Regents Exams and Answers: Algebra II Revised Edition

REGENTS EXAMS AND ANSWERS

Algebra II

Revised Edition

Gary M. Rubinstein

B.A. Mathematics

Tufts University

M.S. Computer Science

University of Colorado

Table of Contents

Barron’s Regents Exams and Answers: Algebra II

Cover

Title Page

Copyright Information

Preface

How to Use This Book

Test-Taking Tips and Strategies

Some Key Algebra II Facts and Skills

1. Polynomial Expressions and Equations

2. Rational Expressions and Equations

3. Exponential and Logarithmic Expressions and Equations

4. Radical Expressions and Equations

5. Trigonometric Expressions and Equations

6. Systems of Equations

7. Functions

8. Sequences

9. Probability

10. Normal Distribution

11. Inferential Statistics

Glossary of Terms

Regents Examinations, Answers, and Self-Analysis Charts

June 2017 Exam

August 2017 Exam

June 2018 Exam

August 2018 Exam

June 2019 Exam

August 2019 Exam

Guide

Table of Contents

Copyright © 2021, 2020, 2018, 2017 by Kaplan, Inc., d/b/a Barron’s Educational Series

All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this eBook on screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.

Published by Kaplan, Inc., d/b/a Barron’s Educational Series

750 Third Avenue

New York, NY 10017

www.barronseduc.com

ISBN: 978-1-5062-7186-6

10 9 8 7 6 5 4 3 2 1

Kaplan, Inc., d/b/a Barron’s Educational Series print books are available at special quantity discounts to use for sales promotions, employee premiums, or educational purposes. For more information or to purchase books, please call the Simon & Schuster special sales department at 866-506-1949.

Preface

In 2009 New York State adopted the Common Core Standards in order to qualify for President Obama’s Race To The Top initiative. The Common Core math is more difficult than the previous math curriculum, and the new state tests, including the Common Core Algebra II Regents, are more difficult as well.

Certain topics that had been in the Algebra II curriculum for decades have been removed for not being rigorous enough. Other topics have been added with the goal of making 21st-century American students more career and college ready than their predecessors.

The main topics that have been cut from the curriculum are permutations, combinations, Bernoulli trials, binomial expansion, and the majority of the trigonometric identities like the sine sum, cosine sum, sine difference, cosine difference, and the different double and half angle formulas. Other topics have been moved into earlier grades like the Law of Sines and the Law of Cosines. More than half the Trigonometry that had once been part of Algebra II is no longer part of the course.

Other topics have been added to fill the gaps left by those topics now considered obsolete. Primarily these are topics often taught in AP statistics as part of inferential statistics.

Aside from the change in topics, there is a change in the style of questions. There is a lot more need now for students to think more deeply about the topics as questions are intentionally phrased in a less straightforward way than they had been in the past.

Conquering the Algebra II test, something that was never an easy feat beforehand, has gotten that much more difficult and will require more test preparation than before. Getting this book is a great first step toward that goal. In addition to a review of all the topics that can appear on this test, there are nearly 1,000 practice questions of various levels of difficulty. The book can serve as a review or even as a way to learn the material for the first time. Teachers can also use this book to guide their pacing and to focus on the types of questions that are most likely to appear on the test while spending less focus on complicated aspects of the Common Core curriculum that are unlikely to be on the test.

The Common Core is part of a grand plan that is intended to propel our country to the top of the international rankings in math and reading. Good luck. We are all counting on you!

Gary Rubinstein

Math Teacher

2016

How to Use This Book

This book is designed to help you get the most out of your review for the new Regents exam in Algebra II. Use this book to improve your understanding of the Algebra II topics and improve your grade.

TEST-TAKING TIPS

The first section in this book contains test-taking tips and strategies to help prepare you for the Algebra II Regents exam. This information is valuable, so be sure to read it carefully and refer to it during your study time. Remember: no single problem-solving strategy works for all problems—you should have a toolbox of strategies to pick from as you’re facing unfamiliar or difficult problems on the test.

PRACTICE WITH KEY ALGEBRA II FACTS AND SKILLS

The second section in this book provides you with key Algebra II facts, useful skills, and practice problems with solutions. It provides you with a quick and easy way to refresh the skills you learned in class.

REGENTS EXAMS AND ANSWERS

The final section of the book contains actual Algebra II Regents. These exams and thorough answer explanations are probably the most useful tool for your review, as they let you know what’s most important. By answering the questions on these exams, you will be able to identify your strengths and weaknesses and then concentrate on the areas in which you may need more study.

