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

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra Regents

Part I

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


9. When the formula p = 2l + 2w is solved for w, the result is

(1) w = (2l + p) / 2
(2) w = (p - 2l) / 2
(3) w = p/2 + l
(4) w = l - p/2

Answer:(2) w = (p - 2l) / 2

Use inverse operations to isolate w.

p = 2l + 2w

p - 2l = 2w

(p - 2l) = w

The correct answer is Choice (2).

10. Market Street Pizza kept a record of pizza sales for the month of February. The results are shown in the table below.

Of all the pizzas sold in February, what percent were plain, deep-dish pizzas?

(1) 20%
(2) 30%
(3) 40%
(4) 50%

Answer: (1) 20%

There were 200 plain, deep-dish pizzas sold. The percentage would be 200 divided by the total number of pizzas sold. Find the grand total of all pizzas in the table.

300 + 200 + 80 + 25 + 120 + 105 + 100 + 70 = 1000 pizzas sold.

200/1000 = 0.20 = 20%, which is Choice (1).

11. When solving -2(3x - 5) = 9/2 x - 2 for x, the solution is

(1) 8/7
(2) 10/11
(3) -16/21
(4) -16/3

Answer: (1) 8/7

You can solve this with the calculator but be sure to use parentheses when substituting.

-2(3x - 5) = 9/2 x - 2

-6x + 10 = 9/2 x - 2

-12/2 x + 10 = 9/2 x - 2

-21/2 x = -12

-21x = -24

x = -24/-21

x = 8/7

The correct Choice is Choice (1).

12.The expression x^{2a + b} is equivalent to

(1) x^2a + x^b
(2) x^a + x^{a + b}
(3) x^a * x^{a + b}
(4) x^{a + b} * x^{a + b}

Answer: (3) x^a * x^{a + b}

Terms with different exponents CANNOT be combined because they are not Like Terms. Terms with the same base can be multiplied. When multiplying terms with the same base, the exponents are added.

The first two choices cannot be added together because they are not like terms. Eliminate Choices (1) and (2).

In Choice (3), x^a * x^{a + b} = x^{2a + b}, which is the correct answer.

In Choice (4), x^{a + b} * x^{a + b} = x^{2a + 2b}. Eliminate Choice (4).

13. The inputs and outputs of a function are shown in the table below.

This function can best be described as

(1) linear
(2) quadratic
(3) exponential
(4) absolute value

Answer: (3) exponential

There is no common difference but there is a common ratio. You can see from the bottom three lines that 0.5 times 2 is 1 and 1 times 2 is 2. You can check the others as well. Going up the table, each decimal is half as large as the one below it.

Since the table has a common ratio, it is exponential, which is Choice (3).

14. Stephanie is solving the equation x² - 12 = 7x - 8. Her first step is shown below. Given: x² - 12 = 7x - 8
Step 1: x² - 4 = 7x

(1) associative property
(2) commutiative property
(3) distributive property
(4) addition property of equality

Answer: (4) addition property of equality

To get from the Given equation to the Step 1, Stephanie added 8 to both sides of the equation. This is possible because of the addition property of equality.

The correct answer is Choice (4).

15. What is the sum of 8√(3) and √(3)?

(1) 8√(6)
(2) 9√(6)
(3) 7√(3)
(4) 9√(3)

Answer: (4) 9√(3)

The sum of 8x and x is 9x. This is true whether x is apples, kittens, -27, or √(3).

Choice (4) is the correct answer.

16. The dot plots below represent test scores for 20 students on a math test.

The mode for this math test is 80 and the median is 85. Which dot plot correctly represents this data?

(1) I
(2) II
(3) III
(4) IV

Answer: (1) I

The mode of a dot plot is the number with the highest columns of dots. That eliminates Choices (2) and (4).

The median of 20 students is the average of the 10th and 11th score. In plot I, the 10th and 11th dots are both 85. In plot III, the 10th and 11th dots are in the 80 column.

Eliminate Choice (3).

Choice (1) is the correct answer.

More to come. Comments and questions welcome.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra Regents

Part I

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


1. When factored, the expression x³ - 36x is equivalent to

(1) (x + 6)(x - 6)
(2) (x + 18)(x - 18)
(3) x(x + 6)(x - 6)
(4) x(x + 18)(x - 18)

Answer: (3) x(x + 6)(x - 6)

If the expression is third-degree, that is, it has x³ in it, then there must be three factors of x when factored. Eliminated choices (1) and (2).

The Greatest Common Factor of the two terms is x. When you factor the x, you get x(x² - 36).

That binomial is the difference of two perfect squares which factors into (x + 6)(x - 6) because (6)(-6) equals -36 and (6) + (-6) = 0.

The correct answer is Choice (3).

You can put y = x³ - 36x into your graphing calculator and then graph the four choices. You will see that the Choices (1) and (2) are parabolas and that Choice (4) does not match the original graph. You can also look at the Table of Values if the graph goes off the screen.

2. Which equation represents the line that passes through the points (-1,8) and (4,-2)?

(1) y = -2x + 6
(2) y = -2x + 10
(3) y = -0.5x + 7.5
(4) y = -0.5x + 8.5

Answer: (1) y = -2x + 6

Find the slope of the line and then check which choice has an equation that works for both points.

The slope of a line is (y2 - y1) / (x2 - x1) = (-2 - 8) / (4 - (-1)) = 10 / (-5) = -2. Eliminate Choices (3) and (4).

At this point, you can graph the first two lines and check the Table of Values. Or you can do it algebraically.

y - (-2) = -(2)(x - 4)
y - (-2) = -2x + 8
y = -2x + 6

Once again, you could have graph all four lines and checked the Table of Values to find a match.

