Akwankyerɛ

From the New York State Education Department. The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA II. Internet. Available from https://www.nysedregents.org/algebratwo/123/algtwo12023-exam.pdf; accessed 23, June, 2023.

Part I

Answer all 24 questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question.

2

Which expression is equivalent to (x+2)^{2}-5(x+2)+6?

x(x-1)

(x-3)(x-2)

(x-4)(x+3)

(x-6)(x+1)

2

To the nearest tenth, the solution to the equation 4300e^{0.07x}-123=5000 is

1.1

2.5

6.3

68.5

2

The value of an automobile t years after it was purchased is given by the function V=38,000(0.84)^t . Which statement is true?

The value of the car increases 84% each year.

The value of the car decreases 84% each year.

The value of the car increases 16% each year.

The value of the car decreases 16% each year.

2

Which function represents exponential decay?

p(x)=(\frac{1}{4})^{-x}

q(x)=1.8^{-x}

r(x)=2.3^{2x}

s(x)=4^{\frac{x}{2}}

2

The expression \frac{x^4-5x^2+4x+14}{x+2} is equivalent to

x^3-2x^2-x+6+\frac{2}{x+2}

x^3-5x+4-\frac{14}{x+2}

x^3+2x^2-x+2+\frac{18}{x+2}

x^3+2x^2-9x+22-\frac{30}{x+2}

2

The sum of the first 20 terms of the series -2+6-18+54- ... is

-610

-59

1,743,392,200

2,324,522,934

2

If f(x)=2x^{4}-x^3-16x+8, then f(\frac{1}{2})

equals 0 and 2x+1 is a factor of f(x)

equals 0 and 2x-1 is a factor of f(x)

does not equal 0 and 2x+1 is not a factor of f(x)

does not equal 0 and 2x-1 is a factor of f(x)

2

If (6-ki)^{2}=27-36i, the value of k is

-36

-3

3

6

2

What is the solution set of the equation \frac{x+2}{x}+\frac{x}{3}=\frac{2x^{2}+6}{3x} ?

{-3}

{-3,0}

{3}

{0,3}

2

How many real solutions exist for the system of equations below?
y=\frac{1}{4}x-8
y=\frac{1}{2}x^{2}+2x

1

2

3

0

2

Which equation represents a polynomial identity?

x^3+y^{3}=(x+y)^3

x^3+y^{3}=(x+y)(x^2-xy+y^2)

x^3+y^{3}=(x+y)(x^2-xy-y^2)

x^3+y^{3}=(x-y)(x^2+xy+y^2)

2

Given x>0, the expression \frac{x^{\frac{1}{5}}}{x^{\frac{1}{2}}} can be rewritten as

\sqrt[3]{x}

-{\sqrt[10]{x^3}}

\frac{1}{\sqrt[10]{x^3}}

\sqrt[3]{x^{10}}

2

A cyclist pedals a bike at a rate of 60 revolutions per minute. The height, h, of a pedal at time t, in seconds, is plotted below.
The graph can be modeled by the function h(t)=5\mathrm{sin}(kt), where k is equal to

1

2\pi

60

\frac{\pi}{30}

2

Which statement about data collection is most accurate?

A survey about parenting styles given to every tenth student entering the library will provide unbiased results.

An observational study allows a researcher to determine the cause of an outcome.

Margin of error increases as sample size increases.

A survey collected from a random sample of students in a school can be used to represent the opinions of the school population.

2

If f(x)=\frac{1}{2}x+2, then the inverse function is

f^{-1}(x)=-\frac{1}{2}x-2

f^{-1}(x)=\frac{1}{2}x-1

f^{-1}(x)=2x-4

f^{-1}(x)=2x+2

2

Given f(x)=x^{4}-x^3-6x^2, for what values of x will f(x)>0?

x<-2, only

x<-2 or x>3

x<-2 or 0\leq{x}\leq{3}

x>3, only

2

For which approximate value(s) of x will \mathrm{log}(x+5)=\vert{x-1}-3?

5,1

-2.41,0.41

-2.41,5

5, only

2

Consider a cubic polynomial with the characteristics below.
exactly one real root
as x\rightarrow\infin, f(x){\rightarrow}-\infin
Given a>0 and b>0, which equation represents a cubic polynomial with these characteristics?

f(x)=(x-a)(x^{2}+b)

f(x)=(a-x)(x^{2}+b)

f(x)=(a-x^2)(x^{2}+b)

f(x)=(x-a)(b-x^{2})

2

Betty conducted a survey of her class to see if they like pizza. She gathered 200 responses and 65% of the voters said they did like pizza. Betty then ran a simulation of 400 more surveys, each with 200 responses, assuming that 65% of the voters would like pizza. The output of the simulation is shown below.
Considering the middle 95% of the data, what is the margin of error for the simulation?

