New York State Algebra I (Common Core) Regents, June 2015 Part 1

Sorry for the rush job, but there were TWO Algebra tests back-to-back, one day after the other. Here is Part 1 for the Common Core exam. As always, images will be added as time permits.

Part 1

1. The cost of airing a commercial on television is modeled by the function C(n) = 110n + 900, where n is the number of times the commercial is aired. Based on this model, which statement is true?

(3) The commercial costs $900 to produce and $110 each time it is aired. None of the other choices fit the function.

2. The graph below represents a jogger's speed during her 20-minute jog around her neighborhood.

Which statement best describes what the jogger was doing during the 9-12 minute interval of her jog?

(4) She was jogging at a constant rate. The y-axis is speed, and it doesn't change during that interval -- it's flat.

3. If the area of a rectangle is expressed as x⁴ - 9y², then the product of the legnth and the width of the rectange could be expressed as

(2) (x² - 3y)(x² + 3y) is the only choice with the correct product. Use the Difference of Squares Rule.

4. Which table represents a function?

(3) Only table three doesn't repeat any x values. No x value can have multiple f(x) values associated with it.

5. Which inequality is represented in the graph below?

(1) y > -3x + 4. The graph is shaded above the line, and the y-intercept is 4. That is enough information to give you the correct choice.

6. Mo's farm stand sold a total of 165 pounds of apples and peaches. She sold apples for $1.75 per pound and peaches for $2.50 per pound. If she made $33750, how many pounds of peaches did she sell?

(3) 65. If you set up a system of equations and solve, you will see that she sold 100 pounds of apples and 65 pounds of peaches.

7. Morgan can start wrestling at age 5 in Division 1. He remains in that division until he next odd birthday when he is required to move up to the next division level. Which graph correctly represents this information.

(1) The step graph with the solid endpoint on the left and the open endpoint on the right. Note that choices (2) and (4) don't make any sense.

8. Which statement is not always true?

(2) The product of two irrational numbers is rational. The square root of 2 times the square root of 3 is the square root of 6, which is NOT rational. Not true.

9. The graph of the function f(x) = (x + 4)^(1/2) is shown below:

The domain of the function is

(4) {x | x > -4}. Only values of x from -4 and above make any sense in that function. Lower numbers would not have real number results.

10. What are the zeroes of the function f(x) = x² - 13x - 30?

4. 15 and -2. Set the function equal to 0 and factor.
x² - 13x - 30 = 0
(x - 15) (x + 2) = 0
x - 15 = 0 or x + 2 = 0
x = 15 or x = -2

11.Joey enlarged a 3-inch by 5-inch photograph on a copy machine. he enlarged it four times. The table below shows the area of the photograph after each enlargement.

What is the average rate of change of the area from the original photograph to the fourth enlargement, to the nearest tenth?

(3) 5.4. Calculate the rate of change: (36.6 - 15) / (4 - 0) = 21.6 / 4 = 5.4

12. Which equation(s) represent the graph below?
I. y = (x + 2)(x² - 4x - 12)
II. y = (x - 3)(x² + x - 2)
III. y = (x - 1)(x² - 5x - 6)

(2). II, only. The zeroes are at -2, 1 and 3. Each equation shows one of those zeros. You have to factor the trinomials to find out if they will reveal the other two zeroes. If you factor you get
I. (x + 2)(x - 6)(x + 2)
II. (x - 3)(x + 2)(x - 1)
III. (x - 1)(x - 6)(x + 1)
Only choice II fits the graph.
You also could have put the three equations in your graphing calculator and graphed them or checked the Tables of Values.

13. A laboratory technician studied the population growth of a colony of bacteria. He recorded the number of bacteria every other day, as shown in the partial table below.

Which function would accurately model the technician's data?

(2) f(t) = 25^{t + 1}. It is exponential, so choices (3) and (4) are out. For choice (1), if t = 0, 25⁰ = 1, not 25.

14. Which quadratic function has the largest maximum?

(3) k(x) = -5x² - 12x + 4

Again, if you graph (1) and (3), you will find out that that maximum value is more than 11. It's actually 11.2 when x = -1.2. You can find x using the formula for the axis of symmetry if you don't know how to find the maximum with your calculator. This is obviously bigger than (4). In choice (2), the maximum is more than 9, but it's much less than 11. (It's 9.5.) If you are really worried, you can do a quadratic regression to find the equation, which is f(x) = -2x² + 6x + 5.

