August 2015 Algebra 1 (Common Core) Regents: Part 1, Multiple-Choice

Here are the questions, with answers and explanations, for the New York State Algebra 1 (Common Core) Regents exam. There were 24 questions, each worth 2 credits. No partial credit. No work needed to be shown (but it would still be a good idea to work out the answers, even if no one will see it). Fifteen correct answers on this part -- which is roughly two-thirds of the questions -- results in 30 credits, which curves to a grade of 65 (which is roughly two-thirds of the points).

Part 1

1. Given the graph of the line represented by the equation f(x) = -2x + b, if b is increased by 4 units, the graph of the new line would be shifted 4 units

(2) up. Adding 4 increasing the value of f(x), which is graphed as y. So the graph is higher.

2. Rowan has $50 in a savings jar and is putting in $5 every week. Jonah has $10 in his own jar and is putting in $15 every week. Each of them plots his progress on a graph with time on the horizontal axis and amount in the jar on the vertical axis. Which statement about their graphs is true?

(3) Jonah's graph has a steeper slope than Rowan's. Rowan's has a slope of 5 while Jonah has a slope of 15, which is greater. The two graphs will intersect at some point, so neither graph will always be above the other.

3. To watch a varsity basketball game, spectators must buy a ticket at the door. The cost of an adult ticket is $3.00 and the cost of a student ticket is $1.50. If the number of adult tickets sold is represented by a and student tickets sold by s, which expression represents the amount of money collected at the door from the ticket sales?

(4) 3.00a + 1.50s. The number of adult tickets X the price of the adult tickets plus the number of the student tickets X the prices of the student tickets.

4. The graph of f(x) is shown below. (diagram omitted)
Which function could represent the graph of f(x)?

(1) f(x) = (x + 2)(x² + 3x - 4). The graph has zeros at -4, -2, and 1, which gives us the factors (x + 4)(x + 2)(x - 1). Remember: you flip the sign because (-4) + 4 = 0. You can eliminate choices (2) and (4) immediately. Multiplying the other two factors, you will get a constant of -4, not +4, so the answer is 1. Note: You could also have solved this by factoring the choices and finding the correct zeroes.

5. The cost of a pack of chewing gum in a vending machine is $0.75. The cost of a bottle of juice in the same machine is $1.25. Julia has $22.00 to spend on chewing gum and bottles of juice for her team and she must buy seven packs of chewing gum. If b represents the number of bottles of juice, which inequality represents the maximum number of bottles she can buy?

(4) 0.75(7) + 1.25b < 22. This is similar to question 3. The price of the gum times the number of packs (7, in this case) plus the price of the juice times the number of bottles must be less than or equal to $22 because that's all that Julia has.

6. Which graph represents the solution of y < x + 3 and y > -2x - 2? (image omitted)

(3). The lines are all graphed correctly, so you don't need to check slopes or y-intercepts. The only difference is the shading. The first inequality (less than) is shaded below the line. The second (greater than) is shaded above the line. That is choice (3).

7. The country of Benin in West Africa has a population of 9.05 million people. The population is growing at a rate of 3.1% each year. Which function can be used to find the population 7 years from now?

(3) f(t) = (9.05 x 10⁶)(1 + 0.031)⁷. All four operations show the correct formula for exponential growth and decay. The question is about growth, so eliminate the choices with minus signs (decay). You need to turn 3.1% into a decimal, and that is 0.031, not 0.31, which is 31%.
Fun fact: there actually is a country in West Africa named Benin, and it's current population is about 10.3 million, so I guess the question was written a couple years ago.

8. A typical cell phone plan has a fixed base fee that includes a certain amount of data and an overage charge for data use beyond the plan. A cell phone plan charges a base fee of $62 and an overage charge of $30 per gigabyte of data that exceed 2 gigabytes. If C represents the cost of g represents the total number of gigabytes of data, which equation could represent this plan when more than 2 gigabytes are used?

(4) C = 62 + 30(g - 2). The fixed rate is 62, which doesn't change. The overage charge, 30, gets the variable. However, since the first 2 gigs are free, you have to subtract 2 from the total. Note: if you subtract g from 2, (2 - g), you will get a negative number. Also note that the question stated that this equation only makes sense if you used more than 2 gigabytes.

