Table of Contents. Unit 3: Rational and Radical Relationships. Answer Key...AK-1. Introduction... v

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2 These materials may not be reproduced for any purpose. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored, or recorded in any form without written permission from the publisher ISBN Copyright 014 J. Weston Walch, Publisher Portland, ME Printed in the United States of America WALCH EDUCATION

3 Table of Contents Introduction... v Unit 3: Rational and Radical Relationships Lesson 1: Operating with Rational Epressions U3-1 Lesson : Solving Rational and Radical Equations... U3-37 Lesson 3: Graphing Rational Functions... U3-84 Lesson 4: Graphing Radical Functions... U3-146 Lesson 5: Comparing Properties of Functions... U3-174 Answer Key...AK-1 iii Table of Contents

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5 Introduction Welcome to the CCGPS Advanced Algebra Student Resource Book. This book will help you learn how to use algebra, geometry, data analysis, and probability to solve problems. Each lesson builds on what you have already learned. As you participate in classroom activities and use this book, you will master important concepts that will help to prepare you for the EOCT and for other mathematics assessments and courses. This book is your resource as you work your way through the Advanced Algebra course. It includes eplanations of the concepts you will learn in class; math vocabulary and definitions; formulas and rules; and eercises so you can practice the math you are learning. Most of your assignments will come from your teacher, but this book will allow you to review what was covered in class, including terms, formulas, and procedures. In Unit 1: Inferences and Conclusions from Data, you will learn about summarizing and interpreting data and using the normal curve. You will eplore populations, random samples, and sampling methods, as well as surveys, eperiments, and observational studies. Finally, you will compare treatments and read reports. In Unit : Polynomial Functions, you will begin by eploring polynomial structures and operations with polynomials. Then you will go on to prove identities, graph polynomial functions, solve systems of equations with polynomials, and work with geometric series. In Unit 3: Rational and Radical Relationships, you will be introduced to operating with rational epressions. Then you will learn about solving rational and radical equations and graphing rational functions. You will solve and graph radical functions. Finally, you will compare properties of functions. In Unit 4: Eponential and Logarithmic Functions, you will start working with eponential functions and begin eploring logarithmic functions. Then you will solve eponential equations using logarithms. In Unit 5: Trigonometric Functions, you will begin by eploring radians and the unit circle. You will graph trigonometric functions, including sine and cosine functions, and use them to model periodic phenomena. Finally, you will learn about the Pythagorean Identity. v Introduction

6 In Unit 6: Mathematical Modeling, you will use mathematics to model equations and piecewise, step, and absolute value functions. Then, you will eplore constraint equations and inequalities. You will go on to model transformations of graphs and compare properties within and between functions. You will model operating on functions and the inverses of functions. Finally, you will learn about geometric modeling. Each lesson is made up of short sections that eplain important concepts, including some completed eamples. Each of these sections is followed by a few problems to help you practice what you have learned. The Words to Know section at the beginning of each lesson includes important terms introduced in that lesson. As you move through your Advanced Algebra course, you will become a more confident and skilled mathematician. We hope this book will serve as a useful resource as you learn. vi Introduction

7 UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions Common Core Georgia Performance Standards MCC9 1.A.APR.6 MCC9 1.A.APR.7 (+) Essential Questions 1. What does a rational epression look like?. How can a rational epression be rewritten into an equivalent form? 3. How is it possible to find the sum or difference of rational epressions? 4. How is it possible to find the product of rational epressions? 5. How is it possible to find the quotient of rational epressions? WORDS TO KNOW common denominator least common denominator (LCD) rational epression reciprocal a quantity that is a shared multiple of the denominators of two or more fractions the least common multiple of the denominators of two or more fractions an epression made of the ratio of two polynomials, in which a variable appears in the denominator a number that, when multiplied by the original number, has a product of 1 U3-1 Lesson 1: Operating with Rational Epressions

8 Recommended Resources Math.com. Quadratic Equation Solver. After entering the coefficients of a quadratic equation into this straightforward site, users can press the Solve button to generate appropriate roots. Detailed factors are not provided, and users must use the zeros to determine their own factors. Mathway. Math Problem Solver. This resource is an online symbolic algebraic manipulator, capable of simplifying epressions. It can be a useful tool for tech-savvy individuals. WolframAlpha. Simplify Rational Epressions. A powerful online symbolic manipulator, this resource can simplify comple rational epressions. It requires precise data entry to achieve valuable results. U3- Unit 3: Rational and Radical Relationships

9 Lesson 3.1.1: Adding and Subtracting Rational Epressions Introduction Epressions come in a variety of types, including rational epressions. A rational epression is a ratio of two polynomials, in which a variable appears in the 3 denominator; for eample, is a rational epression. Working with rational + 1 epressions can often be made easier by analyzing them to uncover more familiar (and sometimes less comple) structures within them. In this section, you will eplore some of those structures. Where one rational epression eists, another may as well. Etracting meaning from the contet may require these epressions to be combined in order to determine a sum or difference. Combining rational epressions through addition or subtraction is not comple, though it does demand attention to detail. Rewriting equivalent fractions can often simplify the solution process. Key Concepts Before adding or subtracting rational epressions, you must find a common denominator. A common denominator is a quantity that is a shared multiple of the denominators of two or more fractions. A common denominator can be determined by finding the product of the denominators. For eample, a common denominator of the rational 3 1 epression is found by multiplying the two denominators, + 1 and + : ( 1)( + ) Once a common denominator has been found, it can be used to write equivalent rational epressions for each term of the sum (or difference, if subtracting). Using common denominators, the same rational epression as before, , can be rewritten in an equivalent form: U3-3 Lesson 1: Operating with Rational Epressions

10 Epressed with a common denominator, the sum of rational epressions is the sum of the numerators: y y + +. Thus, we can rewrite the epressions z z z over a single denominator, as shown Epressed with a common denominator, the difference of rational epressions is the difference of the numerators: y y. z z z The least common denominator (LCD) is the least common multiple of the denominators of two or more fractions. In other words, it s the smallest possible common denominator. The LCD can be determined by finding the product of all the unique factors of the denominator. Our sample rational 3 1 epression, 1 + +, is equivalent to the rational epression The LCD is ( 1)( + ), or. ( 1)( + ) Recall that a rational epression cannot include a value in the denominator that causes it to equal 0, since 0 is undefined in the denominator. U3-4 Unit 3: Rational and Radical Relationships

11 Guided Practice Eample 1 Simplify the rational epression Identify any invalid values of the epression. The first term, 3, is composed entirely of constants; there are no 5 invalid values. With an in the denominator of the second term,, we have to ensure that 0 to keep this epression valid.. Find a common denominator. The two denominators are 5 and. Their product, 5, is a useful common denominator. 3. Rewrite each term of the epression using the new denominator. If the fraction s denominator is not the common denominator, then multiply each fraction s numerator and denominator by the missing factor of the common denominator. Remember that multiplication is commutative: that means 3 3, and Original epression Multiply 3 5 by and 5 by Simplify each term Multiply Rewrite the epression by adding the 5 numerators over the common denominator The rewritten epression is. 5 U3-5 Lesson 1: Operating with Rational Epressions

12 4. Check to see if the result can be written in a simpler form. The numerator does not have either 5 or as a factor; therefore, it cannot be written in any other format that will further simplify the result. Thus +, where 0, is the simplest way to rewrite the original epression, +. 5 Eample Simplify the rational epression Identify any invalid values of the epression. 5 With an in the denominator of the first term,, we have to + 1 ensure that in order to keep this epression valid. Set the denominator equal to 0 and solve this resulting equation for to determine values of that make the epression invalid An -value of 1 invalidates the epression, so 1.. Find a common denominator. The second term, 3, does not have an apparent denominator. Yet any number can be rewritten as a ratio over 1; therefore, the second term is equivalent to 3. Thus, if we rewrite the rational epression as , it can be seen that the two denominators are + 1 and Their product, + 1, is a useful common denominator. U3-6 Unit 3: Rational and Radical Relationships

13 3. Rewrite each term of the epression using the new denominator. The denominator of the first term is already + 1, so it doesn t need to be multiplied by anything Rewritten epression from the previous step Multiply 3 1 by ( + 1) ( 1) The rewritten epression is Simplify each term. Rewrite the epression by adding the numerators over the common denominator. Simplify Check to see if the result can be written in a simpler form. There are no common factors of the numerator or denominator. This epression cannot be written in any other format that will simplify the result. Thus, where 1, is the simplest way to rewrite the original epression, U3-7 Lesson 1: Operating with Rational Epressions

14 Eample 3 5 Simplify the rational epression Identify any invalid values of the epression. In the first rational term, 0. Taking the square root of each side results in 0. This same value also makes the other two denominators invalid. We only need to avoid this one value, so the domain of the epression is 0.. Find a common denominator. As in previous eamples, we can determine a common denominator by finding the product of the denominators. That gives us This would work just fine (it is most certainly a common denominator), however, it does seem large. We can likely determine a smaller common denominator. Each of the three denominators in the original epression has as a factor, so we only need to include it once. The first term has an additional factor of, and the last term has an additional term of 4. The product of these three values the common, another from the first term, and the 4 from the last term is 4. This is a smaller and perhaps easier to handle denominator. Thus, we can use 4 as the common denominator. U3-8 Unit 3: Rational and Radical Relationships

15 3. Rewrite each term of the epression using the new denominator Original epression Multiply + + by 4 4, 3 4 by 4, 4 1 and 4 by Simplify each term Rewrite the epression by adding the numerators over the 4 common denominator Combine like terms The rewritten epression is Check to see if the result can be written in a simpler form. There are no common factors of the numerator or denominator. This epression cannot be written in any other format that will simplify the result. Therefore,, where 0, is the simplest way to rewrite 4 the original epression. You can use this strategy to combine many different rational epressions, including those that appear to be more comple. The epressions may involve more terms, or perhaps subtraction here and there. Yet the process outlined in this eample remains the same. Carefully following these same steps will yield successful results. U3-9 Lesson 1: Operating with Rational Epressions

16 Eample 4 Simplify the rational epression Identify any invalid values of the epression. The first rational term has a denominator of + 1. Set the denominator equal to 0 and then solve the resulting equation for to determine values that make this epression invalid An -value of invalidates the epression, so. The second rational term has a denominator of 3. Again, set the denominator equal to 0 and then solve the resulting equation for to determine values that make this epression invalid An -value of 3 invalidates the epression, so The rational epression is only valid when 1 and 3.. Find a common denominator. The two denominators are + 1 and 3. They share no factors, so their product will be a good common denominator. ( + 1)( 3) ( 3) + 1( 3) The common denominator is 5 3. U3-10 Unit 3: Rational and Radical Relationships

17 3. Rewrite each term of the epression using the new denominator ( 3) 8( 1) ( 3)(+ 1) + + (+ 1)( 3) (16 + 8) Original epression 3 Multiply + 1 by and 3 by Simplify each term. Multiply. Rewrite the epression by adding the numerators over the common denominator. Combine like terms. 4. Check to see if the result can be written in a simpler form. We already know the denominator has 3 and + 1 as factors. The numerator, however, has no integer factors. This epression cannot be written in any other format that will simplify the result Thus 5 3, where 1 and 3, is the simplest way to 3 8 rewrite the original epression, U3-11 Lesson 1: Operating with Rational Epressions

18 Eample 5 Simplify the rational epression Identify any invalid values of the epression. There are three terms in the epression and they each have a denominator. Because it appears to be the most comple, start with the second denominator, 3. This epression is quadratic, so try to factor it. 3 ( 3)( + 1) The factors are ( 3) and ( + 1). Set each of these equal to 0 and solve for to determine values that make the epression invalid An -value of 3 invalidates the epression, as does an -value of 1. Therefore, we can now more clearly see that 3 and 1 for the second term s denominator. That takes care of the third term s denominator, + 1, since it is one of the factors of the second denominator; in other words, 1 is likewise a restriction of the third term in the rational epression. We can verify this by setting the denominator of the third term equal to 0 and solving for Thus, an -value of 1 invalidates the epression, and 1 for the third term s denominator. Now we need to determine invalid values for the first term s denominator, 3. Because 3 ( 3), we can rewrite the first term as follows Now we can see that the first term s denominator, rewritten as 3, is a factor of the second denominator as well. That means that 3 and 1 for all three terms of the epression. U3-1 Unit 3: Rational and Radical Relationships

19 . Find a common denominator. Because the second term s denominator, 3, has the denominators of the first term and third term as factors, the second term s denominator is our common denominator. 3. Rewrite each term of the epression using the new denominator. The middle term needs no adjustment, since it already includes the common denominator ( 1) ( 3) + ( 3)( + 1) + 3 ( + 1)( 3) ( 1 1) (10 6) (4 1) Original epression 1 Multiply 3 by and + 1 by 3 3. Simplify each term. Multiply. Rewrite the epression by adding the numerators over the common denominator. Simplify. Combine like terms. U3-13 Lesson 1: Operating with Rational Epressions

20 4. Check to see if the result can be written in a simpler form. We already know the denominator has 3 and + 1 as factors. The numerator can be factored, because both terms include a factor of 5. Factor the epression to simplify it even further Epression from the previous step 5( + 1) ( + 1)( 3) Factor the numerator and denominator. 5( + 1) ( + 1)( 3) Cancel out common factors. 5 3 Simplify. The factor + 1 cancels out of the numerator and the denominator. Therefore, the simplest way to rewrite the epression is 5, where 3. 3 U3-14 Unit 3: Rational and Radical Relationships

21 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions Practice 3.1.1: Adding and Subtracting Rational Epressions For problems 1 7, simplify each rational epression. Whenever possible, reduce the epression to its lowest terms. State any restrictions on continued U3-15 Lesson 1: Operating with Rational Epressions

22 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions Use your knowledge of rational epressions to complete problems State any restrictions on. 8. From a package containing 1 cookies, Jodi s class removes ( + ) cookies for 3 quality control eperiments. What epression represents the remaining cookies? A triangle has three sides described by the rational epressions 1,, and, with all units in centimeters. What simplified epression + 1 describes the perimeter of this triangle? Colby climbs a steep cliff that rises meters above the beach. On 5 the way down, he stops to rest at a ledge that is meters below the summit. 6 What simplified rational epression represents his height above the beach when he is resting on the ledge? U3-16 Unit 3: Rational and Radical Relationships

23 Lesson 3.1.: Multiplying Rational Epressions Introduction Like with multiplying fractions, finding the product of rational epressions requires that the values in the numerator be handled separately from the values in the denominator. As the epressions grow more comple, manipulating these values is similar to steps taken previously when working with polynomials. Looking for possible invalid values and opportunities to simplify the epression can make the task less comple. Key Concepts Multiplication of rational epressions does not require a common denominator. The product of a collection of rational epressions is the product of the numerators divided by the product of the denominators. Written symbolically, a( ) b c a b c p( ) z( ) ( ) y( ) ( ) ( ) ( ) ( ). w( ) z( ) y( ) w( ) As previously encountered, any values that make the denominator equal to 0 must be identified and removed from the domain. For eample, the value 1 must be ecluded from the rational epression, since that value makes 1 (1) the rational epression equivalent to 1, an undefined form. (1) 1 0 Similar to fractions, rational epressions can (and often should) be reduced by eliminating common factors from both the numerator and denominator. U3-17 Lesson 1: Operating with Rational Epressions

24 Guided Practice 3.1. Eample 1 3 Simplify the rational epression Identify any invalid values of the epression. The denominator of the first term, + 1, cannot equal 0. Set this epression equal to 0 and then solve for to determine values that make this epression invalid An -value of 1 invalidates the epression, so 1.. Rewrite the second term as a fraction. The number 8 can be rewritten in equivalent form as Multiply the epressions to form a single rational epression ( + 1) Check for any factors that might make it possible to further simplify the resulting epression. The individual terms from the original epression are both simple. There is no common factor. 3 The final result of is when U3-18 Unit 3: Rational and Radical Relationships

25 Eample Simplify the rational epression Identify any invalid values of the epression. The denominator of the first term,, cannot equal 0. Therefore, 0.. Rewrite the second term as a rational epression. 5 The epression 5 can be rewritten as Multiply the epressions to form a single rational epression ( 3) Check for any factors that might make it possible to further simplify the resulting epression. The numerator shares a factor,, with the denominator (you might have noticed this in the last step, and adapted then). Factored, the result looks like this: The final result of 3 5 ( ) 5 15 is 5 15 when 0. U3-19 Lesson 1: Operating with Rational Epressions

26 Eample Simplify the rational epression. 1. Identify any invalid values of the epression. Both epressions are invalid when 0. Therefore, 0.. Multiply the epressions to form a single rational epression ( + 1) Check for any common factors that might make it possible to further simplify the resulting epression. There is no common factor. Though the numerator and denominator both have an term, is not a common factor; we cannot factor out of The final result of is 3 when 0. Eample 4 Simplify the rational epression Identify any invalid values of the epression. The denominator of the first term cannot equal 0. That means 0. In the second term, 1 0, which means 1. Therefore, 0 and 1. U3-0 Unit 3: Rational and Radical Relationships

27 . Multiply the epressions to form a single rational epression ( 1)( 3) ( 1) 3. Check for any factors that might make it possible to further simplify the resulting epression. There are no common factors. The final result of is and when 0 3 Eample 5 Simplify the rational epression Identify any invalid values of the epression. The denominator of the first term means + 1 0, or 1. The second term is a quadratic, and must be factored to determine invalid values for ( + 1)( + 8) The factors are ( + 1) and ( + 8). Set each of these equal to 0 and solve for to determine values that make the epression invalid An -value of 1 invalidates the epression, as does an -value of 8. Therefore, for the second denominator, 1 and 8. Notice that 1 is also a restriction on the denominator of the first term. Thus, for the entire rational epression, 1 and 8. U3-1 Lesson 1: Operating with Rational Epressions

28 . Factor the numerator of the first term. Since the numerator of the term 4 5 appears to be quadratic, + 1 factoring this numerator may aid in simplifying the epression. The numerator of the first term is ( 5)( + 1) Notice that ( + 1) is one of the factors of the denominator of the second term. 3. Using the previously determined factors, multiply the epressions to form a single rational epression. Replace the quadratic in each term of the original epression with its factored form ( 5)( 1) ( + 1)( + 8) Multiply the rewritten terms. ( 5)( + 1) + 8 ( 5)( 1)( 8) + 1 ( + 1)( + 8) + + ( + 1)( + 1)( + 8) 4. Simplify the rational epression. Cancel out common factors. ( 5)( + 1)( + 8) ( 5)( + 1) ( + 8) ( + 1)( + 1)( + 8) ( + 1)( + 1) ( + 8) The final result of 1 and is when U3- Unit 3: Rational and Radical Relationships

29 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions Practice 3.1.: Multiplying Rational Epressions For problems 1 5, simplify each rational epression. Whenever possible, reduce the epression to its lowest terms. If necessary, factor the terms in each epression before multiplying. State any restrictions on (+ 5) continued U3-3 Lesson 1: Operating with Rational Epressions

30 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions Use your knowledge of rational epressions and factoring to complete problems State any restrictions on. 6. For what values of is 3 1 an invalid epression? Show that the product of the epressions 9+ 3 the epression + 3. and is equivalent to 8. The net force applied to an object is the product of the object s mass and its acceleration. If the epression 1 describes the mass of a plastic ball in kilograms and the acceleration of the ball is meters per second, what simplified epression represents the net force in Newtons (N) applied to the ball? 9. A triangle has a height of centimeters and a base of 3 centimeters. The area of a triangle is equal to 1 the product of its base and its height. What simplified epression represents the area of this triangle? 10. A rectangular bo has a length of meters, a width of + 4 meters, and a height of meters. What simplified rational epression represents the 16 volume of the bo? U3-4 Unit 3: Rational and Radical Relationships

