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

The Problems of the Day are BACK! Let's see for how long. Life, y'know?

This exam was adminstered in August 2025.

More Regents problems.

August 2025 Algebra Regents

Part I

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


9. The tables below show the input and output values of four different functions.

(1) f(x)
(2) g(x)
(3) h(x)
(4) j(x)

Answer: (4) j(x)

The output values repeat in f(x) and g(x), so eliminate Choices (1) and (2) as they change direction and can't be linear.

The rate of change in h(x) is changing, not constant. It goes down 1, down 2, down 4, etc. Eliminate Choice (3).

The rate of change in j(x) is constant. It goes up 4 from -11 to -7, then up 4 again to -3, etc. This is the correct choice.

10. What is the solution set to the equation 3x² = 24x?

(1) {8}
(2) {0,8}
(3) {0,-8}
(4) {0,8,-8}

Answer: (2) {0,8}

You can solve this by factoring:

3x² = 24x
x² = 8x
x² - 8x = 0
x(x - 8) = 0
x = 0 or x - 8 = 0

x = 0 or x = 8

A quadratic equation can have at most two solutions, so Choice (4) could've been eliminated immediately.

Likewise, since x = 0 is "obviously" a solution, Choice (1) could have been eliminated as well.

If you didn't wish to factor, you could've checked 0, 8, and -8 in a calculator.

11. The table below shows the radioactivity level of a substance after the given time, t, in seconds.

What is the average rate of change in radioactivity level over the interval 1 < t < 3?

(1) 3.75
(2) -3.75
(3) 4.6875
(4) -4.6875

Answer: (2) -3.75

Average rate of change is the slope of the line from t = 1 to t = 3.

(2.5 - 10) / (3 - 1) = -7.5 / 2 = -3.75

That is Choice (2).

12. Fred recorded the number of minutes he read each day, from Monday through Friday. His results are shown in the table.

What is the correlation coefficient, to the nearest thousandth, and strength of the linear model of these data?

(1) 0.984 and strong
(2) 0.968 and strong
(3) 0.984 and weak
(4) 0.968 and weak

Answer: (1) 0.984 and strong

This is an annoying question for the multiple-choice portion of the exam because of the work involved. You can't reason out the final answer because 0.984 and 0.968 are so close together. There's no short cut.

However, both 0.984 and 0.968 indicate strong relationship, so Choices (3) and (4) can be eliminated.

You need to put the data into Lists in your graphing calculation and run a Linear Regression.

You will then find that r² = 0.968 and r = 0.984. You want the r value. (You'll never want the r² value for this level of algebra.

The correct answer is Choice (1).

13. Given f(x) = x², which function will shift f(x) to the left 3 units?

(1) g(x) = x² + 3
(2) h(x) = x² - 3
(3) j(x) = (x - 3)²
(4) k(x) = (x + 3)²

Answer: (4) k(x) = (x + 3)²

If you don't remember the rules, you can graph each of these and check the graphs or Tables of Values.

In Choice (1), the "+ 3" shifts the graph up 3 units. Eliminate Choice (1).

In Choice (2), the "- 3" shifts the graph down 3 units. Eliminate Choice (2).

In Choice (3), the "- 3" inside the parentheses shifts the graph to the right 3 units. Eliminate Choice (3).

In Choice (4), the "+ 3" inside the parentheses shifts the graph to the left 3 units. Choice (4) is the correct answer.

14. A class of 20 students was surveyed to determine the number of pets each student owned. The data are represented in the dot plot below.

Which statement about the data is correct?
(1) The mean and the median are the same.
(2) The median and the mode are the same.
(3) The mean and the mode are the same.
(4) The mean, median, and mode are all the same.

Answer: (2) The median and the mode are the same.

The mode is 2 because it occurs the most. The median is 2 because there are 20 students and both the 10th and 11th numbers are 2.

Choice (2) says "the median and the mode are the same", which is true. Unfortunately, Choice (4) says all three are the same, so you still have to calculate the mean.

The mean is the Sum / 20, which is 50 / 20 = 2.5. So Choices (1), (3), and (4) are incorrect.

15. The range of f(x) = |x + 2| - 5 is

(1) y > -5
(2) y > 2
(3) x > -5
(4) x > 2

Answer: (1) y > -5

Range deals with the y values, so eliminate Choices (3) and (4).

The vertex of this absolute value function is (-2,-5). That is the lowest point of the function. Therefore, the range is every value of y that is greater than -5. That is Choice (1).

16. Which equation is always correct?

(1) a³ • a^x = a^3x
(2) (a⁴)^x = a^{4 + x}
(3) (ab)^x = a^xb^x
(4) a^x • b^y = ab^{x + y}

Answer: (3) (ab)^x = a^xb^x

These are illustrating the laws of exponents, but only one of them is correct.

If you don't remember the rules, pick some prime numbers like a = 2, b = 3, x = 5 and y = 7 and check which are correct. If more than one choice is correct, pick different numbers and check again.

In Choice (1), a³ • a^x = a^{3 + x} not a^3x. Eliminate Choice (1).

In Choice (2), (a⁴)^x = a^4x not a^{4 + x} . Eliminate Choice (2).

In Choice (3), (ab)^x = a^xb^x. This is the correct answer. Both a and b will get the exponent.

In Choice (4), a^x • b^y cannot be combined, and ab^{x + y} is just crazy. (For one thing, the exponent only applies to b while a no longer has any exponent.) Eliminate Choice (4).

More to come. Comments and questions welcome.

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