## Regents Exams: Mathematics – Algebra I Test Guide

The Board of Regents (BOR) adopted the Common Core State Standards in July 2010 and in January 2011, the Board adopted the NYS P-12 Common Core Learning Standards (CCLS), which include the Common Core State Standards and a small amount of additional standards uniquely added by New York State. The CCLS for Mathematics make up a broad set of mathematics understandings for students, defined through the integration of the Standards for Mathematical Content and the Standards for Mathematical Practice.

Any student who in the 2013-14 school year or thereafter, regardless of grade of enrollment, begins his or her first commencement-level math course culminating in a Regents Exam in June 2014 or later must take the New York State CCLS Regents Exam in mathematics that corresponds to that course, as available, and be provided with Common Core instruction. Most typically, this first course will be Algebra I (Common Core). In June 2014, the Regents Examination in Algebra I (Common Core) measuring the CCLS will be administered for the first time.

The Guide for the Regents Examination in Algebra I (Common Core) has been designed to support educators by providing an overview of the new test design. Information about how the assessment shifts has informed test development and how the CCLS will be measured on the new Regents Exam is specified. This test guide supplements the other Common Core Regents Exam implementation resources found on EngageNY.

### 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: 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*

II. 12x

III. 2x

^{4}- 6x + 3x^{2}- 1II. 12x

^{3}+ 8x - 4III. 2x

^{5}+ 8x^{2}+ 10x

Which student's repsonse is correct?

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 **

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: (4) I, II, and III ****Update:** Correction. I misread choice (1). There things happen. That's why I welcome corrections.

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: (3) 120 ****Update:** I had my answer for 23 pasted in the above slot. The work below was correct.

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.

## JMAP REGENTS BY COMMON CORE STATE STANDARD: Algebra I Regents Exam Questions by Common Core State Standard:

JMAP REGENTS BY COMMON CORE

STATE STANDARD: TOPIC

NY Algebra I Regents Exam Questions from Fall 2013 to August 2015 Sorted by CCSS: Topic

www.jmap.org

TABLE OF CONTENTS TOPIC CCSS: SUBTOPIC QUESTION NUMBER

NUMBERS, OPERATIONS AND PROPERTIES

N.RN.3: Classifying Numbers ...........................................................1-5 A.REI.1: Identifying Properties ............................................................ 6

GRAPHS AND STATISTICS

S.ID.5: Frequency Histograms and Tables ........................................... 7 S.ID.1: Box Plots .................................................................................. 8 S.ID.2-3: Central Tendency and Dispersion ....................................9-13 S.ID.6: Regression ......................................................................... 14-19 S.ID.6, 8: Correlation Coefficient and Residuals...........................20-25 S.ID.9: Analysis of Data ..................................................................... 26

RATE F.IF.6: Rate of Change................................................................... 27-32 N.Q.1: Conversions ............................................................................ 33

LINEAR EQUATIONS

A.SSE.1: Modeling Expressions ......................................................... 34 A.REI.3: Solving Linear Equations ............................................... 35-36 A.CED.1, 3: Modeling Linear Equations....................................... 37-40 A.SSE.1, F.LE.5, F.BF.1, A.CED.1-2: Modeling Linear Functions ............................................................ 41-50 A.CED.2, A.REI.10, F.IF.4: Graphing Linear Functions .............. 51-56 A.CED.4: Transforming Formulas ................................................ 57-61

INEQUALITIES A.REI.3: Solving Linear Inequalities ............................................62-66 A.CED.3: Modeling Linear Inequalities ........................................67-70 A.REI.12: Graphing Linear Inequalities ........................................ 71-72

ABSOLUTE VALUE F.IF.7: Graphing Absolute Value Functions ....................................... 73

QUADRATICS

A.SSE.3, A.REI.4: Solving Quadratics ......................................... 74-92 A.CED.1: Modeling Quadratics.......................................................... 93 A.CED.1: Geometric Applications of Quadratics ..........................94-99 A.REI.10, F.IF.4, 8, 9: Graphing Quadratic Functions .............. 100-107 A.REI.4: Using the Discriminant ...................................................... 108

POWERS

A.APR.1: Operations with Polynomials .................................... 109-115 A.SSE.2: Factoring Polynomials ............................................... 116-120 A.APR.3: Zeros of Polynomials ................................................ 121-127 F.IF.8: Evaluating Exponential Expressions ..................................... 128 A.SSE.1, A.CED.1, F.BF.1, F.LE.2, F.IF.8, F.LE.5: Modeling Exponential Functions ............................................... 129-136 A.SSE.3: Solving Exponential Equations ......................................... 137 F.LE.1-3: Comparing Linear and Exponential Functions .......... 138-143 F.LE.3: Comparing Quadratic and Exponential Functions ............... 144

RADICALS F.IF.7: Graphing Root Functions ............................................... 145-146

