Unit 4 TE Exponential and Log

Unit 4

Planning the Unit

I

n this unit, students study arithmetic and geometric sequences and implicit and explicit rules for defining them. Then they analyze exponential and logarithmic patterns and graphs as well as properties of logarithms. Finally, they solve exponential and logarithmic equations.

AP / College Readiness Unit 4 continues to developstudents’understanding of functions and their inverses by:

Vocabulary Development The key terms for this unit can be found on the Unit Opener page. These terms are divided into Academic Vocabulary and Math Terms. Academic Vocabulary includes terms that have additional meaning outside of math. These terms are listed separately to help students transition from their current understanding of a term to its meaning as a mathematics term. To help students learn new vocabulary: Have students discuss meaning anduse graphic organizers to record their understanding of new words. Remind students toplace their graphic organizers in their math notebooks and revisit their notes as their understanding of vocabulary grows. As needed, pronounce new words and place pronunciation guides and definitions on the class Word Wall.

. d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

Graphing exponentialand logarithmic functions. Applying properties of exponents to develop properties of logarithms. Solving exponentialand logarithmic equations.

Unpacking the Embedded Assessments The following are the key skills and knowledge students will need to know for each assessment.

Embedded Assessment 1 Sequences and Series,The Chessboard Problem Identifying termsin arithmetic and geometric sequences Identifying common differencesand common ratios Writing implicit andexplicit rules forarithmetic and geometric sequences

Embedded Assessments

Embedded Assessment 2

Embedded assessments allow students to do the following:

Exponential Functions and Common Logarithms, Whether or Not

Demonstrate their understanding ofnew concepts.

Examining exponential patterns and functions

Integrate previous andnew knowledge by solving real-world problems presented in new settings.

Identifying and analyzing exponential graphs Transforming exponential functions

They also provide formative information to help you adjust instruction to meet your students’ learning needs.

Graphing and transforming natural base exponential functions

Prior to beginning instruction, have students unpack the first embedded assessment in the unit to identify the skills and knowledge necessary for successful completion of that assessment. Help students create a visual display of the unpacked assessment and post it in your class. As students learn new knowledge and skills, remind them that they will be expected to apply that knowledge to the assessment. After students complete each embedded assessment, turn to the next one in the unit and repeat the process of unpacking that assessment with students.

Examining commonlogarithmic functions Understanding properties of logarithms

Embedded Assessment 3 Exponential and Logarithmic Equations, Evaluating Your Interest Solving exponential equations Solving logarithmic equations Solving real-world applications of exponential and logarithmic functions Unit 4 • Series, Exponential and Logarithmic Functions

293a

Planning the Unit

continued

Suggested Pacing The following table provides suggestions for pacing using a 45-minute class period. Space is left for you to write your own pacing guidelines based on your experiences in using the materials. 45-MinutePeriod Unit Overview/Getting Ready

1

Activity 19

3

Activity 20

3


1

Activity 21

5

Activity 22

4


1

Activity 23

3

Activity 24

4


1

Total45-MinutePeriods

YourCommentsonPacing

26

Additional Resources Additional resources that you may find helpful for your instruction include the following, which may be found in the Teacher Resources at SpringBoard Digital. Unit Practice (additional problemsfor each activity) Getting Ready Practice (additional lessons andpractice problems for the prerequisite skills) Mini-Lessons (instructional support forconcepts related tolesson content)

293b

SpringBoard® Mathematics Algebra 2

. d e rv e s e r s t h g ri ll A . d r a o B e g e ll o C 5 1 0 2 ©

Unit Overview

Series, Exponential and Logarithmic Functions ESSENTIAL QUESTIONS

Unit Overview In this unit, you will study arithmetic and geometric sequences and series and their applications. You will also study exponential unctions and investigate logarithmic unctions and equations.

Key Terms As you study this unit, add these and other terms to your math notebook. Include in your notes your prior knowledge o each word, as well as your experiences in using the word in dierent mathematical examples. I needed, ask or help in pronouncing new words and add inormation on pronunciation to your math notebook. It is important that you learn new terms and use them correctly in your class discussions and in your problem solutions.

Math Terms • sequence • arithmetic sequence • common dierence • recursive ormula • explicit ormula • series • partial sum • sigma notation • geometric sequence • common ratio • geometric inite seriesseries • ininite series • sum o the ininite . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©



• • • • • • • • • •

exponential unction exponential decay actor exponential growth actor asymptote logarithm common logarithm logarithmic unction natural logarithm Change o Base Formula exponential equation

• compound logarithmicinterest equation • extraneous solution

geometric series

Ask students to read the unit overview and mark the text to identiy key phrases that indicate what they will learn in this unit.

Key Terms As students encounter new terms in this unit, help them to choose an appropriate graphic organizer or their word study. As they complete a graphic organizer, have them place it in their math notebooks and revisit as needed as they gain additional knowledge about each word or concept.

Essential Questions How are unctions that grow at a constant rate distinguished rom those that do not grow at a constant rate? How are logarithmic and exponential equations used to model real-world problems?

EMBEDDED ASSESSMENTS This unit has three embedded assessments, ollowing Activities 20, 22, and 24. By completing these embedded assessments, you will demonstrate your understanding o arithmetic and geometric sequences and series, as well as exponential and logarithmic unctions and equations. Embedded Assessment 1:

Sequences and Series p. 321

Read the and essential questions with students ask them to share possible answers. As students complete the unit, revisit the essential questions to help them adjust their initial answers as needed.

Unpacking Embedded Ass essm ent s Prior to beginning the irst activity in this unit, turn to Embedded Assessment 1 and have students unpack the assessment by identiying the skills and knowledge they will need to complete the assessment successully. Guide students through a close reading o the assessment, and use a graphic organizer or other means to capture their identiication o the skills and knowledge. Repeat the process or each Embedded Assessment in the unit.

Embedded Assessment 2:

Exponential Functions and Common Logarithms

p. 357

Embedded Assessment 3:

Exponential and Logarithmic Equations p. 383

293

Developing Math Language As this unit progresses, help students make the transition rom general words they may already know (the Academic Vocabulary) to the meanings o those words in mathematics. You may want students to work in pairs or small groups to acilitate discussion and to build conidence and luency as they internalize

As needed, pronounce new terms clearly and monitor students’ use o words in their discussions to ensure that they are using terms correctly. Encourage students to practice luency with new words as they gain greater understanding o mathematical and other terms.

new language. Ask students to discuss new academic and mathematics terms as they are introduced, identiying meaning as well as pronunciation and common usage. Remind students to use their math notebooks to record their understanding o new terms and concepts.

293

UNIT 4 Getting Ready Use some or all o these exercises or ormative evaluation o students’ readiness or Unit 4 topics.

Prerequisite Skills • Pattern recognition (Items 1, 2, 3) 7.NS.A.3 • Properties o exponents (Items 4, 5, 6) 8.EE.A.1 • Solving equations (Item 7) HSA-REI.B.3 • Writing and graphing unctions (Item 8) HSA-IF.B.4, HSA-BF.A.1a

UNIT 4

Getting Ready Write your answers on notebook paper. Show your work.

1. Describe the pattern displayed by

5. Evaluate the expression.

1, 2, 5, 10, 17, . . . .

3327 3323

2. Give the next three terms o the sequence 0, −2, 1, −3, . . . . 3. Draw Figure 4, using the pattern below. Then explain how you would create any igure in the pattern. Figure 1

Figure 2

6. Express the product in scientiic notation. (2.9 × 103)(3 × 102) 7. Solve the equation or x.

Figure 3

19 = −8x + 35

Ans wer Key

1. The numbers increase by consecutive increasing odd numbers. 2. 2, −4, 3 3. Figure 4

8. Write a unction C(t) to represent the cost o a

4. Simpliy each expression.

taxicab ride, where the charge includes a ee o $2.50 plus $0.50 or each tenth o a mile t. Then give the slope and y-intercept o the graph o the unction.

 2 2 a.  6x   y 3 

b. (2a2b)(3b3) 12 6 c. 10a b

4.

5. 6. 7. 8.

Sample explanation: Each igure has the same number o columns o dots as the igure number, and the number o rows o dots is always one more than the igure number. 4 a. 36x6 y 2 4 b. 6a b c. 2a9b8 34 = 81 8.7 × 105 x=2 C(t) = 2.5 + 0.5t; slope = 0.5; y-intercept = 2.5

5a3b−2

Getting Ready Practice For students who may need additional instruction on one or more o the prerequisite skills or this unit, Getting Ready practice pages are available in the Teacher Resources at SpringBoard Digital. These practice pages include worked-out examples as well as multiple opportunities or students to apply concepts learned.

294

SpringBoard® Mathematics Algebra 2, Unit 4 • Series, Exponential and Logarithmic Functions


   

ACTIVITY

ACTIVITY 19

Arithmetic Alkanes Lesson 19-1 Arithmetic Sequences

Ac tiv it y St and ard s F ocu s My Notes

Learning Targets:

In Activity 19, students learn to identiy arithmetic sequences and to determine the nth term o such sequences using recursive and explicit ormulas. They also write ormulas or the sum o the terms in arithmetic sequences, known as an arithmetic series, and calculate the nth partial sums o arithmetic series. Finally, they represent arithmetic series using sigma notation and determine the sums. There is a lot o notation in this activity, and students may get lost in the

• Determine whether a given sequence is arithmetic. • Find the common dierence o an arithmetic sequence. • Write an expression or anarithmetic sequence, and calculate thenth term. SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look or a Pattern, Summarizing, Paraphrasing, Vocabulary Organizer

Hydrocarbons are the simplest organic compounds, containing only carbon and hydrogen atoms. Hydrocarbons that contain only one pair o electrons between two atoms are called alkanes. Alkanes are valuable as clean uels because they burn to orm water and carbon dioxide. The number o carbon and hydrogen atoms in a molecule o the irst six alkanes is shown in the table below.

A l ka n e

C a r b on A t om s

methane

1

ethane

2

propane

3

butane

4

H y d r o g en A t om s 4 6 8 10

pentane

5

12

hexane

6

14

1. Model with mathematics. Graph the data in the table. Write a unction f, where f(n) is the number o hydrogen atoms in an alkane with n carbon atoms. Describe the domain o the unction. f(n) = 2n + 2; n is a positive integer.

y 15 14 13 12 11

symbols. Encourage students to read careully, and check oten to be sure they can explain the meanings o the ormulas and the variables within the ormulas.

s 10 m 9 to A 8 n e g 7 o r d 6 y H 5

Lesson 19-1 PLAN

4 3 2 1

Pacing: 1 class period 12

34 56 78 Carbon Atoms

91 0

MATH TERMS A sequenceis an ordered list of items.

x

Chunking the Lesson #1 #2–3 #4–5 Check Your Understanding #9–10 #11–14 #15–16 Example A Check Your Understanding Lesson Practice

Any unction where the domain is a set o positive consecutive integers orms a sequence. The values in the range o the unction are the terms o the sequence. When naming a term in a se quence, subscripts are used rather than traditional unction notation. For example, the irst term in a sequence would be called a1 rather than f(1).

TEACH

Consider theirst sequence {4, 6, 8, 10, 12, 14} ormed by the number o hydrogen atoms in the six alkanes.

sequences. 1. −16, −14, −12, … [The difference between any term and the previous term is 2; −10, −8, −6.] 2. 6, −2, −10, … [The difference between any term and the previous term is −8; −18, −26, −34.] Discuss with students the methods they used to determine the patterns.

Bell-Ringer Activity Ask students to describe the pattern and give the next 3 terms o the ollowing

2. What is a1? What isa3? a1 = 4; a3 = 8 . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

19

Guided

3. Find the dierences a2 − a1, a3 − a2, a4 − a3, a5 − a4, and a6 − a5. Each difference is 2.

Sequences like the one above are called arithmetic sequences. An arithmetic sequence is a sequence in which the dierence o consecutive terms is a constant. The constant dierence is called the common difference and is usually represented by d.

WRITING MATH If the fourth term in a sequence is 10, then a4 = 10. Sequences may have a finite or an infinite number of terms and are sometimes written in braces { }.

Common Core State Standards for Activity 19 HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.  HSF-BF.A.1

Write a function that describes a relationship between two quantities.



HSF-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a HSF-BF.A.2

context. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 

1 Activating Prior Knowledge, Create Representations, Look for a Pattern, DebriefingThis is an entry-level item. Students plot data and use their knowledge o linear unctions to write a rule or f(n). When debrieing, be sure that students understand that the domain o the unction must be discrete because n represents the number o carbon atoms. 2–3 DebriefingUse Item 2 to assess whether students understand the meaning o subscripts. Also, note that inding that the dierences are constant in Item 3 will help students attach meaning to the deinition o common dierence that ollows.

