The following are some of the multiple questions from the recent January 2018 New York State Common Core Algebra I Regents exam.

### June 2018 Algebra I, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.

**1.** *The solution to 4p + 2 < 2(p + 5) is
*

**Answer: (4) p < 4 **

Distributive property and inverse operations.

4p + 2 < 2(p + 5)

4p + 2 < 2p + 10

2p + 2 < 10

2p < 8

p < 4

There is no multiplication or division by a negative, so there is no need to flip the inequality symbol.

**2.** *If k(x) = 2x*^{2} - 3*sqrt(x), the k(9) is

**Answer: (4) 153**

Substitution and Order of Operations.

2(9)^{2} - 3(9)^(.5) = 2(81) - 3(3) = 162 - 9 = 153

**3.** *The expression 3(x*^{2} + 2x - 3) - 4(4x^{2} - 7x + 5) is equivalent to

**Answer: (2) -13x**^{2} + 34x - 29

Distributive property (including distributing a minus sign) and Combining Like Terms.

3(x^{2} + 2x - 3) - 4(4x^{2} - 7x + 5)

3x^{2} + 6x - 9 - 16x^{2} + 28x - 20

-13x^{2} + 34x - 29

Once you got -13, you could have eliminated choices 3 and 4. If you didn't get either choice 1 or 2, go back and check your signs.

**4.** *The zeros of the function p(x) = x*^{2} - 2x - 24 are

**Answer: (3) -4 and 6**

Factoring to find zeros / roots / x-intercepts. What two factors of -24 have a sum of -2?

x^{2} - 2x - 24 = 0

(x - 6)(x + 4) = 0

x - 6 = 0 or x + 4 = 0

x = 6 or x = -4

I hope you didn't jump the gun after factoring and answer the question without solving for x, which flipped the signs.

**5.** *The box plot below summarizes the data for the average monthly high temperatures in degrees Fahrenheit for Orlando, Florida.
*

*The third quartile is
*

**Answer: (2) 90**

Box-and-whisker plots. Five-Number Summary.

The five-number summary for the plot shown are: Minimum is approximately 71. Q1 is 75. Median is approximately 83. Q3 is 90. Maximum is approximately 92.

"Approximately" because those numbers aren't labeled, but they can be inferred from the choices.

Not that the incorrect choices line up with another key portion of the plot.ut solving for x, which flipped the signs.

**6.** *Joy wants to buy strawberries and raspberries to bring to a party. Strawberries cost $1.60 per pound and raspberries cost $1.75 per pound. If she only has $10 to spend on berries, which inequality represents the situation where she buys x pounds of strawberries and y pounds of raspberries?
*

**Answer: (1) 1.60x + 1.75y **__<__ 10

Modeling inequalities

If x in the number of pounds of strawberries, which cost $1.60 per pound, then the first term is *1.60x*. Eliminate choices 3 and 4.

She can't spend *more than* 10 dollars but and spend *less than* or *exactly* 10 dollars. So choice 1.

**7.** *On the main floor of the Kodak Hall at the Eastman Theater, the number of seats per row increases at a constant rate. Steven counts 31 seats in row 3 and 37 seats in row 6. How many seats are there in row 20?
*

**Answer: (1) 65 **

Sequences. Rate of Change.

If there are 6 more seats (37 - 31) when you go back 3 rows (6 - 3), then there is a rate of change of 2 seats per row.

If you go back another 14 rows (20 - 6), then there should be an additional 28 seats (14 * 2).

37 + 28 = 65 seats.

If x in the number of pounds of strawberries, which cost $1.60 per pound, then the first term is *1.60x*. Eliminate choices 3 and 4.

She can't spend *more than* 10 dollars but and spend *less than* or *exactly* 10 dollars. So choice 1.

**8.** *Which ordered pair below is not a solution to f(x) = x*^{2} - 3x + 4?

**Answer: (4) (-1, 6) **

Graphing. Substitution. Evaluation.

Quickest way is to put the function into the graphing calculator and check the table of values. You will see that (-1, 8) is a solution, not (-1, 6).

