Algebra 2 Problems of the Day (Algebra 2, June 2025 Part I)

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


17. Consider the system of equations below.

3x + 2y = 1
2y + z = 2
2x - 2z = -6 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

(1) 1
(2) -1
(3) -4
(4) 4

Answer: (1) 1

Subtract the second equation from the first equation and the y variable will disappear. Now you can solve for x.

3x + 2y = 1
2y + z = 2
2x - 2z = -6

3x - z = -1
2x - 2z = -6

6x - 2z = -2
2x - 2z = -6

4x = 4

x = 1

This is Choice (1).

Checking: 2(1) - 2z = -6; -2z = -8, z = 4

Then, 2y + 4 = 2, 2y = -2, y = -1

Finally, 3(1) + 2(-1) = 1. Check!

18. The point (2,–3) lies on the graph of the equation y = f(x). Which point must lie on the graph of the equation y = f(x - 4) + 1?

(1) (1,1)
(2) (-2,-2)
(3) (3, 7)
(4) (6,-2)

Answer: (4) (6,-2)

The -4 inside the parentheses shifts the graph four units to the right. The +1 outside the parentheses shifts the graph on unit up.

If you translate the point (2,-3) four units to the right and one unit up, you end up at (6,-2).

The correct answer is Choice (4).

19. Which statement best describes the end behavior of the function y = log(x - 3)?

(1) As x → -∞, y → -∞, and as x → ∞, y → ∞.
(2) As x → 3, y → -∞, and as x → ∞, y → ∞.
(3) As x → -∞, y → 0, and as x → ∞, y → ∞.
(4) As x → 3, y → 0, and as x → ∞, y → ∞.

Answer: (2) As x → 3, y → -∞, and as x → ∞, y → ∞.

Graph the function. One the left side, the graph goes to negative infinity as it gets closer to 3. On the right side, both x and y tend toward infinity.

Choice (2) is the correct answer.

20. The black bear population for a certain area of the Adirondacks can be modeled by the equation

B = 5835.943(1.026)^t, where t is measured in years since 2010. Kieran would like to rewrite this model in terms of a 5-year growth rate. Kieran’s model is best represented by

(1) B = 5835.943(1.005147)^t/5
(2) B = 5835.943(1.005147)^5t
(3) B = 5835.943(1.136938)^t/5
(4) B = 5835.943(1.136938)^5t

Answer: (3) B = 5835.943(1.136938)^t/5

If there is a 5-year growth rate, then the exponent will be divided by 5, or t/5. Eliminate Choices (2) and (4).

(1.026)^t is equal to (1.026⁵)^t is equal to (1.136938)^t/5.

This is Choice (3).

21. Which expression or expressions are equal to 0 for all real numbers?

I. (x² + y²)² + (x² + y²)² - 2(x² + y²)² II.(x² + y²)² - (x² + y²)² III. (x² + y²)² - (x² + y²)² - (2xy)²

(1) I, only
(2) III, only
(3) I and II, only
(4) I and III, only

Answer: (4) I and III, only

The first expression is "obviously" equal to 0 (see below), but you need to expand the second and third choices to see if all the terms will cancel out. Again, it is "obvious" that they both can't be true. However, it is possible that neither one is true.

In the first expression, let z = (x² + y²)². So z + z - 2z = 0, which is true. Eliminate Choice (2).

In the second expression, the first term is not the same as the second term, so subtracting them cannot equal 0. Eliminate Choice (3).

Expand the third expressiong.

(x² + y²)² - (x² + y²)² - (2xy)²
(x⁴ + 2x²y² + y⁴) - (x⁴ - 2x²y² + y⁴) - (4x²y²)
x⁴ + 2x²y² + y⁴ - x⁴ + 2x²y² - y⁴ - 4x²y²
x⁴ - x⁴ + 2x²y² + 2x²y² + y⁴ - 4x²y² - y⁴ = 0

Choice (4) is the correct answer.


