June 2022 Algebra 2 Regents, Part I (multiple choice)

This exam was adminstered in June 2022.

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June 2022 Algebra 2 Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


17. The inverse of f(x) = -6x + 1/2 is

(1) f^-1(x) = 6x - 1/2
(2) f^-1(x) = 1/(-6x + 1/2)
(3) f^-1(x) = -1/6 x + 1/12
(4) f^-1(x) = -1/6 x + 2

Answer: (3) f^-1(x) = -1/6 x + 1/12

Swap x and f(x), relabeling the second one as f¹(x), and then solve for f^-1(x).

f(x) = -6x + 1/2
x = -6f^-1(x) + 1/2
x - 1/2 = -6f^-1(x)
(x - 1/2) / -6 = f^-1(x)

Therefore, f^-1(x) = -1/6 x + 1/12, which is Choice (3).

18. The expression (x² + 12) / (x² + 3) can be rewritten as

(1) 10/(x² + 3)
(2) 1 + 9/(x² + 3)
(3) x + 9
(4) 4

Answer: (2) 1 + 9/(x² + 3)

Neither expression can be factored, so the expression cannot be simplified. However, you can split the fraction as follows:

(x² + 12) / (x² + 3) = (x² + 3) / (x² + 3) + 9 / (x² + 3)

= 1 + 9 / (x² + 3)

This is Choice (2).

19. IAn angle, θ , is rotated counterclockwise on the unit circle, with its terminal side in the second quadrant, as shown in the diagram below.

Which value represents the radian measure of angle θ?

(1) 1
(2) 2
(3) 65.4
(4) 114.6

Answer: (2) 2

I'm confused by this question because the answer is in the diagram, 2. The radian measure is the length of the arc, not the number of degrees, which looks to be 114.6 (from the choices).

The radius is 1.

A semicircle is pi (about 3.14) radians. And 114.6 degrees is a little less than two thirds of 180 degrees. Moreover, 2 is a little less than 2/3 of pi.

20. The depth of the water, d(t), in feet, on a given day at Thunder Bay, t hours after midnight is modeled by d(t) = 5 sin(π/6 (t - 5)) + 7. Which statement about Thunder Bay tide is false?

(1) A low tide occurred at 2 a.m.
(2) The maximum depth of the water was 12 feet.
(3) The water depth at 9 a.m. was approximately 11 feet.
(4) The difference in water depth between high tide and low tide is 14 feet.

Answer: (4) The difference in water depth between high tide and low tide is 14 feet.

Put the function into your graphing calculator and check the choices.

At t = 2 (2 am), the tide is at a minimum, which is 2 feet. The two 2s might be confusing, but this condition is true.

The maximum depth will be 12 feet. At some points, the sine function will return a value of 1, and then 5(1) + 7 = 12. So this statement is true.

The rate of change of t(x) over the interval [2,4] is (3 - -5)/(4 - 2) = 8/2 = 4.

The rate of change of t is twice that of g, which means that the rate of change of g is half the change of t.

The difference between the high and the low points is 12 - 2 = 10, which is 5 * 2. It is NOT 14 feet (which is 12 + 2 -- don't do that!). Choice (4) is the correct response.

21. A function is defined as a_n = a_n-1 + log_n+1 (n − 1), where a₁ = 8. What is the value of a₃ ?

(1) 8
(2) 8.5
(3) 9.2
(4) 10

Answer: (2) 8.5

Substitute to find a₂ and then a₃.

a₂ = a₁ + log₃ (2 − 1) = 8 + 0 = 8

a₃ = a₂ + log₄ (3 − 1) = 8 + 0.5 = 8.5

This is Choice (2).

22. Which function has a maximum y-value of 4 and a midline of y = 1?

Answer: (2) g(x) = −3cos(x) + 1

Choice (3) and (4) do not have maximums. Eliminate them.

Choice (1) doesn't have a "midline" because it extends to infinity.

Choice (2) has a midline of 1 and an amplitude of 3. That gives it a maximum of 1 + 3 = 4. Choice (2) is the answer.

23. WWhich expression is equivalent to (x + yi)(x² − xyi − y²), where i is the imaginary unit?

(1) x³ + y³i
(2) x³ − xy² - (xy² + y³)i
(3) x³ − 2xy² - y³i
(4) x³ − y³i

Answer: (4) x³ − y³i

Use the Distributive Property and then simplify.

(x + yi)(x² − xyi − y²)

= x(x² − xyi − y²) + yi(x² − xyi − y²)

= x³ − x²yi − xy² + x²yi − x(yi)(yi) − y³i

= x³ − xy² − xy²(-1) − y³i

= x³ − y³i

This is Choice (4).

24. The growth of a $500 investment can be modeled by the function P(t) = 500(1.03)^t, where t represents time in years. In terms of the monthly rate of growth, the value of the investment can be best approximated by

(1) P(t) = 500(1.00247)^12t
(2) P(t) = 500(1.00247)^t
(3) P(t) = 500(1.03)^12t
(4) P(t) = 500(1.03)^t/12

Answer: (1) P(t) = 500(1.00247)^12t

To calculate monthly rate of growth, use the following formula:

P(t) = 500(1.03^1/12)^12t = 500(1.00247)^12t

This is Choice (1).

End of Part I.



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