15. If f(x) = 3^x and g(x) = 2x + 5, at which value of x is f(x) < g(x)?

(1) -1. Plug the values in or enter the equations in your calculator. Or try this trick on the calculator
3^{-3, -1, 2, 4} [ENTER]
2*{-3, -1, 2, 4}+5 [ENTER]
Compare the results. Make sure you type the choices in the same order, so you can compare them easily.

16. Beverly did a study this past spring using data she collected from a cafeteria. She recorded data weekly for ice cream sales and soda sales. Beverly found the line of best fit and the correlation coefficient as shown in the diagram below.

Given the information, which statement(s) can correctly be concluded?
I. Eating more ice cream causes a person to become thirsty.
II. Drinking more soda causes a person to become hungry.
III. There is a strong correlation between ice cream sales and soda sales.

(2) III, only. The correlation constant is .96, which shows a strong connection. There's nothing to indicate increased hungry or thirst. (When I'm hungry, I don't think "ice cream!", for example.) This is one of those cases that fits the expression "correlation does not prove causation". Something else (like rising or falling temperatures) could be causing sales of each to rise or fall at the same time.

17. The function V(t) = 1350(1.017)^t represents the value V(t), in dollars, of a comic book t years after its purchase. The yearly rate of appreciation of the comic book is

(4) 0.017%.
(2) 1.7%. This is a straight definition question. Keep this question in mind in PART 2 when you have to write a compound interest formula. It's a good example with numbers, as opposed to the formal definition in the reference table. Move the decimal two places over to get the percent.

18. When directed to solve a quadratic equation by completing the square, Sam arrived at the equation (x - 5/2)² = 13/4. Which equation could have been the original equation given to Sam?

(4) x² - 5x + 3 = 0. The term - 5/2 comes from being half of -5, so choices (1) and (2) are out. Squaring 5/2 yields 25/4, which had to be added to both sides of the equation. Subtract 13/4 - 25/4 = -12/4 = -3. Which means that the original equation ended in + 3.

19. The distance a free falling object has traveled can be modeled by the equation d = 1/2 at², where a is acceleration due to gravity and t is the amount of time the object has fallen. What is t in terms of a and d?

(2) t = the square root of (2d / a). As shown, multiply both sides by 2, divide by a, and take the square root.

20. The table below shows the annual salaries for the 24 member of a professional sports team in terms of millions of dollars. [table omitted] The team signs an additional player to a contract worth 10 million dollars per year. Which statement about the median and the mean is true?

(3) Only the mean will increase. DO NOT CALCULATE THE MEAN!! Seriously. Look at the table. The mean must be between 0.5 and 7.2. The number 10 MUST be higher than the mean, so the new average will go up.

On the other hand, check the median. With 24 players, the median in the average of the 12th and 13th, both of which are 1.4, so the median is 1.4. When you add a 25th player, the median is just the 13th player, which is still 1.4. So, in this case, the median does not change.

21. A student is asked to solve the equation 4(3x - 1)² - 17 = 83
The student's solution to the problem starts as

4(3x - 1)² = 100
(3x - 1)² = 25

A correct next step in the solution of the problem is

(1) 3x - 1 = +5. Take the square root of both sides. You could also expand the binomial by squaring it, but the only choice to do this, (4), inexplicably rooted the other side of the equation. If (4) said 25 instead of 5, it would have been correct.

22. A pattern of blocks is shown below.

If the pattern continues, which formula(s) could be used to determine the number of blocks in the nth term?

(3) II and III. II shows the recursive formula and III shows the explicit formula. The first formula makes no sense -- it increases by 1 instead of 4.

23. What are the solutions to the equation x² - 8x = 24?

(1) x = 4 +2(10)^(.5) You have two choices. The easiest is to complete the square again. If you forgot how, then you can use the Quadratic formula, which is on the reference sheet. Obviously, from the choices, straight factoring is not an option.

24. Natasha is planning a school celebration and wants to have live music and food for everyone who attends. She has found a band that will charge her $750 and a caterer who will provide snacks and drinks for $2.25 per person. If her goal is to keep the average cost per person between $2.75 and $3.25, how many people, p, must attend?

(4) 750 < p < 1500. See the image below.

UPDATE: Part 2 is now online.

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