## Thursday, July 08, 2021

### June 2021 Algebra 1 Regents (v202), Part I (multiple choice)

This exam was adminstered in June 2021. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

Part II is located here.

### Part I

1. A high school club is researching a tour package offered by the Island Kayak Company. The company charges \$35 per person and \$245 for the tour guide. Which function represents the total cost, C(x), of this kayak tour package for x club members?

(1) C(x) = 35x
(2) C(x) = 35x + 245
(3) C(x) = 35(x + 245)
(4) C(x) = 35 + (x + 245)

Answer: (2) C(x) = 35x + 245

\$35 per person is the cost that is multiplied by the x number of people going on the tour. The \$245 is paid one time before anyone has even signed up for the tour, which makes it the y-intercept.

2. The expression 3(x + 4) - (2x + 7) is equivalent to

(1) x + 5
(2) x - 10
(3) x - 3
(4) x + 11

Distribute the 3 and distribute the minus sign (-1).
3(x + 4) - (2x + 7) = 3x + 12 - 2x - 7 = x + 5

3. A function is defined as K(x) = 2x2 - 5x + 3. The value of K(-3) is

(1) 54
(2) 36
(3) 0
(4) -18

K(-3) = 2(-3)2 - 5(-3) + 3 = 2(9) - (-15) + 3 = 18 + 15 + 3 = 36

4. Which relation is not a function?

In a function map, each member of the domain (on the left) can only have one arrow coming out of it because each input can only can associated with one output.

In Choice (4), there are two arrows coming out of the number six, meaning that (6, 10) and (6, 12) are in the same relation. This would cause it to fail the vertical line test.

5. The value of Tony's investment was \$1140 on January 1st. On this date three years later, his investment was worth \$1824. The average rate of change for this investment was \$19 per

(1) day
(2) month
(3) quarter
(4) year

1824 - 1140 = 684, which is 684 / 3 = 228 per year, which is 228 / 12 = 19 per month.

This was an interesting question in that you had to work through the choices. However, number sense should tell you that "per month" was the most likely, so you could have tried 684 / 36 first.

6. The solution to 3(x - 8) + 4x = 8x + 4 is

(1) 12
(2) 28
(3) -12
(4) -28

Use the Distributive Property, and then solve the equation for x.
3(x - 8) + 4x = 8x + 4
3x - 24 + 4x = 8x + 4
7x - 24 = 8x + 4
-28 = x

7. An ice cream shop sells ice cream cones, c, and milkshakes, m. Each ice cream cone costs \$1.50 and each milkshake costs \$2.00. Donna has \$19.00 to spend on ice cream cones and milkshakes. If she must buy 5 ice cream cones, which inequality could be used to determine the maximum number of milkshakes she can buy?

(1) 1.50(5) + 2.00m > 19.00
(2) 1.50(5) + 2.00m < 19.00
(3) l.50c + 2.00(5) > 19.00
(4) l.50c + 2.00(5) < 19.00

Answer: (1) 1.50(5) + 2.00m > 19.00

If she must buy 5 ice cream cones, then substitute c = 5, not m = 5. If she can't spend more than \$19, than you need to have a < symbol.

8. When written in standard form, the product of (3 + x) and (2x - 5) is

(1) 3x - 2
(2) 2x2 + x - 15
(3) 2x2 - llx - 15
(4) 6x - 15 + 2x2 - 5x

Answer: (2) 2x2 + x - 15

First of all, you know that the answer will be a quadratic expression, so Choice (1) is out. And Choice (4) is NOT is standard form, so eliminate that.

(x + 3)(2x - 5) = 2x2 + 6x - 5x - 15 = 2x2 + x - 15

9. If x = 2, y = 3 SQRT(2), and w = 2 SQRT(8), which expression results in a rational number?

(1) x + y
(2) y - w
(3) (w)(y)
(4) y / x

The product of SQRT(2) and SQRT(8) is SQRT((2)(8)) = SQRT(16) = 4, which is a rational number. When that result is multiplied by 3 and 2, it will still be rational. (24)

10. Which product is equivalent to 4x2 - 3x - 27?

(1) (2x + 9)(2x - 3)
(2) (2x - 9)(2x + 3)
(3) (4x + 9)(x - 3)
(4) (4x - 9)(x + 3)

Answer: (3) (4x + 9)(x - 3)

You can multiply these or graph them or factor the given expression.