22. The equation 1/x - 1/5 = x/5 has

(1) rational solutions
(2) irrational solutions
(3) imaginary solutions
(4) no solutions

Answer: (2) irrational solutions

Solve for x.

1/x - 1/5 = x/5

1/x = x/5 + 1/5

1/x = (x + 1)/5

x(x + 1) = 5

x² + x = 5

x² + x - 5 = 0

If you check the discriminant, b² - 4ac, then 1² - (4)(1)(-5) = 21, which is positive but not a perfect square. This means that there are two solutions but they will be irrational.

This is Choice (2).

23. For x ≠ +4y, the expression (x² + 3xy - 28y²) / (16y² - x² is equivalent to
(1) -1 - (7/4) y
(2) (x - 7y) / (4y - x)
(3) (x + 7y) / (x + 4y)
(4) (-x - 7y) / (x + 4y)

Answer: (4) (-x - 7y) / (x + 4y)

Factor the numerator and the denominator and simplify. (x² + 3xy - 28y²) / (16y² - x²
( (x + 7y)(x - 4y) ) / ( (4y - x)(4y + x) )
( (x + 7y)(-1) ) / (4y + x)
( -x - 7y) / (x + 4y)

This is Choice (4).

24. Which equation represents a parabola with a focus of (-2,1) and directrix of y = 5?,

(1) (x + 2)² = -8(y - 3)
(2) (x + 2)² = 5(y - 1)
(3) (x + 2)² = -8(y - 1)
(4) (x + 2)² = 8(y - 3)

Answer: (1) (x + 2)² = -8(y - 3)

The formula for finding a parabola from the focus and directrix is (x - h)² = 4p(y - k), where p is the distance from the vertex to the focus. The vertex is halfway between the focus and directrix, or (-2, 3), which makes p = -2.

Substituting what we know, we get (x - (-2))² = 4(-2)(y - 3), or (x + 2)² = -8(y - 3).

This is Choice (1).

End of Part I.

Questions, comments, and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2, June 2025 Part I)

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


9. The probabilities that a randomly selected teenager uses social media websites F and I are shown below.

P(F) = 0.71
P(I) = 0.52
P(F or I) = 0.77 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

(1) 0.06
(2) 0.19
(3) 0.46
(4) 0.96

Answer: (3) 0.46

The union (OR) of two probabilities is the equal to the sum of the independent probabilities minus their intersection (AND).

P(F or I) = P(F) + P(I) - P(F and I)
P(F) + P(I) - P(F or I) = P(F and I)
0.71 + 0.52 - 0.77 = P(F and I)
0.46 = P(F and I)

This is Choice (3).

10. Consider f(x) = (x - 2)²(x + 3), and g(x) as strictly defined in the table below.
Which statement or statements must be true, based on the information given?
I. Both f(x) and g(x) have the same x-intercepts.
II. Both f(x) and g(x) have a y-intercept at y = -6

(1) I, only
(2) II, only
(3) I and II
(4) neither I or II

Answer: (1) I, only

The x-intercepts of f(x) are 2 and -3. The x-intercepts of g(x) are -3 and 2. These are the same.

The y-intercept of f(x) = (-2)²(3) = 12. The y-intercept of g(x) is -6. These are not the same.

The correct answer is Choice (1).

11. Josie examines the graphs of f(x) = 3^x - 8 and g(x) = 1 / (x² - 4). The number of solutions to f(x) = g(x) is

(1) 1
(2) 2
(3) 3
(4) 0

Answer: (3) 3

Just graph them. The answer is 3.

Choice (3) is the correct answer.

12. Which binomial is a factor of g³ + 6g² + g - 14?

(1) g - 1
(2) g - 2
(3) g + 1
(4) g + 2

Answer: (4) g + 2

Once again: graph it. Check the zeroes. The roots are x = -2, or x = -2 + √(11).

