NYSED High School Regents Algebra II

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.

Required

2

Suppose two sets of test scores have the same mean, but different standard deviations, σ_1 and σ_2, with σ_2 > σ_1. Which statement best describes the variability of these data sets?

Required

2

If f(x) = \log_3 x and g(x) is the image of f(x) after a translation five units to the left, which equation represents g(x)?

Required

2

When factoring to reveal the roots of the equation x^{3} + 2x^{2} - 9x - 18 = 0, which equations can be used?

I. x^{2}(x + 2) - 9(x + 2) = 0 II. x(x^{2} - 9) + 2(x^{2} - 9) = 0 III. (x - 2)(x^{2} - 9) = 0

Required

2

When a ball bounces, the heights of consecutive bounces form a geometric sequence. The height of the first bounce is 121 centimeters and the height of the third bounce is 64 centimeters. To the nearest centimeter, what is the height of the fifth bounce?

Required

2

The solutions to the equation 5x^{2} - 2x + 13 = 9 are

Required

2

Julia deposits $2000 into a savings account that earns 4% interest per year. The exponential function that models this savings account is y = 2000(1.04)^t, where t is the time in years. Which equation correctly represents the amount of money in her savings account in terms of the monthly growth rate?

Required

2

Tides are a periodic rise and fall of ocean water. On a typical day at a seaport, to predict the time of the next high tide, the most important value to have would be the

Required

2

An estimate of the number of milligrams of a medication in the bloodstream t hours after 400 mg has been taken can be modeled by the function below.

I(t) = 0.5t^{4} + 3.45t^{3} - 96.65t^{2} + 347.7t, where 0 \leq t \leq 6

Over what time interval does the amount of medication in the bloodstream strictly increase?

Required

2

Which representation of a quadratic has imaginary roots?

Required

2

A random sample of 100 people that would best estimate the proportion of all registered voters in a district who support improvements to the high school football field should be drawn from registered voters in the district at a

Required

2

Which expression is equivalent to (2x - i)^{2} - (2x - i)(2x + 3i) where i is the imaginary unit and x is a real number?

Required

2

Suppose events A and B are independent and P (A and B ) is 0.2. Which statement could be true?

Required

2

The function f(x) = a \cos{bx} + c is plotted on the graph shown below.

What are the values of a, b, and c?

Required

2

Which equation represents the equation of the parabola with focus (-3,3) and directrix y = 7?

Required

2

What is the solution set of the equation \frac{2}{3x+1} = \frac{1}{x} - \frac{6x}{3x+1}?

Required

2

Savannah just got contact lenses. Her doctor said she can wear them 2 hours the first day, and can then increase the length of time by 30 minutes each day. If this pattern continues, which formula would not be appropriate to determine the length of time, in either minutes or hours, she could wear her contact lenses on the nth day?

Required

2

If f(x) = a^x where a > 1, then the inverse of the function is

Required

2

Kelly-Ann has $20,000 to invest. She puts half of the money into an account that grows at an annual rate of 0.9% compounded monthly. At the same time, she puts the other half of the money into an account that grows continuously at an annual rate of 0.8%. Which function represents the value of Kelly-Ann’s investments after t years?

Required

2

Which graph represents a polynomial function that contains x^{2} + 2x + 1 as a factor?

Required

2

Sodium iodide-131, used to treat certain medical conditions, has a half-life of 1.8 hours. The data table below shows the amount of sodium iodide-131, rounded to the nearest thousandth, as the dose fades over time.

What approximate amount of sodium iodide-131 will remain in the body after 18 hours?

Required

2

Which expression(s) are equivalent to \frac{x^{2} - 4x}{2x} , where x \neq 0?

I.\frac{x}{2} - 2
II.\frac{x-4}{2}
III.\frac{x-1}{2} - \frac{3}{2}

Required

2

Consider f(x) = 4x^{2} + 6x - 3, and p(x) defined by the graph below.

The difference between the values of the maximum of p and minimum of f is

Required

2

The scores on a mathematics college-entry exam are normally distributed with a mean of 68 and standard deviation 7.2. Students scoring higher than one standard deviation above the mean will not be enrolled in the mathematics tutoring program. How many of the 750 incoming students can be expected to be enrolled in the tutoring program?

Required

2

How many solutions exist for \frac{1}{1-x^2} = -|3x - 2| + 5?

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.

Required

2

Justify why \frac{\sqrt[3]{x^{2}y^{5}}}{\sqrt[4]{x^{3}y^{4}}} is equivalent to x^{\frac{-1}{12}} y^{\frac{2}{3}} using properties of rational exponents,
where x \neq 0 and y \neq 0.