9. Four expressions are shown below:

I. 2(2x² - 2x - 60L
II. 4(x² - x - 30>
III. 4(x + 6)(x - 5)
IV. 4x(x - 1) - 120
The expression 4x² - 4x - 120 is equivalent to

(3) I, II, and IV. Multiply the factors and three of the expressions are correct. Expression III has +4x as its middle term.

10. Last week, a candle store received $355.60 for selling 20 candles. Small candles sell for $10.98 and large candles sell for $27.98. How many large candles did the store sell?

(2) 8. The set of equations to solve is:

S + L = 20
10.98S + 27.98L = 355.60 Note: because each term has two decimal places, you can remove the decimal points if you want to.
Because this a multiple-choice problem, you have the option to work backward from the answer. That is, you can plug is (14, 6), (8, 12), (10, 10), and (8, 12) for (S, L) and see which works. If you do this, my advice is to start with 10 and 10 -- it's quickest and you can see if you are over and need fewer Large candles or if it's under and you need more. (Or if it's correct, which in this case, it isn't.)
10.98(20 - L) + 27.98L = 355.60
219.60 - 10.98L + 27.98L = 355.60
219.60 + 17L = 355.60
17L = 136
L = 8

11. Which representations are functions? (image omitted)

(2) II and IV. Option I is not a function because 2 appears twice in the x column. Option III is not a function because it fails the vertical-line test.

12. (Image omitted) If f(x) = (the square root of (2x + 3)) over (6x - 5), then f(1/2) =

(3) -1. This looks overly complicated but it's a straight substitute and evaluate problem. It's also one that's difficult to put in most calculators is you aren't careful, so work some of it out before going to a calculator. Once you start, you might not even need it.
Two times (1/2) + 3 = 1 + 3 = 4. The square root of 4 is 2. Six times (1/2) - 5 is 3 - 5 = -2. 2/-2 = -1.

13. The zeros of the function f(x) = 3x² - 3x = 6 are

(4) -1 and 2. Once again, you can work backward from the choices, substituting and evaluating. Or you can factor:

3x² - 3x - 6 = 0
3(x² - x - 2) = 0
3(x - 2)(x + 1) = 0 x = 2 or x = -1

14. Which recursively defined function has a first term equal to 10 and a common difference of 4?

(1) f(1) = 10, f(x) = f(x - 1) + 4. Choices (3) and (4) are silly as they give the first term as 4, instead of 10. The common difference means that each term is 4 more than the previous one, not four times more, so choice (3) is out.

15. Firing a piece of pottery in a kiln takes place at different temperatures for different amounts of time. The graph below shows the temperatures in a kiln while firing a piece of pottery after the kiln is preheated to 200^oF. (image omitted)
During which time interval did the temperature in the kiln show the greatest average rate of change?

(1) 0 to 1 hour. The greatest change will have the steepest slope. The slope is decreasing from point to point. You can check that using the Slope formula.
(700 - 200)/(1 - 0) = 500
(900 - 700)/(1.5 - 1) = 400
(1640 - 1300)/(5 - 2.5) = 136
(1800 - 1640)/8 - 5) = 53.3

16. Which graph represents f(x) = |x|, x < 1 and f(x) = Sqrt(x), x > 1? (image omitted)

(3). This could be put into a graphing calculator and you would see the result.
The left side is the absolute value graph, which is a straight line, decreasing to zero. The right side looks like half of a sideways parabola.

17. If f(x) = x² - 2x - 8 and g(x) = (1/4)x - 1, for which values of x is f(x) = g(x)?

(2) -1.75 and 4. Again, you have a calculator and you have TWO options: graph them or substitute. You can tell from the constants that they will NOT be the same at 0, so you can eliminate (3) and (4) immediately.
If you want to solve the equations, set them equal to each other and make a quadratic equation:

x² - 2x - 8 = (1/4)x - 1
x² - 2x - 8 - (1/4)x + 1 = 0
x² - 2.25x - 7 = 0
Because completing the square would be a bit messy at this point (and if you use the calculator to complete the square, you might as well just graph it!), I'm going to multiply by 4 to get rid of the decimal.
4x² - 9x - 28 = 0
(4x + 7)(x - 4) = 0
x = -7/4 or x = 4

18. Alicia has invented a new app for smart phones that two companies are interested in purchasing for a 2-year contract.
Company A is offering her $10,000 for the first month and will increase the amount each month by $5000.
Company B is offering $500 for the first month and will double their payment each month from the previous month.
Monthly payment are made at the end of the each month. For which monthly payment will company B's payment first exceed company A's payment?