Remember, the answer explanations in this book are more than just simple solutions to the problems—they contain facts and explanations that are crucial to success in the Algebra II course and on the Regents exam. Careful review of these answers will increase your chances of doing well.

SELF-ANALYSIS CHARTS

Each of the Algebra II Regents exams ends with a Self-Analysis Chart. This chart will further help you identify weaknesses and direct your study efforts where needed. In addition, the chart classifies the questions on each exam into an organized set of topic groups. This feature will also help you to locate other questions on the same topic in the other Algebra II exam.

IMPORTANT TERMS TO KNOW

The terms that are listed in the glossary are the ones that have appeared most frequently on past Algebra II exams. All terms and their definitions are conveniently organized for a quick reference.

Test-Taking Tips and Strategies

Knowing the material is only part of the battle in acing the new Algebra II Regents exam. Things like improper management of time, careless errors, and struggling with the calculator can cost valuable points. This section contains some test-taking strategies to help you perform your best on test day.

Tip 1

Time Management

SUGGESTIONS

Don’t rush. The Algebra II Regents exam is three hours long. While you are officially allowed to leave after 90 minutes, you really should stay until the end of the exam. Just as it wouldn’t be wise to come to the test an hour late, it is almost as bad to leave a test an hour early.

Do the test twice. The best way to protect against careless errors is to do the entire test twice and compare the answers you got the first time to the answers you got the second time. For any answers that don’t agree, do a tie breaker third attempt. Redoing the test and comparing answers is much more effective than simply looking over your work. Students tend to miss careless errors when looking over their work. By redoing the questions, you are less likely to make the same mistake.

Bring a watch. What will happen if the clock is broken? Without knowing how much time is left, you might rush and make careless errors. Yes, the proctor will probably write the time elapsed on the board and update it every so often, but it’s better to be safe than sorry.

The TI-84 graphing calculator has a built in clock. Press the [MODE] to see it. If the time is not right, go to SET CLOCK and set it correctly. The TI-Nspire does not have a built-in clock.

Tip 2

Know How to Get Partial Credit

SUGGESTIONS

Know the structure of the exam. The Algebra Regents exam has 37 questions. The first 24 of those questions are multiple-choice worth two points each. There is no partial credit if you make a mistake on one of those questions. Even the smallest careless error, like missing a negative sign, will result in no credit for that question. Parts II, III, and IV are free-response questions with no multiple-choice. Besides giving a numerical answer, you may be asked to explain your reasoning. Part II has eight free-response questions worth two points each. The smallest careless error will cause you to lose one point, which is half the value of the question. Part III has four free-response questions worth four points each. These questions generally have multiple parts. Part IV has one free-response question worth six points. This question will have multiple parts.

Explain your reasoning. When a free-response question asks you to Justify your answer, Explain your answer, or Explain how you determined your answer, the grader is expecting a few clearly written sentences. For these, you don’t want to write too little since the grader needs to see that you understand why you did the different steps you did to solve the equation. You also don’t want to write too much because if anything you write is not accurate, points can be deducted.

Here is an example followed by two solutions. The first would not get full credit, but the second would.

Example 1

Use algebra to solve for x in the equation Justify your steps.

Solution 1 (partial credit):

Solution 2 (full credit):

Computational errors vs. conceptual errors

In the Part III and Part IV questions, the graders are instructed to take off one point for a computational error but half credit for a conceptual error. This is the difference between these two types of errors.

If a four-point question was

x – 1 = 2

and a student did it like this,

the student would lose one point out of 4 because there was one computational error since 2 + 1 = 3 and not 4.

Had the student done it like this,

the student would lose half credit, or 2 points, since this error was conceptual. The student thought that to eliminate the –1, he should subtract 1 from both sides of the equation.

Either error might just be careless, but the conceptual error is the one that gets the harsher deduction.

Tip 3

Know Your Calculator

SUGGESTIONS

Which calculator should you use? The two calculators used for this book are the TI-84 and the TI-Nspire. Both are very powerful. The TI-84 is somewhat easier to use for the functions needed for this test. The TI-Nspire has more features for courses in the future. The choice is up to you. This author prefers the TI-84 for the Algebra Regents. Graphing calculators come with manuals that are as thick as the book you are holding. There are also plenty of video tutorials online for learning how to use advanced features of the calculator. To become an expert user, watch the online tutorials or read the manual.

