(x, why?): Algebra 2 Problems of the Day (Algebra 2, June 2025 Part I)

Wednesday, June 03, 2026

Algebra 2 Problems of the Day (Algebra 2, June 2025 Part I)

This exam was adminstered in June 2025.

June 2025 Algebra 2 Regents

Part I

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

17. Consider the system of equations below.

3x + 2y = 1
2y + z = 2
2x - 2z = -6 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

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

Answer: (1) 1

Subtract the second equation from the first equation and the y variable will disappear. Now you can solve for x.

3x + 2y = 1
2y + z = 2
2x - 2z = -6

3x - z = -1
2x - 2z = -6

6x - 2z = -2
2x - 2z = -6

4x = 4

x = 1

This is Choice (1).

Checking: 2(1) - 2z = -6; -2z = -8, z = 4

Then, 2y + 4 = 2, 2y = -2, y = -1

Finally, 3(1) + 2(-1) = 1. Check!

18. The point (2,–3) lies on the graph of the equation y = f(x). Which point must lie on the graph of the equation y = f(x - 4) + 1?

(1) (1,1)
(2) (-2,-2)
(3) (3, 7)
(4) (6,-2)

Answer: (4) (6,-2)

The -4 inside the parentheses shifts the graph four units to the right. The +1 outside the parentheses shifts the graph on unit up.

If you translate the point (2,-3) four units to the right and one unit up, you end up at (6,-2).

The correct answer is Choice (4).

19. Which statement best describes the end behavior of the function y = log(x - 3)?

(1) As x → -∞, y → -∞, and as x → ∞, y → ∞.
(2) As x → 3, y → -∞, and as x → ∞, y → ∞.
(3) As x → -∞, y → 0, and as x → ∞, y → ∞.
(4) As x → 3, y → 0, and as x → ∞, y → ∞.

Answer: (2) As x → 3, y → -∞, and as x → ∞, y → ∞.

Graph the function. One the left side, the graph goes to negative infinity as it gets closer to 3. On the right side, both x and y tend toward infinity.

Choice (2) is the correct answer.

20. The black bear population for a certain area of the Adirondacks can be modeled by the equation

B = 5835.943(1.026)^t, where t is measured in years since 2010. Kieran would like to rewrite this model in terms of a 5-year growth rate. Kieran’s model is best represented by

(1) B = 5835.943(1.005147)^t/5
(2) B = 5835.943(1.005147)^5t
(3) B = 5835.943(1.136938)^t/5
(4) B = 5835.943(1.136938)^5t

Answer: (3) B = 5835.943(1.136938)^t/5

If there is a 5-year growth rate, then the exponent will be divided by 5, or t/5. Eliminate Choices (2) and (4).

(1.026)^t is equal to (1.026⁵)^t is equal to (1.136938)^t/5.

This is Choice (3).

21. Which expression or expressions are equal to 0 for all real numbers?

I. (x² + y²)² + (x² + y²)² - 2(x² + y²)² II.(x² + y²)² - (x² + y²)² III. (x² + y²)² - (x² + y²)² - (2xy)²

(1) I, only
(2) III, only
(3) I and II, only
(4) I and III, only

Answer: (4) I and III, only

The first expression is "obviously" equal to 0 (see below), but you need to expand the second and third choices to see if all the terms will cancel out. Again, it is "obvious" that they both can't be true. However, it is possible that neither one is true.

In the first expression, let z = (x² + y²)². So z + z - 2z = 0, which is true. Eliminate Choice (2).

In the second expression, the first term is not the same as the second term, so subtracting them cannot equal 0. Eliminate Choice (3).

Expand the third expressiong.

(x² + y²)² - (x² + y²)² - (2xy)²
(x⁴ + 2x²y² + y⁴) - (x⁴ - 2x²y² + y⁴) - (4x²y²)
x⁴ + 2x²y² + y⁴ - x⁴ + 2x²y² - y⁴ - 4x²y²
x⁴ - x⁴ + 2x²y² + 2x²y² + y⁴ - 4x²y² - y⁴ = 0

Choice (4) is the correct answer.

22. The equation 1/x - 1/5 = x/5 has

(1) rational solutions
(2) irrational solutions
(3) imaginary solutions
(4) no solutions

Answer: (2) irrational solutions

Solve for x.

1/x - 1/5 = x/5

1/x = x/5 + 1/5

1/x = (x + 1)/5

x(x + 1) = 5

x² + x = 5

x² + x - 5 = 0

If you check the discriminant, b² - 4ac, then 1² - (4)(1)(-5) = 21, which is positive but not a perfect square. This means that there are two solutions but they will be irrational.

This is Choice (2).

23. For x ≠ +4y, the expression (x² + 3xy - 28y²) / (16y² - x² is equivalent to
(1) -1 - (7/4) y
(2) (x - 7y) / (4y - x)
(3) (x + 7y) / (x + 4y)
(4) (-x - 7y) / (x + 4y)

Answer: (4) (-x - 7y) / (x + 4y)

Factor the numerator and the denominator and simplify. (x² + 3xy - 28y²) / (16y² - x²
( (x + 7y)(x - 4y) ) / ( (4y - x)(4y + x) )
( (x + 7y)(-1) ) / (4y + x)
( -x - 7y) / (x + 4y)

This is Choice (4).

24. Which equation represents a parabola with a focus of (-2,1) and directrix of y = 5?,

(1) (x + 2)² = -8(y - 3)
(2) (x + 2)² = 5(y - 1)
(3) (x + 2)² = -8(y - 1)
(4) (x + 2)² = 8(y - 3)

Answer: (1) (x + 2)² = -8(y - 3)

The formula for finding a parabola from the focus and directrix is (x - h)² = 4p(y - k), where p is the distance from the vertex to the focus. The vertex is halfway between the focus and directrix, or (-2, 3), which makes p = -2.

Substituting what we know, we get (x - (-2))² = 4(-2)(y - 3), or (x + 2)² = -8(y - 3).

This is Choice (1).

End of Part I.

Questions, comments, and corrections welcome.

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