Friday, January 29, 2016

January 2016 New York Algebra I (Common Core) Part 1: Multiple-Choice

Below are the questions with answers and explanations for Part 1 of the Algebra I (Common Core) Regents exam for January 2016, the multiple-choice questions. The open-ended questions will be posted separately.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 1 is worth 2 credits, for a total of 48 credits. Usually, a score of 30 credits on the entire test is worth a score of 65. The curve is steep after that. To achieve a final score of, say, 75, you will need roughly 55 credits. (The exact curve will not be revealed for a few days.)

January 2016 Algebra 1 (Common Core) Regents, Part 1

1. In the function f(x) = (x - 2)2 + 4, the minimum value occurs when x is

(2) 2. The vertex is at the point (h, k) taken from the general form, f(x) = (x - h)2 + k.

2. The graph below was created by an employee at a gas station. (image omitted)
Which statement can be justified by using the graph?

(2) For every gallon of gas purchased, $3.75 was paid. (1) and (3) can be eliminated by checking the graph. Neither (10, 35) nor ((2, 5) are points on the line. You can't see (1, 3.75) but if you multiply by 4, you will see (4, 15) on the graph. Choice (4) is just ridiculous -- the graph has nothing to do with miles driven.

3. For a recently released movie, the function y = 119.67(0.61)x models the revenue earned, y, in millions of dollars each week, x, for several weeks after its release.

Based on the equation, how much more money, in millions of dollars, was earened in revenue for week 3 than for week 5?

(3) 17.06. Substituting 3 for x in the function gives us about 27.16 million, and substitution 5 gives us 10.11, with a difference of 17.05. The difference with the answer is a minor rounding error.

4. Given the following expressions:
I. -(5/8) + (3/5); II. (1/2) + sqrt(2); III. (sqrt(5))*(sqrt(5)); IV. 3*(sqrt(49))
Which expression(s) result in an irrational number?

(1) II only. The first is the sum of two rationals, which is rational. The second is the sum of a rational and an irrational, which is irrational. The third squares the square root of 5, which is 5, a rational number. The last is three times the square root of 49, which equals 3 * 7 which is 21, a rational number.

5. Which inequality is represented by the graph below?

(2) y > 2x - 3. The y-intercept is -3. The slope is 2. The graph is shaded above the line. (The line is also solid, but that doesn't matter for the choices given.)

6. Michael borrows money from his uncle, who is charging him simple interest using the formula I = Prt. To figure out what the interest rate, r, is, Michael rearranges the formula to find r. His new formula is r equals

(3) I/Pt. Divide both sides of the equation by P and t to isolate r.

7. Which equation is equivalent to y - 34 = x(x - 12)?

(4) y = (x - 6)2 + 2.
Distribute the x and you get: y - 34 = x2 - 12x.
Add 34 to both sides: y = x2 - 12x + 34.
There are no integer factors of 34 that have a sum of -12. (Eliminate choices 1 and 2.) This means completing the square, OR working backward from the other choices.
Squaring (-6) gives us +36. To make 36 into 34, we need to subtract 2. Choice (4).

8. The equation A = 1300(1.02)7 is being used to calculate the amount of money in a savings account. What does 1.02 represent in this equation?

(4) 2% growth. It's greater than 1, so it is growth. (And if it is a savings account, it better be growth, or why have the account?) The decimal .02 is 2% as a percentage.

9. The zeros of the function f(x) = 2x2 - 4x - 6 are

(1) 3 and -1. Plug in the numbers and see what gives you 0. This is probably quicker than factoring.
2x2 - 4x - 6 = 0
x2 - 2x - 3 = 0
(x - 3)(x + 1) = 9
x = 3 or x = -1

10. When (2x - 3)2 is subtracted from 5x2, the result is

(3) x2 + 12x - 9.
5x2 - (4x2 - 12x + 9) = 5x2 - 4x2 + 12x - 9 = x2 + 12x - 9.

11. Joe has a rectangular patio that measures 10 feet by 12 feet. He wants to increase the area by 50% and plans to increase each dimension by equal lengths, x. Which equation could be used to determine x?

(2) (10 + x)(12 + x) = 180. Area = Length * Width = (10)(12) = 120. Increasing 120 by 50% means 120 + .5(120) = 180.

12. When factored completely, x3 - 13x2 - 30x is

(3) x(x + 2)(x - 15)
x(x2 - 13x - 30) = x(x - 15)(x + 2)

13. The table below (image omitted) shows the cost of mailing a postcard in different years. During which time interval did the cost increase at the greatest average rate?

(4) 2006-2012. Find the average rate for each interval. From 1898-1971, the change was 5/73. From 1971-1985, it was 8/14. From 1985-2006, it was 10/21. From 2006-2012, it was 11/6. Choice (4) is the only one greater than 1, so it is obviously the greatest.