Required

2

The zeros of a quartic polynomial function are 2, -2, 4, and -4. Use the zeros to construct a possible sketch of the function, on the set of axes below.

Required

2

Erin and Christa were working on cubing binomials for math homework. Erin believed they could save time with a shortcut. She wrote down the rule below for Christa to follow.

(a + b)^{3} = a^{3} + b^{3}

Does Erin’s shortcut always work? Justify your result algebraically.

Required

2

The probability that a resident of a housing community opposes spending money for community improvement on plumbing issues is 0.8. The probability that a resident favors spending money on improving walkways given that the resident opposes spending money on plumbing issues is 0.85. Determine the probability that a randomly selected resident opposes spending money on plumbing issues and favors spending money on walkways.

Required

2

Rowan is training to run in a race. He runs 15 miles in the first week, and each week following, he runs 3% more than the week before. Using a geometric series formula, find the total number of miles Rowan runs over the first ten weeks of training, rounded to the nearest thousandth.

Required

2

The average monthly high temperature in Buffalo, in degrees Fahrenheit, can be modeled by the function B(t) = 25.29 \sin(0.4895t - 1.9752) + 55.2877, where t is the month number (January = 1). State, to the nearest tenth, the average monthly rate of temperature change between August and November.

Explain its meaning in the given context.

Required

2

Point M \left( t, \frac{4}{7} \right) is located in the second quadrant on the unit circle. Determine the exact value of t.

Required

2

On the grid below, graph the function y=\log_2(x-3)+1

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. 

Required

4

Solve the following system of equations algebraically for all values of a, b, and c.

a + 4b + 6c = 23
a + 2b + c = 2
6b + 2c = a + 14

a= _______ 
b=_______ 
c= _______ 

Required

4

Given a(x) = x^{4} + 2x^{3} + 4x - 10 and b(x) = x + 2, determine \frac{a(x)}{b(x)} in the form q(x) + \frac{r(x)}{b(x)}.

Answer: _______ 

Is b(x) a factor of a(x)? Explain. _______ 

Required

4

A radio station claims to its advertisers that the mean number of minutes commuters listen to the station is 30. The station conducted a survey of 500 of their listeners who commute. The sample statistics are shown below.

A simulation was run 1000 times based upon the results of the survey. The results of the simulation appear below.

Based on the simulation results, is the claim that commuters listen to the station on average 30 minutes plausible? Explain your response including an interval containing the middle 95% of the data, rounded to the nearest hundredth.

Required

4

Solve the given equation algebraically for all values of x.

3\sqrt{x} - 2x = -5

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.

Required

6

Tony is evaluating his retirement savings. He currently has $318,000 in his account, which earns an interest rate of 7% compounded annually. He wants to determine how much he will have in the account in the future, even if he makes no additional contributions to the account.

Write a function, A(t), to represent the amount of money that will be in his account in t years.

Answer:_______ 

Graph A(t) where 0 \leq t \leq 20 on the set of axes below.

Tony’s goal is to save $1,000,000. Determine algebraically, to the nearest year, how many years it will take for him to achieve his goal.

Answer: _______ years

Explain how your graph of A(t) confirms your answer. 

Answer: _______

NYSED High School Regents Algebra II

Part I

Suppose two sets of test scores have the same mean, but different standard deviations, σ_1 and σ_2, with σ_2 > σ_1. Which statement best describes the variability of these data sets?

If f(x) = \log_3 x and g(x) is the image of f(x) after a translation five units to the left, which equation represents g(x)?

When factoring to reveal the roots of the equation x^{3} + 2x^{2} - 9x - 18 = 0, which equations can be used?I. x^{2}(x + 2) - 9(x + 2) = 0 II. x(x^{2} - 9) + 2(x^{2} - 9) = 0 III. (x - 2)(x^{2} - 9) = 0

When a ball bounces, the heights of consecutive bounces form a geometric sequence. The height of the first bounce is 121 centimeters and the height of the third bounce is 64 centimeters. To the nearest centimeter, what is the height of the fifth bounce?

The solutions to the equation 5x^{2} - 2x + 13 = 9 are

Tides are a periodic rise and fall of ocean water. On a typical day at a seaport, to predict the time of the next high tide, the most important value to have would be the

Which representation of a quadratic has imaginary roots?

A random sample of 100 people that would best estimate the proportion of all registered voters in a district who support improvements to the high school football field should be drawn from registered voters in the district at a

Which expression is equivalent to (2x - i)^{2} - (2x - i)(2x + 3i) where i is the imaginary unit and x is a real number?