31 Lesson 3.1.3: Dividing Rational Epressions Introduction Previously, eplorations of ways to recognize and simplify rational epressions have been used to add, subtract, and multiply those epressions. In this section, these skills will be etended to eplore the results of the operation of division on rational epressions. As in previous eamples, familiarity with numerical fractions will be useful in working with rational epressions. Dividing rational epressions will also be made easier using a basic fact learned in dealing with fractions; namely, that division is the equivalent to multiplying by the reciprocal of the divisor. Recall that a reciprocal is a number that, when multiplied by the original number, has a product of 1. For eample, dividing 1 by 3 is the same as multiplying 1 by the reciprocal of 3, which is 1 3. We can verify that 3 and 1 3 are reciprocals by showing that Key Concepts The division epression a( ) b( ) is equivalent to a ( ) 1 b ( ) : the product of the divisor and the reciprocal of the dividend. A rational epression of the form a ( ) c( ) is equivalent to the rational b( ) d( ) a( ) epression ( ) 1 a( ) ( ) ( ) d ( ). b c b c( ) d( ) This epression is the product of the divisor and the reciprocal of the dividend. A rational epression of the form a ( ) is equivalent to the epression a( ) b( ). b( ) If b() is a factor of a(), there is no remainder. If b() is not a factor of a(), then the remainder can be described as r(). If b() is not a factor of a(), a ( ) is equivalent to the sum of the quotient q() b( ) and the ratio r ( ) b( ). U3-5 Lesson 1: Operating with Rational Epressions

32 We can use polynomial division to represent the original rational epression in a different form. For eample, let a() and b() + 7. If a ( ) , then b( ) + 7 the rational epression is equivalent to the epression ( ) + 7 ( + 7). To find the quotient and remainder, use polynomial division: + + 7) In this eample, the quotient, q(), is equal to +, and the remainder, r(), is 0. The original epression a ( ) is equivalent to ( + ) +, b( ) or, more simply, Synthetic division (a shortcut for polynomial division) can be used to ease the task of rewriting rational epressions. The rational epression results in the following synthetic division This result yields a quotient of + 3, with a remainder of 0. This shows that the original epression is equivalent to ( + 3) +, or, more simply, + 3. U3-6 Unit 3: Rational and Radical Relationships

33 Guided Practice Eample 1 Simplify the rational epression Rewrite the division problem as a multiplication problem, using the reciprocal of the divisor. The divisor is 3 ; the reciprocal of 3 is 3. 3 Original epression Multiply by the reciprocal. The rewritten epression is Identify any invalid values of the rewritten epression. The denominator of the first term is a constant, and is therefore valid. The second term requires that Multiply the terms of the rewritten epression Check for any factors that might make it possible to further simplify the resulting epression, and then factor them out. There is a common factor of in both the numerator and the denominator The epression simplifies to when U3-7 Lesson 1: Operating with Rational Epressions

34 Eample Simplify the rational epression Rewrite the division problem as a multiplication problem, using the reciprocal of the divisor. 5 The divisor is + 1 ; the reciprocal of is Original epression Multiply by the reciprocal. The rewritten epression is Identify any invalid values of the rewritten epression. The denominators are + 1 and 5. Since adding 1 to the square of a number will always result in a real number value, and since 5 is a constant, there are no real number values that will make either denominator equal Multiply the terms of the rewritten epression ( + 1) ( + 1) Check for any factors that might make it possible to further simplify the resulting epression, and then factor them out. There are no common factors. 3 5 The epression simplifies to U3-8 Unit 3: Rational and Radical Relationships

35 Eample 3 Simplify the rational epression Rewrite the division problem as a multiplication problem, using the reciprocal of the divisor. The divisor is 3 + ; the reciprocal of 3 + is Original epression The rewritten epression is Multiply by the reciprocal Identify any invalid values of the rewritten epression. The denominators are 6 and 3. Since 6 is a quadratic, see if it can be factored. 6 ( 3)( + ) The factors are 3 and +. Set each factor equal to 0 and solve for to determine the values that make the epression invalid An -value of 3 invalidates the epression, as does an -value of. Therefore, for the first denominator, 3 and. Set the denominator of the second term, 3, equal to 0 to determine values that make the epression invalid Therefore, 3, which was also a restriction determined for the first epression. The invalid values of the rewritten epression are 3 and. U3-9 Lesson 1: Operating with Rational Epressions

36 3. Check for factors that may cancel, then multiply the terms of the rewritten epression. 3 In the rewritten epression, 6 + 3, there are quadratic epressions in both the numerator and denominator of the first term. Both of these quadratic epressions can be factored. We have already determined the factors of 6 are 3 and +. Factor the quadratic in the numerator of the first term, 3. 3 ( 3) The factors are and 3. Notice that in both the numerator and denominator of the first term, 3 is a factor. Use this information to cancel out common factors and multiply the rewritten epression Rewritten epression ( 3) ( 3)( ) Factor the first term of the epression. ( 3) ( 3)( ) Cancel out common factors. 3 Simplify. 3 3 The epression Check for any factors that might make it possible to further simplify the resulting epression, and then factor them out. There are no additional common factors. 3 3 The epression 6 + simplifies to 3 when 3 and. U3-30 Unit 3: Rational and Radical Relationships

37 Eample 4 Simplify the rational epression Rewrite the division problem as a multiplication problem, using the reciprocal of the divisor. The divisor is + ; the reciprocal of is Original epression Multiply by the reciprocal. The rewritten epression is Identify any invalid values of the rewritten epression. The denominators are + 7 and +. Both of these denominators are quadratic, so see if they can be factored. + 7 ( + 7) The factors of the denominator of the first term are and + 7. Set each factor equal to 0 and solve for to determine the values that make the epression invalid An -value of 0 invalidates the epression, as does an -value of 7. Therefore, for the first denominator, 0 and 7. (continued) U3-31 Lesson 1: Operating with Rational Epressions

38 Net, factor the second denominator, +. + ( + )( 1) The factors of the denominator of the second term are + and 1. Set each factor equal to 0 and solve for to determine the values that make the epression invalid An -value of invalidates the epression, as does an -value of 1. Therefore, for the second denominator, and 1. The values of which make the rewritten epression invalid are 0, 7,, and Check for factors that may cancel, then multiply the terms of the rewritten epression In the rewritten epression, , there are quadratic epressions in both the numerator and denominator of both epressions. Rewrite each quadratic epression in factored form to determine if there are common factors that may cancel. We have already determined the factored form of each denominator: + 7 ( + 7) + ( + )( 1) Now, factor the numerators ( + )( + 7) 4 4 4( 1) (continued) U3-3 Unit 3: Rational and Radical Relationships

39 Substitute the factored forms into the rewritten epression to cancel out common factors and multiply the terms ( + )( + 7) 4 ( 1) ( + 7) ( + )( 1) ( + )( + 7) 4 ( 1) ( + 7) ( + )( 1) 4 Simplify. Rewritten epression from step 1 Write the quadratic epressions in factored form to identify common factors. Cancel out common factors. The epression simplifies to when 0, 7,, and 1. Eample 5 Use synthetic division to rewrite a 3 ( ) b( ) + 1 in the form a ( ) r( ) q( ) +. b( ) b( ) 1. Identify any values that make the epression invalid. If + 1 0, the denominator equals 0; this creates an invalid form. Since + 1 cannot equal 0, 1.. Set up the synthetic division problem to divide the given numerator, a(), by the denominator, b(). The numerator is The coefficients of the numerator are 1,, 1, and 3. Because we are dividing by + 1, the number 1 must go on the shelf of the synthetic division. The resulting synthetic division problem is U3-33 Lesson 1: Operating with Rational Epressions

40 3. Solve the synthetic division problem. Perform synthetic division to obtain the following result: The first row, , comes from the eample as shown in step. The net step in solving is to simply bring the first number (1) down to the bottom row. The 1 in the second column is the product of that 1 and 1 (the shelf number). The 1 in the bottom row of the second column is the sum of + ( 1). This process is repeated to determine the values in the third and fourth columns. This yields the bottom row: Identify the remainder. The remainder is the last entry in the bottom row (shown with a bo around it). In this case, the remainder, r(), is 5. Since the denominator of the original epression is + 1, the 5 remainder of the epression is Identify the quotient. The quotient, q(), is described by the coefficients preceding the remainder in the bottom row. Those coefficients are 1, 1, and. Therefore, the quotient is 1 + 1, or Use the results of the synthetic division to rewrite the original epression. The rational epression simplifies to when U3-34 Unit 3: Rational and Radical Relationships

41 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions Practice 3.1.3: Dividing Rational Epressions For problems 1 4, simplify each rational epression. Whenever possible, reduce the epression to its lowest terms. State any restrictions on Use your knowledge of rational epressions to complete problems State any restrictions on Show that the epression is equivalent to Use synthetic division to rewrite the rational epression r( ) form q( ) +. b( ) in the continued U3-35 Lesson 1: Operating with Rational Epressions

42 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 1: Operating with Rational Epressions 7. Use polynomial division to rewrite the rational epression r( ) 1 form q( ) +. b( ) in the 8. One way to determine the time it takes to travel a specified distance is to solve for the quotient of distance traveled and the average velocity. What epression represents the time in hours that it will take Mateo to travel a total of ( 4) km at an average velocity, in km per hour, described by the epression (where > 3)? 3 9. Density is the ratio of an object s mass to its volume. What simplified rational epression represents the density (in grams per meter 3 ) of an alloy ingot whose mass in grams is described by the epression , with a volume, in meters 3, represented by the epression + 8 (where > 3)? On π day (March 14), Tyler s math class shares freshly made pies. If + 1 there are students in Tyler s class who all equally share in the π-day 1 pies, with > 1, what epression represents the fraction of pie each student receives? U3-36 Unit 3: Rational and Radical Relationships

43 UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations Common Core Georgia Performance Standards MCC9 1.A.CED.1 MCC9 1.A.REI. Essential Questions 1. How can you find a solution to a rational equation?. How can you identify etraneous solutions? 3. How can you find a solution to an equation involving radicals? 4. How does the solution to a rational inequality compare to that of a rational equation? WORDS TO KNOW common denominator etraneous solution least common denominator (LCD) proportion radical equation radical epression ratio rational equation a quantity that is a shared multiple of the denominators of two or more fractions a solution of an equation that arises during the solving process, but which is not a solution of the original equation the least common multiple of the denominators of two or more fractions a statement of equality between two ratios an algebraic equation in which at least one term includes a radical epression an epression containing a root the relation between two quantities; can be epressed in words, fractions, decimals, or as a percentage an algebraic equation in which at least one term is epressed as a ratio U3-37 Lesson : Solving Rational and Radical Equations

44 rational inequality rational number a mathematical statement comparing two quantities, including at least one term that is epressed as a ratio any number that can be written as m, where m and n n are integers and n 0; any number that can be written as a decimal that ends or repeats Recommended Resources Math Portal. Rational Equation Solver. This online calculator allows users to solve rational equations. Specific directions and eamples for correctly entering data are included. Math Warehouse. How to Solve Radical Equations. This site first offers the user a simple algorithm for solving radical epressions. After watching a video that shows the process of solving such an equation, users can practice the solution process with several guided samples. Purplemath.com. Solving Rational Inequalities. This site connects solving rational inequalities to solving polynomial inequalities, and provides several worked eamples. This resource is particularly helpful for those who have trouble visualizing the graphs or grids necessary to compare the signs of various factors within a larger epression. U3-38 Unit 3: Rational and Radical Relationships

45 Lesson 3..1: Creating and Solving Rational Equations Introduction Your previous eperiences have helped you understand how to represent situations using algebraic equations, and how solving those equations can reveal something specific about those situations. Most of these equations have involved integer coefficients. However, coefficients can also be epressed as a ratio. A ratio compares two quantities, and is often written as a fraction. Numbers epressed as a ratio are called rational numbers. More formally, a rational number is any number that can be written as m, where m and n are integers and n 0. In this lesson, we will eplore n ways to solve rational equations, and to assess and validate the resulting solutions. Key Concepts A rational equation is an algebraic equation that includes at least one term that is epressed as a ratio. A rational equation defines a specific relationship between numbers and variables; this relationship can be analyzed to find a solution to the equation. Finding the solution to a rational equation may be accomplished by simplifying the rational terms. To simplify rational terms, convert the rational equation into an equivalent equation with integer coefficients. Finding the solution often requires using a common denominator a quantity that is a shared multiple of the denominators of two or more fractions to eliminate any rational terms. Recall that the least common denominator is the least common multiple of the denominators of the fractions in the epression. Like other algebraic equations, a rational equation may have no solutions. An etraneous solution can occur when the solution process results in a value that satisfies the original equation, but creates an invalid equation by creating a denominator that is equal to 0. An etraneous solution can also occur when a solution results in a value that is irrelevant in the contet of the original equation. U3-39 Lesson : Solving Rational and Radical Equations

46 Recall that a proportion is a statement of equality between two ratios. Techniques for solving proportions can often be used to simplify rational equations. The process of solving rational equations can be applied to real-world eamples in which a specific task that may be done alone is to be completed cooperatively. U3-40 Unit 3: Rational and Radical Relationships

47 Guided Practice 3..1 Eample 1 6 Solve the proportion 4 8 for. 1. Determine the least common denominator. The denominator of the left epression is 4, while the denominator of the right epression is 8. The least common denominator (LCD) is 8, the smallest multiple with both denominators as factors.. Multiply each ratio by the common denominator. (8) (8) 6 Multiply each ratio by the LCD, Simplify. 3. Simplify the resulting equation Equation from the previous step Simplify the fractions. The result is an algebraic equation without any remaining rational numbers that can be solved using methods you have already learned. 4. Solve the resulting equation. 6 Simplified equation from the previous step 3 Divide both sides of the equation by. U3-41 Lesson : Solving Rational and Radical Equations

48 5. Verify the solution(s). While it appears that we have a solution, 3, we have to return to the original equation to confirm it Original equation (3) Substitute 3 for The result is a true statement; therefore, 3 is an acceptable result. 6 For the proportion 4 8, 3. Eample Solve the rational equation t 1 3t for t. 1. Determine the least common denominator. The denominators in the equation are t, 1, and 3t. The smallest multiple having all three of these values as a factor is 1t; therefore, the least common denominator is 1t. U3-4 Unit 3: Rational and Radical Relationships

49 . Multiply each term of the equation by the common denominator and simplify. (1 t) 5 (1 t) 3 t t 1 (1 )9 3t (1) 5 t 1 ( )3 1 (4) 9 1 Multiply each term by the LCD, 1t. Simplify each fraction. (1)5 (t)3 (4)9 Continue to simplify. 60 3t 36 The result is an algebraic equation without any remaining rational numbers that can be solved using methods you have already learned. 3. Solve the resulting equation. 60 3t 36 Simplified equation from the previous step t Add 3t to both sides. 4 3t Subtract 36 from both sides. t 8 Divide both sides by Verify the solution(s). While it appears that we have a solution, t 8, we have to return to the original equation to confirm it. Because there are variables in the denominators of the first and last terms, we must determine what value(s) would make the denominator equal 0, and thereby invalidate the equation (since division by 0 yields an undefined result and is not acceptable). 5 The epression is undefined when t 0; therefore, t 0. t 9 Similarly, the epression is also undefined when t 0; once again, 3t t 0. Our solution, t 8, does not violate this condition; therefore, it is an acceptable result For the equation t 1 3 t, t 8. U3-43 Lesson : Solving Rational and Radical Equations

50 Eample Solve the rational equation y y + 3 for y. 1. Determine the least common denominator. The denominators are y, y +, and 3. The smallest multiple having all three of these values as a factor is 3y(y + ), the irreducible product of all three denominators. The least common denominator is 3y(y + ).. Multiply each term by the common denominator. 1 3 y( y+ ) 3 y ( y+ ) y y+ y y + 3 ( ) Simplify the resulting equation. Start by rewriting each occurrence of the common denominator in the equation as a ratio: 3 y( y+ ) 3 y( y+ ) 1 3 y( y+ ) 3 y( y+ ) 1 3 y( y+ ) 1 1 y 1 y Common denominator rewritten as a ratio Rewritten equation 3(y + ) 3y 1 y(y + ) 1 Simplify the ratios. 6y + 1 3y y + y Carry out the multiplication. 3y + 1 y + y Subtract 3y from 6y. The algebraic equation that results this time, 3y + 1 y + y, still has only integer coefficients. However, it is no longer linear this is a quadratic equation because y is raised to a power of. We will need to either factor or use the quadratic formula to determine the value of y. U3-44 Unit 3: Rational and Radical Relationships

51 4. Solve the quadratic equation by factoring or using the quadratic formula. Start by setting the equation equal to 0 so that the equation is in the standard form of a quadratic equation. Then, factor the quadratic. 3y + 1 y + y Equation found in the previous step 1 y + ( y) Subtract 3y from both sides. 0 y y 1 Subtract 1 from both sides. 0 (y 4)(y + 3) Factor the resulting quadratic. y 4 or y 3 5. Verify the solution(s). Now that we have potential solutions, y 4 or y 3, we must return to the original equation. We must determine what value(s) would make the denominators equal 0, and thereby invalidate the equation. The epression is undefined when y 0; therefore, y 0. y The epression 1 is also undefined when y ; therefore, y. y+ There is no value for y that invalidates the last term, 1, because the denominator does not contain a variable. 3 Our solutions, y 4 or y 3, do not violate this condition; therefore, they are acceptable results. 1 1 If, then y 4 or y 3. y y + 3 U3-45 Lesson : Solving Rational and Radical Equations

52 Eample 4 Solve the rational equation 1 p + p + 6 for p. 4 p p p 8 1. Determine the least common denominator. The denominators are p 4, p +, and p p 8. The least common denominator is not apparent with these denominators. However, the product of the first two denominators can be determined. (p 4)(p + ) Product of the first two denominators p(p + ) 4(p + ) p + p 4p 8 p p 8 Rewrite the product using the Distributive Property. Distribute. Simplify. Since the product of the first two denominators is equal to the third denominator, we have determined a suitable common denominator: p p 8.. Multiply each term by the common denominator. 1 p ( p p 8) + p p p p p 4 ( 8) p + ( 8) 6 p p 8 U3-46 Unit 3: Rational and Radical Relationships

53 3. Simplify the resulting equation. Think of each term as a ratio, then simplify each ratio by using factors. p p 8 1 p p 8 p + 1 p 4 1 p + p p p p 8 ( p 4)( p+ ) 1 ( p 4)( p+ ) p + 1 p 4 1 p + p p p p 8 ( p 4)( p+ ) 1 ( p 4)( p ) p 1 p p+ p p p p 8 Equation with each term written as a ratio Rewritten equation with factors Divide out like factors. ( p+ ) p 1 ( 4) + p Simplify. (p + )(1) + (p 4)(p) 6 Simplify the ratios. p + + p 4p 6 p 3p + 6 p 3p 4 0 The simplified equation, p 3p 4 0, is quadratic. Simplify the equation. Continue to simplify. U3-47 Lesson : Solving Rational and Radical Equations

54 4. Solve the resulting quadratic equation by factoring or using the quadratic formula. This quadratic equation can be solved using the quadratic formula, b± b 4( a)( c), where a, b, and c are the coefficients of each term. a b± b 4( a)( c) a ( 3) ± ( 3) 4(1)( 4) 3± ± 5 (1) p 4 or p 1 Quadratic formula Substitute 1 for a, 3 for b, and 4 for c. Simplify. 5. Verify the solution(s). In the original equation, there are variables in the denominator of each term. We must determine what value(s) would make those denominators equal 0, and thereby invalidate the equation. 1 The epression is undefined when p 4; therefore, p 4. p 4 The epression 1 is also undefined when p ; therefore, p. p+ (continued) U3-48 Unit 3: Rational and Radical Relationships

