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.
17. In a geometric sequence, the first term is 4 and the common ratio
is 3. The fifth term of this sequence is
1) 324
2) 108
3) 108
4) 324
Answer: 1) 324
Each term is (3) times the term before it.
The first five terms are 4, 12, 36, 108, 324.
18. The amount of energy, Q, in joules, needed to raise the temperature
of m grams of a substance is given by the formula Q = mC(T_{f}  T_{i}),
where C is the specific heat capacity of the substance. If its initial
temperature is T_{i}, an equation to find its final temperature, T_{f} , is
Answer: 2) [See image]
Inverse operations must be used to isolate Tf. Divide both sides by mC, and then add Ti to both sides.
That gives you Choice (2).
Q = mC(T_{f}  T_{i})
Q / (mC) = (T_{f}  T_{i})
Q / (mC) + T_{i} = T_{f}
19. When using the method of completing the square, which equation is
equivalent to x^{2}  12x  10 = 0?
1) (x + 6)^{2} = 26
2) (x + 6)^{2} = 46
3) (x  6)^{2} = 26
4) (x  6)^{2} = 46
Answer: 4) (x  6)^{2} = 46
When completing the square, the number inside the binomial will be HALF of the "b" term in the original polynomial. Since b = 12, the binomial must be (x  6), not (x + 6). Eliminate Choices (1) and (2).
6^{2} = 36, so add 36 to both sides to complete the square. Then add 10 to each side to remove the 10 from the left side.
x^{2}  12x  10 = 0
x^{2}  12x + 36  10 = 0 + 36
x^{2}  12x + 36 + 10  10 = 36 + 10
x^{2}  12x + 36 = 46
(x  6)^{2} = 46
20. Which quadratic function has the smallest minimum value?
Answer: 1) [SEE IMAGE]
The minimum value of a quadratic function occurs at the vertex, which is on the line of symmetry.
The vertex in j(x), Choice (4), is (0,0). The vertex of h(x) is not shown in the table but it is slightly less than zero. You could work out the equation, but it isn't necessary.
Choice (3), g(x), is written in Vertex form, and it's vertex is (2,2).
Choice (1), f(x), has its yintercept at 2, but that is NOT the vertex, which would be even lower on the graph. You have enough information to know that Choice (1) is correct.
The vertex of f(x) is at x = (5)/(2*6) = 5/12.
f(5/12) = 3.041666...
21. Which representation yields the same outcome as the sequence
defined recursively below?
a1 = 3
a_{n} = 4 + a_{n  1}
1) 3, 7, 11, 15, 19,…
2) 3, 1, 5, 9, 13,…
3) a_{n} = 4n  1
4) a_{n} = 4  n
Answer: 2) 3, 1, 5, 9, 13,…
Each number in the sequence, after the first, is 4 less than the one before it because of the 4 being added to the sequence. So Choice (1) is eliminated because it gets bigger, not smaller.
Choice (2) is a sequence of integers decreasing by 4, so this is the correct choice.
In Choice (3), a(1) = 4(1)  1 = 3, but a(2) = 4(2)  1 = 7. This is the same as Choice (1), which was eliminated. If it had 4 instead of 4, it would have been correct.
Choice (4) goes 3, 2, 1, 0, ... which is decreased by 1, not 4. Eliminate it.
22. If the zeros of the function g(x) are {3,0,4}, which function could
represent g(x)?
1) g(x) = (x + 3)(x  4)
2) g(x) = (x  3)(x + 4)
3) g(x) = x(x + 3)(x  4)
4) g(x) = x(x  3)(x + 4)
Answer: 3) g(x) = x(x + 3)(x  4)
The zeroes of the function means that g(3) = 0, g(0) = 0 and g(4) = 0. There will be three terms in the function.
In Choices (1) and (2), g(0) = 12. Eliminate (1) and (2).
In Choice (3), g(3) = (3)(0)(7) = 0, g(0) = 0(3)(4) = 0, and g(4) = (4)(7)(0) = 0.
This is the correct answer.
In Choice (4), g(3) = (3)(6)(7) = 126. This is incorrect. There's no need to check the other two.
In Factored Form, the zeroes follow "(x  " in each binomial. This example should have been:
(x  3)(x  0)(x  4)
But (x  0) is just x, and (x  3) is (x + 3).
23. Morgan read that a snail moves about 72 feet per day. He performs
the calculation
72 feet / 1 day * 1 day / 24 hours * 1 hour / 60 minutes * 12 inches / 1 foot
to convert
this rate to different units. What are the units for the converted rate?
1) hours / inch
2) minutes / inch
3) inches / hour
4) inches / minute
Answer: 4) inches / minute
Units can be multipled and canceled like any other factors in math (and science).
The days cancel. The hours cancel. Feet cancels foot.
You are left with inches on top and minutes on the bottom.
This is Choice (4).
Choice (2) is upside down. Choices (1) and (3) are incorrect because hour(s) are canceled because they form a multiplicative unit.
24. During summer vacation, Ben decides to sell hot dogs and pretzels
on a food cart in Manhattan. It costs Ben $0.50 for each hot dog
and $0.40 for each pretzel. He has only $100 to spend each day on
hot dogs and pretzels. He wants to sell at least 200 items each day.
If h is the number of hot dogs and p is the number of pretzels, which
inequality would be part of a system of inequalities used to determine
the total number of hot dogs and pretzels Ben can sell?
1) h + p ≤ 200
2) h + p ≥ 200
3) 0.50h + 0.40p ≥ 200
4) 0.50h + 0.40p ≤ 200
Answer: 2) h + p ≥ 200
This was is so simple that I read it twice to make sure I hadn't misread it the first time.
The variables h and p stand for the number of items that he's going to sell. He wants to sell at least 200 of these total. So the sum of h and p must be greater than or equal to 200, but not less than 200.
This is Choice (2).
Choices (3) and (4) tell us how much money Ben must spend to buy the product that he sells, and this amount must be less than or equal to $100, not 200. These choices can be eliminated.
More to come. Comments and questions welcome.
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