Suppose events A and B are independent and P (A and B ) is 0.2. Which statement could be true?

The function f(x) = a \cos{bx} + c is plotted on the graph shown below.What are the values of a, b, and c?

Which equation represents the equation of the parabola with focus (-3,3) and directrix y = 7?

What is the solution set of the equation \frac{2}{3x+1} = \frac{1}{x} - \frac{6x}{3x+1}?

If f(x) = a^x where a > 1, then the inverse of the function is

Which graph represents a polynomial function that contains x^{2} + 2x + 1 as a factor?

Sodium iodide-131, used to treat certain medical conditions, has a half-life of 1.8 hours. The data table below shows the amount of sodium iodide-131, rounded to the nearest thousandth, as the dose fades over time.What approximate amount of sodium iodide-131 will remain in the body after 18 hours?

Which expression(s) are equivalent to \frac{x^{2} - 4x}{2x} , where x \neq 0?I.\frac{x}{2} - 2II.\frac{x-4}{2}III.\frac{x-1}{2} - \frac{3}{2}

Consider f(x) = 4x^{2} + 6x - 3, and p(x) defined by the graph below.The difference between the values of the maximum of p and minimum of f is

How many solutions exist for \frac{1}{1-x^2} = -|3x - 2| + 5?

Part II

Justify why \frac{\sqrt[3]{x^{2}y^{5}}}{\sqrt[4]{x^{3}y^{4}}} is equivalent to x^{\frac{-1}{12}} y^{\frac{2}{3}} using properties of rational exponents,where x \neq 0 and y \neq 0.

The zeros of a quartic polynomial function are 2, -2, 4, and -4. Use the zeros to construct a possible sketch of the function, on the set of axes below.

Erin and Christa were working on cubing binomials for math homework. Erin believed they could save time with a shortcut. She wrote down the rule below for Christa to follow.(a + b)^{3} = a^{3} + b^{3} Does Erin’s shortcut always work? Justify your result algebraically.

Rowan is training to run in a race. He runs 15 miles in the first week, and each week following, he runs 3% more than the week before. Using a geometric series formula, find the total number of miles Rowan runs over the first ten weeks of training, rounded to the nearest thousandth.

Point M \left( t, \frac{4}{7} \right) is located in the second quadrant on the unit circle. Determine the exact value of t.

On the grid below, graph the function y=\log_2(x-3)+1

Part III

Solve the given equation algebraically for all values of x.3\sqrt{x} - 2x = -5

Part IV

Part I

Suppose two sets of test scores have the same mean, but different standard deviations, σ_1 and σ_2, with σ_2 > σ_1. Which statement best describes the variability of these data sets?

If f(x) = \log_3 x and g(x) is the image of f(x) after a translation five units to the left, which equation represents g(x)?

When factoring to reveal the roots of the equation x^{3} + 2x^{2} - 9x - 18 = 0, which equations can be used?I. x^{2}(x + 2) - 9(x + 2) = 0 II. x(x^{2} - 9) + 2(x^{2} - 9) = 0 III. (x - 2)(x^{2} - 9) = 0

When a ball bounces, the heights of consecutive bounces form a geometric sequence. The height of the first bounce is 121 centimeters and the height of the third bounce is 64 centimeters. To the nearest centimeter, what is the height of the fifth bounce?

The solutions to the equation 5x^{2} - 2x + 13 = 9 are

Tides are a periodic rise and fall of ocean water. On a typical day at a seaport, to predict the time of the next high tide, the most important value to have would be the

Which representation of a quadratic has imaginary roots?

A random sample of 100 people that would best estimate the proportion of all registered voters in a district who support improvements to the high school football field should be drawn from registered voters in the district at a

Which expression is equivalent to (2x - i)^{2} - (2x - i)(2x + 3i) where i is the imaginary unit and x is a real number?

Suppose events A and B are independent and P (A and B ) is 0.2. Which statement could be true?

The function f(x) = a \cos{bx} + c is plotted on the graph shown below.What are the values of a, b, and c?

Which equation represents the equation of the parabola with focus (-3,3) and directrix y = 7?

What is the solution set of the equation \frac{2}{3x+1} = \frac{1}{x} - \frac{6x}{3x+1}?

If f(x) = a^x where a > 1, then the inverse of the function is

Which graph represents a polynomial function that contains x^{2} + 2x + 1 as a factor?

Sodium iodide-131, used to treat certain medical conditions, has a half-life of 1.8 hours. The data table below shows the amount of sodium iodide-131, rounded to the nearest thousandth, as the dose fades over time.What approximate amount of sodium iodide-131 will remain in the body after 18 hours?