55 When we found our least common denominator, we determined that the epression p p 8 (p 4)(p + ). That means 1 1 p p 8 ( p 4)( p+ ). The same results as before are true: p 4 and p. One of our algebraic solutions is that p 4. This is not a valid solution, because it would result in a 0 in the denominator. This kind of solution is called an etraneous solution, since it satisfies our algebraic problem, but makes no sense in the original equation or problem. However, our other solution, p 1, poses no such problem. It is an acceptable result, and therefore is our only solution to the original problem. 1 If p + p + 6, then p 1. 4 p p p 8 Eample 5 Demarco and his friend Chase have a small computer business. Demarco can set up a computer for a customer in 90 minutes. Chase can do the same job in hours. Wednesday afternoon, a new client contacts them to set up 14 computers by Friday afternoon. Create a rational equation to find the time it will take for Demarco and Chase to complete this task. 1. Identify the variable. In this eample, the total amount of time it will take Chase and Demarco to complete the task is not known. Let t describe the total amount of time it will take for the task to be completed. Notice that there are two time units described in the problem: hours and minutes. Select one unit to use consistently. Using hours, the work times are 1.5 and hours; using minutes, the work times are 90 and 10 minutes. Because rational epressions and decimals do not always mi well, minutes are a better choice (though the choice will not affect the result). U3-49 Lesson : Solving Rational and Radical Equations

56 . Identify the task that is to be completed. Demarco and Chase need to set up 14 computers for their client. 3. For each participant in the problem, create a ratio that describes the proportion of the task to be completed. There are two participants in this problem: Demarco and Chase. t In t minutes, Demarco can set up 90 computers. In t minutes, Chase can set up t 10 computers. 4. Create a rational equation to find the time it will take for Demarco and Chase to complete the task together. Because Demarco and Chase are working together, use the sum of their individual efforts to create an equation. Demarco s efforts + Chase s efforts completed task t t t t The equation represents the time it will take for Demarco and Chase to set up 14 computers. U3-50 Unit 3: Rational and Radical Relationships

57 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations Practice 3..1: Creating and Solving Rational Equations For problems 1 3, solve each rational equation for Use the given information and what you know about rational equations to complete problems The sum of the reciprocals of two consecutive, positive even integers is 1. What are the two integers? 5. Tyler can shovel the driveway in hours. Dakota can complete the job in 90 minutes. What rational equation represents the information presented in this problem? 6. Emily spends her weekends cliff climbing. She can ascend a steep 480-foot cliff near her home with little difficulty. She can descend from the cliff at 5 times the rate she can ascend, completing the descent in 3 hours less time than it takes her to climb up. Create a rational equation that describes how long it will take her to climb up and down the cliff. continued U3-51 Lesson : Solving Rational and Radical Equations

58 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations 7. James and Liu volunteer at the food pantry one Saturday each month, packing boes of food to deliver to families. Working alone, James can pack 10 boes in hours. Liu can pack 160 boes in 4 hours. Create a rational equation to find the time it takes them to work together to pack 600 boes. 8. Working alone, Erica can thoroughly detail a car in 80 minutes. It takes Ale hours to do the same job. One afternoon, Erica and Ale start detailing a customer s car together. Half an hour later, Jacki, the new employee, joins them. If the three of them finish 15 minutes later, how long would it have taken Jacki to detail the same car on her own? 9. A developer needs to hire a crew to pave the parking lot of a new shopping plaza. Randolph and Sons Paving can complete the job in 8 days. Mortimer Brothers Paving can do the job in 5 days, but can t start until days later than Randolph and Sons. The developer is anious to get the job done, and decides to hire both companies. Randolph and Sons works alone for the first days and then is joined by Mortimer. How many days will it take the two companies to pave the parking lot? 10. Dani spends her summers at her grandparents cabin in Canada. The cabin is 4 kilometers upriver from the nearest town. The river has a downstream current of 4 kilometers per hour (km/h). If Dani can canoe from the cabin to town and back in 14 hours, what is her average canoeing speed in still water? U3-5 Unit 3: Rational and Radical Relationships

59 Lesson 3..: Creating and Solving Rational Inequalities Introduction Each solution to a rational equation is a single, unique value. However, this is not the case with rational inequalities. A rational inequality is a mathematical statement comparing two quantities, and includes at least one term that is epressed as a ratio. In this section, we will eplore the process of solving rational inequalities and determining values that make the inequality true. Key Concepts Solutions to a rational equation, if they eist, are a discrete set of values. However, solutions to a rational inequality, if they eist, are a range of values. Before solving a rational inequality, the inequality should be converted into a form comparing a single rational epression to 0, such as a ( ) < 0. b( ) Rational inequalities can be rewritten using previously learned methods of adding and subtracting rational epressions. A rational epression will be positive, or greater than 0, when the signs of the numerator and the denominator are the same. A rational epression will be negative, or less than 0, when the signs of the numerator and the denominator are not the same. Factoring the numerator or denominator can make it easier to identify values that affect the sign of the rational epression. Because division by 0 is undefined, check for any values that make the denominator equal to 0. These values must be eliminated from any potential solution. Identify the interval of values for which the numerator is positive (or negative). Comparing these to the values for which the denominator is positive (or negative) will identify those values for which the inequality is true. The solution to an inequality cannot be checked in the same way that the solution to an equation can be checked, since we have no way of substituting every possible result back into the original inequality. We can get a sense of the accuracy of a solution by selecting a value within the solution interval, and testing it to see if the inequality is true for that value. The result can only give an indication that the inequality is correct; it does not prove the inequality to be true. U3-53 Lesson : Solving Rational and Radical Equations

60 Guided Practice 3.. Eample 1 Solve the rational inequality 4 < 0 for Check for any values of that would make the rational epression undefined. If the denominator is equal to 0, then the rational epression will be undefined and so will the inequality. Because the denominator is a constant, there are no values that are ecluded from the domain.. Analyze the inequality to identify intervals for the variable that will make the epression less than 0. Because the denominator is a constant, the denominator is always positive. If the result of the numerator is less than 0, then the epression will be less than 0, since a negative number divided by a positive number will result in a negative number (a value less than 0). Values for that make the numerator less than 0 (in other words, negative) will satisfy the inequality. 3. Identify intervals that affect the sign of the numerator. We can determine when the numerator is negative by solving the linear inequality 4 < 0 for. 4 < 0 < 4 Numerator set as a linear inequality Add 4 to both sides. The numerator will be negative if < 4. U3-54 Unit 3: Rational and Radical Relationships

61 4. State the possible solution(s). The possible solution to the inequality 4 < 0 is < Check the possible solution(s). Choose a value that satisfies the possible solution, < 4. One value that satisfies the inequality < 4 is 0. Test 0 in the original inequality to see if it yields a true statement. 4 < 0 Original inequality 3 (0) 4 < 0 Substitute 0 for < 0 Simplify. 4 is less than 0; therefore, the solution < 4 is likely valid. 3 The rational inequality 4 < 0 when < 4. 3 U3-55 Lesson : Solving Rational and Radical Equations

62 Eample Solve the rational inequality > for. 1. Check for any values of that would make the rational epression undefined. If the denominator is equal to 0, then the rational epression will be undefined and so will the inequality. The denominator is + 1; therefore, Subtracting 1 from both sides gives the equivalent statement 1 as a limitation on any solution. Therefore, 1 is ecluded from the domain.. Identify domain intervals that affect the sign of the numerator and denominator. If the numerator and denominator are both positive, or if they are both negative, then the resulting ratio will also be positive and, therefore, greater than 0. Determine the intervals for which the original inequality + 1 > 0 is true. Find the values for which the numerator and denominator each equal 0, and then use those values to define the intervals where the signs of the numerator and denominator should be checked. In this eample, the only number that results in a numerator of 0 is 0, while only 1 results in a value of 0 for the denominator. Therefore, the intervals that affect the sign of the numerator and denominator are < 1, 1 < < 0, and > 0. U3-56 Unit 3: Rational and Radical Relationships

63 3. Use these intervals to create a table of sign changes. Use the identified intervals (ecluding 1, the invalid value for described in step 1) to create a table that represents the sign changes for the different values of in the inequality. Intervals Factors < 1 1 < < 0 > The entries in the table correspond to the sign of the factor in the left column when the value of falls within the interval listed at the top of the column. For eample, the shaded cell indicates that the factor + 1 is negative when < Use the table to identify the intervals that satisfy the conditions described in step. In step, we determined that the epression is positive when + 1 both the numerator and denominator are the same sign. The table shows that this happens when < 1 and when > State the possible solution(s). The possible solutions to the inequality > are < 1 and > 0. U3-57 Lesson : Solving Rational and Radical Equations

64 6. Check the possible solution(s). Choose a value that satisfies the first possible solution, < 1. One value that satisfies the inequality < 1 is. Test in the original inequality to see if it yields a true statement > Original inequality ( ) ( ) + 1 > 0 Substitute for. 1 > 0 > 0 Simplify. is greater than 0; therefore, the solution < 1 is likely valid. Choose a value that satisfies the second possible solution, > 0. One value that satisfies the inequality > 0 is. Test in the original inequality to see if it yields a true statement. > Original inequality () > () Substitute for. 3 > 0 Simplify. is greater than 0; therefore, the solution > 0 is likely valid. 3 The rational inequality > when < 1 or > 0. U3-58 Unit 3: Rational and Radical Relationships

65 Eample 3 Solve the rational inequality for. 1. Check for any values of that would make the rational epression undefined. If the denominator is equal to 0, then the rational epression will be undefined and so will the inequality. The denominator is a quadratic equation, and can be factored as 4 ( + )( ). Therefore, the zeros of the quadratic equation are and. These are the values that make the rational epression undefined; in other words, any solution cannot contain ±.. Since the inequality includes 0 as a possible solution, identify any values that make the rational epression equal to 0. If the numerator is equal to 0, the epression will be equal to 0. This occurs when + 3 0, or when Identify domain intervals that affect the sign of the numerator and denominator. If the numerator and denominator are both positive, or if they are both negative, then the resulting ratio will also be positive and, therefore, greater than Determine the intervals for which the original inequality 4 0 is true. Find the zeros of the numerator and denominator, and then use those values to define the possible intervals. In this eample, we previously identified 3 as the only zero for the numerator, and ± as the zeros for the denominator. Therefore, the intervals are 3, 3 <, < <, and >. U3-59 Lesson : Solving Rational and Radical Equations

66 4. Use these intervals to create a table of sign changes. The intervals found in steps 3 and 4 (ecluding ±, the invalid values for described in step 1) result in the following table. Intervals Factors 3 3 < < < > + 3 (numerator) (denominator) As before, the entries in the table correspond to the sign of the factor in the left column when the value of falls within the interval listed at the top of the column. The bottom row represents not a single factor, but the denominator, which is the product of the factors ( ) and ( + ). 5. Use the table to identify the intervals that satisfy the conditions described in step We have already determined that the epression is positive 4 when both numerator and denominator are the same sign. Inspect the table from the previous step. The top row of the table shows the signs of the numerator for various intervals, while the bottom row shows the signs of the denominator. The table shows that both the numerator and denominator have the same sign when 3 < or when >. 6. State the possible solution(s). The possible solutions to the inequality and > are 3 < U3-60 Unit 3: Rational and Radical Relationships

67 7. Check the possible solution(s). Choose a value that satisfies the first possible solution, 3 <. One value that satisfies the inequality 3 < is 3. Test 3 in the original inequality to see if it yields a true statement Original inequality ( 3) + 3 ( 3) 4 0 Substitute 3 for Simplify. 0 is greater than or equal to 0; therefore, the possible solution 3 < is likely valid. Choose a value that satisfies the second possible solution, >. One value that satisfies the inequality > is 4. Test 4 in the original inequality to see if it yields a true statement Original inequality (4) + 3 (4) 4 0 Substitute 4 for Simplify. is greater than 0; therefore, the possible solution is likely valid. The rational inequality when 3 < or >. U3-61 Lesson : Solving Rational and Radical Equations

68 Eample 4 Solve the rational inequality for. 1. Check for any values of that would make the rational epression undefined. If the denominator is equal to 0, then the rational epression will be undefined and so will the inequality. Since there are two rational epressions, both of the denominators need to be eamined for values that would result in a denominator of 0. In the first rational epression,, 7 would result in a value of for the denominator. Therefore, this value must be ecluded from the solution. The second rational epression s denominator, , is a quadratic equation. It must be factored. Factoring results in ( + 7)( + 1). The zeros of this equation are 7 and 1. Therefore, the values 7 and 1 must be ecluded from the solution.. Combine the rational epressions on one side of the inequality, leaving 0 on the other. To get 0 on the right side of the inequality, subtract from both sides of the inequality Original inequality Subtract from both sides ( + 7)( + 1) Rewrite the quadratic denominator in factored form can be rewritten as ( + 7)( + 1) (continued) U3-6 Unit 3: Rational and Radical Relationships

69 In order to carry out the subtraction of the rewritten epression, a common denominator is needed. Since the denominator of the first term is also a factor in the denominator of the second term, the common denominator is ( + 7)( + 1) ( + 7)( + 1) Rewritten epression ( + 7)( + 1) Multiply the epression by ( + 1) 0 ( + 7)( + 1) ( + 7)( + 1) Simplify. + 0 ( + 7)( + 1) ( + 7)( + 1) Distribute. ( + ) ( ) 0 ( + 7)( + 1) ( + 7)( + 1) Simplify. The original inequality ( + 7)( + 1) Rewrite the numerators over the common denominator. can be rewritten as 3. Since the inequality includes 0 as a possible solution, identify any values that make the rational epression equal to 0. If the numerator equals 0, the entire epression will be equal to 0. The numerator, + 3, is a quadratic and can be written in factored form as ( + )( 1). 1 This quadratic has zeros when or when. At these values, the rational epression equals 0. U3-63 Lesson : Solving Rational and Radical Equations

70 4. Identify intervals that affect the sign of the numerator and denominator. If the numerator and denominator have different signs (that is, one is positive and the other is negative), then the resulting ratio will be negative and, therefore, less than 0. Determine the intervals for which this is true. Find the zeros of the numerator and denominator, and then use those values to define the possible interval(s) of numbers. We previously identified 1 and as zeros for the numerator. In step 1, we found that the denominator has the zeros 7 and 1. 1 Therefore the intervals are < 7, 7 <, < 1, 1<, 1 and. 5. Use these intervals to create a table of sign changes. The intervals found in the previous step (ecluding 7 and 1, the invalid values for described in step 1) result in the following table. Intervals Factors < 7 7 < < 1 1 < (numerator) ( + 7)( + 1) (denominator) As before, the entries in the table correspond to the sign of the factor in the left column when the value of falls within the interval listed at the top of the column. The row marked numerator indicates the sign of the product of the numerator s two factors, and the row marked denominator indicates the sign of the product of the denominator s two factors. U3-64 Unit 3: Rational and Radical Relationships

71 6. Use the table to identify the intervals that satisfy the conditions described in step In step 4, we determined that the epression is negative ( + 7)( + 1) when the numerator and denominator have different signs. The table shows that the numerator and denominator have opposite 1 signs when 7 < and when 1<. 7. State the possible solution(s). The possible solutions to <. are 7 < and 8. Check the possible solution(s). Choose a value that satisfies the first possible solution, 7 <. One value that satisfies the inequality 7 < is 5. Test 5 in the original inequality to see if it yields a true statement ( 5) ( 5) ( 5) ( 5) + 7 ( 5) + 8( 5) is less than or equal to < is likely valid. Original inequality Substitute 5 for. Simplify. Rewrite the terms with a common denominator. ; therefore, the possible solution (continued) U3-65 Lesson : Solving Rational and Radical Equations

72 1 Choose a value that satisfies the second possible solution, 1<. 1 One value that satisfies the inequality 1< is 0. Test 0 in the original inequality to see if it yields a true statement (0) (0) (0) (0) + 7 (0) + 8(0) Original inequality Substitute 0 for. Simplify is less than or equal to ; therefore, the possible solution 7 1 1< is likely valid. The rational inequality when 1 7 < or 1<. U3-66 Unit 3: Rational and Radical Relationships

73 Eample 5 On Earth, the height h above ground of a projectile launched upward can be found using the equation h h 0 + vt 16t, where t is the time in seconds after the object is launched, h 0 is the height above ground at the time of launch, and v represents the upward velocity (in feet per second) with which the object is launched. The average velocity is the ratio of the distance (in this case, height) and time. Ariana launched a model rocket from the deck behind her house. The deck is 1 ft high, and the rocket had an initial (launch) velocity of 50 ft/s. Create a rational inequality that can be solved to determine when the average velocity of the rocket is less than 16 ft/s. 1. Identify a general rational epression that represents the given data. This problem includes a polynomial epression for the height of an object launched upward, and describes the average velocity as a ratio of that velocity and time.. Use data from the problem to create a specific rational epression. The problem has already defined the variable t to represent time, and the equation to represent the height above ground as h h 0 + vt 16t. Since the average velocity is the ratio of those values, then the average velocity h t. Use this information and the values given in the problem statement to write a rational epression for the average velocity of the rocket. ( h0 + vt 16t ) averagevelocity t averagevelocity ( ) ( ) Substitute h 0 + vt 16t for h t 16t Substitute 1 for h 0 and 50 t for v. The epression that represents the average velocity of the rocket is 1+ 50t 16t. t U3-67 Lesson : Solving Rational and Radical Equations

74 3. Identify the value to which this epression will be compared. The problem asks for values for which the rocket s average velocity is less than 16 ft/s. 4. Write the resulting rational epression t 16t Comparing the rational epression to an average t velocity that is less than 16 ft/s results in the following epression t 16t t < t 16t < 16 represents a rational inequality that can be solved t to determine when the average velocity of the rocket is less than 16 ft/s. U3-68 Unit 3: Rational and Radical Relationships

75 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations Practice 3..: Creating and Solving Rational Inequalities For problems 1 4, solve each of the given rational inequalities < < For problems 5 and 6, use the given information to create a rational inequality for each scenario. 5. Due to the moon s lower gravitational pull, a projectile launched on the moon would follow a different path than if the projectile were launched from Earth. On the moon, the height h above ground of a projectile launched upward can be found using the equation h h 0 + vt.7t, where t is the time (in seconds) after the projectile is launched, h 0 is the height above ground at the time of launch, and v represents the upward velocity (in meters per second) with which the object is launched. A projectile is launched from a height of 4 ft with an initial (launch) velocity of 5 ft/s. Create a rational inequality that can be solved to determine when the average velocity is less than 30 ft/s The volume of a pyramid is found using the equation V Bh, where B is the 3 area of the base and h is the height. A pyramid is to be designed with a square base and a volume of 4,000 m 3, with a height that is less than the length of one side of the base. Create an inequality using that could be solved to find the possible values of the length of the base. continued U3-69 Lesson : Solving Rational and Radical Equations

76 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations Use the given information and what you know about rational inequalities to solve problems Bobbi and her brother Bentley rake leaves every fall to earn a little etra money. Bobbi can rake all the leaves at one house in hours. Bentley can rake the leaves at one house in 3 hours. There are only 6 houses on the cul-de-sac where they live, but each house has a yard that needs to be raked. If Bobbi and Bentley work together, what is the minimum time they will need to rake all the yards in their cul-de-sac? 8. The epression represents the cost in dollars for Lenny s Lumber to produce a sawhorse. How many sawhorses must be built to keep the average cost below $1? 0.06t The epression represents the concentration (in g/m ) of 4+ t fertilizer sprayed on a suburban lawn, where t is measured in days. When the concentration eceeds 0.01 g/m, the fertilizer can be harmful to pets that walk unprotected over the surface, so signs have to be posted. How many days must pass after spraying a lawn with fertilizer before such safety signs can be removed? 10. When a new car model is released, sales tend to be climb rapidly with the ecitement (and advertising), before slowing down. Angel, the sales manager at a 18t + 1 local dealership, uses a monthly sales model,, to predict new car model t + 1 sales. The variable t represents the number of months since the car s release. For what values of t can Angel epect to sell at least 3 of a particular model car? U3-70 Unit 3: Rational and Radical Relationships