(8) The first equation is 5000x + 10000, and the payments are 10000, 15000, 20000, 25000, 30000, 35000, 40000, 45000, 50000, 55000, 60000, 65000.
The second equation is 500 * 2^(n-1), and the payments are 500, 1000, 2000, 4000, 8000, 16000, 32000, 64000, 128000, 256000, 512000.
At the 8th month, $128,000 is greater than $55,000. Note that Alicia hasn't earned more money total at this point, but which the second plan, she'll earn over a million dollars in a very short time!

19. The two sets of data below represent the number of runs scored by two different youth baseball teams over the course of a season.

Team A: 4, 8, 5, 12, 3, 9, 5, 2
Team B: 5, 9, 11, 4, 6, 11, 2, 7
Which set of statement about the mean and standard deviation is true?

(1) mean A < mean B; standard deviation A > standard deviation B.
TIP: Learn how to use LISTs and 1-Var Stats on your calculator. Calculating standard deviation by hand is a pain in the butt.
First off, the mean of A is 6 and the mean of B is 6.875, so (2) and (4) are out. Using the calculator, the std dev of A is about 3.16 and B is about 3.06. That's (1).

Look at the differences from the mean. In A, it's 2, 2, 1, 6, 3, 3, 1, 4. In B, if we use 7, we get 2, 2, 4, 3, 1, 4, 5, 0. Eliminating the numbers that are the same, A has 1, 6, 3 and B has 4, 5, 0. Because we need to deal with the squares of these differences, we can see that the sum of the squares in set A will be higher than the sum of the squares in set B.

20. If Lylah completes the square for f(x) = x² - 12x + 7 in order to find the minimum, she must write f(x) in the general form f(x) = (x - a)^{2 + b. What is the value of a for f(x)?}

(1) 6. To complete the square, the first step is to take half of the x term. Half of -12 is -6. However, the formula already has the minus sign in it, so it's just 6. You don't need to work out the rest of the equation nor find b.

21. Given the following quadratic functions:

g(x) = -x² - x + 6
and
x, -3, -3, -1, 0, 1, 2, 3, 4, 5
n(x), -7, 0, 5, 8, 9, 8, 5, 0, -7 Which statement about these functions is true?

(4) The sum of the roots of n(x) = 0 is greater than the sum of the roots of g(x) = 0.
In choice (1), the rate of change for g(x) is 2/2 = 1; the rate of change for n(x) is 4/2 = 2, which is greater. In choice (2), g(0) = 6, n(0) = 8, so g(0) is not greater. In choice (3), g(x) has a maximum value when x = 1/2 and g(x) is a little more than 6, but n(1) = 9, so this statement is not true, either.

22. For which value of P and W is P + W a rational number?

(2) (image omitted) P = 1/SQRT(4) and W = 1/SQRT(9). The only way that the sum of two positive numbers will be a rational number is if both of the numbers are rational. (If two irrational numbers which are inverses will add up to zero.) Choice (2) is the only choice with two rational numbers in the denominators of the fractions. The sum of 1/2 and 1/3 is 5/6, which is rational.

23. The solution of (x + 3)² = 7 is

(3) -3 + Sqrt(7). Inverse operations: take the square root of both sides, then subtract 3.

24. Which trinomial is equivalent to 3(x - 2)² - 2(x - 1)?

(4) 3x² - 14x + 14.

3(x - 2)² - 2(x - 1)
3(x² - 4x + 4) - 2x + 2
3x² - 12x + 12 - 2x + 2
3x² - 14x + 14

That's the end of Part 1. Part 2 will hopefully be uploaded soon, and there will be a link here.

Link to Part 2