Activity 19 • Arithmetic Sequences and Series

295

ACTIVITY 19 Continued Universal Access

  Arithmetic Sequences

ACTIVITY 19 continued

Understanding that a sequence represents a unction may be diicult or students. To help with this concept, ask students to write a set o ordered pairs that shows the unctional relationship between the term number and the terms in the sequence {4, 6, 8, 10, 12, 14}.

My Notes

dierence d.

d = an+1 − an

5. Determine whether the numbers o carbon atoms in the irst six alkanes {1, 2, 3, 4, 5, 6} orm an arithmetic sequence. Explain why or w hy not. Yes; The sequence is arithmetic because there is a common difference of 1 between each pair of consecutive terms.

4–5 Look for a Pattern, Debriefing In Item 4, students must generalize, which is a diicult concept or some students.

Check Your Understanding

Direct students to the Math Tip. I students need additional guidance understanding that these expressions represent consecutive terms in the sequence, try asking these questions: What do an+1 and an represent when n = 1? When n = 2? When n = 3? Item 5 connects the concepts o sequence and common dierence back to the opening context and shows that consecutive integers orm an arithmetic sequence with d = 1. To extend this item, ask students whether it is possible to have an arithmetic sequence where d = 0. Any sequence where the terms remain constant satisies this condition.

Check Your Understanding Debrie students’ answers to these items to ensure that they understand concepts related to arithmetic sequences. For the sequence in Item 7, each term ismultiplied by 2 to ind the next term; be sure students can verbalize why this pattern does not determine an arithmetic sequence.

Ans wer s 6. arithmetic; 5 7. not arithmetic 8. 37, 46, 64 9–10 Activating Prior Knowledge, Create Representations, Look for a Pattern In Item 9, students use their ability to work with literal equations to solve the ormula rom Item 4 or an+1. In Item 10, students identiy each term using math terminology. In order to more easily compare the recursive ormula and the explicit ormula, students need to understand that the expression an = an−1 + d is equivalent to the expression in Item 9. Elicit rom students the act that an is the term that ollows an−1, just as an+1 is the term that ollows an.

4. Use an and an+1 to write a general expression or the common

Determine whether each sequence is arithmetic. I the sequence is arithmetic, state the common dierence.

6. 3, 8, 13, 18, 23, . . . 7. 1, 2, 4, 8, 16, . . . 8. Findthe missing terms in the arithmetic sequence 19, 28, , 55,

MATH TIP In a sequence, an+1 is the term that follows an.

9. Write a ormula or an+1 in Item 4. an+1 = an + d

10. What inormation is needed to ind an+1 using this ormula? The value of the common difference and the value of the previous term are needed.

Finding the value o an+1 in the ormula you wrote in Item 9 requires knowing the value o the previous term. Such a ormula is called a recursive formula , which is used to determine a term o a sequence using one or more o the preceding terms. The terms in an arithmetic sequence can also be written as the sum o the irst term and a multiple o the common dierence. Such a ormula is called an explicit formula because it can be used to calculate any term in the sequence as long as the irst term is known. 11. Complete the blanks or the sequence {4, 6, 8, 10, 12, 14, . . .} ormed by the number o hydrogen atoms. a1 =

4

d=

a2 = 4 +

1

a3 = 4 +

2

a4 = 4 +

3

a5 = 4 +

4

a6 = 4 +

5

a10 = 4 +

9

2

⋅2=6 ⋅2=8 ⋅ 2 = 10 ⋅ 2 = 12 ⋅ 2 = 14 ⋅ 2 = 22

Developing Math Language Watch or students whoand interchange terms recursive formula explicit the formula. As students respond to questions or discuss possible solutions to problems, monitor their use o these terms to ensure their understanding and ability to use language correctly and precisely.

296

,

.





12. Write a general expression an in terms o n or inding the number o hydrogen atoms in an alkane molecule with n carbon atoms. an = 4 + (n − 1)

⋅2

13. Use the expression you wrote in Item 12 to ind the number o hydrogen atoms in decane, the alkane with 10 carbon atoms. Show your work. a10 = 4 + (10 − 1)

⋅ 2 = 4 + 9 ⋅ 2 = 22

14. Find the number o carbon atoms in a molecule o an alkane with 38 hydrogen atoms. n = 18

15. Model with mathematics. Use a1, d, and n to write an explicit ormula or an, the nth term o any arithmetic sequence. an = a1 + (n − 1)

⋅d

16. Use the ormula rom Item 15 to ind the speciied term in each arithmetic sequence. a. Find the 40th term when a1 = 6 and d = 3.

123

b. Find the 30th term o the arithmetic sequenc e 37, 33, 29, 25, .. . . −79

Example A Hope is sending invitations or a party. The cost o the invitations is $5.00, and postage or each is $0.45. Write an expression or the cost o mailing the invitations in terms o the number o invitations mailed. Then calculate the cost o mailing 16 invitations. Step 1: Identiy a1 and d. The cost to mail the irst invitation is equal to the cost o the invitations and the postage or that one invitation. a1 = 5.00 + 0.45 = 5.45. The postage per invitation is the common dierence, d = 0.45. Step 2: Use the inormation rom Step 1to write a general expression or an. I n equals the number o invitations mailed, then the expression or the cost o mailing n invitations is: an = a1 + (n −1)d an = 5.45 + (n −1)(0.45) an = 5.45 + 0.45n − 0.45 an = 5.00 + 0.45n Step 3: Use the general expression to evaluate a16. The cost o mailing 16 invitations is ound by solving or n = 16. a16 = 5.00 + 0.45(16) = 5.00 + 7.20 = 12.20.

Try These A . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

Write an expression or the nth term o the arithmetic sequence, and then ind the term. a. Find the 50th term when a1 = 7 and d = −2. an = 7 + (n − 1)

⋅ −2; a

50

= −91

My Notes

ACTIVITY 19 Continued 11–14 Look for a Pattern, Create Representations, Debriefing Notice that Item 11 skips rom inding the sixth term to the tenth term. This provides scaolding to help students generalize in Item 12. Students who do not make the connection between Items 11 and 12 may try to use the act that to get any term in the sequence, you add 2 to the previous term. However, this leads to a recursive deinition:a1 = 4, an = an−1 + 2, while the item asks or the explicit deinition in terms o n. Suggest to those students that they use their work in Item 11 to answer Item 12.

Item 13 gives students the opportunity to veriy the ormula rom Item 12. I a student gives an incorrect answer, ask the student to compare the answer or the eleventh term in the sequence to the answer or the tenth term in Item 11 to see whether it seems reasonable. Some students may simply write the eleventh term without using the ormula. The purpose o this activity is to derive and use ormulas. Ask any student who does not see the necessity or a ormula to ind the 500th term o the sequence. 15–16 Think-Pair-Share, Create Representations, Look for a Pattern, Debriefing Writing an explicit ormula or an in Item 15 is the culmination o work done in Items 11 through 14. Have students share answers on whiteboards and veriy that all students have correct

responses. Then, in Item 16, students practice using the ormula they derived.

Example A Think-Pair-Share, DebriefingStudents can discuss how they determined the values o a1 and d or this real-world scenario.

b. Find the 28th term o the arithmetic sequence 3, 7, 11, 15, 19, . . . . an = 3 + (n − 1)

⋅ 4; a

28

= 111

c. Which term in the arithmetic sequence 15, 18, 21, 24, . .. is equal to 72? an = 15 + (n − 1)

⋅ 3; n = 20


297

ACTIVITY 19 Continued Check Your Understanding



Debrie students’ answers to these items to ensure that they understand how to use both explicit and recursive ormulas to calculate the nth term o an arithmetic sequence. Ask students to think o situations in which one ormula might be more useul t han the other.

My Notes

Check Your Understanding 17. Show that the expressions or an in Item 12 and f(n) in Item 1 are equivalent.

18. Find the 14th term or the sequence deined below.

Ans wer s 17. an = 4 + (n − 1) 2 = 4 + (2n − 2) = 2n + 2 = f(n) 18. −3.5 19. the 8th term 20. Shontelle can do this because

⋅

term value

Explicit: aand n = 1 + ( − 1) a4 = 7 + (4 − 1)5 a4 = 7 + 3 × 5

Students’ answers to Lesson Practice problems will provide you with a ormative assessment o their understanding o the lesson concepts and their ability to apply their learning.

Recursive: an = an −1 + d a4 = a3 + 5 a4 = (a2 + 5) + 5 a4 = ((a1+ + 5) +5) 5 a4 = ((7+ + 5) +5) 5

  

LESSON 19-1 PRACTICE

298

0.5

Explain why Shontelle can substitute (a2 + 5) or a3 and (a1 + 5) or a2. Compare the result that Shontelle ound when using the recursive ormula with the result o the explicit ormula. What does this tell you about the ormulas?

See the Activity Practice or additional problems or this lesson. You may assign the problems here or use them as a culmination or the activity.

Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to arithmetic sequences and inding the nth term o such a sequence. Be sure students understand that d can be negative, as in Item 22. Students may substitute incorrectly into the ormulas. Encourage them to begin each problem by writing the general ormula and identiying each value in the ormula. For example, or Item 23b, students can write an = a1 + (n − 1)d ollowed by a1 = 3, d = 4. Correctly identiying and substituting these values will be important in upcoming lessons.

0.9

Shontelle used 20. Express regularity inormulas repeated ng. the explicit and recursive to reasoni calculate the ourth term inboth a sequence where a1 = 7 and d = 5. She wrote the ollowing:

ASS ESS

ADA PT

1.3

19. Determine which term in the sequence in Item 18 has the value −1.1.

an = an−1 + d. The results are equivalent; both ormulas give the same result.

21. not arithmetic 22. a.−3; b. an = 23 − 3n c. an = an−1 − 3 23. a.d = 4 b. an = 3 + 4(n − 1) c. an = an−1 + 3 24. 13, 18, 23, 28, 33 25. a1 = 9 8

1234 1.7

For Items 21–23, determine whether each sequence is arithmetic. I the sequence is arithmetic, then a. state the common dierence. b. use the explicit ormula to write a general expression oran in terms o n. c. use the recursive ormula to write a general expression oran in terms o an−1. CONNECT

TO HISTORY

Item 21 is a famous sequence known as the Fibonacci sequence. Find out more about this interesting sequence. You can find its pattern in beehives, pinecones, and flowers.

21. 1, 1, 2, 3, 5, 8, . . . 22. 20, 17, 14, 11, 8, . . . 23. 3, 7, 11, . . . 24. A sequence is deined by a1 = 13, an = 5 + an−1. Write the irst ive terms in the sequence.

25. Make sense o f problems. Find the irst term. n

3456

an

7 8

3 4


5 8

1 2


ACTIVITY 19 Continued

  Arithmetic Series


Lesson 19-2 PLAN

My Notes

Learning Targets:

Pacing:1 class period

• Write a ormula or the nth partial sum o an arithmetic series. • Calculate partial sums o an arithmetic series.

Chunking the Lesson #1–2 #3–4 #5–6 Example A Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Look or a Pattern, ThinkPair-Share, Create Representations A series is the sum o the terms in a sequence. The sum o the irst n terms o a series is the nth partial sum o the series and is denoted by Sn.

TEACH

1. Consider the arithmetic sequence {4, 6, 8, 10, 12, 14, 16, 18}. a. Find S4.

Bell-Ringer Activity

S4 = 28

Ask students to determine the sum o the irst 4 terms o each sequence. 1. 1, 4, 7, 10, … [22] 2. 3, 5, 7, 9, 11, … [ 24] 3. 2, 9, 16, 23, 30, 37, … [50]

b. Find S5. S5 = 40

c. Find S8.

Discuss the methods students used to ind the sums.

S8 = 88

d. How does a1 + a8 compare to a2 + a7, a3 + a6, and a4 + a5?

Developing Math Language

Each sum is 22.

e. Make use of structure. Explain how to indS8 using the value o a1 + a8. Since there are 4 pairs of numbers with the same sum, S8 = 4(a1 + a8) = 4(22) = 88.

2. Consider the arithmetic series 1 + 2 + 3 + . . . + 98 + 99 + 100. a. How many terms are in this series? 100

b. I all the terms in this series are paired as shown below, how many pairs will there be? 50

1 + 2 + 3 + 4 + . . . + 97+ 98 + 99 + 100


c. What is the sum o each pair o numbers?

CONNECT

TO HISTORY

A story is often told that in the 1780s, a German schoolmaster decided to keep his students quiet by having them find the sum of the first 100 integers. One young pupil was able to name the sum immediately. This young man, Carl Friedrich Gauss, would become one of the world’s most famous mathematicians. He reportedly used the method in Item 2 to find the sum, using mental math.

As you guide students through their learning o the terms series and partial sum, explain meanings in terms that are accessible or your students. For example, point out that the word part is included in the word partial. Use the concrete examples in the items to help students gain understanding. Encourage students to make notes about these terms and their understanding o what they mean and how to use them to describe precise mathematical concepts and processes. 1–2 Look for a PatternThese items provide the scaolding or students to develop the ormula or the partial sum o an arithmetic series, which students will do in Item 3.