77 Lesson 3..3: Solving Radical Equations Introduction Using the Pythagorean Theorem to solve problems provides a familiar eample of a relationship between variables that involve radicals (or roots). Recall that the word root in mathematics has several definitions. You have seen and worked with function roots, or zeros; in this lesson, you will work with a different kind of root that is, square roots, cube roots, and other radicals. Radical epressions, or epressions containing a root, follow rules that may seem more complicated than those of more common numbers. While you can easily add the numbers 4 and 9 to get 13, there is no similar simple way to add the radicals 4 and 9 without simplifying each individual term first. Since radical epressions require more comple methods in order to be simplified, radical equations also involve more comple methods in order to be solved. However, some common real-life situations can be more readily solved by applying radical equations, making fluency with radical equations a useful skill to learn. In this lesson, we will eplore ways to solve radical equations, as well as how to assess and validate the resulting solutions. Key Concepts A radical equation is an algebraic equation in which at least one term includes a radical epression. A radical equation defines a specific relationship between numbers and variables that can be analyzed to find a solution. It is possible to find the solution to a radical equation by isolating the radical, then raising the epression to an appropriate power to eliminate the radical. Like other algebraic equations, a radical equation may have no solutions. Recall that an etraneous solution can occur when the solution process results in a value that satisfies the original equation, but creates an invalid equation; in the case of radical equations, etraneous solutions occur when a negative value results under the radical. An etraneous solution can also occur when a solution results in a value that is irrelevant in the contet of the original equation. U3-71 Lesson : Solving Rational and Radical Equations

78 Guided Practice 3..3 Eample 1 Solve the radical equation 5 for. 1. Isolate the radical epression. The radical epression ( ) equation. 5 is already isolated on one side of the. Raise both sides of the equation to a power to eliminate the radical. Because we have a square root on the left side of the equation, we need to raise both sides to the power of to eliminate the radical epression. In other words, square both sides. 5 Original equation ( ) 5 Square both sides of the equation. 5 4 Simplify. 3. Solve the resulting equation Equation found in the previous step Add 5 to both sides. The solution appears to be 9. However, we need to confirm that this is not an etraneous solution. 4. Verify the solution(s). Substitute the potential solution into the original equation, and check the result. 5 Original equation (9) 5 Substitute 9 for. 4 Simplify. The solution satisfies the original equation, and is therefore valid. The solution to the radical equation 5 is 9. U3-7 Unit 3: Rational and Radical Relationships

79 Eample Solve the radical equation 5 for. 1. Isolate the radical epression. At first glance, this looks much like the equation given in Eample 1. However, 5 is not under the radical. This small change creates a big difference, for in this eample, the radical epression ( ) is not already isolated on one side of the equation. The first step is to isolate the radical. 5 Original equation 7 Add 5 to both sides.. Raise both sides of the equation to a power to eliminate the radical, then simplify the result. Square both sides of the equation to eliminate the radical epression, then solve the equation algebraically. 7 Equation found in the previous step ( ) 49 7 Square both sides. Simplify. The solution appears to be Verify the solution(s). Substitute the potential solution into the original equation, and check the result. 5 Original equation (49) 5 Substitute 49 for. 7 5 Simplify. Since the solution satisfies the original equation, the solution is valid. The solution to the radical equation 5 is 49. U3-73 Lesson : Solving Rational and Radical Equations

80 Eample 3 Solve the radical equation for. 1. Isolate the radical epression. The radical epression ( ) the equation. is already isolated on one side of. Raise both sides of the equation to a power to eliminate the radical, then simplify the result. Square both sides of the equation to eliminate the radical epression. Solve one side of the equation for 0. Original equation ( ) ( ) Simplify. Square both sides. + Add to both sides. 0 + Subtract from both sides. The result is a quadratic equation. 3. Solve for by factoring, or by using the quadratic formula. The quadratic appears to be factorable ( 1)( + ) The solutions appear to be or 1. U3-74 Unit 3: Rational and Radical Relationships

81 4. Verify the solution(s). Substitute the potential solutions into the original equation, and check the result. Let s start with. Original equation ( ) ( ) Substitute for. 4 Simplify. The result,, is a false statement. This solution does not satisfy the original equation. is an eample of an etraneous solution, one that results from the process of our solution, but does not satisfy the original equation. Check 1. Original equation (1) (1) Substitute 1 for. 1 1 Simplify. 1 1 The result is a true statement; therefore, this solution satisfies the original equation. Since only the second solution satisfies the original equation, only that solution is valid. The solution to the radical equation is 1. U3-75 Lesson : Solving Rational and Radical Equations

82 Eample 4 Solve the radical equation 7 3 for. 1. Isolate the radical epressions. The equation contains two distinct radical epressions. Each radical epression is alone on one side of the equation, so it should be possible to eliminate the radicals using the methods previously demonstrated.. Raise both sides of the equation to a power to eliminate the radicals, then simplify the result. Square both sides of the equation, then solve algebraically. 7 3 Original equation ( 7 ) ( 3 ) 7 9() The solution appears to be 5. Square both sides. Simplify. 3. Verify the solution(s). Substitute the potential solution into the original equation, and check the result. 7 3 Original equation (5) 7 3 Substitute 5 for Simplify Since the solution satisfies the original equation, the solution is valid. The solution to the radical equation 7 3 is 5. U3-76 Unit 3: Rational and Radical Relationships

83 Eample 5 Solve the radical equation for. 1. Isolate the radical epressions. With three different radical epressions, this equation offers no simple path to isolating the radicals. It is not possible to get each radical on one side of the equation because there just aren t enough sides. In its current format, the most comple radical is alone on one side. The other two simpler radicals are on the other side. While this radical equation is not as ideally isolated as those in the previous eamples, it is as isolated as we need it to be.. Raise both sides of the equation to a power to eliminate the radicals, then simplify the result. Square both sides of the equation, then simplify as much as possible Original equation ( + 5 ) ( 3+ 5) ( + )( + ) + ( ) ( ) Simplify. Square both sides Distribute. ( ) ( ) ( ) ( ) Write each term as the product of the distribution Simplify. 5 5 While we may not have solved the equation yet, we have managed to rewrite it in a simpler format that we can solve algebraically using previously learned processes. U3-77 Lesson : Solving Rational and Radical Equations

84 3. Eliminate the radical from the simplified equation. As in the previous step, square both sides of the equation, then simplify the result. 5 Simplified equation from the previous step ( ) 5 5 Square both sides. Simplify. 0 5 Subtract 5 from both sides. 0 ( 5) Factor the resulting quadratic. We can determine the potential solutions by setting each factor of the equation 0 ( 5) equal to 0: Our two potential solutions are 0 and 5. U3-78 Unit 3: Rational and Radical Relationships

85 4. Verify the solution(s). Substitute the potential solutions into the original equation, and check the result. Let s start with Original equation 5 + (0) 3(0) + 5 Substitute 0 for Simplify. 5 5 The result, 5 5, is a true statement. The solution 0 appears to be valid. Now check Original equation 5 + (5) 3(5) + 5 Substitute 5 for Simplify The result, 5 5, is a true statement. The solution 5 also appears to be valid. Both potential solutions satisfy the original equation; therefore, both are valid. The solutions to the radical equation are 0 and 5. U3-79 Lesson : Solving Rational and Radical Equations

86 Eample 6 Solve the radical equation for. 1. Isolate the radical epression. Not every radical represents a square root. Higher roots are represented with a radical symbol and a root on the shelf of the radical symbol. The solution process, however, is essentially the same Original equation Subtract 10 from both sides.. Raise both sides of the equation to a power to eliminate the radical, then simplify the result. Eliminating the radical this time requires raising each side to the third power (i.e., cubing each side), rather than squaring each side. 3 ( ) ( 3) Cube both sides. Simplify. The solution appears to be Verify the solution(s). Substitute the potential solution into the original equation, and check the result Original equation 3 3( 11) Substitute 11 for Simplify Since the solution satisfies the original equation, the solution is valid. The solution to the radical equation is 11. U3-80 Unit 3: Rational and Radical Relationships

87 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations Practice 3..3: Solving Radical Equations For problems 1 3, solve each radical equation for. Round answers to the nearest thousandth Use the given information and what you know about radical equations to solve problems Round answers to the nearest thousandth. 4. A right triangle has a hypotenuse measuring 05 mm. If one leg measures 133 mm, what is the length of the other leg? 5. If you attach a ball to a piece of string and whirl it over your head, a tension force in the string will resist the ball s tendency to fly away from you. The magnitude of that force depends on the mass of the ball, the length of the string, and the speed at which you move the ball. This relationship is described by the formula for mv force: F. The mass m of the ball is measured in kilograms, and the length r of the string is r (meters), since the string serves as the radius of a circle. If the velocity, v, is then measured in meters per second, the force, F, will be in Newtons, a commonly used metric unit of force represented by the letter N. With what velocity would you be spinning a 1.5-kilogram ball if the force on a 1.5-meterlong string is 4.8 N? continued U3-81 Lesson : Solving Rational and Radical Equations

88 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations 6. As a sailboat hull pushes the water aside, a wave forms along the hull. If that wave becomes too big, it pulls back on the boat, slowing it down. There is a maimum speed that any hull can maintain before the force eerted by the wave becomes too large to overcome, and the boat capsizes. This maimum speed can be determined using the formula v 1.34 L, where v represents the sailboat s velocity through the water in knots (nautical miles per hour) and L is the length along the waterline where the boat and the water surface are in contact. If a team of designers wants their newest sailboat to be able to sail at least 6.5 knots, what is the minimum waterline length the boat can have? 7. The velocity, v, at which a falling object hits the ground (without air resistance) can be found using the formula v gh, where v is the velocity of the object in meters per second (m/s), h is its height in meters, and g is the object s acceleration due to gravity (in m/s ). Find an object s acceleration due to gravity if it falls 40 meters and hits the ground at a velocity of 8 m/s. 8. The volume, V, of a pyramid with a square base is related to the height, h, of the pyramid, and the length of one side of the square base, s, by the formula 3V s. Find the length of one side of the square base of a pyramid that is h 5 cm high and has a total volume of 540 cm. U3-8 Unit 3: Rational and Radical Relationships continued

89 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson : Solving Rational and Radical Equations 9. The amount of time it takes a pendulum to swing back and forth along its path is known as its period, and is often symbolized by the lowercase Greek letter tau, τ. The value of τ is dependent on the pendulum s acceleration due to gravity, g, L and the pendulum s length, L, and is found using the formula τ π. g A large science and industry museum features a multi-story pendulum that has a period of 9 seconds. That is, the pendulum takes 9 seconds to swing from one end of its arc to the other. If the pendulum s acceleration due to gravity, g, is 9.8 m/s, how many meters long is the museum s pendulum? 10. The amount of time it takes a planet to complete one orbit around its central star is called a period of revolution, τ. The period of revolution depends on the planet s distance from its star, measured in astronomical units (AU); 1 AU 93 million miles (the mean distance between the Earth and the Sun). Kepler s Third Law of Planetary Motion quantifies this relationship, stating that the ratio of the square of the period of revolution and the cube of the planet s distance from its star, r, is a constant. Venus has an orbital period of just years, and orbits the Sun at a distance of 0.7 AU. If Mars has an orbital period of ( 1.88 years, how far is Mars from the Sun? Use the proportion τ ) τ V ( M ) 3 3 ( rv ) ( rm ), where the subscripts V and M represent the planets Venus and Mars, to find the answer. U3-83 Lesson : Solving Rational and Radical Equations

90 UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions Common Core Georgia Performance Standards MCC9 1.A.CED. MCC9 1.F.IF.4 MCC9 1.F.IF.5 MCC9 1.F.IF.7d (+) MCC9 1.A.REI.11 Essential Questions 1. How can you determine from a data set whether an equation relating two variables can be written as a rational equation?. What are the characteristics of the graph of a rational function and what is the significance of each characteristic? 3. How do you calculate the zeros for a rational function? 4. How can a system of two rational equations have no common solution? WORDS TO KNOW asymptote degree domain etreme value factor factoring U3-84 Unit 3: Rational and Radical Relationships a line that a function gets closer and closer to, but never crosses or touches the greatest eponent attached to the variables in the polynomial the set of all input values (-values) that satisfy the given function without restriction a point on the graph of a rational function that occurs at or near the endpoints of its domain or range one of two or more numbers or epressions that when multiplied produce a given product the process of separating an epression into two or more prime factors

91 horizontal asymptote quadratic formula a straight line of the form y a which occurs at the maimum or minimum of the graph of a rational function; the graph of the rational function approaches this line but never touches it a formula that states that the solutions of a quadratic equation of the form a + b + c 0 are given by b b ac ± 4 a range the set of all outputs of a function; the set of y-values that are valid for the function p( ) rational function a function that can be written in the form f ( ) q( ), where p() and q() are polynomials and q() 0 real zero root slant asymptote solution(s) to a system vertical asymptote zero a solution to an equation that can be graphed on a coordinate plane of real numbers the -intercept of a function; also known as zero an asymptote of a graph of the form y a + b, where a 0 for a system of two equations f() and g(), the value(s) of for which f() g() and that are in the domains of the functions; on a graph, the solutions are the points at which the equations intersect a straight line of the form b which the graph of a rational function approaches but never touches the -intercept of a function; also known as root U3-85 Lesson 3: Graphing Rational Functions

92 Recommended Resources Coolmath.com. Graphing Rational Functions. This site includes accessible but etensive coverage of asymptotes, including slant asymptotes. Mathway. Math Problem Solver. This graphing utility allows users to plot a variety of mathematical functions, including rational functions. Users can choose from a variety of mathematical symbolic-manipulation tools and a pop-up menu of activities to perform with functions, such as finding zeros. Purplemath.com. Graphing Rational Functions: Introduction. This site provides a detailed eplanation of how to graph rational functions, with graphing calculator simulations that are helpful for visual learners. ThatTutorGuy. Asymptotes and Rational Functions. This site offers streaming video for instruction on horizontal, vertical, and slant asymptotes, as well as eamples of how to graph rational functions. U3-86 Unit 3: Rational and Radical Relationships

93 Lesson 3.3.1: Creating Rational Equations in Two Variables Introduction Though we may be unaware of the role they play, rational functions are an important part of daily life. Picture a patient in a doctor s office who needs to be put on blood pressure medication. How does the doctor know the correct dosage of medication, and how frequently it should be taken, for that person? Or, imagine a factory that manufactures bicycles. How does the production manager determine the optimal amount of bicycles to produce that will yield the highest amount of profit and yet minimize production costs? Across many fields, rational functions are used regularly to help make important decisions. Key Concepts A rational function is the quotient of two polynomial functions. p( ) In mathematical terms, the rational function f() is defined as f ( ) q( ), where p() and q() are polynomial functions and q() 0. 1 For eample, the rational function f ( ) 4 consists of the polynomial functions g() 1 and h() 4. Recall that the domain of a function is the set of all input values (-values) that satisfy the given function without restriction. The degree of a polynomial is the greatest eponent attached to the variables in the polynomial. The domain of a first-degree rational function is found by identifying the asymptotes. An asymptote is a line that a function gets closer and closer to, but never crosses or touches. A vertical asymptote is a straight line of the form b which the graph of a rational function approaches but never touches. A horizontal asymptote is a straight line of the form y a that occurs at the maimum or minimum of the graph of a rational function. The domain will contain every real number ecept for the point at which the vertical asymptote falls. Recall that the domain of a function can be epressed in interval notation. That is, the domain is written in the form of (a, b), where a and b are the endpoints of the interval. Depending on the values of the interval, the notation may change. U3-87 Lesson 3: Graphing Rational Functions

94 The table that follows summarizes how to note the intervals for various situations. Notice that a parenthesis net to a number indicates that the number is not included in the solution set; however, a bracket indicates that the number is part of the solution set. Interval notation Eample Description (a, b) (, 10) All numbers between and 10; endpoints are not included. [a, b] [, 10] All numbers between and 10; endpoints are included. (a, b] (, 10] All numbers between and 10; is not included, but 10 is included. [a, b) [, 10) All numbers between and 10; is included, but 10 is not included. The zeros of a rational function are the -intercepts of the function; in other words, the domain value(s) for which f() 0. Zeros are also known as roots. 1 In the eample f ( ) 4, the domain values are found by assuming that the denominator 4 0. This means that the numerator 1 0 if f() 0. Therefore, the zero for this function is found by solving the equation 1 0 for. The result is 1. Thus, the graph intersects the -ais at the point 1,0. The values of the domain for which a rational function is undefined are found by determining the values of the domain for which the denominator is equal 1 to 0. In the eample f ( ) 4, the values of the domain for which the function is undefined are found by solving the equation 4 0 for, which gives ±. Therefore, the function is undefined at the domain values of and. U3-88 Unit 3: Rational and Radical Relationships

95 The values of the domain at which a rational function is undefined are also the equations of its vertical asymptotes. The vertical asymptotes do not include the 1 points that make the function true. In the eample, f ( ), the asymptotes 4 are and, because () () 1 () 4 3 f, which is undefined, and ( 0 f ) 1 ( ) 5, which is also undefined. ( ) 4 0 Another way to identify the asymptotes and zeros of a rational function is to look at a table of values generated on a graphing calculator. Follow the steps appropriate to your calculator model. On a TI-83/84: Step 1: Press [Y]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y 1, use your keypad to enter values for the function. Use [X, T, θ, n] for and [ ] for any eponents. Step 3: To view a table of values for the function, press [ND][GRAPH]. Step 4: Arrow up and down the list of ordered pairs to find the zeros and the domain values for which the function is undefined. On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function to the right of f1() and press [enter]. Step 4: To see a table of values, press [menu], then use the arrow keys to select : View, then 9: Show Table. Step 5: Arrow up and down the list of ordered pairs to find the zeros and the domain values for which the function is undefined. U3-89 Lesson 3: Graphing Rational Functions

96 Guided Practice Eample 1 Show that the function f ( ) is a rational function. 1. Determine if the given function is a rational function in its current form. In order for f() to be a rational function, it has to be a quotient of two polynomials. The given function consists of a constant, 1, and a rational epression, ; therefore, the function is not presented in the + 3 form of a rational function.. If possible, rewrite the function so that it is a rational function. A rational function must be a quotient of two polynomials. Rewrite the constant, 1, as a quotient of two polynomials. f ( ) Original function 1 f ( ) Rewrite the constant, 1, as a fraction. 1( + 3) f ( ) + 1( 3) + 3 Multiply by the common factor. ( + 3) f ( ) + ( 3) + 3 Simplify. f + 3 ( ) + 3 Subtract the fractions. f ( ) Simplify. U3-90 Unit 3: Rational and Radical Relationships

97 3. Verify that the rewritten function is a rational function. In order for f() to be a rational function, it must be a quotient of two polynomial functions. The binomial + 1 is a polynomial function. The binomial + 3 is a polynomial function. 4. Draw conclusions. The original function can be rewritten as a rational function. Therefore, the original function f ( ) 1 is a rational + 3 function. Eample Write a rational function, h(), using the functions f() 1 and g() Write the definition of a rational function using f(), g(), and h(). There are two possible ways to write the rational function, h(), in terms of f() and g(). f ( ) One way is h( ), where f() is the numerator and g() is the g( ) g( ) denominator.the second way is h( ), where the positions of f ( ) f() and g() have been swapped that is, g() is the numerator and f() is the denominator. U3-91 Lesson 3: Graphing Rational Functions

