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

This exam was adminstered in January 2025.

January 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. Anna plans to spend $30 on balloons and party hats for her daughter’s birthday party. Including tax, balloons cost $2 each and party hats cost $1.50 each. The number of party hats Anna needs is twice as many as the number of balloons.

If x represents the number of balloons and y represents the number of party hats, write a system of equations that can be used to represent this situation.

Graph your system of equations on the set of axes below.

State the coordinates of the point of intersection of your lines.

Explain what each coordinate means in the context of the problem.

Answer:

Write the system of equations for the number of balloons and hats and for the price of the balloons and hats. Then rewrite them solving for y in terms of x (isolate the y variable) so you can put them into your graphing calculator or graph using the slope and intercept.

This question may be confusing to anyone who did a lot of test prep because it seems like it should be an inequality question, but it is NOT.

y = 2x
2x + 1.50y = 30

1.5y = -2x + 30
y = -4/3x + 20

The graph would look as follows:

The coordinates of the solution are (6,12).

In the context of the problem, there will be 12 hats and 6 balloons.

End of Part Exam

How did you do?

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 older 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, January 2025 Part III)

This exam was adminstered in January 2025.

January 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. Alex had $1.70 in nickels and dimes on his desk. There were 25 coins in all. Write a system of equations that could be used to determine both the number of nickels, n, and the number of dimes, d, that Alex had.

Use your system of equations to algebraically determine both the number of nickels and the number of dimes that he had.

Answer:

Write the system of equations for the number of nickels and dimes and the value of the nickels and dimes. Then solve the system.

.05n + .10d = 1.70
n + d = 25

.5n + d = 17.0
n + d = 25

.5n = 8
n = 16

16 + d = 25
d = 9

Alex has 16 nickels and 9 dimes.

32. The table below shows the average heart rate, x, and Calories burned, y, for seven men on an Olympic rowing team during a one-hour workout class.

Write the linear regression equation that models these data, rounding all values to the nearest tenth.

State the correlation coefficient, rounded to the nearest tenth.

State what the correlation coefficient suggests about the linear fit of these data.

Answer:

Put the data into the L1 and L2 in your graphing calculator and run a linear regressions.

Rounding to the nearest tenth, you'll get y = 9.1x - 527.6. Remember to write the equation. Put just write a = 9.1 and b = 527.6.

If the correlation coefficient didn't appear on your screen, go to the Catalogue and run "DiagnosticOn" and do the regression a second time.

To the nearest tenth, r = 0.9.

This suggests that there is a strong positive correlation between average heart rate and calories burned.

33. Using the quadratic formula, solve x² + 4x - 3 = 0.
Express your solution in simplest radical form.

Answer:

The equation is in the back of the booklet. (Lucky you -- back in my century, we had to memorize it!)

x = (-b + √(b² - 4ac)) / (2a)

x = (-4 + √(4² - 4(1)(-3)) / (2(1))

x = (-4 + √(16 + 12)) / 2

x = (-4 + √(28)) / 2

x = (-4 + (√(4)*√(7)) / 2

x = (-4 + 2√(7)) / 2

x = -2 + √(7)

The next to last line would have been an acceptable answer.

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

y = x² - 7x + 12
y = 2x - 6

Answer:

Set the two expressions equal to each other and then solve the quadratic equation. Then solve for y for each value of x.

x² - 7x + 12 = 2x - 6
x² - 9x + 18 = 0
(x - 3)(x - 6) = 0
x - 3 = 0 or x - 6 = 0
x = 3 or x = 6

If x = 3: y = 2(3) - 6 = 0
If x = 6: y = 2(6) - 6 = 6.

End of Part III

How did you do?

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 older 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, January 2025 Part II)

This exam was adminstered in January 2025.

January 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. The graph below models Sally’s drive to the store.

State an interval when Sally is traveling at a constant speed. Explain your reasoning

Answer:

Sally's spped is the y-coodinate, so a constant speed means a horizontal line, not a line with a constant slope.