The correct answer is Choice (1).

3. A geometric sequence is shown below.

1/2, 2, 8, 32, ...

What is the common ratio?

(1) 1/4
(2) 2
(3) 1/2
(4) 4

Answer: (4) 4

The common ratio is what you multiply each term by to get the next term. There are four 1/2's in 2, and 2 goes into 8 four times, etc.

The correct Choice is Choice (4).

If the common ratio was a fraction, like, 1/2 or 1/4, then the sequence would be getting smaller.

If the common ratio was 2, the sequence would've been 1/2, 1, 2, 4, 8, 16, 32, ...

4. What is the constant term of the polynomial 2x³ - x + 5 + 4x²?

(1) 5
(2) 2
(3) 3
(4) 4

Answer: (1) 5

The constant term is the one without a variable. It's a constant because its value doesn't change!

The constant in this polynomial is the + 5.

Choice (1) is the correct answer.

5. A landscaping company charges a set fee for a spring cleanup, plus an hourly labor rate. The total cost is modeled by the function C(x) = 55x + 80. In this function, what does the 55 represent?

(1) the set fee for the cleanup
(2) the hourly labor rate for a cleanup
(3) the profit earned by the company for one cleanup
(4) the number of hours of labor required for one cleanup

Answer: (2) the hourly labor rate for a cleanup

The variable represents the number of hours worked. That is the only thing unknown. The variable is multiplied by the hourly labor rate, which in this function is 55.

The correct answer is (2).

6. Which expression is equivalent to (5x² - 2x + 4) - (3x² + 3x - 1)?

(1) 2x² + x + 3
(2) 2x² - 5x + 5
(3) 2x⁴ + x² + 3
(4) 2x⁴ - 5x² + 5

Answer: (1) 2x² + x + 3

When adding or subtracting polynommials, the exponents DO NOT change. You can only combine like terms, like counting puppies and kittens. Eliminate Choices (3) and (4).

Distribute the minus sign and combine like terms.

(5x² - 2x + 4) - (3x² + 3x - 1)

5x² - 2x + 4 - 3x² - 3x + 1

2x² - 5x + 3

The correct answer is Choice (1).

7. A system of inequalities is graphed on the set of axes below.

Which point is a solution to this system?

(1) (1,1)
(2) (2,-2)
(3) (1,8)
(4) (4,2)

Answer: (4) (4,2)

The solution to the system of inequalities is the set of points in the double-shaded (criss-cross) area as long as they are NOT on the broken line. The broken line is not in the solution set. It represents a boundary that isn't part of the solution.

Choice (1) is the intersection of the broken and solid lines. Eliminate Choice (1).

Choice (2) is on the broken line. Eliminate Choice (2).

Choice (3) is a solution to only one of the two inequalities. Eliminate Choice (3).

Choice (4) is in the double-shaded area. This is the correct answer.

8. In an arithmetic sequence, the first term is 25 and the third term is 15. What is the tenth term in this sequence?

(1) -20
(2) -25
(3) 70
(4) 75

Answer: (1) -20

The sequence is getting smaller, not bigger, so 70 and 75 are just silly. Eliminate Choices (3) and (4).

The difference is -10 after 2 terms, so each term is going down by 5. You could easily write this out:

25, 20, 15, 10, 5, 0, -5, -10, -15, -20.

The correct choice is Choice (1).

Algebraically, if each term subtracts 5 and the tenth term is nine terms after the first term, then the tenth term must be:

25 - 5(9) = -20

This is useful if you need to find a bigger number or if the common difference isn't an "easy" number like 5.

More to come. Comments and questions welcome.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! 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 Problems of the Day (Algebra Regents, June 2025 Part IV)

This exam was adminstered in June 2025.

June 2025 Algebra, Part IV

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

35. Sarah earns $6 per hour babysitting and $12 per hour tutoring. Her goal is to earn at least $120 per week. Sarah is allowed to work a maximum of 14 hours per week doing both jobs.

If x represents the number of hours Sarah babysits and y represents the number of hours she tutors, write a system of inequalities that could model this situation.

On the set of axes below, graph the system of inequalities that you wrote.

State a combination of hours babysitting and tutoring that would satisfy this situation. Justify your answer.

Answer:

Write two inequalities in Standard form. If you can graph from Standard Form, GREAT! If not, solve each inequality for y and use your graphing calculator to assist you.

Sarah can work a maximum of 14 hours, so the total of x and y must be less than or equal to 14.

x + y < 14

The amount she earns is $6 times x hours plus $12 times y hours, and she wants this to be at least (that is, greater than or equal to) $120.

6x + 12y > 120

Both inequalities are solid lines. The first will be shaded beneath the line, and the second will be shaded above the line.

Sarah can work anywhere from 14 hours babysitting and 0 hours tutoring to 0 hours babysitting and 14 hours tutoring. You can plot these two points and draw the line.
Or you can rewrite the inequality as y = 14 - x, and graph a line with a slope of -1 and a y-intercept of 14.

The second inequality has an x-intercept (y=0) of 6x = 120 or x = 20 and a y-intercept (x=0) of 12y = 120 or y = 10.
You can rewrite the inequality as 12y > -6x + 120 or y > -1/2 x + 10

Your graph should look something like this:

You can pick any solution in the double-shaded area, but you have to answer the question. DO NOT give a pair of coordinates.

For example: 2 hours of babysitting and 10 hours of tutoring.

Justification: 2 + 10 < 14; 12 < 14; check.

6(2) + 12(10) > 120; 132 > 120; check.

End of 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!