101

d. Construct viable arguments. Find the sum o the series. Explain how you arrived at the sum. There are 50 pairs of numbers, each with a sum of 101, so the sum is 50(101)= 5050.


299

ACTIVITY 19 Continued 3–4 Think-Pair-ShareIn Item 3, students develop the general ormula sn = n (a1 + an ) or the partial sum o an 2 arithmetic series.

Some students may be concerned about the case when n is odd. I n = 5, students can consider this as representing 5, or 2 1, pairs o terms. 2 2 The expression n represents the middle 2 term in the series, which will be equal to hal the sum o the irst and last terms. Thus the ormula holds or the case when n is odd. It is important that students share their answers to Item 3 to veriy that all students understand how to derive the ormula.



My Notes

3. Consider thearithmetic seriesa1 + a2 + a3 + . . . + an−2 + an−1 + an. a1 + a2 + a3 + . . . + an − 2 + an−1 + an

a. Write an expression or the number o pairs o terms in this series. n 2

b. Write a ormula orSn, the partial sum o the arithmetic series. Sn = n (a1 + an) 2

In Item 4, students veriy that the ormula they just ound gives the correct sums, the same ones they ound in Item 1.

ELL Support Students may use the termseries incorrectly because it sounds like a plural word and because it is sometimes used in everyday language to reer to a sequence. In act, a common error is to useseries interchangeably with sequence. Monitor students’ use oseries careully to be sure they understand that the term reers to a single sum.

4. Use the ormula rom Item 3b to ind each partial sum o the arithmetic sequence {4, 6, 8, 10, 12, 14, 16, 18}. Compare your results to your answers in Item 1. Results in Items 4a–c should be the same as 1a–c.

a. S4 S4 = 4 (4 + 10) = 2(14)= 28 2

b. S5 S5 = 5 (4 + 12) = 5 (16)= 40

2

2

c. S8 S4 = 8 (4 + 18) = 4(22) = 88 2

300






My Notes

5. A second orm o the ormula or inding the partial sum o an arithmetic series is Sn = n [2a1 + (n − 1)d ]. Derive this ormula,

2 starting with the ormula rom Item 3b o this lesson and the nth term ormula, an = a1 + (n − 1)d, rom Item 15 o the previous lesson. Sn = n (a1 + an) = n [a1 + (a1 + (n − 1) 2 2

⋅ d )] = n2 [2a

1

+ (n − 1)

⋅ d]

6. Use the ormula Sn = n [2a1 + (n − 1)d ] to ind the indicated partial sum 2 o each arithmetic series. Show your work. a. 3 + 8 + 13 + 18 + . . .; S20 S20 = 20[2 2

⋅ 3 + (20 − 1) ⋅ 5] = 10(6+ 19 ⋅ 5) = 10(101)= 1010

b. −2 − 4 − 6 − 8 − . . .; S18 S18 = 18[2 (−2) + (18 − 1)(−2)] = 9[−4 + 17(−2)] = 9(−38) = −342 2

5–6 Think-Pair-Share, Create Representations, Look for a Pattern In Item 5, students show how to derive an alternate ormula or the partial sum o an arithmetic series, Sn = n [2a1 + (n − 1)d]. Point out to 2 students that they will use one or the other ormula, depending upon which values they are given or a particular series. To use the ormula rom Item 3, they need to know n, a1, and an. For this alternate ormula, they need n, a1, and d.

In Item 6, students apply the alternate ormula to ind partial sums.

Example A Think-Pair-Share, DebriefingStudents can discuss the steps they took to determine the solution. They should explain which o the two partial sum ormulas they chose or each item in the Try These, and why.

Example A Find the partial sum S10 o the arithmetic series with a1 = −3, d = 4. Step 1: Find a10. The terms are −3, 1, 5, 9, . . . . a 1 = −3 a10 = a1 + (n − 1)d = −3 + (10 − 1)(4) = −3 + (9)(4) = −3 + 36 = 33 Step 2: Substitute or n, a1, and a10 in the ormula. Simpliy. 10 S10 = n (a+ =a− 33 =) 5(30) 150 n )( + = 3 2 1 2 Or use the ormula Sn = n [2a1 + (n − 1):d] 2 S10 = 10 [2(−3) + (10 − 1) 4] = 5[−6 + 36] = 150 2

Try These A Find the indicated sum o each arithmetic series. Show your work.

a. Find S8 or the arithmetic series with a1 = 5 and a8 = 40. . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

180

b. 12 + 18 + 24 + 30 + . . .; S10 390

c. 30 + 20 + 10 + 0 + . . .; S25 −2250


301

ACTIVITY 19 Continued Check Your Understanding Debrie students’ answers to these items to ensure that they understand the concepts related to inding partial sums o arithmetic series. Ask students whether they could ind the sum o all terms in an arithmetic sequence.

Ans wer s 7. n = 3, n = 6; a1 = 12; a6 = 37

2 8. 12, 17, 22, 27, 32, 37; 12+ 17 + 22 + 27 + 32 + 37 = 147 9. Sample answer: Use Sn = n [2a1 + (n − 1)d] when an 2 is unknown.

ASS ESS Students’ answers to Lesson Practice problems will provide you with a ormative assessment o their understanding o the lesson concepts and their ability to apply their learning. See the Activity Practice or additional problems or this lesson. You may assign the problems here or use them as a culmination or the activity.



My Notes

Check Your Understanding 7. Explain what each term o the equation S6 = 3(12 + 37) = 147 means in terms o n and an.

8. Find each term o the arithmetic series in Item 7, and then veriy the given sum.

9. When would the ormula Sn = n [2a1 + (n − 1)d ] be preerred to the 2 ormula Sn = n (a1 + an )? 2

   10. Find the partial sum S10 o the arithmetic series with a1 = 4, d = 5. 11. Find the partial sum S12 o the arithmetic series 26 + 24 + 22 + 20 + . . . . 12. Find the sum o the irst 10 terms o an arithmetic sequence with an eighth term o 8.2 and a common dierence o 0.4.

13. Model with mathematics. An auditorium has 12 seats in the irst row, 15 in the second row, and 18 in the third row. I this pattern continues, what is the total number o seats or the irst eight rows?

LESSON 19-2 PRACTICE 10. S10 = 265 11. S12 = 180 12. S10 = 72 13. 180 seats

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to inding the nth partial sum o an arithmetic series. The goal o this lesson is or students to develop and use the ormulas; watch or students who ind sums by simply adding terms in the series. Insist that they show their work using the ormulas.

302




  Sigma Notation


Lesson 19-3 PLAN

My Notes

Learning Targets:


the index, lower and upper limits, and general term in sigma • Identiy notation. • Express the sum o a series using sigma notation. • Find the sum o a series written in sigma notation.

Chunking the Lesson Example A Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Look or a Pattern, ThinkPair-Share, Create Representations

TEACH

In the Binomial Theorem activity in Unit 3, you were introduced to a shorthand notation called sigma notation (Σ). It is used to express the sum o a series. 4

Bell-Ringer Activity Ask students to use one o the ormulas or Sn rom Lesson 19-2 to ind the

MATH TIP

The expression ∑ n=1 (2n + 5) is read “the sum rom n = 1 to n = 4 o 2n + 5.” To expand the series to show the terms o the series, substitute 1, 2, 3, and 4 into the expression or the general term. To ind the sum o the series, add the terms.

4

4

∑ (2n + 5) = (2 ⋅ 1 + 5) + (2 ⋅ 2 + 5) + (2 ⋅ 3 + 5) + (2 ⋅ 4 + 5)

∑ (2n + 5) n=1

n=1

index of summation

= 7 + 9 + 11 + 13 = 40

general term

lower limit of summation

1. 2, 4, 6, 8, …; S4 2. −16, −6, 4, 14, 24, 34, …;S6 3. 5, 12, 19, 26, 31, …;S5

[20] [54] [90]

Discuss the ormulas students used to determine their answers.

TEACHER to TEACHER

Example A

MATH TIP

6

Evaluate ∑( 2 j − 3) .

To find the first term in a series written in sigma notation, substitute the value of the lower limit into the expression for the general term.

j=1

Step 1: The values o j are 1, 2, 3, 4, 5, and 6. Write a sum with six addends, one or each value o the variable. = [2(1) − 3] + [2(2) − 3] + [2(3) − 3] + [2(4) − 3] + [2(5) − 3] + [2(6) − 3]

To find subsequent terms, substitute consecutive integers that follow the lower limit, stopping at the upper limit.

Step 2: Evaluate each expression. = −1 + 1 + 3 + 5 + 7 + 9 Step 3: Simpliy. = 24

Students were introduced to summation notation, or sigma notation, in Unit 3. Note that i the lower limit is 1, the upper limit is equal to the number o terms in the series. I the lower limit is an integer other than 1, sayi, and the upper limit is j, then the number o terms in the series will be j − i + 1.

Example A Marking the Text, Simplify the Problem, Think-Pair-

Try These A a. Use appropriate tools strategically. Write the terms in the 8


indicated partial sum o each arithmetic sequence.

upper limit of summation

series ∑ (3n − 2) . Then ind the indicated sum. n=1

1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 = 92

b. Write the sum o the irst 10 terms o 80 + 75 + 70 + 65 + . . . using sigma notation. 10

Share, DebriefingYou might start by inding the value o 2 j − 3 when j = 1 and j = 2. Now have students work in small groups to ind the values o 2 j − 3 or j = 3 through j = 6. Elicit the act rom students that they now must add all o these values because sigma notation represents their sum.

Sample answer:∑ ( 85 − 5n ) n=1

Differentiating Instruction Supportstudents who struggle with sigma notation by providing this additional practice.

Expand the series and find the sum. 5

1.

∑ (3n + 1)

[50]

n=1 8

Expand the series and find the sum. 5

4.

2.

∑ (5 − 2n)

[−32]

3.

∑ 2n

[110]

(3n + 1) ∑ n=0

[3876]

80

5.

n=1 10 n=1

Then assign the following three items, in which the lower limit is not 1 and the upper limit is not the number of terms in the sequence.

∑ (5 − 2n)

[−6080]

∑ 2n

[991,100]

n=5 1000

6.

n=100


303

ACTIVITY 19 Continued Check Your Understanding Debrie students’ answers to these items to ensure that they know and understand how to calculate d, an, and Sn. Ask them to explain the meaning o each term in their ormulas.

Ans wer s 1. d = an+1 − an 2. an = a1 + (n − 1) d 3. Sn = n (a1 + an ) or 2 n n = [2a1 + (n − 1)d] 2

⋅

  Sigma Notation


My Notes

Check Your Understanding Summarize the ollowing ormulas or an arithmetic series.

1. common dierence

d=

2. nth term

an =

3. sum o irst n terms

Sn = or Sn =

Technology Tip Many graphing calculators and computer algebra systems can calculate sums like the ones in this lesson. Typically, Sum or Sequence unctions return the sum when the general term, index o summation, and upper and lower limits are entered; consult your manual or speciic instructions. You may wish to allow students to check their answers using technology until they gain conidence working with sigma notation. For additional technology resources, visit SpringBoard Digital.

   Find the indicated partial sum o each arithmetic series. 15

4.

∑ (3n − 1) n=1 20

5.

∑ (2k + 1) k =1 10

6.

∑3j j=5

7. Identiy the index, upper and lower limits, and general term o Item 4. 8. Attend to precision. Express the ollowing sum using sigma notation: 3 + 7 + 11 + 15 + 19 + 23 + 27

ASS ESS Students’ answers to Lesson Practice problems will provide you with a ormative assessment o their understanding o the lesson concepts and their ability to apply their learning. See the Activity Practice or additional problems or this lesson. You may assign the problems here or use them as a culmination or the activity. LESSON 19-3 PRACTICE

4. S15 = 345 5. S20 = 440 6. S10 = 135 7. index: n; upper limit: 15; lower limit: 1; general term: 3n − 1 7

8. Sample answer: ∑ −1 + 4n n=1

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand how to ind a sum written in sigma notation. I students need more practice, ind arithmetic sequences rom the previous lessons in this activity and have students use sigma notation to write partial sums or these sequences. (Students may choose which partial sums, or you may wish to assign speciic partial sums.) Students can then exchange papers and ind each indicated sum.

304




    Arithmetic Alkanes

   Write your answers on notebook paper. Show your work.

Lesson 19-1 1. Determine whether or not each sequence is arithmetic. I the sequence is arithmetic, state the common dierence. a. 4, 5, 7, 10, . . . b. 5, 7, 9, 11, . . . c. 12, 9, 6, 3, . . .

2. Determine whether or not each sequence is arithmetic. I theto sequence is arithmetic, use the explicit ormula write a general expression or an in terms o n. a. 4, 12, 20, 28, . . . b. 5, 10, 20, 40, . . . c. 4, 0, −4, −8, . . .