If you "borrow & payback" then (4)(-27) = -108. What two factors of -108 have a sum of -3? Running through the list of factors will get you to (9)(-12). You have to "pay back" the 4.

Write out 4x2 - 12x + 9x - 27, and factor by grouping: 4x(x - 3) + 9(x - 3) = (4x + 9)(x - 3)

11. Given: f(x) = 2/3 x - 4 and g(x) = 1/4x + 1

I. f(4) = g(4)
II. When x = 12,f(x) = g(x).
III. The graphs of f(x) and g(x) intersect at (12,4).
IV. The graphs of f(x) and g(x) intersect at (4,12).
Which statement(s) are true?

(1) II, only
(2) IV, only
(3) I and IV
(4) II and III

Statement I says that f(4) = 2/3(4) - 4 = -4/3 = g(4) = 1/4(4) + 1 = 2. This is not true.
Statement II says that f(12) = 2/3 (12) - 4 = 8 - 4 = 4 = g(12) = 1/4 (12) + 1 = 4. This is true.
Statement III restates statement II with extra information, which if that not only is f(12) = g(12) but they both equal 4.
Statement IV is not true because statement I is not true.

12. Which sketch represents the polynomial function f(x) = x(x + 6)(x + 3)?

Choices (3) and (4) are not functions. They fail the vertical line test.

If f(x) = x(x + 6)(x + 3) then there are zeroes at x = 0, x + 6 = 0 and x + 3 = 0, or x = 0, x = -6 and x = -3.

That's Choice (1).

13. If the parent function of f(x) is p (x) = x2, then the graph of the function f(x) = (x - k)2 + 5, where k > 0, would be a shift of

(1) k units to the left and a move of 5 units up
(2) k units to the left and a move of 5 units down
(3) k units to the right and a move of 5 units up
(4) k units to the right and a move of 5 units down

Answer: (3) k units to the right and a move of 5 units up

Basic definition. You can try it in your graphing calculator by picking any value (e.g. 2) for k and observing how the graph changes.

14. Which expression is equivalent to (-4x2)3?

(1) -12x6
(2) -12x5
(3) -64x6
(4) -64x5

The exponent of 2 in (-4x2)3 goes only with the x. Everything in the parentheses gets the 3 exponent.

(-4)3 x2 * 3 = -64x6. Notice that when you take a power to another power, you have to multiply.

15. Which function has the smallest y-intercept?

The y-intercept in Choice (1) is -6, as you can see from y = mx + b.

The y-intercept in Choice (2) is 1, as you can see from the row where x = 0.

The y-intercept in Choice (3) is -2, as you can see if you calculate f(0).

The y-intercept in Choice (4) is -2, as you can see on the graph at the y-axis.

16. Which domain would be the most appropriate to use for a function that compares the number of emails sent (x) to the amount of data used for a cell phone plan (y)?

(1) integers
(2) whole numbers
(3) rational numbers
(4) irrational numbers

If you are comparing two whole numbers, you will have a ratio, and you would want rational numbers.

The domain refers to the x values. The number of emails is a countable number, so whole numbers would be the most appropriate.

17. Eric deposits \$500 in a bank account that pays 3.5% interest, compounded yearly. Which type of function should he use to determine how much money he will have in the account at the end of 10 years?

(1) linear
(3) absolute value
(4) exponential

If it is compounded interest, then you need an exponential function. Simple interest is linear.

18. Given: the sequence 4, 7, 10, 13,. ..

When using the arithmetic sequence formula an = a1 + (n - 1)d to determine the 10th term, which variable would be replaced with the number 3?

(1) a1
(2) n
(3) an
(4) d

The common difference, d, is 3 from term to term.

19. Below are two representations of data.
A: 2,5,5,6,6,6,7,8,9

Which statement about A and B is true?
(1) median of A > median of B
(2) range of A < range of B
(3) upper quartile of A < upper quartile of B
(4) lower quartile of A > lower quartile of B

The medians of A and B are both 6, so eliminate Choice (1).

The range of A is 7 and the range of B is 6, so eliminate Choice (2).

