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.
9. The asymptote of the graph of f(x) = 5 log(x + 4) is
(1) y = 6
(2) x = -4
(3) x = 4
(4) y = 5
Answer: (2) x = -4
First of all, you can graph this and look. Second, the asymptote will be a vertical line, which means it starts with "x = ", so eliminate Choices (1) and (4).
The asymptote of f(x) = log x is the y-axis, which is x = 0. Multiplying by 5 will not change the asymptote. However, adding 4 inside the parentheses, that is log(x + 4), will move the asymptote 4 units to the left, to x = -4. Choice (2) is the correct answer.
10. The probability of having math homework is 1/3 and the probability of having English homework is 1/7. The probability of having math
homework or having English homework is 9/21. What is the probability of having math homework and having English homework?
(1) 19/21
(2) 1/5
(3) 1/21
(4) 10/21
Answer: (3) 1/21
P(M) + P(E) - P(M|E) = 1/3 + 1/7 - 9/21 = 7/21 + 3/21 - 9/21 = 1/21
Choice (3) is the correct choice.
11. The solution set to the equation x - 1 = √(2x + 6) is
(1) {5, -1}
(2) {5}
(3) {-1}
(4) { }
Answer:(2) {5}
You can check each of the listed answers to see if either, or both work.
For x = 5, 5 - 1 = 4 and √(2(5) + 6) = 4, so the answer is either Choice (1) or (2).
For x = -1, -1 -1 = -2, which cannot be the result of a square root. Eliminate Choices (1) and (3).
Choice (1) is the correct answer.
If you tried to solve it algebraically:
x - 1 = √(2x + 6)
x2 - 2x + 1 = 2x + 6
x2 - 4x - 5 = 0
(x - 5)(x + 1) = 0
x = 5 or x = -1
However, when we test the answers, we see that x = -1 must be discarded as extraneous, and only x = 5 is correct.
12. Given x > 0, the expression (1 / x-2)(-3/4) is equivalent to
(1) x √(x)
(2) 1 / (x √(x))
(3) ∛x2
(4) 1 / (∛x2)
Answer: (2) 1 / (x √(x))
Use the rules for exponents to evaluate the expression.
(1 / x-2)(-3/4)
(x2)(-3/4)
(x(-3/2))
1 / x3/2
1 / √(x3 = 1 / x √(x)
Choice (2) is the correct answer.
13. The graph of which function has a period of 3?
(1) y = -7sin(2π/3 x) - 5
(2) y = -7sin(3π/2 x) + 9
(3) y = -7sin(3x) - 5
(4) y = 3sin(π x) + 9
Answer: (1) y = -7sin(2π/3 x) - 5
Once again, you can graph these equations and look to see which is correct.
The period of y = A sin(x) + C is 2π. The period of y = A sin(Bx) + C is 2π/B.
In Choice (1), 2π/ (2π/3) = 3. This is the correct answer.
In Choice (2), 2π/ (3π/2) = 4/3. Eliminate Choice (2).
In Choice (3), 2π/ 3 is not equal to 3. Eliminate Choice (3).
In Choice (4), 2π/ (π) = 2. Eliminate Choice (4).
14. Which graph could represent a 4th degree polynomial function with a positive leading coefficient, 2 real zeros, and 2 imaginary zeros?
Answer: (1) [See image]
A positive leading coefficient on a fourth-degree polynomial means that the end behavior is toward positive infinity on both ends. Eliminate Choices (3) and (4).
Two real zeroes means that the graph crosses the x-axis in two locations, which eliminates Choice (4) and leaves Choice (1).
Since there are only two real zeroes, there must be two imaginary zeroes as well.
15. Given i is the imaginary unit, which expression is equivalent to 5i(2x + 3i) - x√(-9)?
(1) 15 + 13xi
(2) -15 + 13xi
(3) 15 + 7xi
(4) -15 + 7xi
Answer: (4) -15 + 7xi
Use the Distributive Property and the Order of Operations. Don't forget that i2 = -1 and that √(-1) = i.
5i(2x + 3i) - x√(-9)
10xi + 15i2 - 3xi
-15 + 7xi
The correct answer is Choice (4).
16. 6 What is the focus of the parabola 8(y + 2) = (x + 5)2?
(1) (-5,0)
(2) (-5,-4)
(3) (5,0)
(4) (5,4)
Answer: (1) (-5,0)
The vertex form of a parabola is y = a(x - h) + k, and the focus of the parabola is at point (h, k + 1/(4a)). Rewrite the equation in vertex form.
8(y + 2) = (x + 5)2
y + 2 = (1/8)(x + 5)2
y = (1/8)(x + 5)2 - 2
If a = 1/8, then 1/(4a) = 1/(4(1/8)) = 1/(1/2) = 2.
So (h, k + 1/(4a)) = (-5, -2+2) = (-5,0).
More to come. Comments and questions welcome.
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