Clearing the memory. You may be asked at the beginning of the test to clear the memory of your calculator. When practicing for the test, you should clear the memory too so you are practicing under test-taking conditions.

This is how you clear the memory.

For the TI-84:

Press [2ND] and then [+] to get to the MEMORY menu. Then press [7] for Reset.

Use the arrows to go to [ALL] for All Memory. Then press [1].

Press [2] for Reset.

The calculator will be reset as if in brand new condition. The one setting that you may need to change is to turn the diagnostics on if you need to calculate the correlation coefficient.

For the TI-Nspire:

The TI-Nspire must be set to Press-to-Test mode when taking the Algebra Regents. Turn the calculator off by pressing [ctrl] and [home]. Press and hold [esc] and then press [home].

While in Press-to-Test mode, certain features will be deactivated. A small green light will blink on the calculator so a proctor can verify the calculator is in Press-to-Test mode.

To exit Press-to-Test mode, use a USB cable to connect the calculator to another TI-Nspire. Then from the home screen on the calculator in Press-to-Test mode, press [doc], [9], and select Exit Press-to-Test.

Use parentheses

The calculator always uses the order of operations where multiplication and division happen before addition and subtraction. Sometimes, though, you may want the calculator to do the operations in a different order.

Suppose at the end of a quadratic equation, you have to round to the nearest hundredth. If you enter (–) (1) (+) (2ND) (x^2) (5) (/) (2), it displays

which is not the correct answer.

One reason is that for the TI-84 there needs to be a closing parenthesis (or on the TI-Nspire, press [right arrow] to move out from under the radical sign) after the 5 in the square root symbol. Without it, it calculated . More needs to be done, though, since

still is not correct. This is the solution to

To get this correct, there also needs to be parentheses around the entire numerator,

This is the correct answer.

On the TI-Nspire, fractions can also be done with [templates].

Using the ANS feature

The last number calculated with the calculator is stored in something called the ANS variable. This ANS variable will appear if you start an expression with a +, –, ×, or ÷. When an answer has a lot of digits in it, this saves time and is also more accurate.

If for some step in a problem you need to calculate the decimal equivalent of , it will look like this on the TI-84:

For the TI-Nspire, if you try the same thing, it leaves the answer as . To get the decimal approximation, press [ctrl] and [enter] instead of just [enter].

Now if you want to multiply this by 3, just press [×], and the calculator will display Ans*; press [3] and [enter].

The ANS variable can also help you do calculations in stages. To calculate without using so many parentheses as before, it can be done by first calculating and then pressing [÷] and [2] and Ans will appear automatically.

The ANS variable can also be accessed by pressing [2ND] and [–] at the bottom right of the calculator. If after calculating the decimal equivalent of you wanted to subtract from 5, for the TI-84 press [5], [–], [2ND], [ANS], and [ENTER]. For the TI-Nspire press [5], [–], [ctrl], [ans], and [enter].

Tip 4

Use the Reference Sheet

SUGGESTIONS

In the back of the Algebra II Regents booklet is a reference sheet that contains 17 conversion facts, such as inches to centimeters and quarts to pints, and also 17 formulas. Many of these conversion facts and formulas will not be needed for an individual test, but the quadratic formula and the arithmetic sequence formula are the two that will come in the handiest.

HOW MANY POINTS DO YOU NEED TO PASS?

The Algebra II Regents exam is scored out of a possible 86 points. Unlike most tests given in the year by your teacher, the score is not then turned into a percent out of 86. Instead each test has a conversion sheet that varies from year to year. For the June 2016 test, the conversion sheet looked like this.

On this test, 30 points became a 70, 57 points became an 82, and 73 points became a 90. This means that for this examination a student who got 30 out of 86, which is just 35% of the possible points, would get a 73 on this exam. 57 out of 86 is 66%, but this scaled to an 86. 64 out of 86, however, is actually 85% and became an 85. So in the past there has been a curve on the exam for lower scores, though the scaling is not released until after the exam.

1

Some Key Algebra II Facts and Skills

1. Polynomial Expressions and Equations

1.1 Polynomial Arithmetic

A polynomial is an expression that combines numbers and variables raised to powers. 3x² + 5x + 1 is an example of a polynomial.

The terms of this polynomial are 3x², 5x, and 1. Each term has a coefficient, which is the number in front of the term. In the term 3x², the coefficient is 3.

Since the highest exponent in this polynomial is 2, we say that the degree of this polynomial is 2. Polynomials that have a degree of 2