The upper quartile of A is 7.5 and the upper quartile of B is 9. This is the correct statement.

The lower quartile of A and B are both 5.

20. Which system has the same solution as the system below?
x + 3y = 10
-2x - 2y = 4

(1) -x + y = 6, 2x + 6y = 20
(2) -x + y = 14, 2x + 6y = 20
(3) x + y = 6, 2x + 6y = 20
(4) x + y = 14, 2x + 6y = 20

Answer: (2) -x + y = 14, 2x + 6y = 20

This was an interesting question. Uusually when they ask something like this, the choices are multiples of the given equation (although 3 of the choices will not be multiplied correctly). This question, however, is just which two separate systems of equations happen to have the same solution, totally by coincidence.

Put these all in your graphing calulcator to find which ones intersect at the same point.

The given system has a solution of (-8, 6). Now you can graph the others, or you can plug in the values to see which gives you a true statement.

-(-8) + (6) = 14, which is =/= 6. Eliminate Choice (1).

-(8) + 6 = 14, (check), and 2(-8) + 6(6) = -16 + 36 = 20 (check). This is the solution.

If you wanted to solve the original system of equations algebraically

x + 3y = 10
-2x - 2y = 4

2x + 6y = 20
-2x - 2y = 4

4y = 24
y = 6
x + 3(6) = 10
x + 18 = 10 x = -8

21. Given the pattern below, which recursive formula represents the number of triangles in this sequence?

(1) y = 2x + 3
(2) y = 3x + 2
(3) a1 = 2, an = an - 1 + 3
(4) a1 = 3, an = an - 1 + 2

Answer: (4) a1 = 3, an = an - 1 + 2

Notice that Choices (1) and (2) are not recursive formulas, so both can be eliminated.

The first term has 3 triangles and each term after adds 2 more. This is Choice (4).

22. Students were asked to write an expression which had a leading coefficient of 3 and a constant term of -4. Which response is correct?

(1) 3 - 2x3 - 4x
(2) 7x3 - 3x5 - 4
(3) 4 - 7x + 3x3
(4) -4x2 + 3x4 - 4

Answer: (4) -4x2 + 3x4 - 4

Were my students to give me any of these answers, I'd have a problem. None of these is in standard form, which I might not care about in some instances, but if I'm asking for a specific leading coefficient, write it first!

The leading coefficient of (1) is 2. The leading coefficient of (2) is -3. The leading coefficient of (3) is 3, but the constant is 4, not -4. The leading coefficient of (4) is 3, and the constant is -4.

23. Sarah travels on her bicycle at a speed of 22.7 miles per hour. What is Sarah's approximate speed, in kilometers per minute?

(1) 0.2
(2) 0.6
(3) 36.5
(4) 36.6

Common sense should tell you that Choices (3) and (4) might be approximate answers in kilometers per hour, not minute.

To change from miles to kilometers, multiply by 1.609 (as per the back of the booklet). To change hours to minutes, divide by 60.

22.7 * 1.609 / 60 = 0.608..., which is approximately 0.6.

24. Which ordered pair does not fall on the line formed by the other three?

(1) (16,18)
(2) (12,12)
(3) (9,10)
(4) (3,6)

Another interesting problem in that you are required to look at all for points to see with one doesn't belong.

The easiest thing to do is plot the four points on graph paper and see which one doesn't line up.

If you checked the slopes, you would have seen that lines from (16, 18) to the other three all would have different slopes.

(18 - 12) / (16 - 12) = 6/4 or 3/2, (18 - 10) / (16 - 9) = 8/7, (18 - 6) / (16 - 3) = 12 / 13. Since all three are different, you can be sure that (16, 18) is the odd point out.

More to come. Comments and questions welcome.

More Regents problems.

Unknown said...

#20 2x +6y = 20 IS a multiple; choice (1) -x +y is the sum of the originals

Leon Gerber gerberl@stjohns.edu

Unknown said...

Yes, that equation is a multiple. Usually, in this type of problem, almost all of the 8 equations in the four choices are multiples, except that most of them are multiplied incorrectly so that they aren't the correct choice.

My point being that you can usually just *look* at the equations and know which answer is correct without solving anything.

Thanks for stopping by the blog and for commenting.