If g - 1 is a factor, then the polynomial will eqaul zero when g = 1. However, it is -6. Eliminate Choice (1).

If g - 2 is a factor, then the polynomial will eqaul zero when g = 2. However, it is 20. Eliminate Choice (2).

If g + 1 is a factor, then the polynomial will eqaul zero when g = -1. However, it is -10. Eliminate Choice (3).

If g + 2 is a factor, then the polynomial will eqaul zero when g = -2. And it is 0. Choice (4) is the correct answer.

13. Consider the recursively defined sequence below.

a₁ = 8
a_n = 2a_n-1 Which explicit formula represents the same sequence?

(1) a_n = 2ⁿ
(2) a_n = 2(4ⁿ)
(3) a_n = 2⁽ⁿ⁺²⁾
(4) a_n = 8ⁿ

Answer: (3) a_n = 2⁽ⁿ⁺²⁾

If the answer doesn't jump out at you, write out the first few terms of the sequence. Then check each one.

The recursive formula is doubling, starting with 8: 8, 16, 32, 64, ...

Choice (1) is 2, 4, 8, ... Eliminate Choice (1).

Choice (2) is 8, 32, 128, ... Eliminate Choice (2).

Choice (3) is 8, 16, 32, 64. ... Choice (3) is the correct answer.

Choice (4) is 8, 64 ... Eliminate Choice (4).


14. What is the exact value of tan(-5π/6)?

(1) 1 / √(3)
(2) -1 / √(3)
(3) √(3)
(4) -√(3)

Answer: (1) 1 / √(3)

Tangent is positive in Quadrants I and III and this is Quadrant III. Eliminate Choices (2) and (4).

The coordinates for that point on the unit circle are (-√(3)/2, -1/2).

Tangent is sin/cos = (-1/2) / (-√(3)/2) = 1 / √(3)

This is Choice (1).

15. Given m ≠ 0 and (17^1/m)ⁿ = 17², what is n in terms of m?
(1) 2m
(2) 2/m
(3) m/2
(4) 2^m

Answer: (1) 2m

Multiplying the exponents give you 17^n/m = 17². This means n/m = 2.

Therefore, n = 2m.

This is Choice (1).

16. In order to qualify for a college tennis scholarship, Joe needs to win 90% of the matches he plays during his senior year of high school. If he has won 8 of the 10 matches that he has played, which equation can be used to determine how many more consecutive matches, x, Joe must win in order for his winning percentage to equal 90%?

(1) (8 + x) / x = 0.90
(2) 8 / (10 + x) = 0.90
(3) 8/10 + x = 0.90
(4) (8 + x) / (10 + x) = 0.90

Answer: (4) (8 + x) / (10 + x) = 0.90

The number of games won divided by the number of games played must equal 0.90.

Currently, Joe is at 8/10 = .80. If he keeps winning, his average will be 9/11, 10/12, 11/13, etc.

Both the numerator and the denominator will increase.

This means that x games will be added to both the 8 and the 10, so Choice (4) is the correct answer.

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2, June 2025 Part I)

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


1. Which expression is equivalent to 2c³√(c) ?

(1) 2c^4/3
(2) 2c^3/4
(3) (2c)^4/3
(4) (2c)^3/4

Answer: (1) 2c^4/3

When there's a fraction as an exponent, the numerator is the power the base is raised to while the denominator is the nth root of the base.

2c³√(c) = 2c (c)^1/3

When you multiply variables, keep the base and add the exponents. If you add 1 and 1/3, the sum is 4/3. Therefore,

2c³√(c) = 2c (c)^1/3 = 2c^4/3

The correct answer is Choice (1).

2. Which investigation technique is most often used to determine the cause and effect of a medication?

(1) observational study
(2) survey
(3) controlled experiment
(4) census

Answer: (3) controlled experiment

To determine cause and effect, you need to conduct a controlled experiement. A survey, census or observational study are insufficient.

