June 2022 Algebra 2 Regents, Part II

This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

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Algebra 2 June 2022

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Does the equation x² - 4x + 13 = 0 have imaginary solutions? Justify your answer.

Answer:

You can determine the number of solutions by using the discriminant, or by graphing and stating the number of x-intercepts. A negative discriminant means imaginary solutions. No x-intercepts means imaginary solutions.

The discriminant is b² - 4ac = (-4)² - 4(1)(13) = (-4)² - 4(1)(13) = 16 - 52, which is < 0. So the discriminant is negative and the solutions are imaginary.

If graphed, the range of the function would be entirely above the x-axis, so there would only be imaginary solutions.

26. The initial push of a child on a swing causes the swing to travel a total of 6 feet. Each successive swing travels 80% of the distance of the previous swing. Determine the total distance, to the nearest hundredth of a foot, a child travels in the first five swings.

Answer:

You can use the formula for the sum of a finite series, or you can manually calculate 80% for the second through the fifth swings. Either is acceptable but the latter leaves more room for errors to creep in. Two errors is 0 credits, regardless of the work done.

Using the formula from the back of the booklet:

S_n = a₁(1 - (r)ⁿ) / (1 - r)

S₅ = (6)(1 - (0.80)⁵)) / (1 - 0.80) = 20.1696

The total amount is about 20.17 feet.

Doing this in steps:

S1 = 6
S2 = 6(.8) = 4.8
S3 = 4.8(.8) = 3.84
S4 = 3.84(.8) = 3.072
S5 = 3.072(.8) = 2.4576

S1 + S2 + S3 + S4 + S5 = 6 + 4.8 + 3.84 + 3.072 + 2.4576 = 20.1696 = 20.17

27. Solve algebraically for n: 2/n² + 3/n = 4/n²

Answer:

Excuse me for saying that this is ridiculously easy, but this question could appear on an Algebra 1 exam. You don't even need to worry about n = 0 in the problem unless you somehow introduce 0 as a possible solution.

2/n² + 3/n = 4/n²

(n²(2/n² + 3/n) = (4/n²)n²

2 + 3n = 4>

3n = 2

n = 2/3

That's it. Two credits.

28. Factor completely over the set of integers:

-2x⁴ + x³ + 18x² - 9x

Answer:

Remove the GCF, which is x (or -x), then factor by grouping. Then check if anything else can be factored.

Personally, I prefer for the leading coefficient to be positive when I'm factoring, so I would factor out (-1) along with the GCF of x.

-2x⁴ + x³ + 18x² - 9x

(-x) (2x³ - x² - 18x + 9)

(-x) ((x²)(2x - 1) - 9(2x - 1))

(-x) (x² - 9)(2x - 1)

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

Anything equivalent is acceptable. For example, (x)(x + 3)(x - 3)(-2x + 1).

29. The relative frequency table shows the proportion of a population who have a given eye color and the proportion of the same population who wear glasses.

	Wear Glasses	Don't Wear Glasses
Blue Eyes	0.14	0.26
Brown Eyes	0.11	0.24
Green Eyes	0.10	0.15

Given the data, are the events of having blue eyes and wearing glasses independent? Justify your answer.

Answer:

Two events are independent if P(A and B) = P(A)P(B).

Let A be the probability of Wearing Glasses. Let B be the probability of Blue Eyes.

P(A) = 0.14 + 0.11 + 0.10 = 0.35.

P(B) = 0.14 + 0.26 = 0.40

P(A)P(B) = (0.35)(0.40) = 0.14

P(A and B) = 0.14 from the table.

Since the amounts are the same, the events are independent.

30. For x ≠ 0 and y ≠ 0, ∛(8lx¹⁵y⁹) = 3^ax⁵y³3. Determine the value of a.

Answer:

You can simplify the cube root or you can focus on the 81 portion of it.

∛(8l) = ∛(3 * 3 * 3 * 3) = 3 ^4/3 = 3 ^a

Therefore, a = 4/3.

31. Graph y = 2cos(1/2 x) + 5 on the interval [0,2π], using the axes below.

Answer:

The midline is 5 and the amplitude is 2, so the range is from 3 to 7. The period of cos(nx) is 2π/n, which is 2π/(1/2) = 4π. So over the range of [0,2π], only half of the wave will be seen.

Your graph should look something like this:

Do NOT continue past 2π outside of the interval you were given.

Do NOT put Arrows on either end of the line because that indicates going beyond the interval.

DO number the axes, especially the X-axis.

DO make it look like a curve and NOT a linear function.

Remember that this is only a TWO credit problem, and you lose one point for each graphing error. That means that 2 (or more) "little" mistakes means you will get zero credit, regardless of the amount of work you point into it.

32. A cup of coffee is left out on a countertop to cool. The table below represents the temperature, F(t), in degrees Fahrenheit, of the coffee after it is left out fort minutes.

Based on these data, write an exponential regression equation, F(t), to model the temperature of the coffee. Round all values to the nearest thousandth.

Answer:

Put the information into L1 and L2 in your calculator. Run an Exponential Regression. Write the answer using F(t) and t. Do NOT use y or x in your final answer.

If you entered the data correctly, you should have gotten a = 169.136 and b = 0.971. This means that your equation should have been:

F(t) = 169.136(0.971)^t

I scored this problem on literally hundreds of exams. The biggest mistake among those who did the work correctly was to use y and x, or F(t) and x, either of which cost a point. This mistake was far more common than rounding to the wrong number of decimals, or even misentering the data and getting a and b values which were slightly off.

End of Part II

How did you do?

More to come. Comments and questions welcome.

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