3. Determine whether or not each sequence is arithmetic. I the sequence is arithmetic, use the recursive ormula to write a general expression or an in terms o an−1. a. 7, 7.5, 8, 8.5, . . . b. 6, 7, 8, 9, . . . c. −2, 4, −8, . . .

4. Find the indicated term o each arithmetic sequence. a. a1 = 4, d = 5; a15 b. 14, 18, 22, 26, . . .; a20 c. 45, 41, 37, 33, . . .; a18

5. Find the sequence or which a8 does NOT equal 24. A. 3, 6, 9, . . . B. −32, −24, −16, . . . C. 108, 96, 84, . . . D. −8, −4, 0, . . .

6. A radio station oers a $100 prize on the irst day o a contest. Each day that the prize money is not awarded, $50 is added to the prize amount. I a contestant wins on the 17th day o the contest, how much money will be awarded? . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

7. I a4 = 20 and a12 = 68, ind a1, a2, and a3. 8. Find the indicated term o each arithmetic sequence. a. a1 = −2, d = 4; a12 b. 15, 19, 23, 27, . . .; a10 c. 46, 40, 34, 28, . . .; a20


9. What is the irst value o n that corresponds to a positive value? Explain how you ound your answer. 5 4 3 2n 1 an

−42.5 −37.8 −33.1 −28.4 −23.7

10. Find the irst our terms o the sequence with a1 = 2 and an = an − 1 + 1 . 3

6

11. I a = 3.1 and a = −33.7, write an expression or 1the sequence5and ind a2, a3, and a4.

Lesson 19-2 12. Find the indicated partial sum o each arithmetic series. a. a1 = 4, d = 5; S10 b. 14 + 18 + 22 + 26 + . . .; S12 c. 45 + 41 + 37 + 33 + . . .; S18

13. Find the indicated partial sum o each arithmetic series. a. 1 + 3 + 5 + . . .; S6 b. 1 + 3 + 5 + . . .; S10 c. 1 + 3 + 5 + . . .; S12 d. Explain the relationship between n and Sn in parts a–c.

14. Find the indicated partial sum o the arithmetic series. 0 + (x + 2) + (2x + 4) + (3x + 6) + . . .; S10

A. 9x + 18 B. 10x + 20 C. 45x + 90 D. 55x + 110 15. Two companies oer you a job. Company A oers a $40,000 salary withyou an a annualyou raise o $1500.irst-year Company B oers $38,500 irst-year salary with an annual raise o $2000. a. What would your salary be with Company A as you begin your sixth year? b. What would your salary be with Company B as you begin your sixth year? c. What would be your total earnings with Company A ater 5 years? d. What would be your total earnings with Company B ater 5 years?

ACT IV IT Y PR ACT ICE 1. a.no b. yes; d = 2 c. yes; d = −3 2. a.yes; an = −4 + 8n b. no c. yes; an = 8 −4n 3. a.yes; an = an−1 + 0.5 b. yes; an = an−1 + 1 c. no 4. a. a15 = 74 b. a20 = 90 c. a18 = −23 5. D 6. $900 7. a1 = 2, a2 = 8, and a3 = 14 8. a. a12 = 42 b. a10 = 51 c. a20 = −68 9. n = 11. Sample explanation: I divided −23.7 by the common dierence o 4.7; the result is between 5 and 6, so it will take 6 more terms to get a positive value. 10. 2 , 5 , 1, 7 3 6 6 11. an = −3.1 + (n −1)9.2; a2 = −6.1, a3 = −15.3, a4 = −24.5 12. a.S10 = 265 b. S12 = 432 c. S18 = 198 13. a.S6 = 36 b. S10 = 100 c. S12 = 144 d. Sn = n2 14. C 15. a.$47,500

b. $48,500 c. $215,000 d. $212,500


305

ACTIVITY 19 Continued 16. d = 4 17. d = −10; S7 = 119 18. a1 = 6.24; S9 = 70.56 19. a.9 prizes b. $2700 20. B 21. S = 75 (1 + 150) = 11,325 n+ 22. S= (a = a5 ) + 5= (20 12) n 2 1 2

    Arithmetic Alkanes


16. I S12 = 744 and a1 = 40, ind d. 17. In an arithmetic series, a1 = 47 and a7 = −13, ind d and S7.

80

5

23. a.∑ (5 − 6 j) = −65 j =1 20

b. ∑ 5 j = 1050 j =1 15

26. Which statement is true or the partial sum n

∑ (4 j + 3)? j=1

18. In an arithmetic series, a9 = 9.44 and d = 0.4, ind a1 and S9.

19. The irst prize in a contest is $500, the second prize is $450, the third prize is $400, and so on. a. How many prizes will be awarded i the last prize is $100? b. How much money will be given out as prize money?

A. For n = 5, the sum is 35. B. For n = 7, the sum is 133. C. For n = 10, the sum is 230. D. For n = 12, the sum is 408. 27. Evaluate. 6

a.

∑ ( j + 3) j=1 15

c. ∑ j =5 (5 − j) = −55 10

24. yes; ∑ (2 j + 1) = 120; j =1

5

∑ (2 j + 1) = 35; j=1 10

∑ (2 j + 1) = 85; 120 = 35 + 85 j =6

9

25. yes; ∑ ( j − 7) = −3; j=4

9

∑ ( j − 7) = −18 ; j =1 3

the sum o 13 + 25 + 37 + . . . + 193. 20. Find A. 1339 B. 1648 C. 1930 D. 2060

b.

( j − 12) ∑ j=10

c.

∑ ( 4 j)

8 j=1

21. Find the sum o the irst 150 natural numbers. 22. A store puts boxes o canned goods into a stacked display. There are 20 boxes in the bottom layer. Each layer has two ewer boxes than the layer below it. There are ive layers o b oxes. How many boxes are in the display? Explain your answer.

Lesson 19-3 23. Find the indicated partial sum o each arithmetic series.

8

28.Whichisgreater:

j= 4

a.

−3 = −18 − (−15) 26. B 27. a.∑ ( j + 3) = 39 j=1 15

7

A. ∑ 4 + 3 j j=1 7

B. ∑ (4 − 3 j) 7

C.

20

7

D. ∑ (4 + 3 j )

j=1

j=1

15

c.

8

c. ∑ (4 j) = 144

⋅

∑ (5 − j)

j=1

8

28. ∑ (−3 j + 29) = 55 and j=4 8

29 = − 61;

24.Does (

∑ (3 + 4 j) j=1

j=5

( j − 12) = 3 b. ∑ j=10

∑ −3+j

∑ (5 − 6 j)

b. ∑ 5 j

6

10

5

j =1

j=1

∑ 2) j + 1= ( ∑+2)+j

10

1 (+∑ 2) j

1?

j=6

9

3

25.Does ( ∑ )j − 7= ( ∑− )j− 7 − ( ∑ )j 7 ? Veriy j=4

j=1

MATHEMATICAL PRACTICES Look For and Make Use of Structure

Veriy your answer. 9

5  j π . 30. Evaluate ∑   2  j =1  

j=1

31. How does the common dierence o an arithmetic sequence relate to inding the partial sum o an arithmetic series?

your answer.

j=4 8

∑ (−3 j + 29) is greater. j=4

29. D 5  j ⋅ π  15π 30. ∑  =  2  2 j =1 

31. Sample answer: Because there is a constant dierence between sequential terms o a sequence, terms can be paired to represent a constant sum. Thereore, the partial sum is the product o the number o pairs times the sum. ADD ITI ONA L PR ACT ICE I students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital or additional practice problems.

306

j= 4

j=1

j=1

j=1

8

or ∑ −3 j + 29?

29. Which expression is the sum o the series 7 + 10 + 13 + . . . + 25?

5

∑ ( j − 7) = −15;

( −3 j + )29 ∑



ACTIVITY    

ACTIVITY 20

Squares with Patterns Lesson 2 0-1 Geometric Sequences

20

Guided Act iv it y St and ard s F ocu s In Activity 20, students learn about geometric sequences and series. First they learn to identiy and deine a geometric sequence, including identiying the common ratio. Then they will examine and ind sums o inite and ininite geometric series. Students will use inormation they learned in the previous activity about writing sequences and series in explicit and recursive orms.

My Notes

Learning Targets:

• Determine whether a given sequence is geometric. • Find the common ratio o a geometric sequence. • Write an expression or a geometric sequence, and calculate the nth term. Summarizing, Paraphrasing, SUGGESTED LEARNING STRATEGIES: Create Representations Meredith is designing a mural or an outside wall o a warehouse that is being converted into the Taylor Modern Art Museum. The mural is 32 eet wide by 31 eet high. The design consists o squares in ive dierent sizes that are painted black or white as shown below.

Lesson 20-1 PLAN Pacing: 1 class period Chunking the Lesson

Taylor Modern Art Museum

#1 #2–3 #4–5 Check Your Understanding #9 #10–12 Check Your Understanding Lesson Practice

TEACH Bell-Ringer Activity Ask students to ind the next number in each sequence. 1. 1, 4, 7, 10, … [13] 5 2. 1 , 1, 3 , 2,… 2 2  2  3. −50, 40, −30, 20, … [−10] 1 Create Representations, Debriefing The purpose o this item is to generate several sequences o numbers that will

1. Let Square 1 be the largest size and Square 5 be the smallest size. For each size, record the length o the side, the number o squares o that size in the design, and the area o the square.


Square #

Side of Square (ft)

1

16

2

256

2

8

4

64

3

4

8

16

4

2

16

4

5

1

32

1

be used throughout the activity. I students have diiculty completing the table, reer them to inormation in the opening paragraph about the overall dimensions o the mural. Using the dimensions along with the visual representations will allow students to determine the inormation needed to complete the table. Debrie ater this item to be cert ain all students have the correct answers beore they move on to the next items.

Number of Area of Squares Square (ft2)

Differentiating Instruction

Common Core State Standards for Activity 20 HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.  HSF-BF.A.1




Supportstudents in completing Item 1 by having them label squares in the diagram as 1, 2, 3, 4, 5. This will help them complete the table without having to reread the problem statement several times.



Activity 20 • Geometric Sequences and Series

307

ACTIVITY 20 Continued 2–3 Look for a Pattern, Vocabulary Organizer, Quickwrite, Debriefing Students may notice t he multiplicative patterns in the last three columns o the table. They should be able to identiy the irst column as an arithmetic sequence with a common dierence o 1.

  Geometric Sequences


My Notes

Sample answer: As the square number increases by 1, the side length of a square is divided by 2. The number of squares doubles each time the square number increases by 1. The areas are perfect squares. Each product of the side of a square and the number of squares is 32.

Use Item 3 to assess student understanding o a geometric sequence and a common ratio. Be sure that students understand that the common ratio is the number multiplied by the previous item to generate the next term. So, while students may say that each

b. Each column o numbers orms a sequence o numbers. List the our sequences that you see in the columns o the table. 1, 2, 3, 4, 5; 16, 8, 4, 2, 1; 2, 4, 8, 16, 32; 256, 64, 16, 4, 1

term in the o squares is divided by 2side to generate thesequence next term, the common ratio is in act 1. 2

c. Are any o those sequences arithmetic? Why or why not?

Developing Math Language Geometric sequence and common ratio are deined or the student. The Math Tip may urther help with the understanding o these new concepts. As you guide students through their learning o these terms, explain meanings in terms that are accessible or your students. Students should add the new terms to their math notebooks, including notes about their meanings and how to use them to describe precise mathematical concepts and processes.

Universal Access Watch or students who write the reciprocals o the common ratios or Item 3b. Remind students o the deinition o a common ratio—it is the ratio o consecutive terms. This means the next term is the numerator and the previous term is the denominator. Identiyingr correctly will be important when students study geometric series in the upcoming lessons.

308

2. Reer to the table in Item 1. a. Describe any patterns that you notice in the table.

The only sequence that is arithmetic is 1, 2, 3, 4, 5, because it is the only sequence with a common difference. The common difference is 1.

MATH TIP To find the common difference in an arithmetic sequence, subtract the preceding term from the following term. To find the common ratio in a geometric sequence, divide any term by the preceding term.

A geometric sequence is a sequence in which the ratio o consecutive terms is a constant. The constant is called the common ratio and is denoted by r.

3. Consider the sequences in Item 2b. a. List those sequences that are geometric. Side of square: 16, 8, 4, 2, 1 Number of squares: 2, 4, 8, 16, 32 Area of square: 256, 64, 16, 4, 1

b. State the common ratio or each geometric sequence. Side of square: r= 1 2 Number of squares: r=2 r= 1 Area of square: 4





4. Use an and an−1 to write a general expression or the common ratior. r=


My Notes

an an−1

5. Consider the sequences in the columns o the table in Item 1 that are

y

labeled Square # and Side o Square. a. Plot the Square # sequence by plotting the ordered pairs (term number, square number). b. Using another color or symbol, plot the Side o Square sequence by plotting the ordered pairs (term number, side o square).