If you eliminate choice 1 as obviously very flat, you can sketch out the other points and see that 2006-2012 is the steepest line.

14. When solving the equation x2 - 8x - 7 = 0 by completing the sqaure, which equation is a step in the process?

(2) (x - 4)2 = 23. Half of 8 is 4, so eliminate choices (3) and (4). (-4)2 = +16. You have to add 16 to both sides AND add 7 to both sides to get the 7 to the other side of the equation. 16 + 7 = 23.

15. A construction company uses the function f(p), where p is the number of of people working on a project, to model the amount of money it spends to complete a project. A reasonable domain for this function would be

(1) positive integers. The number of people has to be a counting number. It cannot be negative nor a fraction. People can also be zero, but in the context of the problem, if zero people are working on a project, the company won't make any money. (None of the options include zero.)

16. Which function is shown in the table below? (image omitted)

(4) f(x) = 3x. Substitute 0 into the functions and only choice (4) works. For that matter, choice (4) is the only one that can produce fractions as output when the input is integers.

17. Given the functions h(x) = 1/2x + 3 and f(x) = |x|, which value of x makes h(x) = f(x)?

(1) -2. It is quicker to work backward from the answers given. 1/2(-2) + 3 = -1 + 3 = 2 = |-2|

18. Which recursively defined function represents the sequence 3, 7, 15, 31 ... ?

(3) f(1) = 3, f(n + 1) = 2f(n) + 1. Each term after the first is one more than twice the previous term.

Note: I don't know if there are typos in choices (1) and (2) or if it was intended to write f(n) as an exponent. It's just odd-looking. As an exponent or not, the answers are incorrect, so it doesn't matter.

19. The range of the function defined as y = 5x is

(2) y > 0. A positive number to any exponent will be positive, never zero or negative.

20. The graph of y = f(x) is shown below. (image omitted)
What is the graph of y = f(x + 1) - 2?

(1). The graph will move 1 space to the left and 2 down because h = -1 and k = -2.

21. Which pair of equations could not be used to solve the following equations for x and y?

4x + 2y = 22
-2x + 2y = -8

(4) 8x + 4y = 44; -8x + 8y = -8. In choices (1) - (3), one or both of the equations is multiplied by some constant. In (4), the -8 was not multiplied by 4 but the left side of the equation was.

22. The graph representing a function is shown below. (image omitted)
Which function has a minimum that is less than the one shown in the graph?

The graph has a minimum at (3, -7). This eliminates choices (2) (-3, -6) and (4) (8, 2).
In choice (1), the axis of symmetry is 3, and the y-coordinate of the vertex is y = (3)2 - 6(3) + 7 = 9 - 18 + 7 = -2
In choice (3), the axis of symmetry is 1, and the y-coordinate of the vertex is y = (1)2 - 2(1) - 10 = 1 - 2 - 10 = -11.
Alternatively, you could have put these into your graphing calculator and just observed the correct answer.

23. Grisham is considering the three situations below.
I. For the first 28 days, a sunflower grows at a rate of 3.5 cm per day.
II. The value of a car depreciates at a rate of 15% per year after it is purchased.
III. The amount of bacteria in a culture triples every two days during an experiment.
Which of the statements describe a situation with an equal difference over an equal interval?

(1) I, only. The first situation is a linear function, growing the same amount every day. (Constant slope.) The other two are exponential functions with the amount changing from interval to interval, even if the percentage remains the same.

24. After performing analyses on a set of data, Jackie examined the scatter plot of the residual values for each analysis. Which scatter plot indicates the best linear fit for the data? (images omitted)

(3). The residual plot should contain randomly-scattered points above and below the x-axis. It should not have a pattern to it. Choices (1), (2) and (4) show curve-like patterns to the plotting of their residuals. These indicate a poor fit for the data.

End of Part I


Heather said...

Thank you for taking the time to post this. It is hard waiting for the grades to be posted, but I just went over these questions with my daughter and she is feeling positive about the outcome!

(x, why?) said...

You're welcome. Thanks for leaving a comment. Feedback is the only way I know if I'm doing a service and not wasting my time.

The rest of the test is up in other posts. I hope your daughter did well. On the one hand, another feeling confident about the multiple-choice most likely passed the exam. On the other hand, anyone feeling that confident about the multiple-choice most likely is settling for *just* passing the exam.

Anonymous said...

Thank u do much for posting this. I had to do test corrections (we took it as a practice regents) and was so confused on everything. U were a big help!

(x, why?) said...

You're welcome. I hope the explanations helped with more than just making corrections.

I have this fear that my students will start using my blog against me!