*The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the*

**Comments**section.Part II was posted here. Part III was posted here. Part IV was posted here.

### January 2017, Algebra I (Common Core), Part I

*This question was worth 6 credits*

**1.***Which expression is equivalent to 16x ^{2} - 36?*

(2) 4(2x + 3)(2x - 3). This question was a giveaway. It the **Difference of Squares** which can be factored into **Conjugates**, that is, the same **binomial** except that one has a *plus* and the other a *minus*. Only one choice has that!

Additionally, if you forgot how to factor, you could just multiply the choices and see which gives you the original expression.

You could factor this by dividing by 4, getting 4(4x^{2} - 9) which becomes 4(2x + 3)(2x - 3).

Or you could have factored it into (4x + 6)(4x - 6), each of which could have a factor of 2 taken out of them.

**2.***What is the solution set of the equation (x - 2)(x - a) = 0?
*

(3) 2 and a. Flip the signs: 2 - 2 = 0 or *a* - *a* = 0. Don't let the *a* throw you off.

**3.***Analysis of data from a statistical study shows a linear relationship in the data with a correlation coefficient of -0.524. Which statement best summarizes this result?
*

(4) There is a moderate negative correlation between the variables. Hopefully, you immediately eliminated the "positive" choices. A coefficient of -0.5 (or +0.5) can best be described as "*moderate*", as opposed to "*strong*" or "*weak*".

**4.***Boyle's Law involves the pressure and volume of gas in a container. It can be represented by the formula
P _{1}V_{1} = P_{2}V_{2}. When the formula is solved for P_{2}, the result is
*

(3) P_{1}V_{1} / V_{2}. Divide both sides by V2 to isolate P2

**5.***A radio station did a survey to determine what kind of music
to play by taking a sample of middle school, high school, and
college students. They were asked which of three different types of
music they prefer on the radio: hip-hop, alternative, or classic rock
The results are summarized in the table below.
*

Hip-Hop | Alternative | Classic Rock | |

Middle School | 28 | 18 | 4 |

High School | 22 | 22 | 6 |

College | 16 | 20 | 14 |

*What percentage of college students prefer classic rock?*

(2) 28%. The first two rows are irrelevant. Only the college row is important. There were a total of 50 college students surveyed and only 14 prefer classic rock. That's 28%.

**6.***Which function has zeros of -4 and 2?*

(4) The graph shows the function crossing the x-axis at -4 and 2. Not a trick question.

The equation for the function would involve multiplying the factors (x + 4)(x - 2), which would have a middle term of 2x, not 7x nor -7x.

Likewise, if the zeros are -4 and 2, then the **Axis of Symmetry** must be -1, which is exactly in the middle. If you use the formula
x = (-b/2a), it is obvious that b cannot be either 7 or -7.

**7.***Which expression is equivalent to 2(3g - 4) - (8g + 3)? *

(4) -2g - 11.

6g - 8 - 8g - 3

-2g - 11

**8.***In 2014, the cost to mail a letter was 49¢ for up to one ounce. Every additional ounce cost 21¢. Which recursive function could be used to determine the cost of a 3-ounce letter, in cents?
*

(1) a_{1} = 49; a_{n} = a_{n-1} + 21. The first ounce is 49 cents, and each additional is 21.

**9.***A car leaves Albany, NY, and travels west toward Buffalo, NY. The equation D = 280 - 59t can be used to represent the distance, D, from Buffalo after t hours. In this equation, the 59 represents the
*

(2) speed of the car. Every hour, the car is 59 miles closer because it is traveling at 59 miles per hour.

**10.*** Faith wants to use the formula C(f) = (5/9) (f - 32) to convert degrees Fahrenheit, f, to degrees Celsius, C(f). If Faith calculated C(68), what would her result be?
*

(1) 20^{o} Celsius. First of all, you are converting from Fahrenheit to Celsius, so choices (2) and (4) are right out. Second, when it comes to regular temperatures (not extreme cold), Celsius are lower than Fahrenheit. (They are the same at *40 below zero*!) So you didn't even have to do the math to answer this. If you've been to the Caribbean in the winter, it's been in the 20s there and it's beach weather!

If you didn't realize that, there's the mathematical answer:

C(68) = (5/9)(68 - 32) = (5/9)(36) = (180)/(9) = 20

**11.***Which scenario represents exponential growth? *

(3) A species of fly doubles its population every month during the summer. Doubling is exponential. The rest are linear.

**12.***What is the minimum value of the function y = |x+ 3| - 2? *

(1) -2. The vertex of this **absolute value** graph is (-3, -2). The minimum value is -2.

**13.***What type of relationship exists between the number of pages printed
on a printer and the amount of ink used by that printer? *

(2) positive correlation, and causal. The more pages printed, the more ink is used. And one causes the other.

And non-causal positive relationship would be something like "ice cream sales go up and swimsuit sales go up". Both go up in the warmer weather.

**14.***A computer application generates a sequence of musical notes using the function f(n) = 6(16) ^{n}, where
n is the number of the note in the sequence and f(n) is the note frequency in hertz. Which function will
generate the same note sequence as f(n)? *

(2) h(n) = 6(2)^{4n}. Forget everything in this question about music. You don't need to know it.

What you need to know is that *6(16) ^{n} = 6(2)^{4n}*.

16 is the same as 2

^{4}

So (16)

^{n}is the same as (2

^{4})

^{n}

Rules of exponents say that you multiply the 4 and the

*n*, which gives (2)

^{4n}.

**15.*** Which value of x is a solution to the equation 13 - 36x ^{2} = - 12?*

(4) -5/6

You can rewrite the equation as 36x^{2} - 25 = 0, which is a **Difference of Squares**.