0.01

0.02

0.05

0.07

2

If \mathrm{cos}A=\frac{\sqrt{5}}{3} and \mathrm{tan}A<0, what is the value of \mathrm{sin}A?

\frac{2}{3}

-\frac{\sqrt{5}}{3}

-\frac{2}{3}

\frac{3}{\sqrt{5}}

2

A tree farm initially has 150 trees. Each year, 20\% of the trees are cut down and 80 seedlings are planted. Which recursive formula models the number of trees, a_n, after n years?

a_1=150

a_n=a_{n-1}(0.2)+80

a_1=150

a_n=a_{n-1}(0.8)+80

a_n=150(0.2)^{n}+80

a_n=150(0.8)^{n}+80

2

Which equation represents a parabola with a focus of (4,-3) and directrix of y=1?

(x-1)^{2}=4(y+3)

(x-1)^{2}=-8(y-3)

(x+4)^{2}=4(y-3)

(x-4)^{2}=-8(y+1)

2

Mia has a student loan that is in deferment, meaning that she does not need to make payments right now. The balance of her loan account during her deferment can be represented by the function f(x)=35000(1.0325)^{x}, where x is the number of years since the deferment began. If the bank decides to calculate her balance showing a monthly growth rate, an approximately equivalent function would be

f(x)=35,000(1.0027)^{12x}

f(x)=35,000(1.0027)^\frac{x}{12}

f(x)=35,000(1.0325)^{12x}

f(x)=35,000(1.0325)^\frac{x}{12}

1

Which graph shows a quadratic function with two imaginary zeros?

null

Part II

Answer all 8 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.

2

Algebraically determine the zeros of the function below.

r(x)=3x^{3}+12x^2-3x-12

and . Show your work.

2

Given a>0, solve the equation a^{x+1}=\sqrt[3]{a^2} for x algebraically:

Show your work.

2

Given P(A)=\frac{1}{3} and P(B)=\frac{5}{12}, where A and B are independent events, determine P(A\cap{B}).

Show your work.

2

The scores on a collegiate mathematics readiness assessment are approximately normally distributed with a mean of 680 and a standard deviation of 120.

Determine the percentage of scores between 690 and 900, to the nearest percent.

Show your work.

2

Consider the data in the table below.

null

State an exponential regression equation to model these data, rounding all values to the nearest thousandth.

2

Write the expression A(x)\sdot B(x)-3C(x) as a polynomial in standard form.

A(x)=x^{3}+2x-1

B(x)=x^{2}+7

C(x)=x^{4}-5x

Show your work.

2

Over the set of integers, completely factor x^{4}-5x^{2}+4.

Show your work.

2

Natalia's teacher has given her the following information about angle \theta.
\pi<\theta<2\pi
\mathrm{cos}\theta=\frac{\sqrt{3}}{4}
Explain how Natalia can determine if the value of \mathrm{tan}\theta is positive or negative.

Part III

Answer all 4 questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided for each question to determine your answer. Note that diagrams are not necessarily drawn to scale. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.

4

Solve the equation \sqrt{49-10x}+5=2x algebraically.

Show your work.

4

Joette is playing a carnival game. To win a prize, one has to correctly guess which of five equally sized regions a spinner will land on, as shown in the diagram below.

null

She complains that the game is unfair because her favorite number, 2, has only been spun once in ten times she played the game.

State the proportion of 2's that were spun.

Show your work.

State the theoretical probability of spinning a 2.

Show your work.

The simulation output below shows the results of simulating ten spins of a fair spinner, repeated 100 times.

null

Does the output indicate that the carnival game was unfair? Explain your answer.

4

Graph c(x)=-9(3)^{x-4}+2 on the axes in the Show Your Work space.
Describe the end behavior of c(x) as x approaches positive infinity.
Describe the end behavior of c(x) as x approaches negative infinity.

4

The monthly high temperature (\degreeF) in Buffalo, New York can be modeled by B(m)=24.9\mathrm{sin}(0.5m-2.05)+55.25, where m is the number of the month and January = 1.

Find the average rate of change in the monthly high temperature between June and October, to the nearest hundredth.