From 5 hours to 9 hours, Sally is driving at a constant speed of 35 mph as indicated by the horizontal line. The rate of change is 0.

26. Graph the function f(x) = x² + 4x + 3.

Sate the equation of the axis of symmetry of f(x).

Answer:

Use your calculator to find the table of values for the parabola. The axis of symmetry is the vertical line that goes through the vertex of the graph and has the equation x = -b/(2a).

The following points are on the graph (-4,3), (-3,0), (-2,-1), (-1,0) and (0,3). Plot these and draw the curve. Remember to put arrows at the end because the graph continues upward.

The access of symmetry is x = -4/(2(1), which is x = -2.

This question is only worth two points, so if you gave the vertex instead of the axis, you didn't get any credit.

27. The function f(x) is shown in the table below.

State an appropriate value for m in the table, so that f(x) remains a function. Explain your reasoning.

Answer:

The relation is a function as long as the x values do not repeat. (Again! I wrote this very same comment in June 2024 and August 2024 for Question 27!)

Therefore, m can have any value except 0, 1, 2, 3, 4, 5, or 6, which are already used. Valid values include 7, 8, 9, or larger; -1, -2, -3, or lower; 1.5, 6 1/2, π.

You are not being asked to define the function or predict what the next value should be.

Reason: the number you selected has not already been used in the table as a value of x. The values of x cannot be repeated in a function.

28. Solve x² + 8x = 33 for x by completing the square.

Answer:

You must complete the square. Any other method, so long as it yields the correct answer, is only worth 1 point.

The square of a binomial, such as (x + n)², looks like x² + 2n + n². So the middle term of the quadratic expression is 2n and the original n is half as much.

Half of 8 is 4, so the binomial that is being squared is (x + 4).

If you square (x + 4), you get x² + 8x + 16, so to "COMPLETE THE SQUARE", we need to add 16 to both sides of the equation: x² + 8x + 16 = 33 + 16.

Factor this into: (x + 4)² = 49.

Now you can solve this:

(x + 4)² = 49
x + 4 = + 7
x = 7 - 4 or x = -7 - 4
x = 3 or x = -11

29. If f(x) = (-3x - 5)/2, algebraically determine the value of x when f(x) = -22

Answer:
Note that the problem does NOT say x = -22, what is f(-22)? The question says if f(x) = -22, then what is x?

Replace f(x) with (-3x - 5)/2 in the equation, like this:

(-3x - 5)/2 = -22
-3x - 5 = - 44
-3x = -39
x = 13

If you evaluated f(-22) = 30.5, you received half credit.

30. Rationalize the denominator of the fraction below. Express the solution in simplest form.

4 / √(2)

Answer:
Multiply the fraction by √(2) / √(2) to rationalize the denominator, and then reduce the fraction.

4 / √(2) * √(2) / √(2) = 4 √(2) / 2 = 2 √(2)

End of Part II

How did you do?

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 older 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, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra 2 Regents

Part I

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


17. Given q(x) = 2log(x) and r(x) = (x - 2)³ - 4, what is a solution of q(x) = r(x) to the nearest tenth?

(1) 1.1
(2) 3.7
(3) 3.9
(4) 4.3

Answer: (2) 3.7

This is a MULTIPLE-CHOICE question. Do not attempt to solve it algebraically. It will NOT be a fun time.

You can graph both equations on your calculator and look for the point where they intersect. (There's actually a function on the calculator that will help you find it if you know how to use it.)

You can also find the values of q(1.1), r(1.1), q(3.7), r(3.7), etc to find the ones that are the same.

Let's do this just for giggles:

q(1.1) = 0.08, r(1.1) = -4.7

q(3.7) = 1.1, r(3.7) = .91

q(3.9) = 1.2, r(3.9) = 2.9

q(4.3) = 1.7, r(4.3) = 8.2

You will notice that when x = 3.7, the values of q(x) and r(x) are the closest. They differ because of rounding. (I did a little experimenting so that you wouldn't have to, and found that x = 3.725 is pretty close but still not exact.)

