Algebra Problems of the Day (Algebra 1 Regents, January 2023)

The following questions appeared on the August 2022 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, August 2022

Part I: Each correct answer will receive 2 credits.


9. Which expression is equivalent to (x + 4)²(x + 4)³?

1) (x + 4)⁶
2) (x + 4)⁵
3) (x² + 16)⁶
4) (x² + 16)⁵

Answer: 2) (x + 4)⁵

The bases are the same, so keep the base and add the exponents.

(x + 4)²(x + 4)³ = (x + 4)⁵.

10. Caitlin graphs the function f(x) = ax², where a is a positive integer. If Caitlin multiplies a by -2, when compared to f(x), the new graph will become

1) narrower and open downward
2) narrower and open upward
3) wider and open downward
4) wider and open upward

Answer: 1) narrower and open downward

If a is positive, then -2a is negative. If the coefficient is negative, then the parabola opens downward. Eliminate Choices (2) and (4).

As the coefficient gets larger, the value of f(x) increases faster, so the parabola will by narrower.

The correct choice is Choice (1).

11. Sunny purchases a new car for $29,873. The car depreciates 20% annually.
Which expression can be used to determine the value of the car after t years?

1) 29,873(.20)^t
2) 29,873(20)^tJessica, only
3) 29,873(1 - .20)^t
4) 29,873(1 + .20)^t

Answer: 3) 29,873(1 - .20)^t

The worth of a depreciated object after one year is 100% of the value minus the rate of depreciaton. This is shown in Choice (3).

Choice (1) actually shows an 80% depreciation because 1.00 - .80 = .20. Eliminate Choice (1).

Choice (2) shows exponential growth of 2000% per year. (Thats 20 * 100%.) Eliminate Choice (2).

Choice (4) shows 20% appreciation, not depreciation. It is gaining 20% of value every year.

12. If f(x) = x² + 2x + 1 and g(x) = 7x - 5, for which values of x is f(x) = g(x)?

1) -1 and 6
2) -6 amd -1
3) -3 and -2
4) 2 and 3

Answer: 4) 2 and 3

Set the two expressions equal to each other, and solve for x by factoring.

x² + 2x + 1 = 7x - 5
x² - 5x + 6 = 0
(x - 3)(x - 2) = 0
(x - 3) = 0 or (x - 2) = 0
x = 3 or x = 2

This is Choice (4).

13. Skyler mows lawns in the summer. The function f(x) is used to model the amount of money earned, where x is the number of lawns completely mowed. A reasonable domain for this function would be

1) real numbers
2) rational numbers
3) irrational numbers
4) natural numbers

Answer: 4) natural numbers

The domain is the input and the range is the output.

The input to f(x) is the number of lawns. It is reasonable to believe that the number of lawns should be a natural or whole number. It is reasonable to assume that if he mows half a lawn, he will not get paid for the job (not even half of the job). So the correct answer is Choice (4).

The range for f(x) could be rational numbers because it may deal in dollars and cents.

Irrational or real numbers (which include irrational numbers) aren't reasonable for either.

14. Which expression is equivalent to 2x² + 8x - 10?

1) 2(x - 1)(x + 5)
2) 2(x + 1)(x - 5)
3) 2(x - 1)(x - 5)
4) 2(x + 1)(x + 5)

Answer: 1) 2(x - 1)(x + 5)

Factor out the 2 and then factor the polynomial into two binomials.

2x² + 8x - 10

2(x² + 4x - 5)

2(x + 5)(x - 1)

This is Choice (1).

15. Ian throws a ball up in the air and lets it fall to the ground. The height of the ball, h(t), is modeled by the equation h(t) = -16t² + 6t + 3, with h(t) measured in feet, and time, t, measured in seconds. The number 3 in h(t) represents

1) the maximum height of the ball
2) the height from which the ball is thrown
3) the number of seconds it takes for the ball to reach the ground
4) the number of seconds it takes for the ball to reach its maximum height

Answer: 2) the height from which the ball is thrown

The 3 is the value of f(0), the starting value. This is Choice (2).

The maximum height of the ball if f(x) when x = -(6)/(2(-16)). Eliminate Choice (1).

The number of seconds it takes for for the ball to reach the ground is the value of x when f(x) = 0. Eliminate Choice (3).

The number of seconds it takes for the ball to reach its maximum height is -6/-32 = 3/16 of a second. Eliminate Choice (4).

16. IThirty-two teams are participating in a basketball tournament. Only the winning teams in each round advance to the next round, as shown in the table below.

Which function type best models the relationship between the number of rounds completed and the number of teams remaining?

1) absolute value
2) exponential
3) linear
4) quadratic

Answer: 2) exponential

If you look at the f(x) values, each is half of the number before it. This is an exponential function, which is Choice (2).

Specifically, it is f(x) = 32(1/2)^x.

It is not absolute value because the rate of change would have to be equal, and then change and become linear with the rate of change being additive inverse of the previous slope.

It is not linear becaue the rate of change would have to be equal, but it is decreasing.

It is not a quadratic pattern because then the second differences would be equal. However, the second differences are also decreasing.

More to come. Comments and questions welcome.

More Regents problems.

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