That factors into **conjugates**: (6x + 5)(6x - 5). For either binomial to equal zero under the
**Zero Product Property**, x would have to equal *positive* or *negative* 5/6.

**16.***Which point is a solution to the system below?
*

*2y < -12x + 4*

y < -6x + 4

y < -6x + 4

(4) (-3,2)

Look for a point that fits both inequalities. You can plug the points into both equations, or you can graph the lines in your graphing calculator.

1/2 is NOT less than -6(1) + 4

6 is NOT less that -6(0) = 4

5 IS less than -6(-1/2) + 4, which is 7, BUT 2(5) = 10 is NOT less than -12(-1/2) + 4, which is also 10. (They're equal.)

Choice (4) is left. 2 IS less than (-6)(-3) + 4 which is 22 AND 2(2) = 4 IS less than (-12)(-3) + 4, which is 40.

**17.***When the function f(x) = x ^{2} is multiplied by the value a, where a > 1,
the graph of the new function, g(x) = ax^{2}
*

(2) opens upward and is narrower. Because a > 1, it is positive, so the parabola opens upward. When a > 1, the parabola shoots up faster, and as a result, it will be narrower.

**18.***Andy has $310 in his account. Each week, w, he withdraws $30 for
his expenses. Which expression could be used if he wanted to find
out how much money he had left after 8 weeks?
*

(4) 280 - 30(w - 1). This is a silly answer, but it is mathematically correct.

The answer you would expect is 310 - 30w, because 30 is subtracted each week.

First notice that only two choices have w multiplied by 30, and one of them is *adding* money.

After 1 week, 310 becomes 280. The number of weeks after that is reduced by 1, so *w - 1*.

**19.***The daily cost of production in a factory is calculated using
c(x) = 200 + 16x, where x is the number of complete products manufactured. Which set of numbers best defines the domain of
c(x)?
*

(4) whole numbers. Complete products would have to be positive integers or zero, which is the set of whole numbers. Negative numbers and fractions (real and rational numbers) wouldn't make sense.

**20.*** Noah conducted a survey on sports participation. He created
the following two dot plots to represent the number of students
participating, by age, in soccer and basketball .
*

*Which statement about the given data sets is correct?*

(4) The data for basketball players have a greater mean than the data
for soccer players.

Without doing the calculations, you can see that the data for the soccer players skew left and while the data for the basketball players skew right. So the mean and the median are both going to be higher for the basketball players. (You can check the median by counting the dots quickly enough if you don't believe me.)

**21.***A graph of average resting heart rates is shown below. The average
resting heart rate for adults is 72 beats per minute, but doctors
consider resting rates from 60-100 beats per minute within normal
range.
*

*Which statement about average resting heart rates is not supported by the graph?*

(1) A 10-year-old has the same average resting heart rate as a 20-year-old.

According to the graph, a 10-year-old's rate is higher (approximate 90 beats/min) than a 20-year-old (72 beats/min).

**22.***The method of completing the square was used to solve the equation
2x ^{2} - 12x + 6 = 0. Which equation is a correct step when using this
method? *

(1) (x - 3)^{2} = 6

^{2}- 12x + 6 = 0

x

^{2}- 6x + 3 = 0

x

^{2}- 6x + 3 + 6 = 0 + 6

x

^{2}- 6x + 9 = 6

(x - 3)

^{2}= 6

To complete the square, first divide all the terms by 2 to get rid of the leading coefficient. Next, look at the middle term, which in this case is now 6. Half of 6 is 3, and 3 squared is 9. We need to have 9 as the constant to have a *perfect square*. To get 9, we have to add 6 to both sides of the equation. (I sometimes teach students to *subtract 3* and then *add 9* if they can't figure out the square that they need. It's an extra step, but they can get the correct solution, so it's worth it.)

Once we have *x ^{2} - 6x + 9*, we can factor that into (x - 3)(x - 3) or simply (x - 3)

^{2}, which is the square we wanted to complete.

*Hint:*Once we knew that half of 6 is 3, we also knew that (x - 3)

^{2}would be in our answer.

**23.***Nancy works for a company that offers two types of savings plans.
Plan A is represented on the graph below.
*

*Plan B is represented by the function f(x) = 0.01 + 0.05x*

Her decision is

^{2}, where x is the number of weeks. Nancy wants to have the highest savings possible after a year. Nancy picks Plan B.Her decision is

correct, because Plan B is a quadratic function and will increase at a faster rate .

First, *x ^{2}* is

**quadratic**, not

**exponential**. Second, check to make sure that choice (3), "Plan A will have a higher value after 1 year" is incorrect. Although the function will increase at a faster rate, it starts off with a much lower rate. Check that it will catch up.

f(52) = 0.01 + 0.05(52)^{2} = 135.21. On the graph, Plan A gives an amount between $100 and $110 at 52 weeks.

**24.*** The 2014 winner of the Boston Marathon runs as many as 120 miles
per week. During the last few weeks of his training for an event, his mileage can be modeled by M(w) = 120(.90) ^{w-l}, where w represents the number of weeks since training began. Which statement is true about the model M(w)?
*

(3) M (w) represents the total mileage run in a given week.

The number of miles is decreasing by 10% per week, or is 90% of the previous week. This may sound counterproductive in training if you don't know that a marathon is only about 26 miles. It is stated in the question that *w* is the number of weeks since training began and not the number of weeks left until the marathon.

End of **Part I**

How did you do?

Comments, questions, corrections and concerns are all welcome.

Typos happen.

## 2 comments:

I disagree with problem 20. The basketball distribution clearly skews left. The soccer distribution does have a slight skew to the right (positive). Although answer 4 is clearly a correct statement, answer 1 may also be justified and should have been given full credit.

^ #20 is actually correct

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