Choice (2) is the correct answer.

How would I solve this algebraically? I wouldn't. Simple as that.

If you would like to, please place it in the comments.

18. The volume of a cardboard box can be modeled by V(x), which is the product of the length, width, and height, x. If the length can be represented by L(x) = 18 - 2x and the width can be represented by W(x) = 18 - 2x, then which function represents V(x)?

(1) V(x) = 4x² - 72x + 324
(2) V(x) = 4x³ - 72x² + 324x
(3) V(x) = -3x + 36
(4) V(x) = 4x³ + 324x

Answer: (2) V(x) = 4x³ - 72x² + 324x

It should be "obvious" that since we have to multiply three terms with the variable x in them that the final formula will contain x³ in it.

V = (18 - 2x)(18 - 2x)(x)
= (324 - 36x - 36x + 4x²)(x)
= 324x - 72x² + 4x³

Choice (2) is the correct choice.

Choice (4) is missing the middle term. Choice (1) is the Area of the base. Choice (3) is nonsense because it just added the three expressions.

19. The expression 8^x/2 * 8^x/3 is equivalent to

(1) ⁶√(8^5x)
(2) 64^(5x/6)
(3) ⁵√(8^2x)
(4) 64^(x2/6)

Answer: (1) ⁶√(8^5x)

Keep the base and add the exponents x/2 + x/3.

The common denominator is 6, so 3x/6 + 2x/6 = 5x/6, and the expression will be 8^5x/6, which is not one of the listed choices.

However, the one-sixth power is the same as taking the sixth root of the expressions.

Therefore, 8^5x/6 is the same as (8^5x)^(1/6 which is the same as ⁶√(8^5x). This is Choice (1).

20. If θ is an angle in standard position whose terminal side passes through the point (-3,-4), which statement is true?

(1) sec θ > 0 and tan θ > 0
(2) sec θ < 0 and tan θ < 0
(3) sec θ > 0 and tan θ < 0
(4) sec θ < 0 and tan θ > 0

Answer: (4) sec θ < 0 and tan θ > 0

Point (-3,-4) is in Quadrant III where sin is negative, cos is negative, and tan is positive. Eliminate Choices (2) and (3).

Sec = 1/cos and has the same sign and cos, which is negative in Quadrant III. So the correct answer is Choice (4).

21. What is the value of y for the system shown below? 3x + 4y - 5z = -27
2x + 3y - z = -3
6x - y + 4z = 3

(1) -27
(2) 6
(3) 3
(4) -3

Answer: (3) 3

Solve the system of equations by eliminating the x term first since 2 and 3 are factors of 6.

3x + 4y - 5z = -27
2x + 3y - z = -3
6x - y + 4z = 3

6x + 8y - 10z = -54
6x + 9y - 3z = -9
6x - y + 4z = 3

First 2 equations: y + 7z = 45
Last 2 equations: 10y - 7y = -12
Combined: 11y = 33
y = 3

Choice (3) is the correct answer.

22. The number of employees who work nights and weekends at a department store is summarized in the table below.

Let N represent the event “works nights” and let W represent the event “works weekends.” Based on the table, are N and W independent events?

(1) Yes, because P(N) • P(W) = P(N ∩ W).
(2) Yes, because P(N) • P(W) =/= P(N ∩ W).
(3) No, because P(N) • P(W) = P(N ∩ W).
(4) No, because P(N) • P(W) =/= P(N ∩ W).

Answer: (1) Yes, because P(N) • P(W) = P(N ∩ W).

The answer is either Yes because they are equal or No because they are not equal. Multiply the probabilities.

There are a total of 120 employees. P(N) = 20/120 and P(W) = 48/120, and (20/120)(48/120) = (1/6)(2/5) = 1/15.

P(N ∩ W) = 8/120 = 1/15.