The correct answer is Choice (3).

3. What is the solution to 5(2)^19x = 50?

(1) x = (log(50)) / 19
(2) x = (log₂(10)) / 19
(3) x = (log₂(45)) / 19
(4) x = 5 / 19

Answer: (2) x = (log₂(10)) / 19

Use inverse operations.

5(2)^19x = 50
(2)^19x = 10
log₂ 10 = 19x
x = log₂ 10 / 19

The correct answer is Choice (2).

4. The function P(t) = 256,485(0.965)^t models the decreasing population of a city from 1999 to 2014, where t is the time in years since 1999.

Which statement is not true?

(1) The function estimated the population was 256,485 in 1999.
(2) The decay rate was 0.35%.
(3) The decay factor is 0.965.
(4) The population declined over 15 years.

Answer: (2) The decay rate was 0.35%.

I will say that this was a confusing question, and for a moment, I wondered if there was a typo. However, I would've remembered if a recent question had been invalidated. My trouble comes from them using the term "decay factor", which I have to say, personally, I don't see that term used. Note that I don't normally teach Algebra II, but when talking about equations of this type, we usually talk about the "decay rate" and not the "decay factor".

In any case ...

In Choice (1), the starting population is 256,485 in 1999. This is true, so eliminate Choice (1).

In Choice (2), the decay rate is NOT 0.35%. If you subtract 1.000 - 0.965, you get 0.035, which is 3.5%. This is the correct answer.

In Choice (3), the decay factor is 0.965. This is true. This is different from the decay rate. This is the factor that you are multiply the starting value by. Eliminate Choice (3).

In Choice (4), since the decay factor is less than 1.00, which is why it's decay and not growth, the population will continually shrink. The question says it's decreasing from 1999 to 2014, so it's decreasing for 15 years. This is correct, so eliminate Choice (4).

5. Four different surveys gathered data about the purchasing behaviors of pet owners. Pet owners from the same population were randomly selected. While collecting data, Chris surveyed 942 pet owners, John surveyed 410, Brooke surveyed 800, and Shane surveyed 100. Whose survey will likely have the smallest margin of error?

(1) Brooke
(2) Chris
(3) John
(4) Shane

Answer: (2) Chris

The more pet owners interviewed, the smaller the margin of error that can be expected. More is better.

Chris interviewed the most. This is Choice (2).

6. Given i is the imaginary unit and a = i³, b = i², and c = i, which expression is equivalent to 2ax² + 3bx - cx?

(1) -2ix² - 3x + ix
(2) -2ix² - 3x
(3) -2ix² - 3x - ix
(4) -8ix² - 3x - ix

Answer: (3) -2ix² - 3x - ix

Replace a with (-i). Replace b with (-1). Replace c with (i). Simplify the expression.

2ax² + 3bx - cx = 2(-i)x² + 3(-1)x - (i)x = -2ix² - 3x - ix.

This is Choice (3).

7. Which sequence has a common ratio of 1/2?

(1) -1/4 a, -1/8 a, -1/16 a, -1/32 a ...
(2) 1/32 a, 1/16 a, 1/8 a, 1/4 a, ...
(3) 20a, 39/2 a, 19a, 37/2 a, ...
(4) 22a, 22.5a, 23a, 23.5a, ...

Answer: (1) -1/4 a, -1/8 a, -1/16 a, -1/32 a ...

Multiply the first term by 1/2. Did you get the second term? If not, eliminate that choice. If you did, check the next number.

In Choice (1), if you multiply (-1/4a)(1/2) = -1/8 a, etc. This is the Correct answer.

In Choice (2), if you multiply (1/32 a)(1/2) = 1/64 a. The common ratio in the second choice is 2, not 1/2. Eliminate Choice (2).

In Choice (3), half of 20a is 10a, not 39/2 a. This is an arithmetic sequence with a common difference of -1/2 a.