21 20 19 18 17

either sequence a linear unction? Explain why or why not. c. Is The Square # plot is linear because there is a constant rate of

16 15

Square # Side of Square

Item 5 allows students to look at an arithmetic sequence and a geometric

change. Each time the term number increases by 1, the Square # 14 13 increases by 1. The Side of Square sequence does not increase by12 a constant rate of change. Each time the term number increases by 11 1, the Side of Square decreases by an increasingly smaller amount. 10

Check Your Understanding 6. Determine whether each sequence is arithmetic, geometric, or neither. I the sequence is arithmetic, state the common dierence. I it is geometric, state the common ratio. a. 3, 9, 27, 81, 243, . . . b. 1, −2, 4, −8, 16, . . . c. 4, 9, 16, 25, 36, . . . d. 25, 20, 15, 10, 5, . . .

7. Use an+1 and an+2 to write an expression or the common ratio r. 8. Describe the graph o the irst 5 terms o a geometric sequence with the irst term 2 and the common ratio equal to 1.

9. Reason abstractly. Use the expression rom Item 4 to write a recursive formula or the term an and describe what the ormula means.

an = an−1r; We can nd a term in a geometric sequence by knowing the previous term and the common ratio.


4–5 Think-Pair-Share, Create Representations, Quickwrite, DebriefingWriting a general expression or the common dierence in an arithmetic sequence, which students did in the preceding act ivity, should help them respond to Item 4. Be sure that they write the e xpression or r as an+1 . Other variations may exist, but an this is the most commonly used expression.

9 8 7 6 5 4 3 2 1 12

34 56 78 Term Number

91 0

x

sequence the graphically. It can used to reinorce concept that anbe arithmetic sequence is a linear unction in which the domain is a set o positive, consecutive integers. Students should be able to relate the common dierence o the sequence to the constant rate o change o a linear unction. Students should also note that the terms in the geometric sequence are not linear because the constant related to a geometric sequence is multiplicative rather than additive.

Check Your Understanding Debrie students’ answers to these items to ensure that they can identiy arithmetic and geometric sequences. Ask students to describe their methods or identiying these sequences, and, i time allows, present additional examples o sequences and have students identiy them using their described methods.

Ans wers 6. a. geometric; r = 3 b. geometric; r = −2 c. neither d. arithmetic; d = −5 a 7. r = n+2 an+1 8. The graph will consist o t he points (1, 2), (2, 2), (3, 2), (4, 2), and (5, 2). 9 Create RepresentationsI necessary, review the meaning o recursive formula. Ask students to describe how they used their answer to Item 4 to write the recursive ormula.


309

ACTIVITY 20 Continued 10–12 Think-Pair-Share, Look for a Pattern, Create Representations, DebriefingNotice that Item 10 skips rom inding the sixth term in the sequence to inding the tenth term. This provides scaolding to help students generalize in Item 11.

Item 11 is a c ulminating item or the irst part o t he activity. Have students share answers on whiteboards and veriy that all students have correct responses. In Item 12, students practice using the ormula derived in Item 11. The purpose o this activity is to derive and use



My Notes

The terms in a geometric sequence also can be written as the product o the irst term and a p ower o the common ratio.

10. For the geometric sequence {4, 8, 16, 32, 64, … }, identiy a1 and r. Then ill in the missing exponents and blanks. 4

a1 =

r=

2

⋅2 =8 a = 4 ⋅ 2 = 16 32 a =4⋅2 = 64 a =4⋅2 = 128 a =4⋅2 = 2048 a =4 2 = 11. Use a , r⋅, and n to write an explicit formula or the nth term o any 1

a2 = 4

2

3

3

4

4

5

5

6

9

ormulas. Be sure that students do not use repeated multiplication to ind the missing terms. Students should show how to use the ormula to ind the desired term. They may veriy the results with repeated multiplication i they choose.

Technology Tip

10

1

geometric sequence. an = a1

⋅r

n−1

12. Use the ormula rom Item 11 to ind the indicated term in each

Students can use a calculator to ind the terms o a geometric sequence.

geometric sequence. a. 1, 2, 4, 8, 16, . . . ; a16 32,768

Enter the irst term and press ENTER . Press x and enter the common ratio. Press ENTER repeatedly to ind each subsequent term. For some students, writing this process in their notes will be helpul, as they can reer to it again as they work through the course.

b. 4096, 1024, 256, 64, . . . ; a9 1 or 0.0625 16

For additional technology resources, visit SpringBoard Digital.

Differentiating Instruction Supportstudents who struggle with Item 10 by helping them transition rom recognizing a repeated multiplication pattern to being able to express the pattern using exponents. Have students write each term in Item 10 in the orm below. 

⋅ ⋅ a = 4⋅2⋅ = 4⋅2 = 16 a = 4⋅⋅ 2 ⋅ =⋅4 2 = 32 a2 = 4 2 = 4 2  = 8 3



4

 

310

 

 






My Notes

Check Your Understandin g 13. a. Complete the table or the terms in the sequence with a1 = 3; r = 2. Term

Recursive an = an−1

⋅r

3

3

a1

Explicit an = a1

⋅2

Value of Term

⋅r

1−1

n−1

=3

3

3⋅2 =3⋅2 6 ⋅2 (3 ⋅ 2) ⋅ 2 3⋅2 =3⋅2 a 12 a 24 ((3 2) 2) 2 3 2 ⋅ ⋅ ⋅ ⋅ = 3⋅ 2 a = 3⋅ 2 (((3 ⋅ 2) ⋅ 2)⋅ 2)⋅ 2 3 ⋅ 2 48 b. What does the product (3 ⋅ 2) represent in the recursive expression or a ? a2

2−1

3

3−1

3 4 5

2

4−1

3

5−1

4

3

c. Express regularity in repeated reasoning. Compare the recursive and explicit expressions or each term. What do you notice?

    14. Write a ormula that will produce the sequence that appears on the calculator screen below. 5*3

Check Your Understanding Debrie students’ answers to these items to ensure that they understand the dierence between the explicit orm and the recursive orm o a geometric sequence. Be sure students understand and can verbalize the meaning o every term and variable in the ormulas.

Ans wers 13. a.See student page. b. a1 r c. Sample answer: The number o

⋅

times that 2 is a actor in t he recursive expression matches the value o the exponent in the explicit ormula.


15 Ans*3 45 135 405

b. 5, 15, 45, 135, . . .

14. an = 5(3)n−1 15. a.arithmetic; d = 2 b. geometric; r = 3 c. geometric; r = − 2 3 d. neither 16. a.a8 = −4374

c. 6, −4, 83, − 16 9,... d. 1, 2, 4, 7, 11, . . .

b. a12 = −a0.5 = a 3; 17. recursive: n+1 n explicit: an = 0.25(3)n−1

15. Determine whether each sequence is arithmetic, geometric, or neither. I the sequence is arithmetic, state the common dierence, and i it is geometric, state the common ratio. a. 3, 5, 7, 9, 11, . . .



⋅

16. Find the indicated term o each geometric sequence. a. a1 = −2, r = 3; a8 b. a1 = 1024, r = − 1 ; a12

2 17. Attend to precision. Given the data in the table below, write both a recursive ormula and an explicit ormula or an. n an

1234 0.75 0.25

2.25

6.75

ADA PT Check students’ answers to the Lesson Practice to ensure that they can dierentiate between an arithmetic and a geometric sequence. Students may beneit rom making a table or other graphic organizer that compares and contrasts the two types o sequences and includes examples o each. Encourage students to reer to their graphic organizers as needed.


311


  Geometric Series


Lesson 20-2 PLAN

My Notes

Materials

SUGGESTED LEARNING STRATEGIES: Close Reading, Vocabulary Organizer, Think-Pair-Share, Create Representations

Pacing:1 class period Chunking the Lesson #1

Example A

Check Your Understanding #4a

#4b–d

#5–6

Check Your Understanding Lesson Practice

TEACH

MATH TERMS

1. 4, 16, 64, 256, … [65,536] 2. 1, −3, 9, −27, … [ −16,384] 1 3. 4, 2, 1, 0.5, …  32  1 Close Reading, Think-Pair-Share Look or groups o students who are able to ollow the steps to derive the ormula successully, and have them share their work with the class.

TEACHER to TEACHER Students ocus on a ormula or determining the sum o a inite geometric series: n  Sn = a1  11− −rr  , r ≠ 1.They ollow the steps given to derive the ormula and then apply it. A common student error when applying the ormula is to use n − 1 as the exponent instead on.

Example A Think-Pair-Share, DebriefingThis example veriies that the ormula results in the desired sum.

ELL Support Students have encountered a lot o math terms that sound very similar: arithmetic seriesand arithmetic sequence, arithmetic sequenceand geometric sequence, common difference and common ratio, etc. Provide linguistic support through translations o these terms and other language. As appropriate, remind students to reer to the English-Spanish glossary to aid their comprehension. Finally, monitor classroom discussions or understanding and correct language use.

The sum o the terms o a geometric sequence is a geometric series . The sum o a finite geometric series where r ≠ 1 is given by these ormulas: Sn = a1 + a1r + a1r2 + a1r3 + . . . + a1rn−1

A finite seriesis the sum of a finite sequence and has a specific number of terms. An infinite seriesis the sum of an infinite sequence and has an infinite number of terms. You will work with infinite series later in this Lesson.

Bell-Ringer Activity Ask students to ind the eighth term in each geometric sequence.

312

Learning Targets:

• Derive the ormula or the sum o a inite geometric series. • Calculate the partial sums o a geometric series.

• paper squares • scissors

MATH TIP When writing out a sequence, separate the terms with commas. A series is written out as an expression and the terms are separated by addition symbols. If a series has negative terms, then the series may be written with subtraction symbols.

n  Sn = a1 1 − r   1 − r 

1. To derive the ormula, Step 1 requires multiplying the equation o the sum by −r. Follow the remaining steps on the let to complete the derivation o the sum ormula. Step 1 Sn = a1 + a1r + a1r2 + a1r3 + . . . + a1rn − 1 −rSn = −a1r − a1r2 − a1r3 − . . . − a1rn−1 − a1rn Step 2 Combine terms on each side o the equation (most terms will cancel out). Sn − rSn = a1 − a1rn

Step 3 Factor out Sn on the let side o the equation and actor out a1 on the right. Sn(1 − r) = a1(1 − rn)

Step 4 Solve or Sn. Sn =

n  a1(1− rn ) = a1  1− r  (1− r )  1− r 

Example A Find the total o the Area o Square column in the table in Item 1 rom the last lesson. Then use the ormula developed in Item 1 o this lesson to ind the total area and show that the result is the same. Step 1: Add the areas o each square rom the table. 256 + 64 + 16 + 4 + 1 = 341

Square # 12345 64 Area 256

16

4

1

Step 2: Find the common ratio. 64 = 0.25, 16 = 0.25, 4 = 0.25; r = 0.25 64 256 16 Step 3: Substitute n = 5, a1 = 256, and r = 0.25 into the ormula or Sn. 5 n   Sn = a1 1 − r ; S5 = 256  1 − 0.25   1 − r   1 − 0.25  Step 4: Evaluate S5. 5  S5 = 256 1 − 0.25  = 341  1 − 0.25 






My Notes

Try These A Find the indicated sum o each geometric series. Show your work. a. Find S5 or the geometric series with a1 = 5 and r = 2. 155

b. 256 + 64 + 16 + 4 + . . .; S6 341.25 10

c.

∑ 2 ⋅3n − 1

n =1

59,048

MATH TIP Recall thatsigma notationis a shorthand notation for a series. For example: 3

∑ 8⋅2n − 1

n =1

= 8(2)(1−1) + 8(2)(2−1) + 8(2)(3−1)

Check Your Understandin g 2. Reason qu antitatively. How do you determine i the common ratio in a series is negative?

⋅

⋅

⋅

=8 1+8 2+8 4 = 8 + 16 + 32 = 56

Check Your Understanding Debrie students’ answers to these items to ensure that they understand sigma notation and can ind and write a sum using it. Ask students to describe how the sigma notation or a geometric series is similar to and dierent rom the sigma notation or an arithmetic series. For example, both contain a general term, an index o summation, and upper and lower limits; the general term or a geometric series has an exponent containing a variable.

Ans wers 2. The terms will have signs alternating between positive and negative. 5

3. Find the sum o the series 2 + 8 + 32 + 128 + 512 using sigma

3.

notation.

j −1

∑ 2 ( 4) j =1

TEACHER to TEACHER Recall that the sum o the irst n terms o a series is a partial sum. For some geometric series, the partial sums S1, S2, S3, S4, . . . orm a sequence with terms that approach a limiting value. The limiting value is called the sum of the infinite geometric series. To understand the concept o an ininite sum o a geometric series, ollow these steps.

