Algebra Problems of the Day (Algebra 1 Regents, January 2024 Part I)

The following questions appeared on the January 2024 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, January 2024

Part I: Each correct answer will receive 2 credits.


17.If f(x) = 2x + 6 and g(x) = |x| are graphed on the same coordinate computations. plane, for which value of x is f(x) = g(x)?

1) 6
2) 2
3) -2
4) -6

Answer: 3) -2

Set the two functions equal to each other and solver for x.

f(x) = g(x)
2x + 6 = |x|
2x + 6 = -x or 2x + 6 = x
6 = -3x or 6 = -x
-2 = x or -6 = x

Next check to see if the answer is a valid solution.

2(-2) + 6 ?= |-2|
2 = 2

2(-6) + 6 ?= |-6|
-6 =/= 6

Discard x = -6 as a solution. Only x = -2 is a solution, which is Choice (3).

Alternatively, you could've just substituted each choice into both equations until you found a match.

18. What is the solution to the inequality 2x - 7 > 2.5x + 3?

1) x > -5
2) x < -5
3) x > -20
4) x < -20

Answer: 4) x < -20

Use inverse operations to isolate x.

2x - 7 > 2.5x + 3

-.5x > 10

x < -20

Don't forget to flip the inequality symbol with dividing both sides by a negative number.

19. Three expressions are written below.

A. (2xy²)³
B. (2x)³y⁶
C. (2x²y²)(4xy³) Which expressions are equivalent to 8x³y⁶?

1) A and B, only
2) B and C, only
3) A and C, only
4) A, B, and C

Answer: 1) A and B, only

Use the Laws of Exponents and evaluate the three expressions.

(2xy²)³ is 8x³y⁶, which eliminates Choice (2).

(2x)³y⁶ is 8x³y⁶

(2x²y²)(4xy³) is 8x³y⁵, which eliminates Choices (3) and (4).

Choice (1) remains.

20. Joe deposits $4000 into a certificate of deposit (CD) at his local bank. The CD earns 3% interest, compounded annually. The value of the CD in x years can be found using the function

1) f(x) = 4000 + 0.3x
2) f(x) = 4000 + 0.03x
3) f(x) = 4000(1.3)^x
4) f(x) = 4000(1.03)^x

Answer: 4) f(x) = 4000(1.03)^x

Choices (1) and (2) show simple interest, not compounded interest.

Choice (1) and (3) show 30% interest, not 3%, with is 0.03 as a decimal.

The correct answer is Choice (4).

21. When factored completely, -x³ + 10x² + 24x

1) -x(x + 4)(x - 6)
2) -x(x - 4)(x - 6)
3) -x(x + 2)(x - 12)
4) -x(x - 2)(x + 12)

Answer: 3) -x(x + 2)(x - 12)

Factor the -x from each term and then factor the remaining quadratic expression.

-x³ + 10x² + 24x

-x(x² - 10x² - 24)

-x(x - 12)(x + 2)

Notice that when factoring out the -x, the signs of the coefficients of each of the terms were changed.

The correct answer is Choice (3).

22. When the temperature is 59°F, the speed of sound at sea level is 1225 kilometers per hour. Which process could be used to convert this speed into feet per second?

Answer: 2) [See image]

Kilometers needs to be converted to feet, so feet needs to be in the top of a fraction and kilometers in the bottom to cancel out with the given unit.

Hours needs to be converted to seconds, so seconds needs to be in the bottom of a fraction and hours needs to be in the top to cancel out with the given unit.

Doing unit analysis on the choices:

Choice (1) has miles twice on top and both feet and seconds on the bottom. Eliminate Choice (1).

Choice (2) cancels all the units except for feet on top and seconds on the bottom. This is the correct answer.

Choice (3) has kilometers twice on top and miles twice on the bottom, so they won't cancel out. Eliminate Choice (3).

Choice (4) has minutes twice on top and hours twice on the bottom, so they won't cancel. Eliminate Choice (4).

23. The zeros of a polynomial function are -2, 4, and 0. What are all the factors of this function?

1) (x + 2) and (x - 4)
2) (x - 2) and (x + 4)
3) x, (x + 2) and (x - 4)
4) x, (x - 2) and (x + 4)

Answer: 3) x, (x + 2) and (x - 4)

There are three zeroes, so there must be three factors. Remember that you flip the signs when writing the factors. If x = 4, it is because x - 4 = 0.

So the correct answer is x, (x + 2) and (x - 4), which is Choice (3).

24. What is the range of the function f(x) = (x - 4)² + 1?

1) x > 4
2) x > 4
3) f(x) > -1
4) f(x) > -1

Answer: 4) f(x) > -1

The range of the function refers to all of the values that the output of the function could possibly have. That is, the y values or the f(x) values. This means that you can eliminate Choices (1) and (2).

The given function is quadratic. It is a parabola with a minimum point (its vertex) at the point (4,-1). If you didn't realize this by looking at the funtion, you could put it in a graphing calculator and see it.

Since (4,-1) is a point on the graph, Choice (3) must be eliminated. The range is all values greater than or equal to -1.

This is Choice (4).

End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

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