FUNCTIONS

F.IF.2: Functional Notation ....................................................... 147-149 F.IF.1: Defining Functions ........................................................ 150-153 F.IF.2, 5: Domain and Range ..................................................... 154-159 F.LE.1, 2: Families of Functions ............................................... 160-162 F.BF.3: Transformations with Functions and Relations ............ 163-168 F.IF.4: Relating Graphs to Events ............................................. 169-171 F.IF.7: Graphing Piecewise-Defined Functions ......................... 172-175 F.IF.7: Graphing Step Functions................................................ 176-177 F.IF.3, F.LE.2, F.BF.2: Sequences ............................................ 178-184

SYSTEMS

A.REI.5-6: Solving Linear Systems .......................................... 185-187 A.CED.2-3, A.REI.6: Modeling Linear Systems ....................... 188-194 A.CED.2: Graphing Linear Systems ................................................. 195 A.CED.3: Modeling Systems of Linear Inequalities ................. 196-198 A.REI.12: Graphing Systems of Linear Inequalities ................. 199-203 A.REI.7, 11: Quadratic-Linear Systems .................................... 204-207 A.REI.11: Nonlinear Systems ........................................................... 208

Algebra I Regents Exam Questions by Common Core State Standard: Topic www.jmap.org

1

Algebra I Regents Exam Questions by Common Core State Standard: Topic

NUMBERS, OPERATIONS AND PROPERTIES N.RN.3: CLASSIFYING NUMBERS

1 Given: L = 2

M = 3 3

N = 16

P = 9 Which expression results in a rational number? 1 L +M 2 M +N 3 N +P 4 P + L

2 Which statement is not always true? 1 The product of two irrational numbers is

irrational. 2 The product of two rational numbers is

rational. 3 The sum of two rational numbers is rational. 4 The sum of a rational number and an irrational

number is irrational.

3 Ms. Fox asked her class "Is the sum of 4.2 and 2 rational or irrational?" Patrick answered that the sum would be irrational. State whether Patrick is correct or incorrect. Justify your reasoning.

4 Which statement is not always true? 1 The sum of two rational numbers is rational. 2 The product of two irrational numbers is

rational. 3 The sum of a rational number and an irrational

number is irrational. 4 The product of a nonzero rational number and

an irrational number is irrational.

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

1 P = 1 3

and W = 1 6

2 P = 1 4

and W = 1 9

3 P = 1 6

and W = 1 10

4 P = 1 25

and W = 1 2

A.REI.1: IDENTIFYING PROPERTIES

6 When solving the equation 4(3x2 + 2) − 9 = 8x 2 + 7, Emily wrote 4(3x 2 + 2) = 8x2 + 16 as her first step. Which property justifies Emily's first step? 1 addition property of equality 2 commutative property of addition 3 multiplication property of equality 4 distributive property of multiplication over

addition

GRAPHS AND STATISTICS S.ID.5: FREQUENCY HISTOGRAMS AND TABLES

7 The school newspaper surveyed the student body for an article about club membership. The table below shows the number of students in each grade level who belong to one or more clubs.

If there are 180 students in ninth grade, what percentage of the ninth grade students belong to more than one club?

Algebra I Regents Exam Questions by Common Core State Standard: Topic www.jmap.org

2

S.ID.1: BOX PLOTS

8 Robin collected data on the number of hours she watched television on Sunday through Thursday nights for a period of 3 weeks. The data are shown in the table below.

Using an appropriate scale on the number line below, construct a box plot for the 15 values.

S.ID.2-3: CENTRAL TENDENCY AND DISPERSION

9 Christopher looked at his quiz scores shown below for the first and second semester of his Algebra class. Semester 1: 78, 91, 88, 83, 94 Semester 2: 91, 96, 80, 77, 88, 85, 92 Which statement about Christopher's performance is correct? 1 The interquartile range for semester 1 is greater

than the interquartile range for semester 2. 2 The median score for semester 1 is greater than

the median score for semester 2. 3 The mean score for semester 2 is greater than

the mean score for semester 1. 4 The third quartile for semester 2 is greater than

the third quartile for semester 1.

10 Isaiah collects data from two different companies, each with four employees. The results of the study, based on each worker’s age and salary, are listed in the tables below.

Which statement is true about these data? 1 The median salaries in both companies are

greater than $37,000. 2 The mean salary in company 1 is greater than

the mean salary in company 2. 3 The salary range in company 2 is greater than

the salary range in company 1. 4 The mean age of workers at company 1 is

greater than the mean age of workers at company 2.

Algebra I Regents Exam Questions by Common Core State Standard: Topic www.jmap.org

3

11 Corinne is planning a beach vacation in July and is analyzing the daily high temperatures for her potential destination. She would like to choose a destination with a high median temperature and a small interquartile range. She constructed box plots shown in the diagram below.

Which destination has a median temperature above 80 degrees and the smallest interquartile range? 1 Ocean Beach 2 Whispering Palms 3 Serene Shores 4 Pelican Beach

12 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 statements about the mean and standard deviation is true? 1 mean A < mean B

standard deviation A > standard deviation B 2 mean A > mean B

standard de

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## 1 ccss regents exam algebra

Happy. Is there peace in the family. Peace. The girl learns, gets to know herself.

NYS Algebra 1 [Common Core] August 2016 Regents Exam -- Part 1 #'s 1-12 ANSWERSYou can't do that. I'm married. - So what. Your husband fucked my wife and nothing.

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