98 . Write each possibility for the rational function, h(), in terms of the given functions. Use the definition of a rational function and the given polynomials to write each possibility. First possibility: f ( ) h( ) g( ) h( ) ( ) Definition of a rational function, h() 1 Substitute 1 for f() and 4 4 for g(). ( ) Second possibility: g( ) h( ) f ( ) h( ) ( ) 4 1 ( ) Definition of a rational function, h() Substitute 4 for g() and 1 for f(). 1 The two possibilities for the rational function h() are h( ) 4 4 and h( ). 1 U3-9 Unit 3: Rational and Radical Relationships

99 3. Verify that both possible functions for h() are rational functions. Simplify each function for h(). 1 h( ) First possible rational function 4 1 Rewrite the fraction as a subtraction h( ) 4 4 of two fractions. 1 h( ) 4 Simplify. 1 1 h( ) 4 By definition, this possibility is not a rational function because in its simplest form, it is not the quotient of two polynomials. 4 Look at the second possible rational function, h( ). 1 This possibility cannot be simplified any further. By definition, this possibility is a rational function because in its simplest form, it is the quotient of two polynomials. Therefore, the only rational function that can be written for h() 4 using the functions f() 1 and g() 4 is h( ). 1 U3-93 Lesson 3: Graphing Rational Functions

100 Eample 3 Write a rational function that has a zero of 1 and is undefined at. 1. Use the definition of a rational function to write the simplest general polynomials for the numerator and the denominator. f ( ) Recall the definition of a rational function is h( ) g( ). The simplest polynomials for the numerator and the denominator have the form f() + a and g() + b. Therefore, h ( f ) ( ) a g + +. ( ) b. Determine the polynomial for the numerator. Use the zero of the function to determine the numerator. We are given that a zero of this function is at 1; therefore, f( 1) 0. f() + a Numerator of the rational function (0) ( 1) + a Substitute 0 for f() and 1 for. a 1 Add 1 to each side to solve for a. Now that we know a 1, substitute 1 for a in the numerator of the rational function. f() + a Numerator of the rational function f() + (1) Substitute 1 for a. f() + 1 Simplify. Therefore, f() + 1, and + 1 is the polynomial for the numerator. U3-94 Unit 3: Rational and Radical Relationships

101 3. Determine the polynomial for the denominator. The denominator is undefined for values that would result in a 0 in the denominator. We are given that the function is undefined at ; therefore, g() 0. g() + b Denominator of the rational function (0) () + b Substitute 0 for g() and for. b Subtract from each side to solve for b. Now that we know b, substitute for b in the denominator of the rational function. g() + b Denominator of the rational function g() + ( ) Substitute for b. g() Simplify. Therefore, g(), and is the polynomial for the denominator. 4. Write the rational function using the polynomials determined for the numerator and denominator. Substitute the functions determined in steps and 3 for the numerator and denominator of the definition of a rational function. f ( ) h( ) g( ) ( + 1) h( ) ( ) Definition of a rational function Substitute + 1 for f() and for g(). This rational function is already in simplest form; therefore, the rational function, h( ) + 1, has a zero of 1 and is undefined at. U3-95 Lesson 3: Graphing Rational Functions

102 Eample 4 ( + 1) Identify the vertical asymptote(s) for the rational function f ( ) ( )( + 3)( 4). 1. Identify the values of for which the rational function is undefined. The rational function is undefined for values that would result in a 0 in the denominator. Therefore, any factors in the denominator (, + 3, and 4) that are equal to 0 would result in a value of that gives an undefined value of f(). Set each factor equal to 0 and solve the equation for to determine each value that makes f() undefined The values of, 3, and 4 result in an undefined rational function.. List the vertical asymptotes for the rational function. The vertical asymptotes are located at the values of for which the function is undefined and are of the form a. Therefore, the vertical asymptotes of the rational function, ( + 1) f ( ), are, 3, and 4. ( )( + 3)( 4) U3-96 Unit 3: Rational and Radical Relationships

103 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions Practice 3.3.1: Creating Rational Equations in Two Variables For problems 1 3, write a rational function, f(), in simplest form using the two given functions. 1. g() 3; h() + 3. g() 7 ; h() g() 1; h( ) 3 For problems 4 7, complete each problem involving rational functions. 4. Peyton buys a smartphone for $300. She signs a contract for wireless and voice services that will cost $45 per month. Write a rational function for the monthly cost of the smartphone and the data plan, C(m), over m months. 5. Write a rational function, f(n), for 3 times a number n less twice its reciprocal. 6. Risa and Madison are fiber-optic cable installers. Risa takes 1 hour less to install a floor of a new office building than Madison does. The number of hours (y) it takes them to install a floor of the office building when working together is given by the equation +, where is the number of hours Madison takes 1 y when working alone and 1 is how long Risa takes when working alone. Show that y can be written as a rational equation. continued U3-97 Lesson 3: Graphing Rational Functions

104 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions 7. Xavier bikes 100 kilometers in the outbound leg of the Summit to Surf bike ride for t hours at a rate of r 1 kilometers per hour. His time on the return trip is hours longer than on the first leg of the ride at a speed of r. If the speed is distance (100 km) divided by time, show that the average speed r for the round trip can be written as a rational equation. (Hint: Write a fraction for r 1 and r. What is the average of r 1 and r?) g( ) For problems 8 10, write a rational function of the form f ( ) given vertical asymptote(s) and zero(s). h( ) 8. vertical asymptotes: 0, 1; zero: 1 that has the 9. vertical asymptote: ; zeros: 4, vertical asymptotes: 10, 0; zeros: 0.1, 0.01 U3-98 Unit 3: Rational and Radical Relationships

105 Lesson 3.3.: Graphing Rational Functions Introduction The graphs of rational functions can be sketched by knowing how to calculate horizontal and/or vertical asymptotes of the function and its zeros. The sketch of the graph can be improved by plotting points with - and y-values that are at or near the limit of those domain and range values. Graphing calculators can be used to sketch the graph of a rational function, and can also provide the user with a table of - and y-values from which these critical features of the graph can be read. Key Concepts There are several key features of the graph of a rational function that are helpful when creating a sketch, including its zeros, horizontal and vertical asymptotes, domain, range, and etreme values. The range is the set of all outputs of a function; that is, the set of y-values that are valid for the function. Recall that the zeros of the rational function are the domain value(s) for which g( ) f() 0. If the rational function f() 0 is represented by f ( ) h( ), then g() 0 and h() 0. ( a)( b) If a rational function f() 0 is represented by f ( ) ( c)( d), then ( a)( b) 0 and ( c)( d) 0. The zeros are represented by the points (a, 0), (b, 0), etc., on the graph. Asymptotes are straight lines on the graph of a rational function that form the boundaries for parts of the graph. In some cases, the domain values of the asymptotes are the domain values for which the rational function is undefined. In other cases, the domain values of the asymptotes form the maimum and/ or minimum value(s) of the rational function. Vertical asymptotes are straight lines of the form a. For eample, the graph 4 of the rational function f ( ) 4 has vertical asymptotes at ±, which are solutions to the equation 4 0. The function f() is undefined at these domain values. U3-99 Lesson 3: Graphing Rational Functions

106 Horizontal asymptotes are straight lines of the form y b. For eample, the 4 7 graph of the function g( ) has a horizontal asymptote at y 4. As seen in the following graph, g() approaches y y U3-100 Unit 3: Rational and Radical Relationships

107 Slant asymptotes are asymptotes of a graph of the form y a + b, where a 0. A rational function that has a numerator with a degree that is eactly 1 more than the degree of the denominator will have a slant asymptote. 10 y To find the slant asymptote, use polynomial long division. The result of dividing the numerator by the denominator is the equation of the slant asymptote. Rational functions may or may not have a vertical asymptote, but will always have either a horizontal or a slant asymptote. A rational function will never have both a horizontal asymptote and a slant asymptote. For the graph of a rational function, it can be useful to plot several points that represent values of and y that are far away from zeros and asymptotes. Such a value is known as an etreme value a point on the graph of a rational function that occurs at or near the endpoints of its domain or range. These so-called etreme values will ensure that the sketch will be accurate over the domain and range of the function. For eample, consider the function f ( ). The domain of the function 1 is all numbers ecept for 1 since the function is undefined for this value of. For values of very close to 1, the value of the function is large. For U3-101 Lesson 3: Graphing Rational Functions

108 1.01, f() 101, and for 0.99, f() 99. But, for 100, f() 1.1, and for 100, f() Sketching the points (1.01, 101), (0.99, 99), (100, 1.1), and ( 100, 0.99) shows that f() has a horizontal asymptote at y 1 and a vertical asymptote at y An eample of a function without a vertical asymptote is g( ) + 1 because there are no values of that will result in a denominator that equals 0. However, g() does have horizontal asymptotes at ±1. The etreme values of g() occur at ±1: g (1) 6(1) and g ( 1) 6( 1) ( 1) The etreme values of occur over the domain (, ). If 100, g(100) 0.06; if 100, g( 100) In other words, the graph of g() approaches 0 as approaches ±100. These ideas can be eplored and etended using a graphing calculator. Refer to the previous sub-lesson for directions specific to your calculator model. U3-10 Unit 3: Rational and Radical Relationships

109 Guided Practice 3.3. Eample 1 Sketch the graph of the rational function f ( ). Include points on both sides of any asymptotes Find the value of f(0). The point for which 0 is the y-intercepts of the graph, and has the form (0, f(0)). The function value f(0) may or may not be equal to 0. (0) (0) (0) + f 1 The point on the graph representing f(0) is (0, 1).. Find the value(s) of for which f() 0. The values of for which f() 0 are the zeros of the function. If f() 0, + 0, so 0, which means that ±. Therefore, the zeros of f() are given by (,0 ) and ( ),0. 3. Write the equation(s) for the vertical asymptote(s) of f(), if any. Vertical asymptotes of f(), if any, occur at values of which make the denominator equal 0. In this case, setting the denominator equal to 0 and solving for ( + 0) will not yield values of that are within the domain of the function, which is (, ). Therefore, there are no vertical asymptotes for f(). U3-103 Lesson 3: Graphing Rational Functions

110 4. Write the equation(s) of the horizontal asymptote(s) of f(), if any. The horizontal asymptotes are of the form y b. One way to find b is to solve the rational function for the domain variable,. f ( ) + f ( ) 1 + f() ( + ) 1 ( ) Original function Rewrite the function as two fractions. Cross multiply. f() + f() f() + f() [f() + 1] [1 f()] [ f ] 1 ( ) f ( ) + 1 [ f ] 1 ( ) f ( ) + 1 Simplify. Use subtraction and addition to group like terms ( and f()) on the same sides of the equation. Simplify. Divide. Solve for by taking the square root of both sides. This rational epression for is undefined when the denominator f() + 1 0, so the horizontal asymptote is y To prepare for sketching the graph of f(), pick values for that meet the conditions for f() 0 from step 1, along with some values of <, < <, and <. Then, calculate the value of f() for each value of. List the calculations in a table. Let, 1, 0, 1,, and 3 and solve for the corresponding f() value in the original function f() U3-104 Unit 3: Rational and Radical Relationships

111 6. List the points calculated in steps 1 and for the y-intercepts and zeros of the function in a table. 0 f() Use your results to sketch the graph of the function f ( ). + To sketch the graph, first plot the points listed in the tables in steps 5 and 6. The conditions listed in steps 5 and 6 will ensure that all of the regions on the coordinate plane that contain parts of the rational function s graph will be represented. Use dotted lines to sketch the horizontal asymptote, y 1. Use a graphing calculator to check your work. Refer to the appropriate calculator directions for your calculator model as provided in the Key Concepts in the previous sub-lesson. Your graph should resemble the following: 10 y U3-105 Lesson 3: Graphing Rational Functions

112 Eample Sketch the graph of the rational function f ( ) + 1, which represents the sum of a number and its reciprocal. 1. Find the value of f(0). The point for which 0 is the y-intercept of the graph, and has the form (0, f(0)). f (0) 0+ 1, which is undefined. 0 There is no point on the graph representing f(0). Therefore, 0 is a vertical asymptote.. Find the value(s) of for which f() 0. The values of for which f() 0 are the zeros of the function. Rewrite the function as one fraction. 1 f ( ) + Original function 1 f ( ) + 1 Write the term as a fraction. 1 f ( ) + 1 Multiply 1 by (which is equivalent to 1). 1 f ( ) + Multiply. + 1 f ( ) Simplify. If f(0) 0, then (assuming that 0 from step 1). The equation has no solution on the interval (, ), so this rational function has no zeros. U3-106 Unit 3: Rational and Radical Relationships

113 3. Find any other asymptote(s) by solving the function for. Rearrange and solve the function for to reveal what values of f() are possible for the function. This will also indicate any values that the function cannot have. + 1 f ( ) f ( ) f() () 1 ( + 1) Function from the previous step Rewrite the function as two fractions. Cross multiply. [f()] + 1 Simplify. Use subtraction to collect all like terms 0 [f()] + 1 on one side of the equation. Two values of f() that allow this quadratic equation to be solved are f() ±. Values of f() and f() result in domain values (values of ) that are in the domain of the function. However, values of f() such that < f() < cannot be achieved with the domain values of the function. + 1 Notice that the function f ( ) has a numerator that is eactly one degree higher than that of the denominator. This indicates a slant asymptote. To determine the equation of the slant asymptote, use polynomial long division to divide + 1 by. ) +1 1 The slant asymptote is the polynomial part of the answer,, not the remainder. Therefore, the slant asymptote is y. U3-107 Lesson 3: Graphing Rational Functions

114 4. Use the results of steps 1 3 to create a table of at least 6 points that can be used to sketch the graph. Recall from step 1 that there is a vertical asymptote at 0. There is also a gap in the range of function values between f() and f(). Points on either side of these range values should begin to suggest the shape of the graph. Therefore, choose values of that result in range values close to these function values and solve for f(). Summarize your results in a table f() Use your results to sketch the graph of the function f ( ) + 1. To sketch the graph, first plot the points listed in the table in step 4. Use a dotted line to sketch the vertical asymptote, 0, and the slant asymptote, y. Use a graphing calculator to check your work. Refer to the appropriate calculator directions for your calculator model as provided in the Key Concepts in the previous sub-lesson. Your graph should resemble the following: U3-108 Unit 3: Rational and Radical Relationships

115 Eample 3 5 Sketch the graph of the rational function f ( ). Include points on both sides of any asymptotes Find the value of f(0). The point for which 0 is the y-intercept of the graph, and has the form (0, f(0)). The function value f(0) may or may not be equal to 0. 5 f ( ) 9 Original function (0) 5 f (0) (0) 9 Substitute 0 for. 0 f (0) Simplify. 5 f (0) 9 The value of f(0) is 5 9. Therefore, the point on the graph representing f(0) is 0, Find the value(s) of for which f() 0. The values of for which f() 0 are the zeros of the function. If f() 0, and the denominator ( 9) cannot equal 0, then the numerator ( 5) must equal 0. Determine the values that make the numerator equal to Numerator Add 5 to both sides. 5 Take the square root of both sides. ±5 Simplify. The values of that make the numerator equal 0 are 5 and 5. Therefore, the zeros of f() are given by (5, 0) and ( 5, 0). U3-109 Lesson 3: Graphing Rational Functions

116 3. Write the equation(s) for the vertical asymptote(s) of f(), if any eist. The values of for the vertical asymptote(s) occur when the denominator is equal to 0. Determine the values that make the denominator equal to Denominator Add 9 to both sides. 9 Take the square root of both sides. ±3 Simplify. The values of that make the denominator equal 0 are 3 and 3. Therefore, the vertical asymptotes are 3 and Write the equation(s) of the horizontal asymptote(s) of f(), if any. The horizontal asymptotes are of the form y b. One way to find b is to solve the rational function for the domain variable,. 5 f ( ) Original function 9 f ( ) 5 Rewrite the function as two 1 9 fractions. f() ( 9) 1 ( 5) f() 9 f() 5 f() 9 f() 5 [f() 1] 9 f() 5 9 f ( ) 5 f ( ) 1 9 f ( ) 5 f ( ) 1 Cross multiply. Simplify. Use subtraction and addition to group like terms on the same sides of the equation. Simplify. Divide. Isolate by taking the square root of both sides. This rational epression for is undefined when f() 0, so the horizontal asymptote is y 1. U3-110 Unit 3: Rational and Radical Relationships

117 5. To prepare for the sketch of the graph for f(), pick values for that meet the conditions > 3, 3 < < 3, and < 3. Then, calculate the value of f() for each value of. List the calculations in a table. Let 5, 4,, 1, 0, 1,, 4, and 5. Substitute each value of into the original equation and solve for f() f() Finally, list the points calculated in steps 1 and for the y-intercepts and zeros of the function and in a table f() Use your results to sketch the graph of the function f ( ) 9 To sketch the graph, first plot the points listed in the tables in steps 5 and 6.. The conditions listed in steps 5 and 6 will ensure that all of the regions on the coordinate plane that contain parts of the rational function s graph will be represented. Use dotted lines to sketch the horizontal asymptote, 1, and the vertical asymptotes, y 3 and y 3. Use a graphing calculator to check your work. Refer to the appropriate calculator directions for your calculator model as provided in the Key Concepts in the previous sub-lesson U3-111 Lesson 3: Graphing Rational Functions

118 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions Practice 3.3.: Graphing Rational Functions For problems 1 4, sketch the graph of the given rational function on a coordinate plane. Include asymptotes, zeros, and etreme points, if any f ( ) g( ) f ( ) 3 4. g ( ) + U3-11 Unit 3: Rational and Radical Relationships continued

119 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions For problems 5 and 6, name the horizontal, slant, and/or vertical asymptote(s) and the zero(s) of each graphed rational function continued U3-113 Lesson 3: Graphing Rational Functions

120 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions Use the following information to complete problems A distributor of vegetable oil uses cylindrical containers that have a volume of 0.5 m 3. The distributor would like to find out what container radius would result in the smallest container surface area. 7. Use the formulas for the surface area of a cylinder, S(r, h) πr + πrh, and the volume of a cylinder, V(r, h) πr h, to write a rational function for the surface area in terms of the radius r of the cylindrical containers. 8. Sketch a graph on a coordinate plane for all of the surface areas that are possible for different radii. 9. Describe how the surface area of the container changes as the radius of the container changes from less than 1 meter to greater than 1 meter. 10. What is the significance of the parts of the rational function graph that have domain values given by r 0? U3-114 Unit 3: Rational and Radical Relationships

121 Lesson 3.3.3: Finding the Zeros Introduction The zero(s) of a rational function can be identified in different ways, some of which involve manipulation of the terms in the function. Factoring, finding common denominators, and using the quadratic formula are methods that can be used in this search. A graph of the rational function can also be used to visually identify the location of zeros, or to indicate that they do not eist. A data table accompanying a graph can be used to approimate zeros of rational functions when the calculation techniques are too involved, or when those techniques increase the chances of making algebraic or arithmetic errors. Key Concepts Recall that the zeros of a rational function are the domain value(s) for which g( ) f() 0. If the rational function f() 0 is represented by f ( ) h( ), then g() 0 and h() 0. ( a)( b) If a rational function f() 0 is represented by f ( ) ( c)( d), then ( a)( b) 0 and ( c)( d) 0. The zeros are represented by the points (a, 0), (b, 0), etc., on the graph. Recall that a factor is one of two or more numbers or epressions that when multiplied produce a given product. Factoring, or the process of separating an epression into two or more prime factors, can sometimes be used to simplify the numerator of a rational function so that the zero(s) can be identified. For eample, the numerator of the rational (+ 1)(3 5) function f ( ) can be factored to give f ( ). 9 9 The zeros can be found by solving the equations and for. Sometimes the factoring involves a monomial term. For eample, the 3 + numerator of the function g( ) can be factored by removing the + 5 ( + 1) monomial out of the numerator to yield g( ). Then, to find + 5 the zeros, solve the equations 0 and for. U3-115 Lesson 3: Graphing Rational Functions