• Start with a square piece o paper, and let it represent one whole unit. • Cut the paper in hal, place one piece o the paper on your desk, and keep

• •

the other piece o paper in your hand. The paper on your desk represents the irst partial sum o the series, S1 = 1. 2 Cut the paper in your hand in hal again, adding one o the pieces to the paper on your desk and keeping the other piece in your hand. The paper on your desk now represents the second partial sum. Repeat this process as many times as you are able.

MATH TIP If the terms in the sequence a1, a2, a3, . . . , an, . . . get close to some constant asn gets very large, the constant is the limiting value of the sequence. For example, in the sequence 1, 1 , 1 , 1 , 1 , . . . , 1 , . . ., 2 3 4 5 n the terms get closer to a limiting value of 0 as n gets larger.

4. Use appropria te tools strateg ically.Each time you add a piece o


paper to your desk, the paper represents the next term in the geometric series. a. As you continue the process o placing hal o the remaining paper on your desk, what happens to the amount o paper on your desktop? The amount of paper on the desk gets closer to 1, the amount represented by the srcinal square.

MINI-LESSON:

Partial Sums of Geometric Series

If students need additional help with finding partial sums of geometric series, a mini-lesson is available to provide practice.

The inal part o this lesson deals with ininite sums. The concept o the existence o an ininite sum is diicult or many students. A geometric series can have a inite or ininite number o terms. The sumSn o the irstn terms o an ininite geometric series is called the nth partial sum o the series. I the sequence o partial sumsS1, S2, S3, … Sn, … approaches some speciic number S, then the geometric series is said to have that numberS as its sum. The paper-cutting activity provides a concrete example o partial sums o an ininite series approaching a speciic value.

Developing Math Language Discuss with students that a partial sum of a series is a type o finite series. This will help students to see how some o the new vocabulary terms are related. Reinorce students’ acquisition o all vocabulary through regular reerence to words in the text as well as words you and students place on the classroom Word Wall. 4a Use ManipulativesStudents will realize that the longer they continue the process o adding hal the remaining paper to their desks, the closer the amount o paper on their desks gets to a c omplete square, which represents one whole.

See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.


313

ACTIVITY 20 Continued 4b–d Create Representations, DebriefingStudents will also see the limiting value o 1 by looking at the problem numerically and graphically.



My Notes

4. b. Fill in the blanks to complete the partial sums or the ininite geometric series represented by the pieces o paper on your desk.

CONNECT TO AP

To illustrate convergence and divergence, have students graph the partial sums or each ininite geometric series below. 1 1 1 + 1 ... n= +2+ + 4 8 16 Sn= 1+ + 1+ +1 1 . . . 16 8 4 2 The graphs show that the irst series converges and the second diverges.

CONNECT

TO AP

An infinite series whose partial sums continually get closer to a specific number is said to converge, and that number is called thesum of the infinite series.

S1 = 1 2 1 3 S2 = 1 + ___ 4 = ___ 4 2 1 1 4 8 S3 = 1+ ___ + = ___ 2

7

8 ___ 15

1 1 1 4 +___ 8 = ___ 16 S4 = 1+ ___ + 2

16 ___

1= 1 ___ 81 + ___ 16 S5 = + +41 +___ 2

1 32 ___

1 1 1 8 + ___ 16 S6 = 1+ ___ +4 +___ +

1 32 = ___

2

31 32 ___ 1

63

64 ___

64 ___

c. Plot the irst six partial sums. y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 12

34

56

78

91 0

x

d. Do the partial sums appear to be approaching a limiting value? I so, what does the value appear to be? Yes; the limiting value appears to be one.


314





My Notes

5. Consider the geometric series 2 + 4 + 8 + 16 + 32 + . . . . a. List the irst ive partial sums or this series. S1 = 2; S2 = 6; S3 = 14;S4 = 30; S5 = 62

5–6 Look for a Pattern, Quickwrite, DebriefingThese items provide one example in which the ininite sum o a geometric series does not exist (Item 5) and one in which the sum does exist (Item 6).

TEACHER to TEACHER The study o ininite geometric series lays a oundation or the study o ininite series in calculus. In a calculus course, students will use thenth-term test or divergence, which states that i a sequence {an} does not converge to

b. Do these partial sums appear to have a limiting value? No; there does not appear to be a limiting value.

0, then the series∑ an diverges. In other words, in Item 5, since the terms 2, 4, 8, 16, 32, … do not approach 0, the ininite sum 2+ 4 + 8 + 16 + 32 + … diverges, or does not exist.

c. Does there appear to be a sum o the ininite series? I so, what does the sum appear to be? I not, why not? No; there does not appear to be an in nite sum. Since the terms that are being added in the partial sums are growing larger and larger, there will not be a limiting value on the sums.

6. Consider the geometric series 3 − 1+ −1 +1 − 1 + 1 − 1 3 9 27 81 243 a. List the irst seven partial sums or this series.

...

S1 = 3; S2 = 2; S3 = ≈ 2. 33 3; S4 = 20 ≈ 2. 22 2; S5 = 61 ≈ 2. 25 9; 3 9 27 S6 = 182 ≈ 2. 247; S7 = 547 ≈ 2. 25 1 243 81

b. Do these partial sums appear to have a limiting value? Yes; about 2.25.

WRITING MATH You can write the sum of an infinite series by using summation or sigma, notation and using an infinity symbol for the upper limit. For example, ∞

n −1

∑ 3(− 31)

In general, the converse o this test is not true, and thus it can never be used to prove convergence. In Item 6, you could not use the test to say that because the terms 3, −1,, 1 − 1 , 1 , . . . approach 0, 3 9 27 1 + 1 1 ,... must the sum 3 − 1+ − 3 9 27 exist.

n=1

1 ... = 3− 1 + − 3

c. Does there appear to be a sum o the ininite series? I so, what does the sum appear to be? I not, why not? Yes; the sum appears to be 2.25.



315

ACTIVITY 20 Continued Check Your Understanding Debrie students’ answers to these items to ensure that they can write partial sums o an ininite series. Ask students to explain the relationship between the terms o a sequence and the partial sums.

Ans wer s 7. 1; 3; 7; 15; 31; 63; 127; 255; no limit; no ininite sum 8. 2 ; 8 ; 26 ; 80 ; 242 ; 728 which 5 15 45 135 405 1215 approximately equals 0.4; 0.533; 0.578; 0.593; 0.598; 0.599. There appears to be a limit and ininite sum o about 0.6.




My Notes

Check Your Understanding Find the indicated partial sums o each geometric series. Do these partial sums appear to have a limiting value? I so, what does the ininite sum appear to be?

7. First 8 partial sums o the series 1 + 2 + 4 + 8 + . . . 8. First 6 partial sums o the series 2 + 2+ +2 +2 ... 5

15

45

135

    Find the indicated partial sum o each geometric series.

9. 1 − 3 + 9 −27 + . . .; S7 1 − 1 + 1 − 1 + . . .; S 10. 625 9 125 25 5 Consider the geometric series −1 + 1 − 1 + 1 − 1 + . . .

11. Find S4 and S6. Generalize the partial sum when n is an even number. 12. Find S5 and S7. Generalize the partial sum when n is on odd number. 13. Describe any conclusions drawn rom Items 11 and 12. 14. Construct vi able arguments. What conclusions i any can you draw rom this lesson about the partial sums o geometric series where r ≥ 1 orr ≤ −1?


9. S7 = 547 10. S9 = 781.2496 11. The sum is 0 when n is even. 12. The sum is −1 when n is odd. 13. The sums oscillate between 0 and −1. 14. When r is greater than or equal to 1, or less than or equal to −1, the partial sums do not appear to have a limiting value.

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand how to ind a particular partial sum without inding all the previous sums. I students make errors using the ormula n  Sn = a1  1 − r , be sure they are using  1− r  the proper order o operations. For example, they must ind the value o rn beore subtracting in the numerator. I they are using a calculator, they must be sure that the calculator subtracts in the numerator and in the denominator beore dividing or multiplying; this will require either using parentheses or perorming the calculation in several steps.

316




  Convergence of Series


Lesson 20-3 PLAN

My Notes

Learning Targets:


• Determine i an ininite geometric sum converges. • Find the sum o a convergent geometric series.

Chunking the Lesson #1 Check Your Understanding #5–6 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Create Representations, Look or a Pattern, Quickwrite a1 (1 − r n ) . To ind the 1− r sum o an ininite series, ind the value that Sn gets close to as n gets very large. For any ininite geometric series where −1 < r < 1, as n gets very large, rn gets close to 0. Recall the ormula or the sum o a inite series Sn =

n

Sn =

a (1 − r ) 1 1− r

a1 (1 − 0) 1− r a ≈ 1 1− r

∞

n=0

only i |r| < 1 or −1 < r < 1. I |r| ≥ 1, the ininite sum does not exist.

1. Consider the three series rom Items 4–6 o the previous lesson. Decide whether the ormula or the sum o an ininite geometric seri es applies. I so, use it to ind the sum. Compare the results to your previous answers. 1 +1 + 1 a. 1 + 1+ + 2

4

8

16

... S=

32

b. 2 + 4 + 8 + 16 + 32 + . . .

Does not apply since r = 2 ≥ 1.

1 +1 − 1 + 1 − 1 c. 3 − 1+ − 3

S=

9

27

81

1 2 = 1; the results are the same. 1− 1 2

243

...

3 = 3 = 9 = 2.25; the results are the sa me. 4 4 1− − 1 3 3

( )

Check Your Understanding . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

Find the ininite sum i it exists or tell why it does not exist. Show your work.

2. 64 + 16 + 4 + 1 + . . . 3. 1 + 5+ +25 +125 3

12

∞

4.

48

192

...

n −1

∑ 3 ( 25 )

n =1

Ans wers 2. S = 256 = 85. 3 3 3. does not exist because r = 5 ≥ 1 4 6 + 12 =K 4. S = 3+ + = 3 5 5 25 3 5


1. S6: 3 + 0.3 + 0.03 + …

−1 < r < 1 can be written as|r| < 1.

a1 i and 1− r

TEACH Ask students to ind the indicated sum or each geometric series.

MATH TIP

S≈

An ininite geometric series ∑ a1r n converges to the sum S =

MATH TIP Sn represents the sum of a finite series. Use S to indicate the sum of an infinite series.

As n increases, rn gets close to, or approaches, 0. It is important to realize that asrn approaches 0, you can say that |rn|, but not rn, is getting “smaller.”

[3.33333]

2. S5: 9 − 18 + 36 − 48 + … [75] 3. S6: 1 + 10 + 100 + 1000 + … [11,111] 1 Create Representations, Debriefing Students use the ormula or the sum o an ininite geometric series, a S = 1 , − 1 < r < 1,andvalidatethe 1− r ormula by comparing the results to series or which they have already examined the limit o the part ial sums. Be sure students understand the restriction on r and that a sum does not exist i r ≥ 1.

Some students may understand the ormula better i they look at a numerical example irst. When inding an ininite you are inding the value that Ssum, n gets close to as n gets very large. Consider the series 1 + 1+ + 1 +1 . . . . The nth partial 3 9 27 81 n 1 1 − 1  3  3  sum is Sn = . As n gets very 1− 1 3 n large, 1 gets close to 0, and so 3 1 (1 − 0) 1 Sn ≈ 3 = 3 = 1. 1 1− 1− 1 2 3 3

()

()

Check Your Understanding Debrie students’ answers to these items to ensure that they know how to ind ininite sums and how to determine whether an ininite sum exists. Watch or students who incorrectly determine whether a sum exists because they ind the reciprocal o r instead o the correct value o r.


317

ACTIVITY 20 Continued 5–6 Look for a Pattern, Quickwrite, Think-Pair-Share, Debriefing Item 5 is included in case students question whether or not ininite arithmetic series have ininite sums.

  Convergence of Series


My Notes

5. Consider the arithmetic series 2 + 5 + 8 + 11 + . . . . a. Find the irst our partial sums o the series. S1 = 2; S2 = 7; S3 = 15;S4 = 26

Item 6 gives students the opportunity to relect on all the ormulas derived in the activity. Ater sharing answers with the entire group to make sure all responses are correct, students can record the inormation in their math notebooks.

b. Do these partial sums appear to have a limiting value? No; there does not appear to be a limiting value.

c. Does the arithmetic series appear to have an ininite sum? Explain. No; there does not appear to be an in nite sum. Since the terms that are being added in the partial sums are growing larger and larger, there will not be a limiting value on the sums.

TEACHER to TEACHER All arithmetic series other than the trivial arithmetic series 0+ 0 + 0 + … diverge by thenth-term test.