In Choice (4), half of 22a is 11a, not 22.5a. This is an arithmetic sequence with a common difference of 1/2 a, or .5a.

8. The result of dividing 2x³ + 6x² + 7x + 2 by x + 1 is

(1) 2² + 4x + 3 - 1 / (x + 1)
(2) 2² + 4x + 3 + 1 / (x + 1)
(3) 2² + 8x - 15 + 17 / (x + 1)
(4) 2² + 8x + 15 - 13 / (x + 1)

Answer: (1) 2² + 4x + 3 - 1 / (x + 1)

If 2x³ + 6x² + 7x + 2 were divisible by (x + 1), then f(-1) would be equal to 0. However, f(-1) = -1, which is a remainder.

Only one choice has -1 / (x + 1) in it, so that is the answer.

Doing it the long way:

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra Regents, June 2025 Part IV)

This exam was adminstered in June 2025.

June 2025 Algebra 2, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

37. Cesium-137 decay can be modeled with the formula A(t) = A₀e^kt, where A(t) represents the mass remaining in grams after t years and A₀ represents the initial mass. A sample of 500 grams of cesium-137 takes approximately 60.34 years to decay to 125 grams. Use this sample with the given formula to determine the constant k, to the nearest thousandth.

Use this value for k to write a function, A(t), that will find the mass of the 500-gram sample remaining after any amount of time, t, in years.

Graph A(t) on the graph below from t = 0 to t = 150 years.

Use A(t) to calculate the average rate of change in grams per year, from t = 0 to t = 60 years, to the nearest tenth.

Explain what this value means in the given context.

Answer:

Write the original formula: A(t) = A₀e^kt. Next, substitute what you know to find out what you don't know through inverse operations.

A(t) = A₀e^kt

125 = 500 e^60.34k

125/500 = e^60.34k

.25 = e^60.34k

ln .25 = 60.34k

-1.386294... = 60.34k

-0.23 = k

Subsitute this value into A(t).

A(t) = 500 e^-.023t

Put this into the graphing calculator (using y for A(t) and x for t) and look at the table of values to make the graph.

The following graph came from the nyresgents.org website:

To find the average rate of change, calculate ( A(60) - A(0) ) / 60 - 0, using the values from the Table of Values.

( 125.79 - 500) / (60 - 0) = -374.21 / 60 = -6.2 grams per year.

In this context, the mass of Cesium-137 decays at an average rate of 6.2 grams per year.

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2 Regents, June 2025 Part III)

This exam was adminstered in June 2025.

June 2025 Algebra 2, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

33. Solve algebraically for x:

2/x = (2x + 3) / (x - 4).
Express your answers in simplest a + bi form.

Answer:

Cross-multiply and solve the quadratic equation.

2/x = (2x + 3) / (x - 4)

2(x - 4) = x(2x + 3)

2x - 8 = 2x² + 3x

2x² + x + 8 = 0

Use the Quadratic Formula:

x = (-1 + √((1)² - 4(2)(8)) ) / ( 2(2))

x = (-1 + √(1 - 64) ) / (4)

x = -1/4 + √(-63) / 4

x = -1/4 + i √(63) / 4

x = -1/4 + i √((9)(7)) / 4

x = -1/4 + 3i √(7) / 4

34. A highly selective college reports that the mean score earned by accepted students on the Mathematics Level 2 subject test is 750 with a standard deviation of 20 and that the scores are approximately normally distributed.

Given this information, determine the interval representing the middle 95% of student scores.

To the nearest whole percent, determine the percentage of accepted students who scored a 760 or less.

Answer:

The middle 95% is the mean minus 2 times the standard deviation to the mean plus 2 times the standard deviation. (Note that I copied this sentence from January 2024, question 34. Similar question, same question number.)

750 - 2(20) = 710, 750 + 2(20) = 790.

The interval representing the middle 95% of student scores is 710 < p < 790.

