Algebra Problems of the Day (Algebra Regents, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra Regents

Part I

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


9. The heights, in inches, of eight football players are given below.

76, 70, 72, 70, 69, 71, 78, 74 Which box plot represents these data?

Answer: (3) [see image]

First, put the numbers in order: 69, 70, 70, 71, 72, 74, 76, 78.

The 5-Number Summary is Min 69 and Max 78.
You can eliminate Choices (1) and (2).

The median is (71+72)/5 = 71.5.
Eliminate Choice (4).

Q1 is 70 and Q3 is (74+76)/2 = 75.

The correct answer is Choice (3).

10. A bookstore owner recorded the number of books sold and the profit made selling the books.

What is the average rate of change, in dollars per book, between 100 and 350 books sold?

(1) 0.50
(2) 0.67
(3) 1.50
(4) 2.00

Answer: (3) 1.50

Subtract the profit for 100 books from the profit for 350 books and divide the difference by 250 (which is 350 - 100).

(425 - 50) / 250 = 1.5, which is Choice (3).

11. If f(x) = x², then which function represents a shift of the graph of f(x) 4 units to the right and 3 units down?

(1) g(x) = (x + 4)² + 3
(2) j(x) = (x + 4)² - 3
(3) h(x) = (x - 4)² - 3
(4) k(x) = (x - 4)² + 3

Answer: (3) h(x) = (x - 4)² - 3

Subtracting inside the parentheses moves the parabola to the right. (Adding moves it to the left.)

Subtracting outside the parentheses moves the parabola down. (Adding movies it up.)

Choice (3) is the correct answer.

12. The amount of money a plumber charges is represented by the function p(h) = 45 + 90h. The best interpretation of the y-intercept of this function is that the plumber charges

(1) $45 to come to the house
(2) $45 per hour that he works
(3) $90 to come to the house
(4) $90 per hour that he works

Answer: (1) $45 to come to the house

The plumber charges $45 to come to the house and $90 per hour that he works, so $45 would be the y-intercept and $90 would be the slope of the graph of this function.

Choice (1) is the correct answer.

13. What is the solution to the inequality 2m - 4 < 3(2m + 4)?

(1) m < -2
(2) m > -2
(3) m < -4
(4) m > -4

Answer: (4) m > -4

2m - 4 < 3(2m + 4)
2m - 4 < 6m + 12
-4 < 4m + 12
-16 < 4m
-4 < m
m > -4

Choice (4) is the correct answer.

14. A survey of students at West High School was taken to determine a theme for the prom. The results of the survey are summarized in the table below.

Approximately what percentage of the students who chose the Broadway theme were girls?

(1) 26
(2) 27
(3) 46
(4) 68

Answer: (3) 46

Divide 68 by the number who chose Broadway, which is (68 + 79) = 147.

68/147 = 0.462..., or about 46%.

Choice (3) is the correct answer.

15. The sum of 2√(54) and 2√(6) is

(1) 4√(60)
(2) 8√(15)
(3) 7√(6)
(4) 8√(6)

Answer: (4) 8√(6)

Simplify the radicals so that they can be added together. You need to have a common radical, similar to needing a common denominator for fractions. Basically, you want to combine like terms.

2√(54) + 2√(6)
2√((9)(6)) + 2√(6)
3*2√(6) + 2√(6)
= 8√(6)

Choice (4) is the correct answer.

16. The functions f(x) = x² - 5x - 14 and g(x) = x + 2 are graphed on the same set of axes. What are the solutions to the equation f(x) 5 g(x)?

(1) -14 and 0
(2) 0 and 2
(3) -2 and 8
(4) -2 and 7

Answer: (3) -2 and 8

You can put these functions into the graphing calculator and check the Table of Values to find where the values are equal. Otherwise, you have to solve for x:

x² - 5x - 14 = x + 2
x² - 6x - 16 = 0
(x - 8)(x + 2) = 0
x - 8 = 0 or x + 2 = 0
x = 8 or x = -2

The correct answer is Choice (3).

More to come. Comments and questions welcome.

Writing Update I will be at Philcon this weekend, November 23-34, 2024, in Cherry Hill, NJ, at the DoubleTree by Hilton. I'll be at TWO book launch parties: one that will include my book (see the table below), and another for an anthology that I have a story in. If you're in the area, come on by!

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, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra Regents

Part I

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


1.What is the correct factorization of x² + 4x - 12?

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

Answer: (4) (x - 2)(x + 6)

What two factors of -12 have a sum of +4? One factor must be negative and the other must be positive, and the "larger" number must be positive.

Choice (1) is x² - x - 12.

Choice (2) is x² + x - 12.

Choice (3) is x² - 4x - 12.

Choice (2) is x² + 4x - 12. This is the correct answer.

2. Which situation can be modeled by a linear function?

(1) A printer can print one page every three seconds.
(2) A bank account earns 0.5% interest each year, compounded annually.
(3) The number of cells in an organism doubles every four days.
(4) The attendance at a professional sports team’s games decreases by 1.5% each year.