6. Summarize the ollowing ormulas or a geometric series. common ratio

r=

nth term

an =

an an−1

Check Your Understanding Debrie students’ answers to these items to ensure that they understand and can veriy that the series is geometric. These items make a connection between ininite geometric series and repe ating decimals, which students have prior experience with. Ater Item 8, ask students to ind the decimal value o 2 9 and compare it to the series.

Sum o irst n terms

Sn =

Ininite sum

S = 1− r

if − 1 < r < 1

Consider the series 0.2 + 0.02 + 0.002 + . . . .

7. Find the common ratio between the terms o the series. 8. Does this series have an ininite sum? I yes, use the ormula to ind the sum.

9. Construct vi able arguments. Make a conjecture about the ini nite sum 0.5 + 0.05 + 0.005 + . . . . Then veriy your conjecture with the ormula.


    Find the ininite sum i it exists, or tell why it does not exist. 9 ... 10. 18 − 9+ −9 + 2 4 11. 729 + 486 + 324 + 216 + . . .


12. 81 + 108 + 144 + 192 + . . . 13. −33 − 66 − 99 − 132 − . . . 14. Reason qu antitatively. At the beginning o the lesson it is stated that “or any ininite geometric series where −1 < r < 1, as n gets very


large, rn gets close to 0.” Justiy this statement with an example, using a negative value or r.

10. S = 12 11. S = 2187 12. does not exist because r = 4 ≥ 1 3

318

n−1

Check Your Understandin g

ASS ESS

Check students’ answers to the Lesson Practice to ensure that they understand how to determine whether an ininite sum exists or a geometric series. Watch or students who attempt to ind an ininite sum without irst checking whether it exists. Remind them that ininite sums may not exist, and demonstrate using examples rom this activity.

⋅r

n  a1  1− r   1− r 

a1

Ans wer s 7. r = 0.10 a 8. yes; S = 1 = 0.2 = 0.2 = 2 1 − r 1 − 0.1 0.9 9 a 9. 5 , S = 1 = 0.5 = 5 9 1 − r 1 − 0.1 9

ADA PT

a1

13. does not exist; arithmetic sequence with terms growing larger in absolute value 14. Sample answer: I r is −0.5, r2 is 0.25, which is closer to zero than r. As r is raised to greater and greater powers, the result gets closer to zero.




    Squares with Patterns

   Write your answers on notebook paper. Show your work.

Lesson 20-1 1. Writearithmetic, geometric,or neither or each sequence. I arithmetic, state the common dierence. I geometric, state the common ratio. a. 4, 12, 36, 108, 324, . . . b. 1, 2, 6, 24, 120, . . . c. 4, 9, 14, 19, 24, . . . d. 35, −30, 25, −20, 15, . . . o each geometric series. 2. Find r = −3; aterm a. a1 the = 1,indicated 10 b. a1 = 3072, r = 1 ; a8 4

3. I an is a geometric sequence, express the quotient a o 7 in terms o r. a4

9. A ball is dropped rom a height o 24 eet. The ball bounces to 85% o its previous height with each bounce. Write an expression and solve to ind how high (to the ne arest tenth o a oot) the ball bounces on the sixth bounce.

10. Write the recursive ormula or each sequence. a. 4, 2, 1, 0.5, . . . b. 2, 6, 18, 54, 120, . . . c. 4 , 4 , 4 , . . . 5 25 125

d. −45, 5, − 5 , . . . 9

11. Write the explicit ormula or each sequence. a. 4, 2, 1, 0.5, . . . b. 2, 6, 18, 54, 120, . . . c. 4 , 4 , 4 , . . . 5 25 125

9

1 , 1 , 1 . . . . What is a ? 6 81 27 9

Lesson 20-2 12. Find the indicated partial sum o each geometric series.

+8 . . . ; S7 5 25 1 1 1 b. + + + . . . ; S15 8 4 2

a. 5 + 2+ +4

3 81

B. 3 C. 364 81 D. 9 5. Determine the irst three terms o a geometric sequence with a common ratio o 2 and deined as ollows: x − 1, x + 6, 3x + 4

6. Determine whether each sequence is geometric.


continued

d. −45, 5, − 5 , . . .

4. The irst three terms o a geometric series are

A.

ACTIVITY 20

I it is a geometric sequence, state the common ratio. a. x, x2, x4, . . . b. (x + 3), (x + 3)2, (x + 3)3, . . . c. 3x, 3x+1, 3x+2, . . . d. x2, (2x)2, (3x)2, . . .

7. I a3 = 9 and a5 = 81 , ind a1 and r. 32

512

8. The 5 in the expression an = 4(5)n−1 represents

()

13. For the geometric series 2.9 + 3.77 + 4.90 + 6.37 + . . . , do the ollowing: a. Find S9 (to the nearest hundredth). b. How many more terms have to be added in order or the sum to be greater than 200?

14. George and Martha had two children by 1776, and each child had two children. I this pattern continued to the 12th generation, how many descendants do George and Martha have?

15. A inite geometric series is deined as 0.6 + 0.84 + 1.12 + 1.65 + . . . + 17.36. How many terms are in the series? A. n = 5 B. n = 8 C. n = 10 D. n = 11 6

16.Evaluate

3 2( ∑

ACT IV IT Y PR ACT ICE 1. a.geometric; r = 3 b. neither c. arithmetic; d = 5 d. neither 2. a. a10 = −19,683 b. a8 = 3 , or 0.1875 16 3. r3 4. B 5. x = 8; irst three terms are 7, 14, 28 6. a.not geometric b. yes; r = (x + 3) c. yes; r = 3 d. not geometric 7. a1 = 1 and r = 3 4 2 8. C 6−1 9. a6 = 20.4(0.85) = 9.1 t 10. a.an = 1 an−1 2 b. an = 3an−1 c. an = 1 an−1 5 d. an = − 1 an−1 9 n−1 11. a.an = 4 1 2 n−1 b. an = 2(3) n−1 c. an = 4 1 5 5 n−1 d. an = 45 − 1 9 12. a.S7 = 25, 999 = 8.31968 3125 b. S15 = 4095.875 13. a.S9 = 92.84 b. 3 more terms; S12 = 215.55

() ( )

14. George and Martha are the irst generation; they have 4,096 descendants at the 12th generation. 15. D 16. 378

)j

j=1

which part o the expression? A. n B. a1 C. r D. Sn


319

ACTIVITY 20 Continued 17. a.$80; $5120 b. $10,230 18. a.an = 3 3n−1 = 3n b. 81 c. 121 19. a.S = 48 b. does not exist since r = 2 ≥ 1 c. 7776 ≈ 1110.857 7 2 20. + 2 + 2 + . . . 3 9 27 21. 3 11 22. a.r = 2 ; series converges 3

    Squares with Patterns


⋅

b. r = − 12 ; series converges c. r = 1.5; series diverges 23. B 24. r =

4

25. B 26. True. Sample answer: The terms in an arithmetic series are added to orm partial sums. Since there is a common dierence not equal to 0, the partial sum changes at a constant rate as terms are added, so there is no limiting value on the sums. 27. Sample answer: The common ratio can be ound between any two terms by determining the number o terms between the given terms and rewriting the ratio as a quantity to that power. Working backward rom the irst term will give you a value or a1. ADD ITI ONA L PR ACT ICE I students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital or additional practice problems.

320

17. During a 10-week summer promotion, a baseball team is letting all spectators enter their names in a weekly drawing each time they purchase a game ticket. Once a name is in the drawing, it remains in the drawing unless it is chosen as a winner. Since the number o names in the drawing increases each week, so does the prize money. The irst week o the contest the prize amount is $10, and it doubles each week. a. What is the prize amount in the ourth week o the contest? In the tenth week? b. What is the total amount o money given away during the entire promotion?

18. In case o a school closing due to inclement weather, the high school sta has a calling system to make certain that everyone is notiied. In the irst round o phone calls, the principal calls three sta members. In the second round o calls, each o those three sta members calls three more sta members. The process continues until all o the sta is notiied. a. Write a rule that shows how many sta members are called during the nth round o calls. b. Find the number o sta members called during the ourth round o calls. c. I all o the sta has been notiied ater the ourth round o calls, how many people are on sta at the high school, including the principal?

Lesson 20-3 19. Find the ininite sum i it exists. I it does not exist, tell why. a. 24 + 12 + 6 + 3 + . . . 1 +1+ 2 . . . b. 1 + + 12 6 3 3 c. 1296 − 216 + 36 − 6 + . . .

20. Write an expression in terms oan that means the ∞

()

same as ∑ 2 1 3 j=1

j

21. Express 0.2727 . . . as a raction.

22. Use the common ratio to determine i the ininite series converges or diverges. a. 36 + 24 + 12 + . . . b. −4 + 2 + (−1) + . . . c. 3 + 4.5 + 6.75 + . . .

23. The ininite sum 0.1 + 0.05 + 0.025 + 0.0125 + . . . A. diverges. B. converges at 0.2. C. converges at 0.5. D. converges at 1.0. 24. An ininite geometric series has a1 = 3 and a sum o 4. Find r.

25. The graph depicts which o the ollowing? y 10 9 8 7 6 5 4 3 2 1

x

12345

A. converging arithmetric series B. converging geometric series C. diverging arithmetic series D. diverging geometric series 26. True or alse? No arithmetic series with a common dierence that is not equal to zero has an ininite sum. Explain.

MATHEMATICAL PRACTICES Make Sense of Problems and Solving Them

Persevere in

27. Explain how knowing any two terms o a geometric sequence is suicient or inding the other terms.



Sequences and Series

Embedded Assessment1 Embedded Assessment 1

THE CHESSBOARD PROBLEM

Use after Activity 20

• Identiying terms in arithmetic and geometric sequences • Identiying common differences and common ratios • Writing implicit and explicit rules or arithmetic and geometric sequences

In a classic math problem, a king wants to reward a knight who has rescued him rom an attack. The king gives the knight a chessboard and plans to place money on each square. He gives the knight two options. Option 1 is to place a thousand dollars on the irst square, two thousand on the second square, three thousand on the third square, and so on. Option 2 is to place one penny on the irst square, two pennies on the second, our on the third, and so on.

Ans wer Key 1. a. 1000, 2000, 3000, 4000,5000; arithmetic b. 0.01, 0.02, 0.04, 0.08, 0.16; geometric 2. a. an = 1000 + (n − 1)1000; Sn = n2 (1000 + an) b. an = 0.01(2)n−1; n Sn = 0.01 1 − 2 or 0.01(2 n − 1) 1− 2 3. a. $20,000; $210,000 b. $5242.88; $10,485.75 4. a. $2,080,000 b. about $1.845 × 1017 5. Option 2 is better. The irst term in the arithmetic series is much greater than the irst term in the geometric series. However, the geometric s eries grows aster than the arithmetic series. At some point between the 20th and 64th terms, the corresponding terms in the geometric series will be greater than those in the arithmetic series.

Think about which oer sounds better and then answer these questions.

1. List the irst ive terms in the sequences ormed by the given options. Identiy each sequence as arithmetic, geometric, or neither. a. Option 1 b. Option 2

2. For each option, write a rule that tells how much money is placed on the nth square o the chessboard and a rule that tells the total amount o money placed on squares 1 through n. a. Option 1 b. Option 2

(

3. Find the amount o money placed on the 20th square o the chessboard and the total amount placed on squares 1 through 20 or each option. a. Option 1 b. Option 2

4. There are 64 squares on a chessboard. Find the total amount o money placed on the chessboard or each option. a. Option 1 b. Option 2

5. Which gives the better reward, Option 1 or Option 2? Explain why.


Ass ess ment Foc us

)

Common Core State Standards for Embedded Assessment1 HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.  HSF-BF.A.1






Unit 4 • Series, Exponential and Logarithmic Functions

321

Embedded Assessment1

Sequences and Series

Embedded Assessment 1 T EACHER

to TEACHER

You may wish to read through the scoring guide with students and discuss the differences in the expectations at each level. Check that students understand the terms used.

Unpacking Embedded Ass essm ent 2 Once students have completed this Embedded Assessment, turn to Embedded Assessment 2 and unpack it with them. Use a graphic organizer to help students understand the concepts they will need to know to be successul on Embedded Assessment 2.