So P(N) • P(W) = P(N ∩ W).

The correct answer is Choice (1).

23. Which expression is equivalent to x⁸ - y⁸?

(1) (x - y)⁸
(2) (x² + y²)²(x² - y²)²
(3) (x⁴ + y⁴)(x² + y²)(x + y)(x - y)
(4) (x + y)⁴(x - y)⁴

Answer: (3) (x⁴ + y⁴)(x² + y²)(x + y)(x - y)

The given expression is a Difference of Perfect Squares and canb be factored into two conjugates. One of those factors will still be a Difference of Perfect Squares and can be factored again. And then again.

x⁸ - y⁸
(x⁴ + y⁴)(x⁴ - y⁴)
(x⁴ + y⁴)(x² + y²)(x² - y²)
(x⁴ + y⁴)(x² + y²)(x + y)(x - y)

The correct answer is Choice (3).

If you factor Choice (1), you will find that it is equivalent to Choice (4), so both could be eliminated.

24. A research assistant receives a first year salary of $90,000 and a 2% annual raise throughout the first ten years of employment. In total, how much money will be earned over the first ten years, to the nearest dollar?

(1) $91,837
(2) $109,709
(3) $877,917
(4) $985,475

Answer: (4) $985,475

Using the formula for a Geometric Series: S_n = (a₁ - a₁rⁿ) / (1 - r), we can calculate the answer.

S₁₀ = (90000 - 90000*1.02¹⁰) / (1 - 1.02) = 985474.89, which is Choice (4).

End of Part I. How did you do?

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 older 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, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra 2 Regents

Part I

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


9. The asymptote of the graph of f(x) = 5 log(x + 4) is

(1) y = 6
(2) x = -4
(3) x = 4
(4) y = 5

Answer: (2) x = -4

First of all, you can graph this and look. Second, the asymptote will be a vertical line, which means it starts with "x = ", so eliminate Choices (1) and (4).

The asymptote of f(x) = log x is the y-axis, which is x = 0. Multiplying by 5 will not change the asymptote. However, adding 4 inside the parentheses, that is log(x + 4), will move the asymptote 4 units to the left, to x = -4. Choice (2) is the correct answer.

10. The probability of having math homework is 1/3 and the probability of having English homework is 1/7. The probability of having math homework or having English homework is 9/21. What is the probability of having math homework and having English homework?

(1) 19/21
(2) 1/5
(3) 1/21
(4) 10/21

Answer: (3) 1/21

P(M) + P(E) - P(M|E) = 1/3 + 1/7 - 9/21 = 7/21 + 3/21 - 9/21 = 1/21

Choice (3) is the correct choice.

11. The solution set to the equation x - 1 = √(2x + 6) is

(1) {5, -1}
(2) {5}
(3) {-1}
(4) { }

Answer:(2) {5}

You can check each of the listed answers to see if either, or both work.

For x = 5, 5 - 1 = 4 and √(2(5) + 6) = 4, so the answer is either Choice (1) or (2).

For x = -1, -1 -1 = -2, which cannot be the result of a square root. Eliminate Choices (1) and (3).

Choice (1) is the correct answer.

If you tried to solve it algebraically:

x - 1 = √(2x + 6)
x² - 2x + 1 = 2x + 6
x² - 4x - 5 = 0
(x - 5)(x + 1) = 0
x = 5 or x = -1

However, when we test the answers, we see that x = -1 must be discarded as extraneous, and only x = 5 is correct.

12. Given x > 0, the expression (1 / x^-2)^(-3/4) is equivalent to

(1) x √(x)
(2) 1 / (x √(x))
(3) ∛x²
(4) 1 / (∛x²)

Answer: (2) 1 / (x √(x))

Use the rules for exponents to evaluate the expression.

(1 / x^-2)^(-3/4)
(x²)^(-3/4)
(x^(-3/2))
1 / x^3/2
1 / √(x³ = 1 / x √(x)

Choice (2) is the correct answer.

