Saturday, February 01, 2020

January 2020 Algebra Regents, Part 2

The following are some of the multiple questions from the January 2020 New York State Common Core Algebra I Regents exam.

January 2020 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. Graph f(x) = -(sqrt(x)) + 1 on the set of axes below.

The function will have a domain of x > 0, meaning that the left side of the graph will be blank.
The y-intercept is 1. The negative in front of the radical means that line with curve down to the right, like half of a parabola that has been tipped onto its side.
Plot points using perfect squares for x -- 0, 1, and 4.
We already have (0, 1). The square root of 1 is 1; make it negative, it becomes -1; add 1, it becomes 0. Plot (1, 0). The square root of 4 is 2; make it negative, it becomes -2; add 1, it becomes -1. Plot (4, -1).
See image below.

26. Maria orders T-shirts for her volleyball camp. Adult-sized T-shirts cost $6.25 each and youth-sized T-shirts cost $4.50 each. Maria has $550 to purchase both adult-sized and youth-sized T-shirts. If she purchases 45 youth-sized T-shirts, determine algebraically the maximum number of adult-sized T-shirts she can purchase.

Algebraically means write and equation with a variable, and solve for that variable. You must do this to get full credit because that's what you are instructed to do.

6.25x + 4.50(45) < 550
6.25x + 202.50 < 550
6.25x < 347.50
x < 55.6
The maximum number of shirts that Maria can order is 55.

27. A news report suggested that an adult should drink a minimum of 4 pints of water per day. Based on this report, determine the minimum amount of water an adult should drink in fluid ounces, per week.

Two conversions need to happen here: days to weeks, and pints to fluid ounces.
If you did not know that 1 pint = 16 fluid ounces, the reference table in the back of the booklet tells you that 1 pint = 2 cups, and 1 cup = 8 fluid ounces. Therefore 1 pint = (2)(8) = 16 fluid ounces.

4 pints / day * (7 days / week) * 16 (fluid ounces / pint)
= 4 pints / day * (7 days / week) * 16 (fluid ounces / pint)
= 4 * 7 * 16 (fluid ounces / week)
= 448 fluid ounces / week

You only needed to say 448. However, if you added units, you had to have them stated correctly.

28. Express (3x - 4)(x + 7) - (1/4)x2 as a trinomial in standard form.

Step one: multiply the binomials.
Step two: combine like terms.
Step three: rewrite in standard form, in you didn't do that already.

(3x - 4)(x + 7) - (1/4)x2
3x2 + 21x - 4x - 28 - (1/4)x2
2 3/4 x2 + 17x - 28

29. John was given the equation 4(2a + 3) = -3(a - 1) + 31 - 11a to solve. Some of the steps and their reasons have already been completed. State a property of numbers for each missing reason.

4(2a + 3) = -3(a - 1) + 31 - 11aGiven
8a + 12 = -3a + 3 + 31 - 11a __________________________
8a + 12 = 34 - 14a Combining like terms
22a + 12 = 34 __________________________

The first blank is the Distributive Property. The second is Equally adding to both sides.
These are definitions: you either know these properties or you do not. You can't make up stuff or just write anything or "figure it out".

Note: I originally had "Zero Property of Addition" written in my answer, but that would not be correct as a step would need to be shown adding 14a to both sides, so that they could create a zero.
According to the model responses, just writing "Addition" would be enough.

30. State whether the product of SQRT(3) and SQRT(9) is rational or irrational. Explain.

The product of SQRT(3) * SQRT(9) = SQRT(27), which is equal to 3 SQRT(3), which is irrational because the product of a rational and an irrational number is always irrational.

You could have simplified SQRT(9) = 3 first, and skipped the SQRT(27) part.

31. Use the method of completing the square to determine the exact values of x for the equation x2 - 8x + 6 = 0.

If you are unsure of this, remember: you have the time. You can solve it another way, such as on your graphing calculator. Once you know the answer, that may make completing the square easier for you. If it doesn't, remember that you still get half credit for a correct answer found using an incorrect method.
Also, exact values means that they want you to use radicals for the irrational numbers. Do NOT use approximate decimal values.

x2 - 8x + 6 = 0, the middle term is -8, and half of -8 is -4. So (x - 4) will be in the final answer.
(-4)2 = 16, so we need to add 16 to both sides.
x2 - 8x + 16 + 6 = 16
x2 - 8x + 16 = 10, factor the trinomial into a perfect square.
(x - 4)2 = 10, now take the square root of each side of the equation.
x - 4 = +SQRT(10)
x = 4 + SQRT(10)

32. A formula for determining the finite sum, S, of an arithmetic sequence of numbers is

S = (n / 2) (a + b),

where n is the number of terms, a is the first term, and b is the last term. Express b in terms of a, S, and n.

The only things that are important in this question are the formula and which variable you have to solve for. The rest would only be important if there is a question about the sum of an arithmetic sequence later in the test.

To express b in terms of the other variables means isolating it on one side of the equation. That means using inverse operations.
I'll show each step, although you might combine the first two.

S = (n / 2) (a + b)
2 S = n (a + b)
(2S)/n = a + b
(2S)/n - a = b

End of Part II

How did you do?

Questions, comments and corrections welcome.

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Unknown said...

Thanks for this. It is awesome!!
I just have a questions about #29. Doesn't the Zero Property of Addition mean that when you add zero the result stays the same? In this case 14a is added to both sides, isn't that the Equality of Addition property where you can add the same thing to sides and the equation remains the same?
Thanks for posting this and helping me out with my question.

(x, why?) said...

You have a point there, since the actual work for that isn't shown. (The actual addition to get zero.)

It would be better just to say, Addition or inverse operations.

I'll change that when I get a chance.