This exam was adminstered in August 2024.
More Regents problems.
August 2024 Algebra 2 Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
1. A grocery store owner wonders how many customers bring reusable
bags to the store. An employee stands at the store entrance for two
hours and counts the number of people bringing in reusable bags.
This type of study is best classified as
(1) a census
(2) an experiment
(3) an observational study
(4) a survey
Answer: (3) an observational study
This is a case of an observational study. I hope it's self-explanatory.
No questions are being asked, so it isn't a census or a survey. No experiment is being conducted.
Choice (3) is the correct choice.
2. The graph of y = 2x - 4 is positive on which interval?
(1) (-∞,∞)
(2) (2,∞)
(3) (-0,∞)
(4) (-4,∞)
Answer: (2) (2,∞)
The graph will cross the x-axis (y=0) when 2x = 4, which occurs when x = 2.
When x > 2, the graph will be above the x-axis.
You can also put this equation into your graphing calculator and look at the screen and the table of values.
The correct answer is Choice (2).
3. Tim deposits $300 into a savings account. The annual interest rate is 2.7% and compounds monthly. He uses the equation
A = 300(1 + 0.027/12)12t to determine how much money, A, he will have after t years. Which equation is equivalent to Tim’s equation?
(1) A = 300[(1.00225)12]t
(2) A = 300[(0.08558)12]t
(3) A = 300[1 + (0.027/12)12t)]
(4) A = (300)12t(1)12t + (0.027/12)12t
Answer: (1) A = 300[(1.00225)12]t
Divide 0.027 by 12 and you'll get 0.00225. Add 1 and you have 1.00225. Choice (1) is the correct answer.
The other answers are a bit silly and violate a bunch of rules of exponents.
Choice (2) splits the 12 and t exponents like Choice (1) does, but the growth rate becomes a decay rate. I have no idea where .08558 came from.
Choice (3) applied the exponent 12t only to the fraction. This is like saying that (1 + 2)2 is equal to (1 + 22), which is silly.
In Choice (4), the 300 gets the exponent of 12t, which is crazy even without all the other problems. This should only be selected by students not even reading the question.
4. Which equation is true for all real values of x?
(1) x4 + x = (x + 1)(x3 - x2 + x)
(2) x4 + x = (x + 1)(x3 + x)
(3) x4 + x = (x2 + x)2
(4) x4 + x = (x + 1)(x3 + x2 + x)
Answer: (1) x4 + x = (x + 1)(x3 - x2 + x)
You can multiply the polynomials on the right side to see which one gives you x4 + x. Or you can put each of these into the graphing calculator to see which ones give you the same graph.
y = x4 + x
y = (x + 1)(x3 - x2 + x)
y = (x + 1)(x3 + x)
y = (x2 + x)2
y = (x + 1)(x3 + x2 + x)
Choice (1) has the same graph and table of values.
Algebraically:
(x + 1)(x3 - x2 + x)
(x)(x3 - x2 + x) + (x3 - x2 + x)
x4 - x3 + x2 + x3 - x2 + x
x4 + x
5. The solution of
(1) -3
(2) -7
(3) 3
(4) 7
Answer: (4) 7
You can work backward from the answers to see which one works. You can immediately see that x = -3 is not a real answer.
You can graph y = x/(x + 3) + 2/(x - 4) - (2x + 27)/(x2 - x - 12) and look for a value of y = 0.
Or you can solve this algebraically once you notice that (x + 3)*(x - 4) = x2 - x - 12.
x/(x + 3) + 2/(x - 4) = (2x + 27)/(x2 - x - 12)
(x - 4)/(x - 4) * x/(x + 3) + (x + 3)/(x + 3) * 2/(x - 4) = (2x + 27)/(x2 - x - 12)
(x - 4)(x)/(x2 - x - 12) + (x + 3)(2)/(x2 - x - 12) = (2x + 27)/(x2 - x - 12)
(x - 4)(x) + (x + 3)(2) = 2x + 27
x2 - 4x + 2x + 6 = 2x + 27
x2 - 4x - 21 = 0
(x - 7)(x + 3) = 0
x = 7 or x = -3
We've already discarded x = -3 as an answer, so x = 7 is the only solution, and that is Choice (4).
If you had graphed the above equation, you would have gotten y = 0 for x = 7.
6. The cost, in dollars, of a single-ride fare in the New York City subway in the years since 1904 is listed in the table below.
Which equation best models the cost of a single-ride fare based on these data?
(1) y = 0.0375(1.0392)x
(2) y = 1.0392(0.0375)x
(3) y = 0.0234x - .0487
(4) y = -0.179 + 0.356 ln(x)
Answer: (1) y = 0.0375(1.0392)x
You can see that the fare went up only 10 cents in the first fifty years but then increased 35 cents in the next twenty or so and $1.00 more in the next twenty. Clearly this is an exponential function and not a linear function.
Only Choices (1) and (2) are exponential functions. Choice (2) shows decay, not growth, which leaves Choice (1) as the correct answer.
7. Which expression is equivalent to (6x4 + 4x3 + x + 200) / (x + 2)
(1) 6x2 - 8x + 17 + 166/(x + 2)
(2) 6x2 - 16x + 33 + 266/(x + 2)
(3) 6x3 + 16x2 + 32x + 65 + 330/(x + 2)
(4) 6x3 - 8x2 + 16x - 31 + 262/(x + 2)
Answer: (4) 6x3 - 8x2 + 16x - 31 + 262/(x + 2)
A fourth-order polynomial divided by (x + 2) becomes a third-order polynomial. Eliminate Choices (1) and (2).
Once again, you can put the original equation along with Choices (3) and (4) into your graphing calculator to see which has the same table of values.
Or you can choose a random value for x, such as 8 or 10, and see which expression gives the same value.
Or you can divide:
Once you get this far, you only have one option left, and that is Choice (4).
If you want to continue, just to make sure, the rest of the division looks like this:
There is a remainder of 262 which means that the final term will be 262/(x + 2).
8. The solution to the equation 6(2x + 4) = 36 is
(1) -1
(2) ln 36 / ln 12 - 4
(3) ln(3) - 4
(4) ln 6 / ln 2 - 4
Answer: (4) ln 6 / ln 2 - 4
A quick check can eliminate x = -1 because 23 = 8 and 6(8) is not equal to 36.
Use inverse operations:
6(2x + 4) = 36
2x + 4 = 6
ln 2x + 4 = ln 6
(x + 4) ln 2 = ln 6
x + 4 = ln 6 / ln 2
x = ln 6 / ln 2 - 4
This is Choice (4).
More to come. Comments and questions welcome.
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