If you change the settings, or use the Trace function, you will see that (1.5, 1.75) is a solution.

**9.** *Students were asked to name their favorite sport from a list of basketball, soccer or tennis. The results are in the table below:
*

What percentage of the students chose soccer as their favorite sport?

**Answer: (1) 39.6% **

Statistics. Two-way frequency tables. Marginal frequencies. Percentages.

Find the number of student who prefer soccer. Find the total number of students. Divide the first by the second and multiply by 100%.

58 + 41 = 99 students like soccer

There are 42 + 84 + 58 + 41 + 20 + 5 = 250 total students

99 / 250 = 0.396 = 39.6%

**10.** *The trinomial x*^{2} - 14x + 49 can be expressed as

**Answer: (1) (x - 7)**^{2}

Factoring. Perfect squares. Completing the squares

Even if you didn't recognize that this trinomial is a perfect square, you could have factored it quickly into (x - 7) and (x - 7), which is (x - 7)^{2}.

Incorrect choices: Choice 2 would give + 14x as the middle term. Choice 3 has two conjugates, so there would be NO middle term. Choice 4 is just silly: -7 times 2 is not +49.

**11.** *A function is definied as {(0,1), (2,3), (5,8), (7,2)}. Isaac is asked to create one more ordered pair for the function. Which ordered pair can be add(ed) to the set to keep it a function?
*

**Answer: (4) (1, 3) **

Functions. Relations.

You can't repeat the input (x) with a different output (y). Choices 1, 2, and 3 would cause the function to fail the Vertical-Line Test because they would duplicate x-values that already exist.

**12.** *The quadratic equation x*^{2} - 6x = 12 is rewritten in the form (x + p)^{2} = q, where q is a constant. What is the value of p?

**Answer: (3) -3 **

Quadratic functions. Parabolas. Minimum value. Vertex.

Two notes: first, "q is a constant" means that it will be some number, but we really don't care what that number will be; second, take note of the fact that there is a plus sign (+) in the rewritten form, not the usual minus sing (-). You don't have to "flip the sign" when reading your answer.

To complete the square, take half of -6, and square it. Add that to both sides.

x^{2} - 6x = 12

x^{2} - 6x + 9 = 12 + 9

(x - 3)^{2} = 21

The constant q is 21, but that isn't important. The value of p is -3.

You can check by graphing that these two equations are equivalent.

Final note: most of the above was unnecessary. Once you found b/2, -6/2 = -3, you had the answer. The rest was just checking.

**13.** *Which of the quadratic functions below has the smallest minimum value?
*

**Answer: (2) [graph] **

Quadratic functions. Parabolas. Minimum value. Vertex.

The table in Choice 4 has a minimum of -6, but the graph in Choice 2 has a minimum of -10, so choice 4 is eliminated.

If you graph h(x) and k(x), you will see that neither one has a minimum of less than -10.

You could also have found the axis of symmetry, and plug them into the function.

For choice 1, the axis of symmetry was x = -2 / 2 = -1, and h(-1) = (-1)^{2} + 2(-1) - 6 = -7

For choice 3, the axis of symmetry is (-5 + -2) / 2 = -3.5, and k(-3.5) = (-3.5 + 5)(-3.5 + 2) = (1.5)(-1.5) = -2.25

**14.** *Which situation is not a linear function?
*

**Answer: (4) A $12,000 car depreciates 15% per year.**

Linear functions have a constant rate of change.

Choices 1, 2, and 3 have constant amounts per month, per mile and per hour.

Choice 4 decreases by 15% per year. This is exponential decay. After one year, the value will be smaller, so 15% of that value will be a smaller decrease.

**15.** *The Utica Boilermaker is a 15-kilometer road race. Sara is signed up to run this race and has done the following trains runs:
*

*I. 10 miles *

II. 44,800 feet

III. 15,560 yards*
*

Which run(s) are at least 15 kilometers.
**Answer: (1) I, only**

Unit conversion.

From the back of the test booklet: 1 mile = 5280 feet, 1 mile = 1760 yards, 1 kilometer = 0.62 miles.

