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

This exam was adminstered in June 2025.

June 2025 Algebra, 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.
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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!

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

This exam was adminstered in June 2025.

June 2025 Algebra, Part III

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

31. Graph f(x) = -3x and g(x) = x² + 2 on the set of axes below.

State the values of x that satisfy the equation f(x) = g(x)

Answer:

Use your graphing calculator to find the points that you need to point. Determine for which x values that the graphs intersect. Only answer the x values, do not give coordinates in your answer.

The equation f(x) = g(x) is true when x = -1 or x = -2.

This could've been solved algebraically for partial credit.

32. Using the quadratic formula, solve 6x² + 2x - 1 = 0.
Express the answer in simplest radical form.

Answer:

Use the quadratic formula with a = 6, b = 2, and c = -1.

( -(2) + √( (2)² - 4(6)(-1) ) ) / ( 2(6))
( -2 + √( 4 - (-24) ) ) / 12
( -2 + √(28) ) / 12
( -2 + √( (4)(7) ) ) / 12
( -2 + 2√(7) ) / 12
( -1 + √(7) ) / 6

33. The table below shows the price of a new cell phone and the length of time, in months, since its release.

State the linear regression equation for this set of data. Round all values to the nearest hundredth.

State the correlation coefficient for this data set, to the nearest hundredth.

State what the correlation coefficient indicates about the linear fit of the data.

Answer:

Put the data into Lists L₁ and L₂. Make sure that "DiagnosticOn" has been executed. (It's in the Catalogue.) Run a Linear Regression.

The equation for the linear equation will be y = -23.67x + 1216

The correlation coefficient is r = -0.99.

The correlation coefficient indicates a strong correlation.

There isn't much to say about this. Either you know how to enter the data and run the linear regression, or you do not.

Or you copy the answers from the people next to you, which may or may not get you into trouble.

34. Solve the following system of equations algebraically for all values of x and y.

y = x² + 9x + 4
y - 2x = -6

Answer:

Isolate the y in the second equation. Then set the two expressions equal to each other and then solve the quadratic equation. After that, find the value of y that corresponds to each value of x.

x² + 9x + 4 = 2x - 6
x² + 7x + 10 = 0
(x + 5)(x + 2) = 0
x + 5 = 0 or x + 2 = 0
x = -5 or x = -2

If x = -5: y = 2(-5) - 6 = -16
If x = -2: y = 2(-2) - 6 = -10.

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

This exam was adminstered in June 2025.

June 2025 Algebra, Part II

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

25. A survey was taken to determine whether students preferred to watch videos or listen to music. Of the 100 students surveyed, 44 were seniors. Of the 65 students who preferred to watch videos, 42 were juniors. Use this information to complete the frequency table below.

Answer:

Fill in what you know and the add or subtract to find the remaining boxes.

26. Solve the inequality for y:

5(2 - y) > -11y - 8

Answer:

Solve the same way that you solve an equation with variables on both sides, but by mindful if you have to multiply or divide by a negative number. Isolate the variable on one side of the inequality.

5(2 - y) > -11y - 8
10 - 5y > -11y - 8
10 + 6y > -8
6y > -18
y > -3

27. Express (5x - 3)(-2x + 7) as a trinomial in standard form.

Answer:

Standard form means to combine all like terms and then arrange the three terms in order of the highest exponent to the lowest (which could be 0 if there is no variable).

You can use FOIL or the Box (Lattice/Area) Method:

(5x - 3)(-2x + 7)
-10x² + 35x + 6x - 21

-10x² + 41x - 21

28. The first and fourth terms in an arithmetic sequence are given below.

-20, _____, _____, -2
Determine the eighth term.

Answer:

Find the differnce between each term and then find the eight term.

(-2) - (-20) = 18 over 3 terms, so each term is increasing by +6.

The eighth term is 4 past the fourth term, so 4(6) = 24. Add -2 + 24 = 22 is the eighth term.

Check: -20, -14, -8, -2, 4, 10, 16, 22.

29. Write an equation in slope-intercept form for the line that passes through (-2,5) and has a slope of -3. [Use of the set of axes below is optional.]

Answer:
If you point the plots on the graph, you should have (-2,5), (-1,2), and (0,-1). That gives the equation y = -3x - 1.

If you start with Point-Slope form, it's

y - 5 = -3(x + 2)
y - 5 = -3x - 6
y = -3x - 1

30. Factor the expression x³ - 36x completely.

Answer:
Any time you see the words "factor" and "completely", you can be sure that there will be multiple steps. Usually, it will involve factoring out the Greatest Common Factor first before factoring what remains. Sometimes it's using the Difference of Squares rule multiple times.

x³ - 36x

x(x² - 36)

x(x - 6)(x + 6)

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!

My Other Blogs

I've had many other blogs over the years that come and go as they fall out of favor until I revive them.

Currently, the other two blogs that I update are:

Walkin' in Brooklyn - my walking blog

https://walkininbrooklyn.blogspot.com/

Mr. Burke's Reading List - my book blog, more of a log of books read than actual reviews

https://mrburkereads.blogspot.com/

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

This exam was adminstered in June 2025.

June 2025 Geometry, Part IV

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

35. Given: Quadrilateral CARI with CA ≅ RI and CI ≅ RA, and AEI and LEH are drawn.

Prove: HA * EL = LI * EH

Answer:

You need to write a proof that shows, first of all, that CARI is a parallelogram. You cannot assume that it is. Then you are going to show that there are two triangles that are similar, so their corresponding sides are proportional. THEN you are going to cross-multiply BUT YOU ARE NOT GOING TO SAY "cross multiply" because that is a NO-NO. (I'm not kidding. See below.)

Statement	Reason
1. CA ≅ RI and CI ≅ RA	Given.
2. CARI is a parallelogram.	Opposite sides of the quadrilateral are congruent.
3. ∠HAE ≅ ∠LIE	Alternate interior angles of parallel lines are congruents
4. ∠HEA ≅ ∠LEI	Alternate interior angles of parallel lines are congruents
5. △HEA ~ △LEI	AA Theorem
6. HA / EH = LI / EL	Corresponding sides of similar traingles are proportional.
7. HA * EL = LI * EH	The product of the means equals the product of the extremes.

You could also have stated that "Cross products are equal." However, "cross multiply" was NOT acceptable. The best that I could figure is that the former is a reason while the latter is a command.

End of Part IV

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!

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

This exam was adminstered in June 2025.

June 2025 Geometry, Part IV

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

35. Triangle PET has vertices with coordinates P(26,4), E(6,8), and T(24,22).
Prove △PET is a right triangle.
[The use of the set of axes on the next page is optional.]

State the coordinates of N, the image of P, after a 180° rotation centered at (1,3).

Prove PENT is a rectangle.
[The use of the set of axes below is optional.]

Answer:

Use the graph. First, it counts as work. Second, having a visual representation makes solving the problem easier. Third, if the triangle is graphed, you can count boxes rather than use formuals and risk encountering a sign or other computational error.

The slope of PE is 4/12 = 1/3.

The slope of PT is -6/2 = -3.

The product of (1/3)(-3) = -1, so PE is perpendicular to PT. Therefore PET is a right triangle.

To find the coordinates of N, count the change in the x-value and the y-value from P to the coordinates (1,3). Then count those same numbers again. (Seven right and one down.) The coordinates of N are (8,2).

The slope of EN is -6/2 = -3.

The slope of TN is 4/12 = 1/3.

PE || NT and PT || EN, so PENT is a parallelogram. Angle P is a right angle, so PENT is a rectangle.

End of Part IV

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!