The $x$ coordinates of the points of intersection of the graph are those points for which $P (x) = Q (x) .$ $P (x) = Q (x) x^{2} + 7 x - 14 = - 3 x + 10 x^{2} + 10 x - 24 = 0 (x + 12) (x - 2) = 0$ $x = â 12$ or $x = 2$ If $x = â 12,$ then $y = (â 3) (â 12) + 10 = 36 + 10 = 46$ If $x = 2,$ then $y = (â 3) (2) + 10 = â 6 + 10 = 4$ Therefore, the points of intersection are $(x, y) = (â 12, 46)$ and $(x, y) = (2, 4) .$
Points	Sample Characteristics
4 Points	A four-point response for reasoning items provides evidence of correct, complete, and appropriate mathematical reasoning. The response may: be clear and well developed with logical reasoning communicated by the use of precise and appropriate representations, symbols, drawings, or mathematical vocabulary. demonstrate a thorough understanding of the mathematics. contain minor flaws that do not detract from the correct reasoning or demonstration of a thorough understanding.
3 Points	A three-point response for reasoning items provides evidence of essentially correct, essentially complete, and essentially appropriate mathematical reasoning. The response may: be clear and developed with logical reasoning communicated by the use of essentially precise and appropriate representations, symbols, drawings, or mathematical vocabulary. contain minor flaws that detract from the correct reasoning or demonstration of a thorough understanding.
2 Points	A two-point response for reasoning items provides evidence of partially correct mathematical reasoning. The response may: display an incomplete reasoning process. contain mathematical flaws.
1 Point	A one-point response for reasoning items provides limited evidence of correct mathematical reasoning. The response may: demonstrate the beginning of a valid chain of reasoning. reflect a lack of essential understanding of the underlying mathematical concepts. contain the correct solution, but work is limited or missing. contain errors in the fundamental mathematical procedures or reasoning. contain omissions or irregularities that lead to an inadequate solution.
0 Point	A zero-point response is completely incorrect, incoherent or irrelevant.

The approximate number of cars parked in a garage $x$ hours after $6 : 00$ a.m. can be modeled by the function $y = 180 + 130 \sin (\frac{x}{12} π),$ where $0 is less than or equal to x, which is less than or equal to 16.$

For each time interval, determine whether the number of cars parked in the garage increased or decreased, based on the model.

Select the boxes to identify each description of the function. Select only one box per row.

The half-life of the isotope Francium‑223 is 22 minutes. For a sample that starts with $F$ grams of Francium‑223, the expression $F {(0.5)}^{\frac{m}{22}}$ represents the number of grams remaining after $m$ minutes.

Which expression represents the number of grams of Francium‑223 remaining after $h$ hours?

KEYi2

Student uses $\frac{h}{60}$ as the exponent because there are $60$ minutes in an houri3

Student uses $60 h$ as the exponent because there are $60$ minutes in an houri4

Student uses ${0.5}^{\frac{1}{22}}$ as the base

Angle $θ$ is positioned in a circle with its initial ray on the positive x‑axis and its terminal ray $1 third$ of one complete revolution about the origin, as shown in the following diagram.

What is the measure of angle theta, in degrees and in radians?

Select from the drop-down menus to correctly complete the statement.

The measure of

θ

is degrees, which is equivalent to radians.

Claim: For every value of the constant $c,$ the equation $x^{4} + (1 - c^{2}) x^{2} - c^{2} = 0$ has at least two distinct real solutions.

What value of $c$ refutes the claim?

Enter your answer in the space provided.

$c =$

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

Number of Weeks after Opening	Box Office Revenue (in millions)
Box Office Revenue for a Movie
$0$	$$ 85.63$
$1$	$$ 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

Holistic Rubric for 4-Point Modeling Constructed Response Items
Sample Top Score Response Part A The function $f (x) = 81.67 {(0.67)}^{x}$ 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 $y = 81.67 {(0.67)}^{x}$ 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.

Consider the equation $a = \sqrt{x + b},$ where $a$ and $b$ represent real numbers.

Which statement is true about the number of real solutions to the equation?

KEY i2

Student thinks $b$ affects the number of solutions. i3

Student thinks the relationship between $a$ and $b$ affects the number of solutions. i4

Student thinks the number of solutions does not depend on $a$ and $b .$

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.

Consider the system of equations shown.System of equations. Top row, y equals open parenthesis x minus 3 close parenthesis squared. Bottom row, y equals x minus 1.

On the coordinate plane, plot the points that represent solutions coordinates x, y of the system.

Select the places on the coordinate plane to plot the points.

A kitchen supply store sells a set of $4$ food storage containers.

From largest to smallest, the volumes of the containers are $v sub 1,$ $v sub 2,$ $v sub 3,$ and $v sub 4,$

$v_{n}$ is $20 percent$ less than $v sub n minus 1, end sub,$ where $n$ is $2,$ $3,$ or $4 .$

$v_{n} = k v_{n - 1},$ where $n$ is $2,$ $3,$ or $4 .$

What is the value of $k ?$

Enter your answer as a fraction in the spaces provided.

fraction bar

Interval	The Claim is True	The Claim is False
$x > 1$
$0 is less than x, which is less than 1$
$negative 1 is less than x, which is less than 0$
$x is less than negative 1$

Time Interval	Increased	Decreased
$7:00 a.m.$ to $9:00 a.m.$
$1 : 00 p.m.$ to $4:00 p.m.$
$5:00 p.m.$ to $8:00 p.m.$

This page shows questions in the Algebra II 2024 public release module at MSDE. Algebra 2 "Algebra II 2024"

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This page shows questions in the Algebra II 2024 public release module at MSDE. Algebra 2
"Algebra II 2024"