Answer: (1) A printer can print one page every three seconds.

The linear function never changes. It has a constant rate. One page every three seconds is a constant rate of change for the total number of pages printed. The correct answer is Choice (1).

Choice (2) is not linear because the interest is compounded, meaning interest is earned on the interest.

Choice (3) is not linear because the amount doubles every day, which is not a constant rate of range.

Choice (4) is not linear because the decrease is compounded from year to year, losing 1.5% of the attendance of the year before.

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

(1) -x² + x + 2
(2) -x² - 8x + 7
(3) -x² - 3x + 8
(4) -x² - 9x + 10

Answer: (4) -x² - 9x + 10

Distribute the 3 and the minus sign and then combine like terms.

3(x² - 2x + 3) - (4x² + 3x - 1)
3x² - 6x + 9 - 4x² - 3x + 1
-x² - 9x + 10

This is Choice (4).

4. At Adelynn’s first birthday party, each guest brought $1 in coins for her piggy bank. Guests brought nickels, dimes, and quarters for a total of $28. There were twice as many dimes as nickels and 12 more quarters than nickels. Which equation could be used to determine the number of nickels, x, that her guests brought to her party?

(1) .05x + .10x + .25x = 28
(2) .05x + .10(2x) + .25(x + 12) = 28
(3) .05(2x) + .10x + .25(x + 12) = 28
(4) .05(x + 12) + .10(2x) + .25x = 28

Answer: (2) .05x + .10(2x) + .25(x + 12) = 28

If x represents the number of nickels, then the value of the nickels is .05x. Eliminate Choices (3) and (4).

Choices (1) states that there are the same number of nickels, dimes, and quarters, which is not true. Eliminate Choice (1).

Choice (2) is the correct answer because there are 2 times as many dimes as nickels and 12 more quarters.

5. A student creates a fourth-degree trinomial with a leading coefficient of 2 and a constant value of 5. The trinomial could be

(1) 2x⁴ + 3x² + 5
(2) 2x⁴ + 5x + 3
(3) 4x² - 3x + 5
(4) 4x³ - 5x² + 3

Answer: (1) 2x⁴ + 3x² + 5

Fourth-degree means that the polynomial has x⁴ as the term with the largest exponent. Leading coefficient of 2 means that it will have 2x⁴. The constant has no variable and has to equal 5.

The only possibility is Choice (1).

Choice (2) has a constant of 3, not 5.

Choices (3) and (4) do not have a leading coefficient of 2, nor are they fourth-degree.

6. When solving the equation 4x² - 16 = 0, Laura wrote 4x² = 16 as her first step. Which property justifies Laura’s first step?

(1) distributive property of multiplication over addition
(2) multiplication property of equality
(3) commutative property of addition
(4) addition property of equality

Answer: (4) addition property of equality

Whether or not this is how you would solve this equation, the question is about Laura solving the problem. The first thing she did was add 16 to both sides of the equation. She can do this because of the Addition Property of Equality.

The correct answer is Choice (4).

7. Which expression results in an irrational number?

Answer: (4) 1/3 + √(3)

The sum of a rational and an irrational number is always irrational.

Choice (1) has a value of 3, which is rational.

Choice (2) has a value of 45, which is rational.

Choice (3) has a value of -5/12, which is rational.

8. Which equation has the same solutions as x² + 6x - 18 = 0?

(1) (x + 3)² = 24
(2) (x + 3)² = 27
(3) (x + 6)² = 24
(4) (x + 6)² = 27

Answer: (2) (x + 3)² = 27

Complete the Square:

x² + 6x - 18 = 0
x² + 6x = 18
x² + 6x + 9 = 27
(x + 3)² = 27

Half of 6 is 3 and 3 squared is 9. Also, x² + 6x + 9 is a perfect square, namely (x + 3)².

The correct choice is Choice (2).

More to come. Comments and questions welcome.

Writing Update I will be at Philcon this weekend, November 23-34, 2024, in Cherry Hill, NJ, at the DoubleTree by Hilton. I'll be at TWO book launch parties: one that will include my book (see the table below), and another for an anthology that I have a story in. If you're in the area, come on by!

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!

School Life #43: Two Wrong

(Click on the comic if you can't see the full image.)

(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Did he say he got two wrong ... or that he was just too wrong.

Proprtionality is important. Percentages are important.

Only getting 2 wrong out of 25 isn't a big dead (except to the highly competitive, and I totally respect that). Getting 2 wrong out of 3 or 4 means you need to do more reviewing.

Yes, a regular schedule of updates to this site would be nice. I'm not sure when that will happen. Working on it.

Writing Update I will be at Philcon this weekend, November 23-34, 2024, in Cherry Hill, NJ, at the DoubleTree by Hilton. I'll be at TWO book launch parties: one that will include my book (see the table below), and another for an anthology that I have a story in. If you're in the area, come on by!

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!

Come back often for more funny math and geeky comics.