13. The graph of which function has a period of 3?

(1) y = -7sin(2π/3 x) - 5
(2) y = -7sin(3π/2 x) + 9
(3) y = -7sin(3x) - 5
(4) y = 3sin(π x) + 9

Answer: (1) y = -7sin(2π/3 x) - 5

Once again, you can graph these equations and look to see which is correct.

The period of y = A sin(x) + C is 2π. The period of y = A sin(Bx) + C is 2π/B.

In Choice (1), 2π/ (2π/3) = 3. This is the correct answer.

In Choice (2), 2π/ (3π/2) = 4/3. Eliminate Choice (2).

In Choice (3), 2π/ 3 is not equal to 3. Eliminate Choice (3).

In Choice (4), 2π/ (π) = 2. Eliminate Choice (4).

14. Which graph could represent a 4th degree polynomial function with a positive leading coefficient, 2 real zeros, and 2 imaginary zeros?

Answer: (1) [See image]

A positive leading coefficient on a fourth-degree polynomial means that the end behavior is toward positive infinity on both ends. Eliminate Choices (3) and (4).

Two real zeroes means that the graph crosses the x-axis in two locations, which eliminates Choice (4) and leaves Choice (1).

Since there are only two real zeroes, there must be two imaginary zeroes as well.

15. Given i is the imaginary unit, which expression is equivalent to 5i(2x + 3i) - x√(-9)?

(1) 15 + 13xi
(2) -15 + 13xi
(3) 15 + 7xi
(4) -15 + 7xi

Answer: (4) -15 + 7xi

Use the Distributive Property and the Order of Operations. Don't forget that i² = -1 and that √(-1) = i.

5i(2x + 3i) - x√(-9)
10xi + 15i² - 3xi
-15 + 7xi

The correct answer is Choice (4).

16. 6 What is the focus of the parabola 8(y + 2) = (x + 5)²?

(1) (-5,0)
(2) (-5,-4)
(3) (5,0)
(4) (5,4)

Answer: (1) (-5,0)

The vertex form of a parabola is y = a(x - h) + k, and the focus of the parabola is at point (h, k + 1/(4a)). Rewrite the equation in vertex form.

8(y + 2) = (x + 5)²

y + 2 = (1/8)(x + 5)²

y = (1/8)(x + 5)² - 2

If a = 1/8, then 1/(4a) = 1/(4(1/8)) = 1/(1/2) = 2.

So (h, k + 1/(4a)) = (-5, -2+2) = (-5,0).

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 older 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, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra 2 Regents

Part I

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


1. A grocery store owner wonders how many customers bring reusable bags to the store. An employee stands at the store entrance for two hours and counts the number of people bringing in reusable bags. This type of study is best classified as

(1) a census
(2) an experiment
(3) an observational study
(4) a survey

Answer: (3) an observational study

This is a case of an observational study. I hope it's self-explanatory.

No questions are being asked, so it isn't a census or a survey. No experiment is being conducted.

Choice (3) is the correct choice.

2. The graph of y = 2^x - 4 is positive on which interval?

(1) (-∞,∞)
(2) (2,∞)
(3) (-0,∞)
(4) (-4,∞)

Answer: (2) (2,∞)

The graph will cross the x-axis (y=0) when 2^x = 4, which occurs when x = 2.

When x > 2, the graph will be above the x-axis.

You can also put this equation into your graphing calculator and look at the screen and the table of values.

The correct answer is Choice (2).

3. Tim deposits $300 into a savings account. The annual interest rate is 2.7% and compounds monthly. He uses the equation A = 300(1 + 0.027/12)^12t to determine how much money, A, he will have after t years. Which equation is equivalent to Tim’s equation?

(1) A = 300[(1.00225)¹²]^t
(2) A = 300[(0.08558)¹²]^t
(3) A = 300[1 + (0.027/12)^12t)]
(4) A = (300)^12t(1)^12t + (0.027/12)^12t

Answer: (1) A = 300[(1.00225)¹²]^t

Divide 0.027 by 12 and you'll get 0.00225. Add 1 and you have 1.00225. Choice (1) is the correct answer.