122 Recall that the quadratic formula can be used to solve some seconddegree polynomials, and also when a polynomial will not factor easily. For a quadratic equation of the form a + b + c 0, the value of is given by b b ac ± 4 +. In the rational equation h( ), the quadratic a + 3 formula can be used to show that the value of in the numerator, +, is ± 1 1 4()( 1) 1 17, which is equal to ±. This means that the zeros of + are + and It is also useful to know if all of the zeros of a rational function have been found. This can be determined by looking at the degree of the numerator s polynomial function. The degree of a polynomial is the highest power of the variable(s) in an epression. For eample, in the rational function f ( ), the degree of the numerator s 4 polynomial is 3. Factoring the numerator shows that there are three zeros: ( 3) ( 3) ( 1)( 3) ( 1)( + 1)( 3) f ( ) The zeros are 1, 1, and 3. In other cases of third-degree numerator polynomials, only one zero can be identified. In these cases, the other two zeros are not real numbers. Zeros that are not real numbers occur in pairs, which can help you identify how many real zeros a numerator s polynomial can have. The real zeros are solutions to an equation that can be graphed on a coordinate plane of real numbers. For eample, a numerator s polynomial of degree 4 can have 4 real zeros, real zeros, or no real zeros. The polynomial still has 4 zeros, but not all of them are real and will not appear on the graph of the function. U3-116 Unit 3: Rational and Radical Relationships

123 If calculations of zeros by factoring and/or using the quadratic formula prove to be difficult, then a graphing calculator or graphing utility software can be used. These tools provide a visual approach to identifying zeros and the values of the - and y-coordinates of a rational function. To find the zeros of a rational function, follow the directions appropriate to your calculator model. On a TI-83/84: Step 1: Press [Y]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y 1, use your keypad to enter values for the function. Use [X, T, θ, n] for and [ ] for any eponents. Press [GRAPH]. Step 3: To adjust the graph s aes so the graph is easier to study, press [WINDOW], and change the ais settings for Xmin, Xma, Ymin, and Yma as needed. Step 4: To find the zeros, press [ND][GRAPH] to display a table of values. Step 5: To find where the zeros are located, find the places where the value of Y 1 changes from positive (+) to negative ( ). Step 6: To narrow down the zeros even more, press [TRACE]. Use the arrow keys to move the cursor along the two parts of the graph until the cursor crosses the -ais and the Y 1 values change from positive to negative. U3-117 Lesson 3: Graphing Rational Functions

124 On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function to the right of f1() and press [enter]. Step 4: To adjust the - and y-ais scales on the window, press [menu] and select 4: Window and then 1: Window Settings. Enter each setting as needed. Tab to OK and press [enter]. Step 5: To see a table of values, press [menu] and scroll down to : View, then 9: Show Table. Step 6: To find where the zeros are located, arrow up and down the table to find the places where the value of Y 1 changes from positive (+) to negative ( ). Step 7: To narrow down the zeros even more, use the Trace tool. Press [ctrl][tab] to reactivate the graph, then [ctrl][t] to hide the table. Then, press [menu] and use the arrow keys to select 5: Trace, then 1: Graph Trace. Press [enter]. Step 8: Use the arrow keys to move the cursor along the two parts of the graph until the cursor crosses the -ais. At that point, the symbol zero appears, indicating the closest approimation of one of the solutions. U3-118 Unit 3: Rational and Radical Relationships

125 Guided Practice Eample 1 3 Find the zero(s) of the rational function f + ( ) Try to factor the quadratic function in the numerator before trying other methods of finding the zeros. There are no common terms in the numerator, so the numerator may be a product of two binomials... Determine if the numerator is the product of two binomials. Reorder the numerator and denominator in descending order according to degree. f ( ) can be rewritten as 4 3+ f ( ) Net, list the factors of 3 and 8. The factors of 3 are 1 and 3, and 1 and 3. The factors of 8 are 1 and 8, 8 and 1, and 4, and and 4. Net, try to find binomials with a product that results in a middle term of. For eample, when multiplied, the binomials 3 8 and 1 + yield a middle term of 3 8 or 5. Continue checking binomial pairs until the correct pair is found. The correct pair of factors is 3 4 and 1 +, because when multiplied, the resulting middle term is or. U3-119 Lesson 3: Graphing Rational Functions

126 3. Once the correct pair of binomial factors is found, set each factor equal to 0 and solve. The factors of the binomial are 3 4 and 1 +. Set each factor equal to 0 and solve for Set the first factor equal to Add 4 to both sides Solve for Set the second factor equal to Solve for. Subtract 1 from both sides. Any zeros of the numerator will also be zeros of the rational function if they are not also zeros of the denominator. Check that the zeros of the numerator, found above, do not also make the denominator equal 0. Substitute 0.5 and 0.75 for and simplify ( 0.5) 3( 0.5) + 4 (0.75) 3(0.75) Substituting each of the zeros into the denominator does not make the denominator equal 0; therefore, the zeros of the rational function f ( ) are 0.75 and U3-10 Unit 3: Rational and Radical Relationships

127 Eample 3 Find the zero(s) of the rational function g( ) Try to factor the quadratic function in the numerator before trying other methods of finding the zeros. There are no common terms in the numerator, so the numerator may be a product of two binomials.. Determine if the numerator is the product of the two binomials. First, list the factors of 1 and 3, since the coefficient of the term is 1. The factors of 1 are 1 and 1, and 1 and 1. The factors of 3 are 1 and 3, and 1 and 3. Net, try to find binomials with a product that results in a middle term of. There is no combination of binomials that when multiplied yield a product of. 3. Use the quadratic formula to find the solutions of the equation 3 0. The numerator, 3, is of the form a + b + c. b b ac Use the quadratic formula, ± 4, to solve for. Let a 1, b 1, and c 3. a b b ac ± 4 a ± ( 1) ( 1) 4(1)( 3) (1) 1± ± Quadratic formula Substitute 1 for a, 1 for b, and 3 for c. Simplify. (continued) U3-11 Lesson 3: Graphing Rational Functions

128 Any zeros of the numerator will also be zeros of the rational function if they are not also zeros of the denominator. Check that the zeros of the numerator do not also make the denominator equal 0. Substitute each of the zeros for and simplify Substituting each of the zeros into the denominator does not make the denominator equal 0; therefore, the zeros of the rational 3 function g( ) are and Eample Find the zero(s) of the rational function h( ) Try to factor the quadratic function in the numerator before trying other methods of finding the zeros. Each term of the numerator s trinomial contains the common factor, so the numerator can be factored Original numerator ( 3 + 1) Factor out. A factor of indicates that one of the zeros of the function is 0. U3-1 Unit 3: Rational and Radical Relationships

129 . Determine if there is an obvious zero for the factor of Sometimes a quick inspection of the other factor(s) in the numerator will reveal another zero. It can be seen that a value of 1 results in 0, as shown Set the second factor equal to 0. (1) 3 (1) Substitute 1 for Simplify. 0 0 Therefore, in this case, 1 is also a zero. 3. Look for other factors of the numerator by determining if other binomials can be found. In this eample, the zero 1 can be made into a factor of 1. Net, use polynomial long division to divide by ) The numerator can be rewritten as ( 1)( 1). U3-13 Lesson 3: Graphing Rational Functions

130 4. Use the quadratic formula to check any remaining second-degree factors for zeros. b b ac Use the quadratic formula, ± 4, to solve the factor a 1 for. Let a 1, b 1, and c 1. b b ac ± 4 Quadratic formula a (1) ± ( 1) 4(1)( 1) Substitute 1 for a, 1 for b, and 1 for c. (1) 1± 5 Simplify. Any zeros of the numerator will also be zeros of the rational function if they are not also zeros of the denominator. Check that the zeros of the numerator do not also make the denominator equal 0. Substitute each of the zeros for and simplify Substituting each of the zeros into the denominator does not make the denominator zero, therefore, the zeros of the rational function h( ) are 1+ 5, 1 5, 0, 1 and 1. U3-14 Unit 3: Rational and Radical Relationships

131 Eample 4 Use a graphing calculator to find the zeros of the rational function f ( ) Enter the function on a graphing calculator to determine the number of zeros and their approimate values. Due to the compleity of the constants and the coefficients in this problem, it can best be solved using a graphing calculator. On a TI-83/84: Step 1: Press [Y]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y 1, use your keypad to enter values for the function, as given in the problem statement. Use [X, T, θ, n] for and [ ] for the eponents. Step 3: Press [GRAPH] to see if the function appears in the standard 10-by-10 window. If not, press [ND][GRAPH] to display a table of values. Step 4: Scroll up and down the list of values until a trend appears that suggests how the graph will behave at large values of and y. The magnitude of the and y values will suggest a better scale for the aes so that all of the functions solutions can be seen. Step 5: Press [WINDOW] to change the values of Xmin, Xma, Ymin, and Yma. The Xscl and Yscl values can also be changed to increase or decrease the magnitude of the intervals on the aes. For this function, change the ais settings for Xmin, Xma, Ymin, and Yma to 0, 0, 00, and 00, respectively. These values can be changed again after the zeros are located for a clearer view of the zeros values. Step 6: Press [ND][GRAPH] to display the - and y-values again. (continued) U3-15 Lesson 3: Graphing Rational Functions

132 Step 7: To find where the zeros are located, find the places where the value of Y 1 changes from positive (+) to negative ( ), or vice versa. At 0, y is undefined. At 1, y Therefore, for this function, a zero occurs between 0 and 1. At 19, y At 0, y Therefore, another zero occurs between 19 and 0. For values of < 0 and > 3, the signs of the y-values are the same. Step 8: To narrow down the zeros even more, press [TRACE]. Move the cursor to the locations on the -ais suggested by Step 7. For the location between 0 and 1, the value of 0.6 gives a value of y 0.85 and the value of 0.7 gives a value of y Therefore, one of the zeros is approimately 0.6. For the location between 19 and 0, the value of gives a value of y and the value of gives a value of y Therefore, the other zero is approimately On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function to the right of f1() and press [enter]. For specific values, refer to the function given in the problem statement. Step 4: To adjust the - and y-ais scales on the window, press [menu] and select 4: Window and then 1: Window Settings. Enter [ ][0] at XMin, [0] at XMa, [ ][00] at YMin, and [00] at YMa. Tab to OK and press [enter]. Step 5: To see a table of values, press [menu] and scroll down to : View, then 9: Show Table. Step 6: To find where the zeros are located, arrow up and down the table to find the places where the value of Y 1 changes from positive (+) to negative ( ), or vice versa. (continued) U3-16 Unit 3: Rational and Radical Relationships

133 Step 7: To narrow down the zeros even more, use the Trace tool. Press [ctrl][tab] to reactivate the graph, then [ctrl][t] to hide the table. Then, press [menu] and use the arrow keys to select 5: Trace, then 1: Graph Trace. Press [enter]. Step 8: Use the arrow keys to move the cursor along the two parts of the graph until the cursor crosses the -ais. At that point, the symbol zero appears, indicating the closest approimation of one of the solutions, which gives an -value of approimately 0.6 when y is approimately 1.1E 9 (or 0 for all practical purposes). Step 9: Continuing the trace to the right, the symbol zero appears again when is approimately and y is approimately 6.9E 5 (or 0 for all practical purposes) y U3-17 Lesson 3: Graphing Rational Functions

134 . Summarize your findings about the zeros for the rational function based on the graphing calculator eplorations. The graph produced using a graphing calculator suggests that this fourth-degree rational function has two real zeros: one at 0.6 and another at The graphing calculator gave more precise estimates of the location of the zeros for values of y that were approimately equal to 0. Because the graph of this function intersects the -ais in only two locations, it suggests that the other two zeros for this function are not real numbers. U3-18 Unit 3: Rational and Radical Relationships

135 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions Practice 3.3.3: Finding the Zeros For problems 1 and, how many real zeros eist for the rational function? List any real zeros f ( ) g( ) + For problems 3 7, find the zero(s) of each rational function. ( 5)(+ 7) 3. a( ) ( + 7)( 5) ( + 1) ( + ) 4. b( ) ( + 1)( ) 6 5. c( ) d( ) e( ) Use the following information to complete problems The time required for the Home Town Hikers to make a 30-mile hike depends on the total distance covered and the speed at which each hiker travels. The rational function H(r) gives the total time for the trip in miles per hour if the hikers move at two different speeds: H ( r ) 0 r r What is the zero of the function? 9. How does the zero relate to the real-world conditions of the problem? 10. What would be a reasonable domain for r in order for the rate and time to have magnitudes that make sense in this situation? U3-19 Lesson 3: Graphing Rational Functions

136 Lesson 3.3.4: Solving a System of Rational Equations Introduction A system of rational equations or functions consists of two or more equations or g( ) functions of the form f ( ). A solution of a system of rational functions eists h( ) at domain values for which f() g() h(). It is also possible for a system of rational equations or functions to have no solutions or an infinite number of solutions. The solution(s) to a system of rational equations or functions only eist at values of for which the equations or functions in the system are defined. The solution(s) of a rational equation or function can be calculated eactly using algebraic methods such as by factoring, by using the quadratic formula, and by employing similar techniques, or they can be approimated graphically using technology. Key Concepts The solution(s) to a system of rational functions f() and g() consist of the value(s) of that are in the domains of the functions and that result in function values for which f() g(). For eample, if f ( ) and +1 g( ), the solution for the system of f() and g() is ( 0.5, 1). The + 3 domain of both functions includes 0.5. On a graph, the solutions are the points at which the equations intersect. A system of rational equations for which no solutions eist can best be understood by looking at a graph of the system. The following graph is for the system f ( ) + and g ( ). U3-130 Unit 3: Rational and Radical Relationships

137 As the graph shows, there are no points of intersection of the two functions, which is the indicator that a solution does not eist. A system of rational equations for which an infinite number of solutions eists can be visualized by looking at a table of values for the system. The following table is for the system f 8 ( ) + 10 and g 16 ( ) f() g() As the table shows, for each value of, f() g(). When a system is too complicated to be solved algebraically, graphing technology can show the solutions and provide an approimate value for them. Use the graphing calculator directions appropriate to your model to find the solutions of rational functions. Note: The precision with which the cursor is moved to the intersection point can also affect the results. U3-131 Lesson 3: Graphing Rational Functions

138 On a TI-83/84: Step 1: Press [Y]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y 1, use your keypad to enter values for f(). (Refer to the problem statement for the values of the functions.) Use [X, T, θ, n] for and [ ] for any eponents. Step 3: At Y, use your keypad to enter values for g(). Press [GRAPH]. Step 4: Press [WINDOW] to change the viewing window. Step 5: To approimate the solutions for the system, press [TRACE]. Use the arrow keys to move the cursor over each of the two intersection points. The values for and y will be displayed. On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function f() to the right of f1(). (Refer to the problem statement for the values of the functions.) Press the down arrow to bring up f(). Step 4: Enter the function g() to the right of f() and press [enter]. Step 5: To adjust the - and y-ais scales on the window, press [menu] and select 4: Window and then 1: Window Settings. Enter values for XMin, XMa, YMin, and YMa. Tab to OK and press [enter]. Step 6: To approimate the solutions, press the [menu] key, and select 5: Trace. Arrow over to 1: Graph Trace. Press [enter]. Step 7: Use the arrow keys to move the cursor along the graph to the intersection point of the two graphs. The symbol zero will appear at the closest approimation for each solution. U3-13 Unit 3: Rational and Radical Relationships

139 Guided Practice Eample 1 ( 1) Find the solution(s) to the rational functions f ( ) + and g( ) Set f() g(). f() must equal g() for a value of the domain variable,, to result in a solution of a system of rational equations. f() g() Set the functions equal to each other. ( 1) Substitute ( 1) for f() and for g() The epression for f() g() is ( 1) Simplify the resulting equation. Use any algebraic techniques that will reduce the problem to a simpler one that can be solved for the domain variable. To simplify this problem, begin by cross multiplying. ( 1) ( 1)( + 1) ( + ) Equation from the previous step Cross multiply. Divide each side by and simplify. Subtract from both sides. Subtract from both sides. Simplify. The problem has been reduced to a quadratic equation: 3 0. U3-133 Lesson 3: Graphing Rational Functions

140 3. Use the quadratic formula to solve the quadratic equation. b b ac Use the quadratic formula, ± 4, to solve for. Let a 1, b 1, and c 3. a b b ac ± 4 Quadratic formula a ± ( 1) ( 1) 4(1)( 3) Substitute 1 for a, 1 for b, and (1) 3 for c. 1± 13 Simplify. The zeros of ( 1) are and Since these radical terms would be difficult to use, they can be converted to decimal approimations: 0.5 ± 1.8, which simplifies to.3 or Determine any additional solutions. It is possible to miss solutions to rational functions when dividing both sides of an equation by a term or terms. To find any additional solutions, find the zeros of the term(s). When simplifying f() g(), we divided both sides of the equation by. The zero of is 0. U3-134 Unit 3: Rational and Radical Relationships

141 5. Substitute the -values into the original equations to verify that they result in the same function values for f() and g(). The values of f() and g() will not be eactly equal since approimations for are used. Let.3. ( 1) f ( ) g( ) + +1 (.3)[(.3) 1] (.3) f (.3) g (.3) (.3) + (.3) + 1 f(.3) 0.7 g(.3) 0.7 Let 1.3. ( 1) f ( ) g( ) + +1 ( [ ] f 1.3) ( ( 1.3) 1.3) 1 ( 1.3) g( 1.3) ( 1.3) + ( 1.3) + 1 f( 1.3) 4.3 g( 1.3) 4.3 Let 0. ( 1) f ( ) g( ) + +1 (0)[(0) 1] (0) f (0) g(0) (0) + (0) + 1 f(0) 0 g(0) 0 The solutions to the system of f() and g() are (0, 0) and approimately {(.3, 0.7), ( 1.3, 4.3)}. U3-135 Lesson 3: Graphing Rational Functions

142 Eample + 1 Show that the rational functions f ( ) ( + ) and g ( ) common solution. ( 1) have no 1. Set f() g(). f() must equal g() for a value of the domain variable,, to result in a solution of a system of rational equations. f() g() Set the functions equal to each other Substitute ( ) ( 1) ( + ) for f() and for g(). ( 1) The epression for f() g() is + 1 ( + ) ( 1).. Simplify the resulting equation. Use any algebraic techniques in order to reduce the problem to a simpler one that can be solved for the domain variable. To simplify this problem, begin by cross multiplying. + 1 ( + ) ( 1) ( 1)( + 1) ( + )( ) Equation found in previous step Cross multiply. ( 1)( + 1) ( + )( ) Divide both sides by. 1 ( + )( ) 1 4 Distribute. Distribute. 1 4 Subtract from both sides. Note that 1 does not equal 4; therefore, any value of chosen will not result in a true statement based on the two functions. U3-136 Unit 3: Rational and Radical Relationships