Which expression(s) are equivalent to \frac{x^{2} - 4x}{2x} , where x \neq 0?I.\frac{x}{2} - 2II.\frac{x-4}{2}III.\frac{x-1}{2} - \frac{3}{2}

Consider f(x) = 4x^{2} + 6x - 3, and p(x) defined by the graph below.The difference between the values of the maximum of p and minimum of f is

How many solutions exist for \frac{1}{1-x^2} = -|3x - 2| + 5?

Part II

Justify why \frac{\sqrt[3]{x^{2}y^{5}}}{\sqrt[4]{x^{3}y^{4}}} is equivalent to x^{\frac{-1}{12}} y^{\frac{2}{3}} using properties of rational exponents,where x \neq 0 and y \neq 0.

The zeros of a quartic polynomial function are 2, -2, 4, and -4. Use the zeros to construct a possible sketch of the function, on the set of axes below.

Erin and Christa were working on cubing binomials for math homework. Erin believed they could save time with a shortcut. She wrote down the rule below for Christa to follow.(a + b)^{3} = a^{3} + b^{3} Does Erin’s shortcut always work? Justify your result algebraically.

Rowan is training to run in a race. He runs 15 miles in the first week, and each week following, he runs 3% more than the week before. Using a geometric series formula, find the total number of miles Rowan runs over the first ten weeks of training, rounded to the nearest thousandth.

Point M \left( t, \frac{4}{7} \right) is located in the second quadrant on the unit circle. Determine the exact value of t.

On the grid below, graph the function y=\log_2(x-3)+1

Part III

Solve the given equation algebraically for all values of x.3\sqrt{x} - 2x = -5

Part IV

When factoring to reveal the roots of the equation x^{3} + 2x^{2} - 9x - 18 = 0, which equations can be used?

I. x^{2}(x + 2) - 9(x + 2) = 0 II. x(x^{2} - 9) + 2(x^{2} - 9) = 0 III. (x - 2)(x^{2} - 9) = 0

The function f(x) = a \cos{bx} + c is plotted on the graph shown below.

What are the values of a, b, and c?

Which expression(s) are equivalent to \frac{x^{2} - 4x}{2x} , where x \neq 0?

I.\frac{x}{2} - 2
II.\frac{x-4}{2}
III.\frac{x-1}{2} - \frac{3}{2}

Consider f(x) = 4x^{2} + 6x - 3, and p(x) defined by the graph below.

The difference between the values of the maximum of p and minimum of f is

Justify why \frac{\sqrt[3]{x^{2}y^{5}}}{\sqrt[4]{x^{3}y^{4}}} is equivalent to x^{\frac{-1}{12}} y^{\frac{2}{3}} using properties of rational exponents,
where x \neq 0 and y \neq 0.

Erin and Christa were working on cubing binomials for math homework. Erin believed they could save time with a shortcut. She wrote down the rule below for Christa to follow.

(a + b)^{3} = a^{3} + b^{3}

Does Erin’s shortcut always work? Justify your result algebraically.

Solve the given equation algebraically for all values of x.

3\sqrt{x} - 2x = -5

When factoring to reveal the roots of the equation x^{3} + 2x^{2} - 9x - 18 = 0, which equations can be used?

I. x^{2}(x + 2) - 9(x + 2) = 0 II. x(x^{2} - 9) + 2(x^{2} - 9) = 0 III. (x - 2)(x^{2} - 9) = 0

The function f(x) = a \cos{bx} + c is plotted on the graph shown below.

What are the values of a, b, and c?

Which expression(s) are equivalent to \frac{x^{2} - 4x}{2x} , where x \neq 0?

I.\frac{x}{2} - 2
II.\frac{x-4}{2}
III.\frac{x-1}{2} - \frac{3}{2}

Consider f(x) = 4x^{2} + 6x - 3, and p(x) defined by the graph below.

The difference between the values of the maximum of p and minimum of f is

Justify why \frac{\sqrt[3]{x^{2}y^{5}}}{\sqrt[4]{x^{3}y^{4}}} is equivalent to x^{\frac{-1}{12}} y^{\frac{2}{3}} using properties of rational exponents,
where x \neq 0 and y \neq 0.

Erin and Christa were working on cubing binomials for math homework. Erin believed they could save time with a shortcut. She wrote down the rule below for Christa to follow.

(a + b)^{3} = a^{3} + b^{3}

Does Erin’s shortcut always work? Justify your result algebraically.

Solve the given equation algebraically for all values of x.

3\sqrt{x} - 2x = -5