June 2019 Algebra 1 Regents, Part 2

The following are some of the multiple questions from the June 2019 New York State Common Core Algebra I Regents exam.

June 2019 Algebra I, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. Solve algebraically for x: -2/3 (x + 12) + 2/3 x = -5/4 x + 2

Answer:
You can do a few things here. You can multiply each term in the equation by 12 (3 times 4) to get rid of the denominators, or you can combine the terms on the left and deal with the fractions later.

First method

-2/3 (x + 12) + 2/3 x = -5/4 x + 2
(12)(-2/3 (x + 12)) + (12)(2/3 x) = (12)(-5/4 x) + (12)(2)
-8 (x + 12) + 8x = -15x + 24
-8x - 96 + 8x = -15x + 24
-96 = -15x + 24
-110 = - 15x
8 = x

Second method

-2/3 (x + 12) + 2/3 x = -5/4 x + 2
-2/3x - 8 + 2/3 x = -5/4 x + 2
-8 = -5/4 x + 2
-10 = -5/4 x
-40 = -5 x
8 = x

26. If C = G - 3F, find the trinomial that represents C when F = 2x² + 6x - 5 and G = 3x² + 4.

Answer: Multiply F by 3 and subtract it from G.
3F = 3(2x² + 6x - 5) = 6x² + 18x - 15

(3x² + 4) - (6x² + 18x - 15) = -3x² - 18x + 19

My guess is one point for multiplying 3F and the other for combining them properly.
One computational error costs a point. Two means no credit.

27. Graph the following piecewise function on the set of axes below.

f(x) = { |x|, -5 < x < 2
{ -2x + 10 , 2 < x < 8

Answer: Important: you are given a specific range. Do NOT add arrows to the endpoints.
Also, remember at x = 2, you need an open circle to mark the end of the absolute value graph, and a closed circle to start the linear function.

28. Solve 5x² = 180 algebraically.

Answer:

5x² = 180
x² = 36
x = SQRT(36)
x = +6

If you forget the -6, you lost a point.
If you divided incorrectly and didn't get a perfect square, you could have salvaged a point if you left your answer in radical form with a + symbol in front of it. However, if you estimated the radical, you probably lost the other point as well.

29. A blizzard occurred on the East Coast during January, 2016. Snowfall totals from the storm were recorded for Washington, D.C. and are shown in the table below.

Washington, D.C.
Time	Snow (inches)
1 a.m.	1
3 a.m	5
6 a.m	11
12 noon	33
3 p.m	36

Which interval, 1 a.m. to 12 noon or 6 a.m. to 3 p.m., has the greatest rate of snowfall, in inches per hour? Justify your answer.

Answer:
Find the rate of change for both intervals:
1 am to 12 noon: (33 - 1) / (12 - 1) = 32 / 11 = 2.909...
6 am to 3 pm (1500 in military time): (36 - 11) / (15 - 6) = 25 / 9 = 2.777...

1 am to 12 noon had the greater rate of snowfall in inches per hour.

If you made one mathematical error, you could still get a point for the final answer based on your work.

30. The formula for the volume of a cone is V = (1/3)πr²h. Solve the equation for h in terms of V, r, and π.

Answer:
This should be a "gimme" because it's one of the most common examples used in Literal Equations exercises. I'm surprised that they didn't use r so you'd have to deal with a square root!

Isolate the h by dividing by the other terms. (Dividing by 1/3 is the same as multiplying by 3.)

V = (1/3)πr²h
3V = πr²h
3V / (πr²) = (πr²h) / (πr²)
3V / (πr²) = h

31. Given the recursive formula:

a₁ = 3
a_n = 2(a_{n - 1} + 1)
State the values of a₂, a₃, and a₄ for the given recursive formula.

Answer:
Just plug the numbers into the formula to come up with the next number. A consistent error will not be penalized multiple times.

a₁ = 3
a₂ = 2(a₁ + 1) = 2(3 + 1) = 2(4) = 8
a₃ = 2(a₂ + 1) = 2(8 + 1) = 2(9) = 18
a₄ = 2(a₃ + 1) = 2(18 + 1) = 2(19) = 38

32. Determine and state the vertex of f(x) = x² - 2x - 8 using the method of completing the square.

Answer:
Okay, I'll be the first to admit that this is an annoying question. If I wanted to know the vertex, I would use the Axis of Symmetry to find it.
If they want you to "Complete the Square", they should just say that.
It's a two point question, so you will lose a point if you use any other method to find the vertex. You will also get one point for stating the vertex without any work at all. As much as I hate to say this, if you aren't completing the square then your work doesn't matter for this problem.

Look at the middle term: -2x. Half of -2 is -1, so the square you need to complete is (x - 1)².
If you square that binomial, the constant will be + 1, not - 8.
I'll do this is extra steps, but you can do it in fewer.

f(x) = x² - 2x - 8
f(x) + 1 = x² - 2x + 1 - 8
f(x) = (x² - 2x + 1) - 8 - 1
f(x) = (x - 1)² - 9

The vertex is (1, -9). You can check this on your calculator.

End of Part II

How did you do?

Questions, comments and corrections welcome.