143 3. Verify your answer with a graphing calculator. This step will give visual evidence that the graphs of the two functions do not intersect. On a TI-83/84: Step 1: Press [Y]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y 1, use your keypad to enter values for f(). (Refer to the problem statement for the values of the functions.) Use [X, T, θ, n] for and [ ] for any eponents. Step 3: At Y, use your keypad to enter values for g(). Press [GRAPH]. Step 4: Press [WINDOW] to change the viewing window. Step 5: To approimate the solutions for the system, press [TRACE]. Use the arrow keys to move the cursor over each of the two intersection points. The values for and y will be displayed. On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function f() to the right of f1(). (Refer to the problem statement for the values of the functions.) Press the down arrow to bring up f(). Step 4: Enter the function g() to the right of f() and press [enter]. Step 5: To adjust the - and y-ais scales on the window, press [menu] and select 4: Window and then 1: Window Settings. Enter values for XMin, XMa, YMin, and YMa. Tab to OK and press [enter]. Step 6: To approimate the solutions, press the [menu] key, and select 5: Trace. Arrow over to 1: Graph Trace. Press [enter]. Step 7: Use the arrow keys to move the cursor along the graph to the intersection point of the two graphs. The symbol zero will appear at the closest approimation for each solution. (continued) U3-137 Lesson 3: Graphing Rational Functions

144 Both calculators will yield a graph similar to the one below. The graphed functions do not intersect, and therefore have no real solutions. 10 y U3-138 Unit 3: Rational and Radical Relationships

145 Eample 3 Show that the functions f ( ) number of solutions and g( ) have an infinite Show that g() can be reduced to f(), or that a common denominator can be found for f() and g(). The condition that f() and g() have an infinite number of solutions means that the functions are identical; therefore, any value of will result in f() g(). Begin by determining a binomial that can be multiplied by the denominator of f() in order to result in the denominator of g(). Notice that the denominator of g(), , can be factored into + 3 and +. Since the denominator of f() is +, this suggests that + 3 might be the factor that will make the two functions equal. In order for the function value of f() to stay the same, the numerator and denominator have to be multiplied by the same value. f ( ) Original function for f() + ( )( + 3) f ( ) Multiply the numerator and ( + )( + 3) denominator by f ( ) Simplify both the numerator and denominator. The function f ( ) can be rewritten as f ( ) Compare the rewritten form of f() with g(). The rewritten form of f(), , is identical to g(), ; therefore, f() g() U3-139 Lesson 3: Graphing Rational Functions

146 3. Verify that any value of yields the same value for f() and g(). Any value of selected should satisfy the condition that f() g(). For instance, let 1. Substitute this value into the original epression for f(). f ( ) + (1) (1) f (1) (1) + 1 f 1 (1) 1+ f (1) 3 Original function Substitute 1 for. Simplify. Now let 1 for every -value in g() and simplify to see if the same result is produced g( ) (1) + (1) 5(1) 6 g (1) (1) + 5(1) g (1) g (1) 1 g (1) 3 Original function Substitute 1 for. Simplify. The value of f(1) is equal to g(1). We could substitute any value of into both of these equations and arrive at the same answer for both functions. Therefore, this proves f() g() for all values chosen for. U3-140 Unit 3: Rational and Radical Relationships

147 Eample First, show that 1 is a solution to the system f ( ) and g( ) Then, state two different reasons why 1 cannot be used as a solution. 1. Evaluate f( 1) and g( 1). Let f ( ) Given function for f() 1 f 3 ( 1) ( 1) + 1 Substitute 1 for. ( 1) 1 f 1 ( 1) + 1 Simplify f ( 1) 0 The result is undefined for f( 1). g( ) Given function for g() ( 1) ( 1) g( 1) Substitute 1 for. ( 1) + 4( 1) g( 1) + Simplify g( 1) 0 The result is undefined for g( 1). When 1, the result for both functions is the same: undefined... Eplain why the results of step 1 do not represent a solution of the given functions. Although the results are the same for both functions, the value of 1 results in an undefined quantity; therefore, 1 cannot be a solution. The problem statement asks for two different reasons why 1 cannot be used as a solution. To find another reason, start by setting the two functions equal to each other. U3-141 Lesson 3: Graphing Rational Functions

148 3. Set f() g(). This is the condition for a value of the domain () to result in a solution of a system of rational equations. f() g() Set the functions equal to each other The epression for f() g() is Substitute for f() and for g() Simplify the resulting equation. Use any algebraic techniques in order to reduce the problem to a simpler one that can be solved for the domain variable. To simplify this problem, begin by factoring the numerators and denominators ( + 1)( 1) ( )( + 1) ( 1)( + 1) ( 1)( + 1) ( + 1)( 1) ( )( + 1) ( 1)( + 1) ( 1)( + 1) Equation found in the previous step Factor each numerator and denominator. Cancel like factors. Simplify. U3-14 Unit 3: Rational and Radical Relationships

149 5. Eplain the results of step 4. The factor + 1 representing the proposed solution 1 cancels out, which means it cannot be used as a solution for the system. 6. Check the results of steps 1 5 by evaluating the equation found in step 4 for 1. Let ( 1) ( 1) 1 ( 1) ( 1) 1 ( 1) Equation found in step 3 for f() g() Substitute 1 for. Simplify. Note that this is not a true statement; therefore, the domain value 1 is not a solution to the functions. U3-143 Lesson 3: Graphing Rational Functions

150 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions Practice 3.3.4: Solving a System of Rational Equations For problems 1 7, find the solution(s), if any, to each system of rational functions f ( ) ; g( ) 1 3. f ( ) 6 ; g( ) f ( ) ; g( ) 3 3 ( + )( 5) 6( + ) 4. f ( ) ; g( ) f ( ) ; g( ) f ( ) ; g( ) f ( ) ( 1) 3 3 ; g( ) U3-144 Unit 3: Rational and Radical Relationships continued

151 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 3: Graphing Rational Functions For problems 8 10, use the given information to find the solution(s) of the described systems of rational functions, if any solutions eist. 8. The school sports association paid $1,50 for a popcorn machine that costs $55 per month to operate. The association also purchased a hot chocolate drink machine for $75 that costs $80 per month to operate. Write rational functions for the total monthly cost of operating the popcorn machine P(n) and the hot chocolate machine H(n), including the original cost of each machine. Assuming a constant rate of use for each machine, when will the total monthly costs of operating the two machines be the same? 9. Gaby s Lawn Service uses a herd of goats to mow a pasture of a specific size. The number of hours, t, it takes goats to mow the pasture is given by the t( t 1) rational function At ( ). If a third goat is used, the time can be given t 1 t( t 1)( t ) by the function Bt ( ). At what value of t do the goats take the 3t 3 same amount of time to finish the pasture as the 3 goats? 10. Two packages are wrapped for a birthday party. The ratio of the surface area to the volume of each package is the same. The rational functions representing the area-to-volume ratios for two different sizes of packages are represented by 4( + 5) the functions A( ) 3( )( 4) and B 5( + 4) ( ). The constants 4( )( 4) and binomial terms represent the dimensions of the packages in inches. If the packages are rectangular solids, what are their dimensions when the ratios are the same? U3-145 Lesson 3: Graphing Rational Functions

152 UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 4: Graphing Radical Functions Common Core Georgia Performance Standards MCC9 1.A.CED. MCC9 1.F.IF.4 MCC9 1.F.IF.5 MCC9 1.F.IF.7b Essential Questions 1. What are the characteristics of the graph of a radical function and what is the significance of each?. What does an etraneous solution to a radical function mean graphically? 3. When does the domain of a radical equation not eist? WORDS TO KNOW cube root function parent function radical function square root function -intercept y-intercept a function that contains the cube root of a variable. The 3 general form is y a ( h) + k, where a, h, and k are real numbers. a function with a simple algebraic rule that represents a family of functions. The graphs of the functions in the family have the same general shape as the parent function. a function with the independent variable under a root. n The general form is f ( ) a ( h) + k, where n is a positive integer root and a, h, and k are real numbers. a function that contains a square root of a variable the point at which a graphed function crosses the -ais; written in the form (, 0) the point at which a graphed function crosses the y-ais; written in the form (0, y) U3-146 Unit 3: Rational and Radical Relationships

153 Recommended Resources Khan Academy. Graphing Radical Functions. This 10-minute video shows an eample of setting up a table of values for a radical function, calculating the domain and range, and then plotting and graphing the function. An interactive script allows users to replay different parts of the video. Math Planet. The Graph of a Radical Function. This site provides a summary of the graph of a radical function and includes a comparison of various radical functions, as well as an instructional video. Purplemath.com. Graphing Radical Functions. This site describes how to graph different types of radical functions using a table of values. U3-147 Lesson 4: Graphing Radical Functions

154 Lesson 3.4.1: Creating Radical Equations in Two Variables Introduction Radical equations and functions are often used to model various real-world scenarios. For eample, a radical equation can be used to determine the distance seen from tall buildings based on the height of the viewing point. The equations of these scenarios are not always given, but can be created based on graphs or verbal descriptions. Key Concepts A radical function is a function with the independent variable under a root. n The general form is f ( ) a ( h) + k, where n is a positive integer root and a, h, and k are real numbers. The most common radical functions are the 3 square root function, f ( ), and the cube root function, g( ). A square root function is a function that contains a square root of a variable. A cube root function is a function that contains the cube root of a variable. 3 The general form is y a ( h) + k, where a, h, and k are real numbers. 3 f ( ) and g( ) are parent functions, meaning f ( ) is the 3 simplest square root function and g( ) is the simplest cube root function. Each parent function preserves the definition and shape of all other square root and cube root functions. Understanding the Shape of a Square Root Function The typical graph of a radical function is one that curves slightly and goes on to either positive infinity or negative infinity for all real -values. For the square root function f ( ), the graph starts at the origin and curves out toward positive -values and goes on to infinity. U3-148 Unit 3: Rational and Radical Relationships

155 Notice how the graph of f ( ) angles upward as the values of increase. Parent Square Root Function, f ( ) 1 y f () A negative sign in front of the radical symbol, such as g( ), causes the function to flip or be reflected over the -ais and go on to positive infinity. On the other hand, a negative sign under the radical, such as h( ), causes the function to flip over the y-ais and go on to negative infinity. 6 y 8 y h() g() 6 4 U3-149 Lesson 4: Graphing Radical Functions

156 A radical function will start at a different point on the -ais when numbers are added or subtracted under the radical sign. For instance, j( ) + 1 would resemble the graph of f ( ), but it would start 1 unit to the left. Thus, the first point would be the endpoint ( 1, 0) instead of (0, 0). Subtracting 1 from ( e.g., k( ) 1 ) moves the graph 1 unit to the right, with the endpoint at (1, 0). 10 y 10 y j() k() Also, if numbers are added to or subtracted from the term containing the radical sign, the graph shifts up and down along the y-ais. So, m( ) + 4 resembles the graph of f ( ), but it has been shifted 4 units up so that it has an endpoint of (0, 4) instead of (0, 0). Similarly, n( ) 4 resembles the graph of f ( ), but it has been shifted 4 units down and thus has an endpoint of (0, 4). 10 y 6 y 8 6 m() n() U3-150 Unit 3: Rational and Radical Relationships

157 If the term containing the radical sign is multiplied by a factor, the graph of the function will be vertically stretched or compressed by that same factor. For instance, the graph of p( ) resembles the graph of f ( ), but 1 it has been stretched vertically by a factor of. The graph of q( ) also resembles the graph of f ( ), but it has been compressed vertically by a factor of y 10 y p() q() If is multiplied by a factor of b, the graph of the function will be stretched or compressed horizontally by that same factor. For instance, r( ) resembles the graph of f ( ), but it has been compressed horizontally by a factor of 1. The graph of s ( ) 1 also resembles f ( ) ; however, it has been stretched horizontally by a factor of. 10 y 10 y r() 4 s() U3-151 Lesson 4: Graphing Radical Functions

158 When determining the equation of a square root function using a graph, note where the graph starts and then determine in which direction the curve travels (toward positive or negative infinity for -values). Also note whether the graph curves above or below the -ais. The following table is a useful guide for interpreting various square root functions and understanding the graph of each. The table describes how the graph of each given square root function differs from that of the parent function, f ( ). Function Relation to the graph of the parent function, f ( ) f ( ) f ( ) is reflected across the -ais. f ( ) f ( ) is reflected across the y-ais. f ( ) + k f ( ) is shifted k units to the left. f ( ) k f ( ) is shifted k units to the right. f ( ) + k f ( ) is shifted k units up. f ( ) k f ( ) is shifted k units down. f ( ) a f ( ) is stretched vertically by a factor of a. f ( ) 1 a f ( ) is compressed vertically by a factor of 1 a. f ( ) a f ( ) is compressed horizontally by a factor of 1 a. f ( ) 1 a f ( ) is stretched horizontally by a factor of a. Understanding the Shape of a Cube Root Function 3 The typical graph of a cube root function, such as f ( ), is similar to a flat letter S that lies on its side. This cube root function travels from negative infinity to positive infinity, with the center at (0, 0) and points (1, 1) and ( 1, 1). U3-15 Unit 3: Rational and Radical Relationships

159 3 Parent Cube Root Function, f ( ) y When a cube root function has a negative in front of the cube root, the graph is reflected over the -ais. 3 Notice that in the following graph of f ( ), the graph still goes through the origin (0, 0) as well as the points (1, 1) and ( 1, 1). y 4 f () As with square root functions, cube root functions can be shifted along the -ais by adding and subtracting numbers underneath the cube root. U3-153 Lesson 4: Graphing Radical Functions

160 The graph of f ( ) resembles the graph of f 3 ( ), but it has been shifted to the left 4 units. Its center is at ( 4, 0), not (0, 0). If the number underneath the radical were subtracted, the graph would shift to the right. y 4 3 f () If values are added to or subtracted from the term containing the cube root, the graph shifts up or down along the y-ais. The graph of f ( ) resembles the graph of f 3 ( ), but it has been shifted up 4 units. Its center is at (0, 4) instead of (0, 0). 8 y 6 3 f () U3-154 Unit 3: Rational and Radical Relationships

161 Subtracting a number from the term containing the radical results in a graph 3 that resembles the graph of f ( ), but it has been shifted down along the y-ais. 3 Function Relation to the graph of the parent function, f ( ) 3 3 f ( ) f ( ) is reflected across the -ais. 3 3 f ( ) f ( ) is reflected across the y-ais. 3 3 f ( ) + k f ( ) is shifted k units to the left. 3 3 f ( ) k f ( ) is shifted k units to the right. 3 3 f ( ) + k f ( ) is shifted k units up. 3 3 f ( ) k f ( ) is shifted k units down. 3 3 f ( ) a f ( ) is stretched vertically by a factor of a. f ( ) 1 3 a f ( ) 3 is compressed vertically by a factor of 1 a. 3 3 f ( ) a f ( ) is compressed horizontally by a factor of 1 a. f ( ) 1 3 a 3 f ( ) is stretched horizontally by a factor of a. U3-155 Lesson 4: Graphing Radical Functions

162 Guided Practice Eample 1 Create the equation of a function in two variables that represents the graphed function. 6 y Note the general shape of the curve and determine the type of function shown. By looking at the graph, we can see that the function is a curved line and not a sideways letter S. Thus, it is a square root function.. Note the endpoint of the graph. The endpoint of the graph is at (3, 0), which means that the parent function has been shifted along the -ais 3 units to the right of the origin. 3. Note the general direction of the curve. Since the curve rises from its endpoint, we know the square root is multiplied by a positive number. Since the curve flows right from its endpoint, we know is also multiplied by a positive number. 4. Create an equation using the information derived from the graph. The parent function of a square root function is f ( ). The graphed function is shifted 3 units to the right of the parent function; therefore, 3 is subtracted from. Based on this information, we can conclude that the radical equation represented in the graph is g( ) 3. U3-156 Unit 3: Rational and Radical Relationships

163 Eample Create the equation of a function in two variables that represents the graphed function. 8 y Note the general shape of the curve and determine the type of function shown. By looking at the graph, we can see that the function is a curved line and not a sideways letter S. Thus, it is a square root function.. Note the endpoint of the graph. The endpoint of the graph is at ( 5, 1), which means that the parent function has been shifted along the -ais 5 units to the left and up 1 unit. 3. Note the general direction of the curve. Since the curve rises from its endpoint, we know the square root is multiplied by a positive number. Since the curve flows right from its endpoint, we know is also multiplied by a positive number. U3-157 Lesson 4: Graphing Radical Functions

164 4. Create an equation using the information derived from the graph. The parent function of a square root function is f ( ). The graphed function is shifted 5 units to the left of the parent function; therefore, 5 is added to, or g( ) + 5. The function is also shifted 1 unit up from the origin; therefore, 1 is added to g( ) + 5 : g( ) Based on this information, we can conclude that the radical equation represented in the graph is g( ) Eample 3 Determine the equation of a cube root function that is shifted units to the left along the -ais. 1. Identify the type of function to be created. As stated, the type of function to be created is a cube root function. 3 The parent function for cube roots is f ( ).. Identify any shifts in the graph. We are given that the function is shifted units to the left along the -ais. No other shifts are described. Because the function is shifting along the -ais, there is only a horizontal shift, not a vertical shift. 3. Use the known information to write the equation of the function. 3 The parent function of a cube root function is f ( ). The function described is shifted units to the left of the parent function; therefore, is added to. Based on this information, we can conclude that the radical 3 equation represented by the verbal description is g( ) +. U3-158 Unit 3: Rational and Radical Relationships

165 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 4: Graphing Radical Functions Practice 3.4.1: Creating Radical Equations in Two Variables For problems 1 5, create an equation in two variables for each graphed function. y y y continued U3-159 Lesson 4: Graphing Radical Functions

166 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 4: Graphing Radical Functions 4. 6 y y For problems 6 9, create an equation in two variables for each function described. 6. A square root function is shifted 6 units to the left and units up from the origin. 7. A cube root function is shifted 3 units down from the origin. 8. A square root function is shifted 4 units to the right of the origin and flipped over the -ais. 9. A cube root function is shifted units up and 3 units to the left of the origin. U3-160 Unit 3: Rational and Radical Relationships continued

167 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 4: Graphing Radical Functions Use your knowledge of radical equations to complete the problem that follows. 10. Eplain why the following graph is not the graph of y y U3-161 Lesson 4: Graphing Radical Functions

168 Lesson 3.4.: Graphing Radical Functions Introduction A radical function can be graphed by identifying some of its key features and characteristics, such as its domain, -intercept(s), and y-intercept(s). By knowing these features of a radical function, you can visualize what the graph will look like prior to graphing by hand or by using technology. Key Concepts Recall that a radical function is a function that includes a radical sign as part of the equation. The parent function of the square root function family is f ( ) and 3 the parent function of the cube root function family is f ( ). All other functions of the square root and cube root function families are based on these basic forms. The key features of a radical function are the characteristics that can be used to sketch the graph of the function without using a table of values. Key features include the - and y-intercepts as well as the domain of the function. The -intercept is the point at which a graphed function crosses the -ais; it is written in the form (, 0). The y-intercept is the point at which a graphed function crosses the y-ais, and is written in the form (0, y). The domain of a function is the set of all input values (-values) that satisfy the given function without restriction. Recall that the domain of a function can be epressed in interval notation. That is, the domain is written in the form of (a, b), where a and b are the endpoints of the interval. Intervals can be written in different ways, depending on whether endpoints are included. For eample, an interval of (, 10) includes all numbers between and 10, but not the endpoints. An interval of [, 10] includes all numbers between and 10, including the endpoints. And, an interval with mied notation, such as (, 10], includes all numbers between and 10; is not included, but 10 is included. U3-16 Unit 3: Rational and Radical Relationships