To answer the second part of the question, you need to use the normalcdf function on your calculator. Use 10^-99 as your minimum number and 760 as the maximum (as stated in the question). Then enter the mean of 750 and standard deviation of 20.

normalcdf(10^-99, 760, 750, 20) = 0.691

69% of accepted students scored 760 or less.

This particular problem could be solved without using a calculator:

Note that 760 is 750 + 10, which is one-half of a standard deviation of above the mean. The normal distribution curve is no longer on the reference chart. However, if you know the percentages, then you know that 19.1% of the data is one-half of a standard deviation above the mean, and, of course, 50% of the data is below the mean. Therefore, 50% + 19.1% = 69.1% of the data.

35. For c(x) = 3x² - 4x + 7 and d(x) = x - 2, determine c(x) • d(x) - [d(x)]³ as a polynomial in standard form.

Answer:

There are three steps to complete. Multiply the two functions. Cube the d(x) function. Then subtract the two products.

First step:

c(x) • d(x) = 3x³ - 10x² + 15x - 14

Second step:

Either multiply (x - 2)(x - 2)(x - 2) or use the fact that (x - a)³ = x³ - 3ax² + 3a²x - a³.

Using the rule, (x - 2)³ = x³ - 6x² + 12x - 8. Or you can do it in two steps:

Note: I checked the official response guide and "f(x) + 5" was acceptable as an answer.

Final step:

3x³ - 10x² + 15x - 14
x³ - 6x² + 12x - 8

2x³ - 4x² + 3x - 6

36. Christopher works for a defense contractor and earned $85,000 his first year. For each additional year he will receive a 2.5% raise.

Write a geometric series formula, Cn, for Christopher’s total earnings over n years.

Use this formula to find Christopher’s total earnings, to the nearest hundred dollars, over his first 10 years of employment.

Answer:

Use the formula for the Reference sheet to answer the first part. Then use that equation, substitute 10 for n, and evaluate.

C_n = (a₁ - a₁ rⁿ) / (1 - r)

C_n = (85000₁ - 85000₁ (1.025)ⁿ) / (1 - 1.025)

Substitute n = 10.

C_n = (85000 - 85000(1.025)¹⁰) / (1 - 1.025) = 952287.45

To the nearest $100, C₁₀ = $952,300.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2 Regents, June 2025 Part II)

This exam was adminstered in June 2025.

June 2025 Algebra 2, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Data collected showing the depth of the water in a bay during a 24-hour period are shown in the graph below.

The depth of the water can be modeled with a trigonometric function of the form d(t) = A sin(π/6 t) + C. Estimate the value of A, to the nearest integer. Justify your answer.

Answer:

The water has a maximum height of 25 and a minimum of 5 feet with the midline at 15 feet. The amplitude, A, is 25 - 15 = 10.

26. Algebraically determine the solution to the equation below.

√(x - 2) + x = 4

Answer:

Use inverse operations to set up a quadratic equation. Solve the quadratic equation and then eliminate extraneous solutions.

√(x - 2) + x = 4

√(x - 2) = 4 - x

x - 2 = 16 - 8x + x²

0 = 18 - 9x + x²

0 = (x - 6)(x - 3)

x - 6 = 0 or x - 3 = 0

x = 6 or x = 3

Check: √(6 - 2) + 6 = 4 -- not true. Eliminate x = 6

Check: √(3 - 2) + 3 = 4, Check.

x = 3 is the only solution.

27. Factor the expression completely.

(x - 1)² + 5(x - 1) - 6

Answer:

Factor completely usually means there will be a couple of steps.

First, let y = x - 1. Factor and then substitute the original expression back.

y² + 5y - 6
(y + 6)(y - 1)
(x - 1 + 6)(x - 1 - 1)
(x + 5)(x - 2)

28. The results of a survey of the students at the local high school regarding the topic “What I Do to Relax” are displayed in the table below.