Show your work.

Explain what this value represents in the given context.

Part IV

Answer the question in this part. A correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. Utilize the information provided to determine your answer. Note that diagrams are not necessarily drawn to scale. A correct numerical answer with no work shown will receive only 1 credit.

6

Objects cool at different rates based on the formula below.

T=(T_{0}-T_{R})e^{-rt}+T_R

T_{0}: initial temperature

T_{R}: room temperature

r: rate of cooling of the object

t: time in minutes that the object cools to a temperature, T

Mark makes T-shirts using a hot press to transfer designs to the shirts. He removes a shirt from a press that heats the shirt to 400°F. The rate of cooling for the shirt is 0.0735 and the room temperature is 75°F. Using this information, write an equation for the temperature of the shirt, T, after t minutes.

Show your work.

Use the equation to find the temperature of the shirt, to the nearest degree, after five minutes.

Show your work.

At the same time, Mark’s friend Jeanine removes a hoodie from a press that heats the hoodie to 450°F. After eight minutes, the hoodie measured 270°F. The room temperature is still 75°F. Determine the rate of cooling of the hoodie, to the nearest ten thousandth.

Show your work.

The T-shirt and hoodie were removed at the same time. Determine when the temperature will be the same, to the nearest minute.

Show your work.

2023 (Jan.): NY Regents - Algebra 2

2023 (Jan.): NY Regents - Algebra 2

Part I

Which expression is equivalent to (x+2)^{2}-5(x+2)+6?

To the nearest tenth, the solution to the equation 4300e^{0.07x}-123=5000 is

The value of an automobile t years after it was purchased is given by the function V=38,000(0.84)^t . Which statement is true?

Which function represents exponential decay?

The expression \frac{x^4-5x^2+4x+14}{x+2} is equivalent to

The sum of the first 20 terms of the series -2+6-18+54- ... is

If f(x)=2x^{4}-x^3-16x+8, then f(\frac{1}{2})

If (6-ki)^{2}=27-36i, the value of k is

What is the solution set of the equation \frac{x+2}{x}+\frac{x}{3}=\frac{2x^{2}+6}{3x} ?

How many real solutions exist for the system of equations below?y=\frac{1}{4}x-8y=\frac{1}{2}x^{2}+2x

Which equation represents a polynomial identity?

Given x>0, the expression \frac{x^{\frac{1}{5}}}{x^{\frac{1}{2}}} can be rewritten as

A cyclist pedals a bike at a rate of 60 revolutions per minute. The height, h, of a pedal at time t, in seconds, is plotted below.The graph can be modeled by the function h(t)=5\mathrm{sin}(kt), where k is equal to

Which statement about data collection is most accurate?

If f(x)=\frac{1}{2}x+2, then the inverse function is

Given f(x)=x^{4}-x^3-6x^2, for what values of x will f(x)>0?

For which approximate value(s) of x will \mathrm{log}(x+5)=\vert{x-1}-3?

Consider a cubic polynomial with the characteristics below.exactly one real rootas x\rightarrow\infin, f(x){\rightarrow}-\infinGiven a>0 and b>0, which equation represents a cubic polynomial with these characteristics?

If \mathrm{cos}A=\frac{\sqrt{5}}{3} and \mathrm{tan}A<0, what is the value of \mathrm{sin}A?

A tree farm initially has 150 trees. Each year, 20\% of the trees are cut down and 80 seedlings are planted. Which recursive formula models the number of trees, a_n, after n years?

Which equation represents a parabola with a focus of (4,-3) and directrix of y=1?

Which graph shows a quadratic function with two imaginary zeros?

Part II

Natalia's teacher has given her the following information about angle \theta.\pi<\theta<2\pi\mathrm{cos}\theta=\frac{\sqrt{3}}{4}Explain how Natalia can determine if the value of \mathrm{tan}\theta is positive or negative.

Part III

Graph c(x)=-9(3)^{x-4}+2 on the axes in the Show Your Work space.Describe the end behavior of c(x) as x approaches positive infinity.Describe the end behavior of c(x) as x approaches negative infinity.

Part IV

Part I

Which expression is equivalent to (x+2)^{2}-5(x+2)+6?

To the nearest tenth, the solution to the equation 4300e^{0.07x}-123=5000 is

The value of an automobile t years after it was purchased is given by the function V=38,000(0.84)^t . Which statement is true?

Which function represents exponential decay?

