Algebra 2 Test

Based on the NY Regents Algebra II Exam (August 2025)

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Part I: Multiple Choice

1. What is the seventh term of the sequence -2, 6, -18, 54, ...?

-1458 -4374 1458 4374

2. Given x ≠ 0, where m(x) = 12x^8a and p(x) = 3x^2a, the expression m(x) / p(x) is equivalent to:

9x^4a 4x^6a 4x⁶ 4x⁴

3. What is the inverse of f(x) = 2x + 6?

f^-1(x) = -2(x + 3) f^-1(x) = x - 3 f^-1(x) = x/2 - 3 f^-1(x) = x/2 + 3

4. The expression ³√(16x⁶) is equivalent to:

4x³ 4x² 2x²³√2 2x³³√2

5. Mary would like to determine if there is an association between a student's height and their shoe size. She measures the height and shoe size of every 10th person entering her school. This is an example of:

a census an observational study a simulation a controlled experiment

6. For all values for which the expressions are defined, which expression can not be rewritten as (x - 6)(x + 2)?

( (x + 2)(x² - 2x - 24) ) / (x + 4) x(x + 2) - 6(x + 2) ( (x - 2)(x² - 4x - 12) ) / (x - 6) (x + 4)(x - 2) - 2(3x + 2)

7. The number of hours in the lifespan of a certain brand of light bulb is normally distributed with a mean of 2387 hours and a standard deviation of 238 hours. To the nearest tenth of a percent, what percent of light bulbs have a lifespan of greater than 2750 hours?

6.4% 15.9% 43.6% 93.6%

8. The solution set to the equation 2/x³ + 1/x = 6/x³ is:

{-2, 0, 2} {2} {-2, 2} {0, 2}

9. What is the solution to 9(e^x-2) = 36?

x = ln(36) / ln(9e) + 2 x = ln(4) - 2 x = ln(4) + 2 x = ln(4/e) + 2

10. Reynaldo got a score of 40 on his first test. If he gets a score of 100 on every additional test, which equation can be used to determine the number of additional tests, x, he would need to take in order to raise his test average to an 80?

(40 + 100x) / (x + 1) = 80 (40 + 100x) / x = 80 (40 + 100 + x) / x = 80 (40 + 100 + x) / (x + 1) = 80

11. Given f(x) = ln(x + 5), what is the smallest integer value of x for which f(x) is defined?

-5 -4 -1 0

12. Which expression is equivalent to (6x³ + 7x² - 9x - 1) / (2x - 1) when x ≠ 1/2?

3x² - 2x - 4 3x² + 5x - 7 - 8/(2x - 1) 3x² + 2x + 5 - 6/(2x - 1) 3x² + 5x - 2 - 3/(2x - 1)

13. A sketch for p(x) is shown (see PDF), where a > 0 and b > 0. The graph has roots at -a (tangent) and b (crosses). An equation for p(x) could be:

p(x) = (x + a)(x - b) p(x) = (x + a)²(x - b) p(x) = (x - a)(x + b) p(x) = (x - a)²(x + b)

14. If f(x) = (1/2)x³ + 3x² - 4x and g(x) = 5 log₃(x + 10), then which value, rounded to the nearest tenth, is not a solution to f(x) = g(x)?

-6.9 -1.4 2.2 9.8

15. The graph of f(x) is shown (see PDF). Which graph represents f(x + 3)?

Graph (1) - Shift Left Graph (2) - Shift Right Graph (3) - Shift Up Graph (4) - Shift Down

16. What is one solution to the system of equations shown below?
x² + y² = 20
y = x - 6

x = 2 (4, -2) y = 4 (4, 2)

17. At a high school, 10th-grade students were surveyed (see PDF table). What is the probability that a randomly selected 10th-grade student from the school walks to school or eats breakfast?