The other answers are a bit silly and violate a bunch of rules of exponents.

Choice (2) splits the 12 and t exponents like Choice (1) does, but the growth rate becomes a decay rate. I have no idea where .08558 came from.

Choice (3) applied the exponent 12t only to the fraction. This is like saying that (1 + 2)² is equal to (1 + 2²), which is silly.

In Choice (4), the 300 gets the exponent of 12t, which is crazy even without all the other problems. This should only be selected by students not even reading the question.

4. Which equation is true for all real values of x?

(1) x⁴ + x = (x + 1)(x³ - x² + x)
(2) x⁴ + x = (x + 1)(x³ + x)
(3) x⁴ + x = (x² + x)²
(4) x⁴ + x = (x + 1)(x³ + x² + x)

Answer: (1) x⁴ + x = (x + 1)(x³ - x² + x)

You can multiply the polynomials on the right side to see which one gives you x⁴ + x. Or you can put each of these into the graphing calculator to see which ones give you the same graph.

y = x⁴ + x
y = (x + 1)(x³ - x² + x)
y = (x + 1)(x³ + x)
y = (x² + x)²
y = (x + 1)(x³ + x² + x)

Choice (1) has the same graph and table of values.

Algebraically:

(x + 1)(x³ - x² + x)
(x)(x³ - x² + x) + (x³ - x² + x)
x⁴ - x³ + x² + x³ - x² + x
x⁴ + x

5. The solution of

x/(x + 3) + 2/(x - 4) = (2x + 27)/(x² - x - 12) is

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

Answer: (4) 7

You can work backward from the answers to see which one works. You can immediately see that x = -3 is not a real answer.

You can graph y = x/(x + 3) + 2/(x - 4) - (2x + 27)/(x² - x - 12) and look for a value of y = 0.

Or you can solve this algebraically once you notice that (x + 3)*(x - 4) = x² - x - 12.

x/(x + 3) + 2/(x - 4) = (2x + 27)/(x² - x - 12)

(x - 4)/(x - 4) * x/(x + 3) + (x + 3)/(x + 3) * 2/(x - 4) = (2x + 27)/(x² - x - 12)

(x - 4)(x)/(x² - x - 12) + (x + 3)(2)/(x² - x - 12) = (2x + 27)/(x² - x - 12)

(x - 4)(x) + (x + 3)(2) = 2x + 27

x² - 4x + 2x + 6 = 2x + 27
x² - 4x - 21 = 0
(x - 7)(x + 3) = 0
x = 7 or x = -3

We've already discarded x = -3 as an answer, so x = 7 is the only solution, and that is Choice (4).

If you had graphed the above equation, you would have gotten y = 0 for x = 7.

6. The cost, in dollars, of a single-ride fare in the New York City subway in the years since 1904 is listed in the table below.

Which equation best models the cost of a single-ride fare based on these data?

(1) y = 0.0375(1.0392)^x
(2) y = 1.0392(0.0375)^x
(3) y = 0.0234x - .0487
(4) y = -0.179 + 0.356 ln(x)

Answer: (1) y = 0.0375(1.0392)^x

You can see that the fare went up only 10 cents in the first fifty years but then increased 35 cents in the next twenty or so and $1.00 more in the next twenty. Clearly this is an exponential function and not a linear function.

Only Choices (1) and (2) are exponential functions. Choice (2) shows decay, not growth, which leaves Choice (1) as the correct answer.

7. Which expression is equivalent to (6x⁴ + 4x³ + x + 200) / (x + 2)

(1) 6x² - 8x + 17 + 166/(x + 2)
(2) 6x² - 16x + 33 + 266/(x + 2)
(3) 6x³ + 16x² + 32x + 65 + 330/(x + 2)
(4) 6x³ - 8x² + 16x - 31 + 262/(x + 2)

Answer: (4) 6x³ - 8x² + 16x - 31 + 262/(x + 2)

A fourth-order polynomial divided by (x + 2) becomes a third-order polynomial. Eliminate Choices (1) and (2).

