January 2024 Algebra 2 Regents, Part I (multiple choice)

This exam was adminstered in January 2024.

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January 2024 Algebra 2 Regents

Part I

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


1. A cafeteria food manager studied the lunchtime eating habits of a group of employees in their office building. The purpose of the study was to determine the proportion of employees who purchased lunch in the cafeteria, brought their lunch from home, or purchased lunch from an outside vendor. This collection of data would best be classified as

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

Answer: (3) an observational study

This is an example of an observational study. There was no experiment or simulation conducted. No interviews were made.

2. Which graph has imaginary roots?

Answer: (2) [See Image]

The roots of a quadratic equation are the x-intercepts of the graph, the points were y = 0.

If the quadratic function does not cross the x-axis then it does not have any real roots and only has imaginary roots. Choice (2) is the only image that doesn't show the graph crossing the x-axis.

3. Given 3 is a root of f(x) = x⁴ − x³ − 21x² + 45x, what are the other unique roots of f(x)?

(1) −5, only
(2) −5 and 0
(3) −3, 1 and 5
(4) −5, −3 and 0

Answer: (2) −5 and 0

You can graph the equation in your calculator and check the table of value to see which inputs give a value of zero.

Note that since every term has a variable in it, then f(0) will equal 0, so Choices (1) and (3) can be eliminated.

If you check x = -3, f(-3) = -216, so Choice (4) can be eliminated.

Choice (2) is the correct answer.

4. Given p ≠ q, p = (1/2)^q, expressed in logarithmic form, is equivalent to

(1) log_p (1/2) = q
(2) log_q (p) = 1/2
(3) log_1/2 (p) = q
(4) log_p (q) = p

Answer: (3) log_1/2 (p) = q

Rewrite the equation using 1/2 as the base.

p = (1/2)^q

log _1/2 p = log _1/2 (1/2)^q

log _1/2 p = q

Choice (3) is the correct answer.

5. Which graph best represents the graph of f(x) = (x + a)²(x − b), where a and b are positive real numbers?

Answer: (1) [See Image]

In a cubic equations without a negative leading coefficient, the graph starts at negative infinity when x is negative and goes to positive infinity when x is positive. Eliminate Choices (3) and (4).

Since both a and b are positive numbers, then the roots of the equation will occur at x = -a and x = b. These are the only two roots. Therefore, Choice (2) can be eliminated because it includes x = 0 as a root.

Moreover, since (x + a) has a exponent of 2, then x = -a will be an inflection point where the graph will decrease again. This is the case in Choice (1), which is the correct answer.

6. The equations y = 3t + 6 and y = (1.82)^t approximately model the growth of two separate populations where t > 0. What is the best approximation of the time, t, at which the populations are the same?

(1) -1.9
(2) 0.3
(3) 5.1
(4) 21.3

Answer: (2) x − 3 is a factor of p(x)

Graph the two equations. You will find that the two eqautions will intersect at approximately t = -1.9 and t = 5.1.

Since t represents time, we can discard the negative outcome, and accept t = 5.1, which is Choice (3).

Note that 0.3 and 21.3 are the y values you will when x = -1.0 and x = 5.1, respecitvely.

7. Given y = -2x and x² + y² = 5, the point of intersection in Quadrant II is

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

Answer: (4) (-1,2)

The points in Quadrant II have a negative x coordinate and a positive y coordinate. Eliminate Choice (1).

If y = -2x, then y will be a positive number when x is an integer. Eliminate Choices (2) and (3).

In Choice (4), 2 = -2(-1) and (-1)² + (2)² = 5. Choice (4) is correct.

8. The rational expression (2x⁴ − 5x² + 3x − 2) / (x - 3) is equivalent to

(1) 2x³ − 5x − 12 − 38/(x - 3)
(2) 2x³ + 6x² + 13x + 42 + 124/(x - 3)
(3) 2x³ − 5x + 18 − 56/(x - 3)
(4) 2x³ - 6x² + 13x − 36 − 106/(x - 3)

Answer: (2) 2x³ + 6x² + 13x + 42 + 124/(x - 3)

Long division will give you the answer.

More to come.


Comments and questions welcome.

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