169 Identifying the Key Features of a Square Root Function The epression under a square root cannot result in a negative value; therefore, the domain of a square root function can be found by setting the epression under the radical to be greater than or equal to 0 and solving for. This indicates the possible values of for the graph of the function. The -intercept can be found by setting the epression under the radical equal to 0 and solving for. The y-intercept can be found by evaluating f(0). This determines the output, or y-value, for an input, or -value, of 0. Identifying the Key Features of a Cube Root Function The domain of a cube root function can be either positive or negative, meaning the domain is always all real numbers or (, ); therefore, the values of are not limited. Like with square root functions, the -intercept can be found by setting the epression under the radical equal to 0 and solving for. Also like with square root functions, the y-intercept can be found by evaluating f(0). This determines the output, or y-value, for an input, or -value, of 0. Graphs of sketched functions can be verified by using a graphing calculator. On a TI-83/84: Step 1: Press [Y]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y 1, use your keypad to enter values for the function. Use [X, T, θ, n] for and [ ] for any eponents. Press [GRAPH]. Step 3: To view a table of values for the function, press [ND][GRAPH]. On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step 3: Enter the function to the right of f1() and press [enter]. Step 4: To see a table of values, press [menu], then use the arrow keys to select : View, then 9: Show Table. Press [enter]. U3-163 Lesson 4: Graphing Radical Functions

170 Guided Practice 3.4. Eample 1 Identify the key features for the graph of f ( ). 1. Determine the domain of the function. To find the domain of the function, set the epression underneath the radical sign to be greater than or equal to 0 and solve for. The epression under the radical sign is ; therefore, 0. The domain of the function is 0. Because 0 is included in the domain, use a bracket for the lower bound. Since the domain continues to infinity, use a parenthesis for the upper bound: [0, ).. Determine the -intercept of the function. To find the -intercept of the function, set the epression under the radical sign equal to 0 and solve for. The epression under the radical is ; therefore, 0. The -intercept is in the form (, 0); therefore, the -intercept of this function is (0, 0). 3. Determine the y-intercept of the function. To find the y-intercept, evaluate f(0). f ( ) Original function f (0) (0) Substitute 0 for. f(0) 0 The y-intercept of this function is (0, 0). 4. Summarize the key features. The domain is 0, or [0, ). The -intercept is (0, 0). The y-intercept is (0, 0). U3-164 Unit 3: Rational and Radical Relationships

171 Eample Identify the key features of the function g( ) +, then use them to create a graph. 1. Determine the domain of the function. To find the domain of the function, set the epression underneath the radical sign to be greater than or equal to 0 and solve for. The epression under the radical sign is + ; therefore, evaluate The domain of the function is, or [, ).. Determine the -intercept of the function. To find the -intercept of the function, set the epression underneath the radical sign to be equal to 0 and solve for. The epression under the radical sign is + ; therefore,. The -intercept is in the form (, 0); therefore, the -intercept is (, 0). 3. Determine the y-intercept of the function. To find the y-intercept, evaluate g(0). g( ) + Original function g (0) (0) + Substitute 0 for. g (0) Simplify. The y-intercept of the function is ( 0, ). U3-165 Lesson 4: Graphing Radical Functions

172 4. Graph the function. Use the key features to create a sketch of the function. Begin by plotting the -intercept at (, 0). Net, plot the y-intercept, ( 0, ). It is often helpful to choose several values for and calculate the corresponding y-values to add to the shape of the graph. Based on the domain, we know the function will continue increasing as approaches positive infinity. Draw a smooth curve through the graphed points, using an arrow at the end of the graphed curve opposite the -intercept to indicate that the function continues toward infinity. The function g( ) + is graphed as follows. 6 y U3-166 Unit 3: Rational and Radical Relationships

173 Eample 3 Identify the key features of the function g( ), then use them to create a graph. 1. Determine the domain of the function. To find the domain of the function, set the epression underneath the radical sign to be greater than or equal to 0 and solve for. The epression under the radical sign is ; therefore, 0. The domain of the function is 0, or [0, ).. Determine the -intercept of the function. To find the -intercept of the function, set the epression under the radical sign equal to 0 and solve for. The epression under the radical is ; therefore, 0. The -intercept is in the form (, 0); therefore, the -intercept of this function is (0, 0). 3. Determine the y-intercept of the function. To find the y-intercept, evaluate g(0). g( ) Original function g (0) (0) Substitute 0 for. g (0) 0 Simplify. The y-intercept of the function is (0, 0). U3-167 Lesson 4: Graphing Radical Functions

174 4. Graph the function. Use the key features to create a sketch of the function. Begin by plotting the -intercept at (0, 0). (0, 0) is also the y-intercept of the function. Notice the negative sign in front of the radical. This indicates that the function will be similar to the parent function f ( ), but will be reflected over the -ais. Based on the domain, we know the function will continue increasing as approaches positive infinity. Choose several other values for to calculate the corresponding y-values. For eample, let 1, 4, and 9. The corresponding y-values are 1,, and 3. These points, (1, 1), (4, ), and (9, 3) can then be plotted. Draw a smooth curve through the graphed points, using an arrow at the end opposite the -intercept to indicate that the function continues toward infinity. The function g( ) is graphed as follows. 6 y U3-168 Unit 3: Rational and Radical Relationships

175 Eample 4 Identify the key features of the function f ( ) + 3, then use them to create a graph. 1. Determine the domain of the function. To find the domain of the function, set the epression underneath the radical sign to be greater than or equal to 0 and solve for. The epression under the radical sign is + 3; therefore, evaluate The domain of the function is 3, or [ 3, ).. Determine the -intercept of the function. To find the -intercept of the function, set the epression underneath the radical sign to be equal to 0 and solve for. The epression under the radical sign is + 3; therefore, 3. The -intercept is in the form (, 0); therefore, the -intercept of this function is at ( 3, 0). 3. Determine the y-intercept of the function. To find the y-intercept, evaluate f(0). f ( ) + 3 Original function f (0) (0) + 3 Substitute 0 for. f (0) 3 Simplify. The y-intercept of the function is ( 0, 3 ), or approimately (0, 3.46). U3-169 Lesson 4: Graphing Radical Functions

176 4. Graph the function. Use the key features to create a sketch of the function. Begin by plotting the -intercept at ( 3, 0). Net, plot the y-intercept, (0, 3.46). Based on the domain, we know the function will continue increasing as approaches positive infinity. If necessary, choose additional values of and calculate the corresponding y-values to obtain additional points. Draw a smooth curve through the graphed points, using an arrow at the end opposite the -intercept to indicate that the function continues toward infinity. The function f ( ) + 3 is graphed as follows. 10 y U3-170 Unit 3: Rational and Radical Relationships

177 Eample 5 Identify the key features of the function f ( ) 3 4, then use them to create a graph. 1. Determine the domain of the function. Since cube roots can be positive or negative, there are no limitations on ; therefore, the domain of is all real numbers or (, ).. Determine the -intercept of the function. To find the -intercept of the function, set the epression underneath the radical sign to be equal to 0 and solve for. The epression under the radical sign is 4; therefore, 4. The -intercept is in the form (, 0); therefore, the -intercept of this function is at (4, 0). 3. Determine the y-intercept of the function. To find the y-intercept, evaluate f(0). f ( ) 3 4 Original function 3 f (0) (0) 4 Substitute 0 for. 3 f (0) 4 Simplify. f(0) 1.59 The y-intercept of the function is (0, 1.59). U3-171 Lesson 4: Graphing Radical Functions

178 4. Graph the function. Use the key features to create a sketch of the function. Begin by plotting the -intercept at (4, 0). Net, plot the y-intercept, (0, 1.59). Based on the domain, (, ), we know the graph will continue toward positive and negative infinity. Choose additional values for to calculate the corresponding y-values. For eample, let,, and 6. The corresponding y-values are approimately 1.87, 1.6, and These points, (, 1.87), (, 1.6), and (6, 1.59) can then be plotted. The function f ( ) 3 4 is graphed as follows. 10 y U3-17 Unit 3: Rational and Radical Relationships

179 PRACTICE UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 4: Graphing Radical Functions Practice 3.4.: Graphing Radical Functions For problems 1 3, identify the domain, -intercept, and y-intercept for each function. Round answers to the nearest tenth. 1. t( ) + 1. f ( ) 3. g( ) 3 5 For problems 4 7, identify the domain, -intercept, and y-intercept for each function. Round answers to the nearest tenth. Then, use the information to graph each function. 4. h( ) 3 5. f ( ) f ( ) g( ) 4 For problems 8 10, identify the domain for each function. Then, use the information to complete each problem. 8. What is the domain of the function representing the speed of sound, S 1.9 5t+ 457? 3 9. Consider the functions f ( ) and g( ). Which function has a domain of all real numbers? 10. How do the graphs of f ( ) and g( ) compare in terms of their domains? U3-173 Lesson 4: Graphing Radical Functions

180 UNIT 3 RATIONAL AND RADICAL RELATIONSHIPS Lesson 5: Comparing Properties of Functions Common Core Georgia Performance Standard MCC9 1.F.IF.9 Essential Questions 1. How can you find the vertical and horizontal asymptotes of a rational function?. How can you find the domains and ranges of rational and radical functions? 3. How can you find the zero(s) and y-intercept(s) of a function? 4. How can you compare functions in different formats, such as tables, graphs, and equations? WORDS TO KNOW asymptote domain range root -intercept y-intercept zero a line that a function gets closer and closer to, but never crosses or touches the set of all input values (-values) that satisfy the given function without restriction the set of all outputs of a function; the set of y-values that are valid for the function the -intercept of a function; also known as zero the point(s) where a function equals 0 and at which the graph crosses the -ais; also called root or zero the point at which the graph crosses the y-ais; written as (0, y) the -intercept of a function; also known as root U3-174 Unit 3: Rational and Radical Relationships

181 Recommended Resources Khan Academy. Asymptotes of Rational Functions. Site visitors can watch any or all of the three provided videos that eplain how to find vertical and horizontal asymptotes and include eamples. Horizontal asymptotes of higher degree polynomials are also discussed. Khan Academy. Domain of a Radical Function. Visitors of this site can watch a video that eplains how to find the domain of a radical function. Other companion videos give additional instruction on how to find the domain and range of a function. Purplemath.com. Graphing Overview: Polynomials, Radicals, Rationals, and Piecewise. This site provides a general overview of the graphs of various functions, including rational and radical functions, as well as links to additional information specific to each type of function. Virtual Math Lab, West Teas A&M University. Tutorial 40: Graphs of Rational Functions. In this enrichment activity, a written tutorial discusses how to find the asymptotes of a rational function and then graph the function. Higherdegree polynomials are discussed, and users are taught how to find oblique asymptotes as well. U3-175 Lesson 5: Comparing Properties of Functions

182 Lesson 3.5.1: Comparing Properties of Functions Introduction Functions have many characteristics, such as domain, range, asymptotes, zeros, and intercepts. These functions can be compared even when given in a different format, such as when provided as an equation, graph, or table. The key to comparing functions is to either find the same information from the different forms or transfer the functions into the same form for a clear comparison. Key Concepts Functions can be described numerically in a table, verbally, algebraically, or graphically. To compare two functions, determine the possible zeros, y-intercepts, domain, and range of each function. Depending on how the function is represented, this information can be found in various ways. Determining the Zeros of a Function The zero of a function is the point where a function equals 0 and at which the graph crosses the -ais. A zero is also called the root or -intercept of a function. To identify the zero of an equation, replace f() or y with 0 and solve for. To identify the zero of a graph, look for the point where the curve crosses the -ais. To identify the zero in a table, look for a point that has 0 as its y-coordinate. Determining the y-intercepts of a Function The y-intercept of a function is the point where the graph crosses the y-ais. It is written as (0, y). To identify the y-intercept of an equation, replace with 0 and solve for f() or y. To identify the y-intercept of a graph, look for the point where the curve or line crosses the y-ais. To identify the y-intercept in a table, look for a point that has 0 as its -coordinate. U3-176 Unit 3: Rational and Radical Relationships

183 Determining the Asymptotes of a Function An asymptote is a line that a function gets closer and closer to, but never crosses or touches. In other words, it is a line that a curve approaches (but does not reach) as its - or y-values become very large or very small. When a rational function is in equation form, find the vertical asymptote by setting the denominator equal to 0 and solving for. To identify the vertical asymptote of a graph, look for a vertical line that the curve approaches but never reaches. To identify the vertical asymptote in a table, look for a grouping of -coordinates with similar values. For a first-degree rational function, the horizontal asymptote can be found by dividing the coefficients of the variables. For eample, in the function 3 1 f ( ), 3 divided by is 1.5, and the asymptote is located at y f() Asymptote y To identify the horizontal asymptote of a graph, look for a horizontal line that the curve approaches but never reaches. To identify the horizontal asymptote in a table, look for a grouping of y-coordinates with similar values. U3-177 Lesson 5: Comparing Properties of Functions

184 Determining the Domain of a Function The domain of a function is the set of all input values (-values) that satisfy the given function without restriction. For a radical function, the domain of an equation is found by solving an inequality in which the radicand is listed as being greater than or equal to 0. The domain of a first-degree rational function is found by identifying the asymptotes. The domain will contain every real number ecept the point where the vertical asymptote is defined. To identify the domain from a graph, determine the -coordinate where the curve starts and in which direction the curve continues. Whichever values are not allowed in the domain will be vertical asymptotes on the graph. Recall that the domain of a function can be epressed in interval notation. That is, the domain is written in the form of (a, b), where a and b are the endpoints of the interval. Depending on the values of the interval, the notation may change, as shown in the following table. Interval notation Eample (a, b) (, 10) [a, b] [, 10] (a, b] (, 10] [a, b) [, 10) Description All numbers between and 10; endpoints are not included. All numbers between and 10; endpoints are included. All numbers between and 10; is not included, but 10 is included. All numbers between and 10; is included, but 10 is not included. Determining the Range of a Function The range is the set of all outputs of a function. It is the set of y-values that are valid for the function. For a radical function, the best way to find the range of an equation is to first identify the domain, and then find the corresponding values for f() or y. The range of a first-degree rational function is found by identifying the asymptotes. The range will contain every real number ecept the point where the horizontal asymptote is defined. To identify the range from a graph, determine the y-coordinate where the curve starts and in which direction the curve continues. Similar to the domain, the range can also be epressed in interval notation. U3-178 Unit 3: Rational and Radical Relationships

185 Guided Practice Eample Given f() as shown in the graph and g( ), which function has the greater zero? y f() Identify the zero of f(). Find the zero of a graph by identifying the point at which the curve crosses the -ais. The curve of the graph crosses the -ais at 5. Therefore, 5 is the zero of f(). U3-179 Lesson 5: Comparing Properties of Functions

186 . Identify the zero of g(). To find the zero of the function g(), set the function equal to 0 and solve for. 3 6 g( ) 7 + Original function 3 6 (0) 7 + Substitute 0 for g(). 0(7 + ) 3 6 Cross multiply Simplify. 6 3 Add 6 to both sides. Divide both sides by 3. The zero of g() is. 3. Determine which function has the greater zero. The zero of f() is 5 and the zero of g() is. 5 is greater than ; therefore, given the two functions f() and g(), f() has the greater zero. U3-180 Unit 3: Rational and Radical Relationships

187 Eample Grady must graph both f() and g() over the interval [ 0, 0]. He has graphed f() as shown below, and now he must graph g( ) Which of these two functions has the larger domain? y f() Identify the domain of f(). To find the domain of a graph, identify the -coordinates where the curve starts and stops. The curve starts at the point (4, 1) and continues towards infinity. The -coordinate is 4, so the domain appears to be [4, ). However, since the graph is only over the interval [ 0, 0], the domain stops at 0. Thus, the domain of f() is [4, 0]. U3-181 Lesson 5: Comparing Properties of Functions

188 . Identify the domain of g(). g() is presented as an equation where the radicand must be greater than or equal to 0. To find the domain of g(), create an inequality in which the radicand is greater than or equal to 0 and solve for. The given function is g( ) , where is the radicand Set the radicand greater than or equal to Subtract 4 from both sides. 4 3 Divide both sides by Convert the improper fraction to a mied fraction. The domain of g() would be 1 1 3,, but since the graph is only over the interval [ 0, 0], the domain of g() is 1 1 3,0. U3-18 Unit 3: Rational and Radical Relationships

189 3. Determine which function has the larger domain. First determine how wide each interval is. The domain of f() is [4, 0]. To determine how wide the interval is, find the difference of the endpoints. For this interval, the endpoints are 0 and The interval [4, 0] is 16 units wide. The domain of g() is 1 1 3,0. To determine how wide this interval is, find the difference of the endpoints. For this interval, the endpoints are 0 and The interval 1 1 3,0 is 11 units wide. 3 f() is 16 units wide. g() is units wide. 1 1 is greater than 16; therefore, g() has the larger domain. 3 U3-183 Lesson 5: Comparing Properties of Functions

190 Eample 3 The curve of the rational function f() passes through the origin and approaches 3 and y as it approaches infinity. Which asymptotes, if any, does f() share with the 6 10 function g( )? Identify the asymptotes of f(). Since the problem statement says that the curve approaches 3 and y as it approaches infinity, this means 3 and y are the asymptotes. Since 3 is a vertical line, it is the vertical asymptote of f(). Since y is a horizontal line, it is the horizontal asymptote of f().. Identify the vertical asymptote of g(). g() is a rational function. To find the vertical asymptote of a rational function, set the denominator equal to 0 and solve for The denominator of the function g( ) is Set the denominator of g() equal to Subtract 9 from both sides. 3 Divide both sides by 3. The vertical asymptote of g() is located at 3. U3-184 Unit 3: Rational and Radical Relationships

191 3. Identify the horizontal asymptote of g(). g() is a first-degree rational function. To find the vertical asymptote of a first-degree rational function, divide the coefficients of the variables In the function g( ), the terms with variables are 6 and The coefficient of the variable in the numerator is 6. The coefficient of the variable in the denominator is 3. 6 y 3 The horizontal asymptote of g() is located at y. 4. Determine which asymptotes the functions share, if any. The vertical asymptote of f() is located at 3, and the vertical asymptote of g() is located at 3. These functions do not share not the same asymptote, since one is positive and one is negative. The horizontal asymptote of f() is located at y, and the horizontal asymptote of g() is located at y. This is the same asymptote. Both functions have the same horizontal asymptote at y. U3-185 Lesson 5: Comparing Properties of Functions

192 Eample 4 The radical function f() has a domain of [ 4, ) and a range of [5, ). The function g() is shown in the following graph. Which function has a higher y-intercept? 1 y g() Estimate the y-intercept of g(). The curve of the graph appears to cross the y-ais at the point (0,.5).. Estimate the y-intercept of f(). Since the range of the function is [5, ), the function only eists at or above y 5. Therefore, this function must intercept the y-ais somewhere above the point (0, 5). 3. Determine which function has a higher y-intercept. Even though the eact y-intercept of f() cannot be determined, it is clear that it is located somewhere above the point (0, 5). Since the point (0,.5) is the y-intercept of g(), and.5 is less than 5, it can be determined that f() has the higher y-intercept. U3-186 Unit 3: Rational and Radical Relationships

193 Eample What are the similarities and differences between the ranges of f ( ) + and g(), a first-degree rational function that contains the following coordinates? g() Identify the range of f(). The range for a first-degree rational function is every real number ecept for the location of the horizontal asymptote. The horizontal asymptote is found by dividing the coefficients of the variables. 4 5 In the function f ( ) +, the terms with variables are 4 and. The coefficient of the variable in the numerator is 4. The coefficient of the variable in the denominator is an understood 1. 4 y 4 1 The horizontal asymptote of f() is located at y 4. Therefore, the range of f() is all real numbers ecept y 4.. Identify the range of g(). The range for a first-degree rational function is every real number ecept for the location of the horizontal asymptote. The values for g(), or y, in the table approach on both ends, so a reasonable estimate for the horizontal asymptote is y. Therefore, the range of g() is all real numbers ecept y. 3. Eplain the similarities and differences between the ranges. For both functions, the range contains all real numbers ecept for one. However, the functions have different eceptions. For f(), the eception is y 4; for g(), the eception has been approimated as y. U3-187 Lesson 5: Comparing Properties of Functions

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