If a student from this survey is selected at random, determine the exact probability that the person claims to relax by listening to music given that the person is female.

Answer:

Add the subtotals and grand total. The probability will be 94 (from the table) divided by the total number of students in the survey.

So the probability is 94/202 or 47/101.

29. The graph of y = f(x) is shown below. The cubic function has a leading coefficient of 1.

Write an equation for f(x).

Answer:
Look for the zeroes of the function.

y = (x + 3)(x - 1)²

You don't need to multiply the factors.

30. Given f(x) = 2/3 x + 6, write the equation of f ^-1(x).

Answer:
Replace f(x) with x and replace x with f ^-1(x). Then solve for f^-1(x). Or you can use y, but replace that before your final answer.

x = 2/3 f ^-1(x) + 6

x - 6 = 2/3 f ^-1(x)

(3/2) (x - 6) = f ^-1(x)

f ^-1(x) = 3/2 x - 9

31. On the coordinate plane below, sketch at least one cycle of the function f(x) = 4cos(2x). Label the axes with an appropriate scale.

Answer:
The midline is y = 0, so the maximum will be at y = 4 and the minimum will be at y = -4. The period is pi instead of 2pi. You can use your graphing calculator if you aren't sure, but they want to see pi on the scale, not 1, 2, 3, 4, etc.

Graph a curve.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Algebra 2 Problems of the Day (Algebra 2, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra 2 Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


17. If 4(10^5x-2) = 12, then x equals

(1) 2.3/5
(2) 1/3( log12/log40 + 5)
(3) (log(3) + 2) / 5
(4) 1/5 (log12/log4 + 2)

Answer: (3) (log(3) + 2) / 5

The rules for logarithms are derived from the rules of exponents. Remember that if you have difficulty. Use inverse operations to find x.

4(10^5x-2) = 12
10^5x-2) = 3
log₁₀3 = 5x - 2
log(3) + 2 = 5x
(log(3) + 2) / 5 = x

This is Choice (3).

18. A random sample of 152 students was surveyed on a particular day about how they got to school. The survey results are summarized in the table below.

Which statement is best supported by the data?

(1) The probability of being late given that a student walked is greater than the probability that a student walked given that the student was late.
(2) The probability of being late given that a student walked is less than the probability that a student walked given that the student was late.
(3) The probability of being late given that a student walked is equal to the probability that a student walked given that the student was late.
(4) The probability of being late given that a student walked cannot be determined.

Answer: (1) The probability of being late given that a student walked is greater than the probability that a student walked given that the student was late.

Write in the subtotals and the grand total to make the calculations easier:

In Choice (1), the probability of being late given that a student walked is 4/22. The probability that a student walked given that the student was late is 4/30. The first probability is greater than the second, so this is the correct answer.

Choice (2) says the first should be less, but it is not. Choice (3) says that the two are equal, but they are not. Choice (4) says the probability cannot be determined, but it can be. (We just did it!)

The correct answer is Choice (1).

19. If f(x) = ³√(x) + 4, then f^-1(x) equals

(1) ³√(x - 4)
(2) (x - 4)³
(3) x³ + 1/4
(4) -³√(x) - 4

Answer: (2) (x - 4)³

Substitute the x with a y, then set the expression equal to x. Finally, rewrite the equation to solve for y.

x = ³√(y) + 4
x - 4 = ³√(y)
(x - 4)³ = y

The correct answer is Choice (1).

20. Given the equation S(x) = 1.7sin(bx) + 12, where the period of S(x) is 12, what is the value of b?

(1) π/6
(2) 24π
(3) π/12
(4) 6π

Answer: (1) π/6

The period of a sine function is 2π/b. We want to find b. The "+ 12" in the function is meaningless to this problem because it has nothing to do with the period, which is also 12.

2π/b = 12
2π/12 = b
π/6 = b

Choice (1) is the correct answer.