Once again, you can put the original equation along with Choices (3) and (4) into your graphing calculator to see which has the same table of values.

Or you can choose a random value for x, such as 8 or 10, and see which expression gives the same value.

Or you can divide:

Once you get this far, you only have one option left, and that is Choice (4).

If you want to continue, just to make sure, the rest of the division looks like this:

There is a remainder of 262 which means that the final term will be 262/(x + 2).

8. The solution to the equation 6(2^{x + 4}) = 36 is

(1) -1
(2) ln 36 / ln 12 - 4
(3) ln(3) - 4
(4) ln 6 / ln 2 - 4

Answer: (4) ln 6 / ln 2 - 4

A quick check can eliminate x = -1 because 2³ = 8 and 6(8) is not equal to 36.

Use inverse operations:

6(2^{x + 4}) = 36
2^{x + 4} = 6
ln 2^{x + 4} = ln 6
(x + 4) ln 2 = ln 6
x + 4 = ln 6 / ln 2
x = ln 6 / ln 2 - 4

This is Choice (4).

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 older 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, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Geometry Regents

Part I

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


17. The line represented by the equation y = 4x + 15 is dilated by a scale factor of 2 centered at the origin. Which equation represents its image?

(1) y = 4x + 15
(2) y = 4x + 30
(3) y = 8x + 15
(4) y = 8x + 30

Answer: (2) y = 4x + 30

The slope of the line will not change because in a dilation the orientation of the image is the same as the pre-image. However, the y-intercept will be twice as far from the origin as the origin line.

Since the original line has a point at (0,15), the image will go through the point (0,30).

Choice (2) is the correct answer.

18. Line segment RH has endpoints R(-4,4) and H(2,-4). Which equation represents a line perpendicular to RH that passes through the point (3,-1)?

(1) y + 1 = 3/4(x - 3)
(2) y + 1 = -3/4(x - 3)
(3) y + 1 = 4/3(x - 3)
(4) y + 1 = -4/3(x - 3)

Answer: (1) y + 1 = 3/4(x - 3)

The only difference in the choices is the slope of the line. So you need to find the slope of the original line and then find the slope of the line that is perpendicular to it.

The slope of line RH is △y/△x = the change in y / the change in x = (-4 - 4) / (2 - (-4)) = -8 / 6 = -4/3.

The slope of a line perpendicular to RH would have a slope that was the inverse reciprocal of the slope of RH, which would be +3/4. This is Choice (1).

Check: (-4/3)(+3/4) = (-12/12) = -1. The lines are perpendicualr.

19. In right triangle SNO below, altitude NW is drawn to hypotenuse SO.

Which statement is not always true?

(1) SO/SN = SN/SW
(2) SW/NS = NS/OW
(3) SO/ON = ON/OW
(4) OW/NW = NW/SW

Answer: (2) SW/NS = NS/OW

There are 3 right triangles in the image: the large one and the two smaller one. All three triangles are similar, meaning that they have the same shape, congruent corresponding angles, and corresponding sides that are proportional. Each triangle has a hypotneuse across from the right angle, and a longer leg and a shorter leg across from the larger and smaller acute angles, respectively.

The proportion in Choice (1) compares the hypotenuse / short leg of SON with the hypotenuse / short leg of NSW. Eliminate Choice (1).

The proportion in Choice (2) compares the short leg / hypotenuse NSW with two legs of two different triangles. There is no reason for these to ratios to be equal to each other. Choice (2) is the correct answer.

The proportion in Choice (3) compares the hypotenuse / long leg of SON with the hypotenuse / long leg of NOW. Elimnate Choice (3).