0.07 0.70 0.77 0.84

18. A vehicle's depreciation rate is 9.2% per year. If a vehicle costs $34,950, then which recursive formula models the value of the vehicle n years after it was purchased?

a_n = 34,950(1.092)ⁿ a_n = 34,950(0.921)ⁿ a₀ = 34,950; a_n = 1.092a_n-1 a₀ = 34,950; a_n = 0.908a_n-1

19. When factored completely, (3x - 1)² - 5(3x - 1) + 6 is equivalent to:

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

20. Given E(t) = 26(2)^t/20 represents the mass, in grams, of a substance after t minutes. Which statement or statements must be true?
I. The initial mass of the substance is 26 grams.
II. The mass of the substance doubles every 20 days.
III. The mass of the substance after 3 hours is approximately 29 grams.

I, only III, only I and II, only I and III, only

21. For x > 0, which expression is equivalent to ³√(9x²) • √(9x)?

9⁵x^7/2 9⁶x³ 9^1/6x^1/3 9^5/6x^7/6

22. The number of people who have read an article grows exponentially... N(t) = 2(1.0098)^t, where t represents minutes. Which equation best represents the number of people who have read the article in terms of the growth rate per second?

N(t) = 2(1.000163)^t/60 N(t) = 2(1.000163)^60t N(t) = 2(1.79524)^t/60 N(t) = 2(1.79524)^60t

23. Which equation represents a parabola with focus (2, -5) and directrix y = 3?

(x - 2)² = -16(y + 1) (x - 2)² = -16(y - 1) (x + 2)² = -16(y - 1) (x - 2)² = 16(y + 1)

24. Which graph (see PDF) shows a fourth-degree polynomial function with exactly two imaginary roots?

Graph (1) Graph (2) Graph (3) Graph (4)

25. Seniors were surveyed (see PDF table). Determine the exact probability that a randomly selected senior from the survey preferred a hoodie, given that the senior wanted a design on the back.

Your answer (as a fraction or decimal):

26. (For Review) Sketch g(x) = -x³ - 7x² + 36 on the axes, including appropriate end behavior and zeros.

This question requires graphing and will not be graded by the script. Please check your graph against the answer key (Zeros: -6, -3, 2. End behavior: Up on left, Down on right).

27. Express 8xi¹⁰ - 4yi¹⁹ + 2yi³ - 6xi in simplest form, where i is the imaginary unit.

Your answer (in a+bi form):

28. The job satisfaction rating at a company is normally distributed with a mean of 12. About 95% of the scores are between 8 and 16. What is the standard deviation of this distribution? Justify your answer.

Your answer (standard deviation):

29. An angle, θ, is drawn in standard position and terminates in Quadrant III. Given cos θ = -(√10) / 10, determine the value of tan θ.

Your answer:

30. Solve algebraically for all values of x: √(x + 5) - x = 3

Your answer (x):

31. Use the geometric series formula to determine the total 30-year earnings for an employee whose first-year salary is $42,000 and earns an annual raise of 3%, rounded to the nearest thousand dollars.

Your answer (total earnings):

32. Algebraically determine the solution(s) to the equation 2x² = 2x - 1, in simplest a + bi form.

Your answer (e.g., 1+2i, 1-2i):

33. The GDP per capita, y, x years after 1990 is listed in the table (see PDF).

(a) Write an exponential regression equation. Round all coefficients to the nearest hundredth. (b) Use the rounded equation to determine, to the nearest tenth of a year, when GDP per capita was $15,000.

34. Consider f(x) = x³ + 3x² - 2x - 6.

Is (x + 3) a factor of f(x)? Justify your answer. Determine all zeros of f(x).

35. Solve the system algebraically:
2a + b - c = -4
4a + b + c = 3
-2a - 3b + 2c = 11

Your answer (e.g., a=1, b=2, c=3):

36. Given: f(x) = 5x² + 3x - 12 and g(x) = 2x - 1. Express 4g(x) - [f(x + 1)] as a polynomial in standard form.

Your answer (polynomial in standard form):

37. The graph of tides B(t) is shown (see PDF).

State the period of B(t), in hours. Write an equation for B(t) in the form B(t) = a cos(bt) + c.

In Derby, Australia, the tide is modeled by D(t) = 8cos(π/6 * t) + 16.5. (Graphing this is for review).

State the height, in feet, of the low tide in Derby.