21. Jin solved the equation √(4 - x) = x + 8 by squaring both sides. What extraneous solution did he find?

(1) -5
(2) -12
(3) 3
(4) 4

Answer: (2) -12

The extraneous value would be one that wouldn't make a valid equation. For examples, if a "solution" makes the expression x + 8 into a negative number, then that solution is extraneous because the value of the square root cannot be negative. The only possible answer from the four choices is (2) -12.

If you wanted to solve it to check (or if there had been more possibilities):

√(4 - x) = x + 8
4 - x = (x + 8)²
4 - x = x² + 16x + 64
0 = x² + 17x + 60
0 = (x + 12)(x + 5)
x + 12 = 0 or x + 5 = 0
x = -12 or x = -5

The solution x = -5 is valid because √(4 - (-5) = (-5) + 8 is a true statement becasue 3 = 3.

However, √(4 - (-12)) = (-12) + 8 is not a true statement because 4 =/= -4.

The correct answer is Choice (2).

22.The expression (x² + y²)² is not equivalent to

(1) (x² - y²)² + (2xy)²
(2) (x + y)⁴ + 2(xy)²
(3) x²(x²+2y²) + (y²)²
(4) (2x² + y²)² - (3x⁴ + 2x²y²)

Answer: (2) (x + y)⁴ + 2(xy)²

If you want a calculator "shortcut", replace x with a value such as 5 and y with a value such as 7 (note that 5 and 7 are mutually prime) and calculate all five expressions to see which doesn't match.

On first look, my "educated guess" would be Choice (2), but let's check all of them:

(x² + y²)² = x⁴ + 2x²y² + y⁴

In Choice (1) (x² - y²)² + (2xy)²
= x⁴ - 2x²y² + y⁴ + (4xy)²
= x⁴ + 2x²y² + y⁴

In Choice (2), (x + y)⁴ is NOT equal to x⁴ + y⁴.
(x + y)⁴ + 2(xy)²
= x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ + 4x²y²
This is totally different from the original expression. Choice (2) is the correct answer.

In Choice (3) x²(x²+2y²) + (y²)²
= x⁴ + 2x²y² + y⁴

In Choice (4) (2x² + y²)² - (3x⁴ + 2x²y²)
= 4x⁴ + 4x²y² + y⁴ - 3x⁴ - 2x²y²
= x⁴ + 2x²y² + y⁴

23. The height of a running trail is modeled by the quartic function y = f(x) shown below, where x is the distance in miles from the start of the trail and y is the height in feet relative to sea level.
If this trail has a minimum height of 16 feet below sea level, which function(s) could represent a running trail whose minimum height is half of the minimum height of the original trail?

I. y = f(1/2 x); II. y = f(x) + 8; III. y = 1/2f(x)

(1) I, only
(2) II, only
(3) I and III
(4) II and III

Answer: (4) II and III

Half of the minimum height would be 8 feet below sea level instead of 16 feet. That mean that the minimum is either raised by 1/2 or raised by 8 feet.

In function I, putting 1/2 in front of x inside the parentheses, the function has changed the period but hasn't changed either the baseline or the amplitude. Function I is no good, so you can eliminate Choices (1) and (3).

Function II must be true because it's in both of the remaining choices. Adding 8 to the y value will raise the minimum from -16 to -8.

Function III shrinks the y value by 1/2, so -16 will become -8. Function III fits the description.

Choice (4) is the correct answer.

24. The crew aboard a small fishing boat caught 350 pounds of fish on Monday. From that Monday through the end of the week on Friday, the weight of the fish caught increased 15% per day. The total weight, in pounds, of fish caught is approximately

(1) 411
(2) 612
(3) 1748
(4) 2360

Answer: (4) 2360

Use the formula for the Geometric Series to find the sum:

S₅ = (350 - 350(1.15)⁵) / (1 - 1.15)

P = 2359.833...

The correct answer is Choice (4).

End of Part I. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!