The proportion in Choice (4) compares the long leg / short leg of NOW with the long leg / short leg of NSW. Eliminate Choice (4).

20. A rectangle has a width of 3 and a length of 4. The rectangle is dilated by a scale factor of 1.8. What is the area of its image, to the nearest tenth?

(1) 3.7
(2) 6.7
(3) 21.6
(4) 38.9

Answer: (4) 38.9

The area of the image is Length X scale factor times Width X scale factor.

A = 3 * 1.8 * 4 * 1.8 = 38.88 = 38.9

The correct answer is Choice (4).

Notice that the original rectangle had an area = 3 * 4 = 12, and the image was bigger, so Choices (1) and (2) are just silly. Choice (3) is the product of 3 * 4 * 1.8.

21. In the diagram below of circle P, diameter MD and chord AL intersect at Q, m∠AQD = 46°, and mLD = 124°.

What is m AD?

(1) 36°
(2) 46°
(3) 51°
(4) 92°

Answer: (1) 36°

You have to work your way around the circle to fill in other things you know. For example, since MD is a diameter, then we know that the sum of arcs ML and LD is 180 degrees. Since LD = 124, then ML = 180 - 124 = 56 degrees.

The sum of arc ML and AD is equal to twice the size of angle AQD, so 56 + AD = 92, and AD = 92 - 56 = 36 degrees, which is Choice (1).

Logically, since Q is not the center of the circle, arc AD was going to be less than 46 degrees, and there was only one choice that made since.

The rule that was used above says, basically, that if two intersecting chords intercept two arcs, the vertical angles were the chords intersect will be equal to the average of the two arc sizes. In this case, the arcs are 56 and 36 degrees, which average to 46 degrees.

22. The right prism with a triangular base shown below is cut by a plane perpendicular to its bases.

The two-dimensional shape of the cross section is always a

(1) triangle
(2) rhombus
(3) pentagon
(4) rectangle

Answer: (4) rectangle

Perpendicular to the bases means you're cutting up and down like you're slicing into a wedge of cheese. When you cut it that way, no matter how you slice it, the cross section will be a rectangle composed of a line from the top and bottom planes (parallel to each other) and two vertical lines down two of the three sides of the triangle. Those two vertical lines are also parallel. Two pairs of parallel sides make a parallelogram, but the sides are horizontal and vertical, making right angles, so it's a rectangle.

The correct answer is Choice (4).

If the question had said parallel to the bases, slicing horizontally between the top and bottom bases, then the cross section would have been a triangle.

The other two choices, pentagon and rhombus, are just silly. Why not an octagon?

23. A rectangular fish tank measures 24 inches long, 12 inches wide, and 16 inches high, as modeled in the diagram below.

If the empty tank weighs 25 pounds and the fish tank is filled with water to a height of 14 inches, what is the approximate weight of the tank and water?
[27.7 in³ = 1 pound of water]

(1) 146
(2) 166
(3) 171
(4) 191

Answer: (3) 171

First of all, the height of the tank doesn't matter because there's only 14 inches of water in the tank, so that's the "height" that we need. Second, you have to find the Volume in cubic inches and then convert that to pounds of water.

Volume = L X W X H = (24)(12)(14) = 4032 in³

Weight of the water = (4032 in³) / (27.7 in³ / pound) = 145.56 pounds

Weight of the tank = 25 pounds

Total weight = 146 + 25 = 171, which is Choice (3).

Notice that 146 was one of the incorrect Choices.

24. A circle has a radius of 4.5. What is the measure of the central angle that intercepts an arc whose length is 6.2, to the nearest degree?

(1) 35°
(2) 42°
(3) 64°
(4) 79°

Answer: (4) 79°

Find the circumference of the circle and then find the part of the circle that the arc represents.

C = 2 (3.141592)(4.5) = 28.274328

Divide 6.2/28.274328 = 0.219

Multiply 0.219 * 360° = 78.84° or 79°, which is Choice (4).

End of Part I.

How did you do?

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 older 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!