The expression \frac{x^4-5x^2+4x+14}{x+2} is equivalent to

The sum of the first 20 terms of the series -2+6-18+54- ... is

If f(x)=2x^{4}-x^3-16x+8, then f(\frac{1}{2})

If (6-ki)^{2}=27-36i, the value of k is

What is the solution set of the equation \frac{x+2}{x}+\frac{x}{3}=\frac{2x^{2}+6}{3x} ?

How many real solutions exist for the system of equations below?y=\frac{1}{4}x-8y=\frac{1}{2}x^{2}+2x

Which equation represents a polynomial identity?

Given x>0, the expression \frac{x^{\frac{1}{5}}}{x^{\frac{1}{2}}} can be rewritten as

A cyclist pedals a bike at a rate of 60 revolutions per minute. The height, h, of a pedal at time t, in seconds, is plotted below.The graph can be modeled by the function h(t)=5\mathrm{sin}(kt), where k is equal to

Which statement about data collection is most accurate?

If f(x)=\frac{1}{2}x+2, then the inverse function is

Given f(x)=x^{4}-x^3-6x^2, for what values of x will f(x)>0?

For which approximate value(s) of x will \mathrm{log}(x+5)=\vert{x-1}-3?

Consider a cubic polynomial with the characteristics below.exactly one real rootas x\rightarrow\infin, f(x){\rightarrow}-\infinGiven a>0 and b>0, which equation represents a cubic polynomial with these characteristics?

If \mathrm{cos}A=\frac{\sqrt{5}}{3} and \mathrm{tan}A<0, what is the value of \mathrm{sin}A?

A tree farm initially has 150 trees. Each year, 20\% of the trees are cut down and 80 seedlings are planted. Which recursive formula models the number of trees, a_n, after n years?

Which equation represents a parabola with a focus of (4,-3) and directrix of y=1?

Which graph shows a quadratic function with two imaginary zeros?

Part II

Natalia's teacher has given her the following information about angle \theta.\pi<\theta<2\pi\mathrm{cos}\theta=\frac{\sqrt{3}}{4}Explain how Natalia can determine if the value of \mathrm{tan}\theta is positive or negative.

Part III

Graph c(x)=-9(3)^{x-4}+2 on the axes in the Show Your Work space.Describe the end behavior of c(x) as x approaches positive infinity.Describe the end behavior of c(x) as x approaches negative infinity.

Part IV

How many real solutions exist for the system of equations below?
y=\frac{1}{4}x-8
y=\frac{1}{2}x^{2}+2x

A cyclist pedals a bike at a rate of 60 revolutions per minute. The height, h, of a pedal at time t, in seconds, is plotted below.
The graph can be modeled by the function h(t)=5\mathrm{sin}(kt), where k is equal to

Consider a cubic polynomial with the characteristics below.
exactly one real root
as x\rightarrow\infin, f(x){\rightarrow}-\infin
Given a>0 and b>0, which equation represents a cubic polynomial with these characteristics?

Natalia's teacher has given her the following information about angle \theta.
\pi<\theta<2\pi
\mathrm{cos}\theta=\frac{\sqrt{3}}{4}
Explain how Natalia can determine if the value of \mathrm{tan}\theta is positive or negative.

Graph c(x)=-9(3)^{x-4}+2 on the axes in the Show Your Work space.
Describe the end behavior of c(x) as x approaches positive infinity.
Describe the end behavior of c(x) as x approaches negative infinity.

How many real solutions exist for the system of equations below?
y=\frac{1}{4}x-8
y=\frac{1}{2}x^{2}+2x

A cyclist pedals a bike at a rate of 60 revolutions per minute. The height, h, of a pedal at time t, in seconds, is plotted below.
The graph can be modeled by the function h(t)=5\mathrm{sin}(kt), where k is equal to

Consider a cubic polynomial with the characteristics below.
exactly one real root
as x\rightarrow\infin, f(x){\rightarrow}-\infin
Given a>0 and b>0, which equation represents a cubic polynomial with these characteristics?

Natalia's teacher has given her the following information about angle \theta.
\pi<\theta<2\pi
\mathrm{cos}\theta=\frac{\sqrt{3}}{4}
Explain how Natalia can determine if the value of \mathrm{tan}\theta is positive or negative.

Graph c(x)=-9(3)^{x-4}+2 on the axes in the Show Your Work space.
Describe the end behavior of c(x) as x approaches positive infinity.
Describe the end behavior of c(x) as x approaches negative infinity.