15 kilometers * (0.62 miles / kilometer) = 9.3 miles

10 miles > 9.3 miles

44,800 feet / (5,280 feet / mile )= 8.48 miles

15,560 yards / (1,760 yards / mile) = 8.8 miles.

Only 10 miles is at least 15 kilometers.

**16.** *If f(x) = x*^{2} + 2, which interval describes the range of this function?

**Answer: (3) [2, infinity)**

Domain and range.

Range is the set of possible y-values. The vertex of this function is (0, 2). The range is all values of y greater than or equal to 2, *y *__>__ 2, or [2, infinity).

**17.** *The amount Mike gets paid weekly can be represented by the expression 2.50a + 290, where a is the number of cell phone accessories he sells that week. What is the constant term in this expression and what does it represent?
*

**Answer: (3) 290, the amount he is guaranteed to be paid each week.**

Linear functions.

The initial value (y-intercept, when graphing) is 290, the constant term. The rate of change is 2.50, which repeats for every accessory sold.

**18.** *A cubic function is graphed on the set of axes below.
*

Which function could represent the graph?

**Answer: (2) g(x) = (x + 3)(x + 1)(x - 1)**

Zeroes of a function. Factored form.

The zeroes of the function are -3, -1 and 1. So the function should have the terms (x + 3)(x + 1)(x - 1).

**19.** *Mrs. Allard asked her students to identify which of the polynomials below are in standard form and explain why.
*

*I. 15x*^{4} - 6x + 3x^{2} - 1

II. 12x^{3} + 8x - 4

III. 2x^{5} + 8x^{2} + 10x*
*

Which student's repsonse is correct?
**Answer: (3) Fred said II and III because the exponents are decreasing**

The Standard form of a polynomial is the term with the highest exponent goes first, then the next highest exponent, and so on.

They are not ordered by coefficients.

**20.** *Which graph does not represent a function that is always increasing over the entire interval -2 < x < 2?
*

**Answer: (3) [graph]**

The function in Choice 3 is decreasing when 0 < x < 2, so it doesn't increase over the entire interval specified in the question.

Choice 4 does not start decreasing until after x > 2.

**21.** *At an ice cream shop, the profit, P(c), is modeled by the function P(c) = 0.87c, where c represents the number of ice cream cones sold. An appropriate domain for this function is *

**Answer: (2) an integer **__>__ 0

The domain should be an integer, not a rational number. Cones are sold as whole units. You wouldn't sell, for example, 3 1/2 cones.

**22.** *How many real-number solutions does 4x*^{2} + 2x + 5 have?

**Answer: (3) zero **__>__ 0

Find the discriminant: b^{2} - 4ac = (2)^{2} - 4(4)(5) = 4 - 80 = -76.

There are no real solutions.

You could also graph this function. You will see that it never touches the x-axis, so it has no solutions. (The minimum occurs at (-.25, 4.75).)

Note that the answer "Infinitely many" is silly. A quadratic equation can only have 0, 1, or 2 solutions.

The only time is could be infinitely many is if both sides of the equation are quadratic expressions which are equivalent.

**23.** *Students were asked to write a formula for the length of a rectangle by using the formula for its perimeter, p = 2L + 2W. Three of their responses are shown below.
*

*Which response are correct?
*

**Answer: (2) II and III, only**__>__ 0

To solve for L in terms of p and W, you need to use inverse operations to isolate L.

In this case, that means subtract 2w and then either divide by 2, or multiply by 1/2. So responses II and III are equivalent.

Response I isn't good because the 1/2 was only applied to the p term and not the W term.

**24.** *If a*_{n} = n(a_{n-1} and a_{1} = 1, what is the value of a_{5}?

**Answer: (2) II and III, only**__>__ 0

a_{1} = 1,

a_{2} = 2(a_{1}) = 2(1) = 2,

a_{3} = 3(a_{2}) = 3(2) = 6,

a_{4} = 4(a_{3}) = 4(6) = 24,

a_{5} = 5(a_{4}) = 5(24) = 120.

Give yourself a pat on the back if you realized that this was the factorial function.

**End of Part I**

How did you do?

Questions, comments and corrections welcome.