THE CHESSBOARD PROBLEM


Scoring Guide Mathematics Knowledge and Thinking (Items 1, 3, 4)

Problem Solving (Items 3, 4)

Mathematical Modeling / Representations (Items 1, 2)

Reasoning and Communication (Item 5)

Exe m p l a r y

Pr o f i ci e n t

Em e r g i n g

I n c o m pl e t e

The solution demonstrates these characteristics:

• Fluency in determining

• A functional understanding• Partial understanding and •

specified terms of a sequence or the sum of a specific number of terms of a series

and accurate identification of specified terms of a sequence or the sum of a specific number of terms of a series

partially accurate identification of specified terms of a sequence or the sum of a specific number of terms of a series

• An appropriate and efficient• A strategy that may include• A strategy that results in • strategy that results in a correct answer

• Fluency in accurately representing real-world scenarios with arithmetic and geometric sequences and series

• Clear and accurate

unnecessary steps but results in a correct answer

some incorrect answers

• Little difficulty in accurately• Some difficulty in representing real-world scenarios with arithmetic and geometric sequences and series

• Adequate explanation of explanation of which option which option provides the provides the better reward better reward

•

representing real-world scenarios with arithmetic and geometric sequences and series

Little or no understanding and inaccurate identification of specified terms of a sequence or the sum of a specific number of terms of a series No clear strategy when solving problems Significant difficulty in representing real-world scenarios with arithmetic and geometric sequences and series

• Misleading or confusing •

Incomplete or inadequate explanation of which option explanation of which option provides the better reward provides the better reward


322


   

ACTIVITY 21

Sizing Up the Situation Lesson 21-1 Exploring Exponenti al Patterns

• Identiy data that grow exponentially. • Compare the rates o change o linear and exponential data. SUGGESTED LEARNING STRATEGIES: Create Representations, Look or a Pattern, Quickwrite Ramon Hall, a graphic artist, needs to make several dierent-sized drat copies o an srcinal design. His srcinal graphic design sketch is contained within a rectangle with a width o 4 cm and a length o 6 cm. Using the oice copy machine, he magniies the srcinal 4 cm × 6 cm design to 120% o the srcinal design size, and calls this his irst drat. Ramon’s second drat results rom magniying the irst drat to 120% o its new size. Each new drat is 120% o the previous drat.

1. Complete the table with the dimensions o Ramon’s irst ive drat versions, showing all decimal places.

Number of Magnifications

W i d th ( cm )

MATH TIP Magnifying a design creates similar figures. The ratio between corresponding lengths of similar figures is called the constant of proportionality, or the scale factor. For a magnification of 120%, the scale factor is 1.2.

L e n gt h ( cm )

0

4

1

4 .8

2

5 .7 6

3

6 .9 1 2

10.3 68

4

8 .2 9 4 4

12 . 4 4 16

5

9 .9 5 3 2 8

14 . 9 2 9 9 2

3.

unctions and to perorm transormations o the parent unction. Review transormations o previous types o unctions, including quadratic and cubic.

Pacing:1 class period Chunking the Lesson

8.6 4

#1–5 #6–8 Check Your Understanding Lesson Practice

TEACH

2. Make sense o f problems. The resulting drat or each magniication


Students will rely on prior knowledge to investigate rates o change o exponential

PLAN

7.2

15 The reasonable domain for both functions is the nonnegative integers, 14 becausen represents a number of magni cations and thus cannot be 13 12 negative or a decimal or fraction. The reasonable range for both 11 functions is the nonnegative real numbers, because W and L represent 10 measurements and so cannot be negative. 9 8 7 6 5 Plot the ordered pairs (n, W) rom the table in Item 1. Use a dierent 4 color or symbol to plot the ordered pairs (n, L). 3 2 1

In Activity 21, students examine exponential unctions and their graphs. They begin by investigating linear growth and decay and compare rates o change in exponential and linear data. Next, they learn to write exp onential unctions. They perorm transormations o the parent exponential unction and, inally, they examine base exponential unctions.

Lesson 21-1

6

has a unique width and a unique length. Thus, there is a unctional relationship between the number o magniications n and the resulting width W. There is also a unctional relationship between the number o magniications n and the resulting length L. What are the reasonable domain and range or these unctions? Explain.

21

Investigative Act iv it y St and ard s F ocu s

My Notes

Learning Targets:

ACTIVITY

Bell-Ringer Activity Ask students to ind the slope o the line represented by each equation.

y

1. y = −3x + 5

[−3]

2. 8 x + 3 y = 24

− 8   3 

3. y − 7 = 5(2 x + 3) [10] 1–5 Create Representations, Look for a Pattern, Quickwrite, Debriefing In Item 1, check to see that students ollow instructions and do not round in their computations. Width Length 1234567 Number of Magnifications

x

Common Core State Standards for Activity 21

As students complete Item 3, they may attempt to plot the points in a straight line. Encourage them to draw their points as accurately as possible. Also, be alert to students who try to connect the points on their graphs with lines. Remind them o their answers to Item 2, in which they explained why a reasonable domain and range or both unctions is discrete.

HSF-IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.


HSF-IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

HSF-IF.C.7

Graph symbolically and complicated show key features simplefunctions cases andexpressed using technology for more cases.ofthe graph, by hand in

Have students who are having diiculty determining whether the points orm a line use a ruler to attempt drawing lines through each set o points in Item 3.

HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Activity 21 • Exponential Functions and Graphs

323

ACTIVITY 21 Continued TEACHER to TEACHER

  Exploring Exponential Patterns


This lesson calls on students’ prior knowledge o linear unctions and their constant rate o change. Students will discover that exponential unctions, on the other hand, do not have a constant rate o change between any two points.

My Notes

Increase in Number of Magnifications

1–5 (continued)Create Representations, Look for a Pattern, Quickwrite, DebriefingMake sure students understand that the table they

complete or Item 4 allows them to determine the rate o change or two unctions. In both cases, the domain is the number o magniications, and the change in the domain value represented in each row is 1. 6–8 Look for a Pattern, Quickwrite, Debriefing The intent o Item 6 is to review linear unctions. In Part a, the data presented are clearly linear, since both ∆x and ∆y are constant. Conirming that the data in Part b are linear is more challenging or the ∆y students. They must veriy that is ∆x constant or all pairs o data presented

4. Use the data in Item 1 to complete the table.

Chan ge in the Width

Ch ang e i n t he Le ngth

0 to 1

4.8 − 4 = 0

.8

− 6 = 1.2 7.2

1 to 2

5.76 − 4.8 =0

.96

8.6− 4 7.2= 1.44

2 to 3

6.912 − 5.76 = 1 .15 2

10 .3 6 8− 8.64 = 1.728

3 to 4

8.2944 − 6.912= 1.3 824

12.4416− 10.368= 2.0736

= 2.48832 9.95328− 8.2944= 1.65888 14.92992− 12.4416

4 to 5

5. From the graphs in Item 3 and the data in Item 4, do these unctions

MATH TIP

appear to be linear? Explain why or why not.

Linear functions have the property that the rate of change of the output variable y with respect to the input variable x is constant, that is, the ratio

y x

linear functions.

is constant for

The functions do not appear to be linear because there is not a constant rate of change and the graphs are not lines.

6. Express regularity

Explain why each in repeated reasoni ng. table below contains data that can be represented by a linear unction. Write an equation to show the linear relationship between x and y.

a.

x

−3

−1

1

3

5

y

8

5

2

−1

−4

Sample answer: The data are linear because there is a constant rate of change;y changes by−3 units for every 2 units of change in x. The linear equation is y = −1.5x + 3.5.

y=

b.

in the table. The intent o Item 7 is or students to work with data that are neither linear nor quadratic. Students should notice that the pattern in this table is a simple multiplicative relationship in which the next entry in the table or y is one-hal o the previous entry. Ater debrieing this item, have any students who thought the unctions were linear go back and reconsider their answers.

x

2

5

11

17

26

y

3

7

15

23

35

y Sample answer: The data are linear because the ratios are x y constant, = 4 , for all pairs of data in the table. The linear x 3 equation isy = 4 x + 1. 3 3

y=

7. Consider the data in the table below. x

01234

y

24

12

6

3

1.5

a. Can the data in the table be represented by a linear unction? Explain why or why not. No, the data cannot be represented by a linear function because the y-values change by a different amount for each unit change x. in

b. Describe any patterns that you see in the consecutive y-values. 1 Eachy-value in the table is 2 of the previous entry.

Common Core State Standards for Activity 21 (continued) HSF-BF.A.1




HSF-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. HSF-BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of cases k (both positive and findofthe ofon k given the graphs. Experiment with and illustrate annegative); explanation thevalue effects the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

324




  Exploring Exponential Patterns


8. Consider the data in the table in Item 1. How does the relationship o the data in this table compare to the relationship o the data in the table in Item 7? Sample answer: Both patterns are formed by multiplying each term by a constant to get the next term. In Item 7 the constant multiplier is 1, and in Item 1 the constant multiplier is 1.2. 2

Check Your Understandin g 9. Complete the table so that the unction represented is a linear unction. x 1 f(x)

23 162

5

40

10. Reason quantitatively. Explain why the unction represented in the table cannot be a linear unction. x f(x)

31 25 4 7

12

16

19

21

   Model with mathematics.Determine whether each unction is linear or nonlinear. Explain your answers.

11. x = number o equally sized pans o brownies; f(x) = number o brownies

12. x = cost o an item; f(x) = price you pay in a state with a 6% sales tax 13. x = number o months; f(x) = amount o money in a bank account with interest compounded monthly

14. 15. . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

x y

2 2.6

4 3.0

6 3.8

10

15

20

25

1.00

0.75

0.50

0.25

16. Identiy i there is a constant rate o change or constant multiplier. Determine the rate o change or constant multiplier.

4.8

3.84

22

28

34

40

10. Sample answer: The dierence between the irst two f(x) values, 12 and 7, is 5. I the unction is linear, then 5 would be the common dierence and the ollowing ordered pairs would be (3, 17), (4, 22), and (5, 27), which is not the case.

ASS ESS Students’ answers to Lesson Practice problems will provide you with a ormative assessment o their understanding o the lesson concepts and their ability to apply their learning.


5

6

12345 16

problems or here this lesson. You may the problems or use them as a assign culmination or the activity.

1.25

1234

x f(x)

10 6.0

y

y

Debrie students’ answers to these items to ensure that they understand the dierence between a linear unction and an exponential unction and can use the inormation in a table to identiy each type o unction.

See the Activity Practice or additional 8 4.8

x

x


Ans wers 9.

4

2

My Notes

6–8 (continued)Look for a Pattern, Quickwrite, DebriefingItem 8 returns to the numerical pattern o Item 1. Students should realize that they cannot calculate consecutive y-values by adding a constant amount each time the x-value increases by 1.

3.072

11. linear 12. linear 13. nonlinear 14. nonlinear 15. linear 16. constant multiplier; 0.8

ADA PT Check students’ answers to the Lesson Practice to ensure that they can dierentiate between linear and nonlinear unctions given a description o a domain and its corresponding range. Encourage students to write the unctions or Items 11–13. Once they have done so, they can create a table o values or graph the unction to help them decide whether it is linear.


325

ACTIVITY 21 Continued Lesson 21-2

  Exponential Functions


PLAN My Notes

Pacing:1 class period #1–2 #3–4 #5–6 #7 Check Your Understanding Lesson Practice

TEACH Bell-Ringer Activity Ask students to ind the value o each expression when x = 2. 1. 4x [16] 1 2. x−4  16  x 3. (x + 2) [16] 4. −3x [−9] 1–2 Close Reading, Vocabulary Organizer, Quickwrite, Look for a Pattern, Create Representations, DebriefingStudents ocus on the concept o an exponential unction and learn its distinguishing eatures. When the change in x is constant, as it is in the table in Item 2, the y-values o an exponential unction change by a constant multiplicative amount.

Completing the table in Item 2 should assist students in recognizing an exponential pattern, and so they can write the exponential unctionW(n) = 4(1.2)n.

TEACHER

Learning Targets:

• Identiy and write exponential unctions. • Determine the decay actor or growth actor o an exponential unction.

Chunking the Lesson

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Create Representations, Look or a Pattern, Quickwrite, Think-Pair-Share

MATH TERMS An exponential function is a function of the form f(x) = a bx, where a and b are constants,x is the domain, f(x) is the range, and a ≠ 0, b > 0, b ≠ 1.

⋅

MATH TERMS

The data in the tables in Items 7 and 8 o the previous lesson were generated by exponential functions . In the special case when the change in the input variable x is constant, the output variable y o an exponential unction changes by a multiplicative constant. For example, in the table in Item 7, the increase in the consecutive x-values results rom repeatedly adding 1, while the decrease in y-values results rom repeatedly multiplying by the constant 12, known as the exponential decay factor . 1. In the table in Item 1 in Lesson 21-1, what is the exponential growth factor ?

In an exponential function, the constant multiplier, or scale factor, is called an exponential decay factorwhen the constant is less than 1. When the constant is greater than 1, it is called an exponential growth factor .

2. You can write an equation or the exponential unction relatingW and n. a. Complete the table below to show the calculations used to ind the width o each magniication.

Number of Magnifications

MATH TIP To compare change in size, you could also use thegrowth rate, or percent increase. This is the percent that is equal to the ratio of the increase amount to the srcinal amount.

TEACHER

to

When students use the Connect to Technology, the issue o domain may come up. I it does not come up here, revisit it in Item 2 o Lesson 21-3. The domain o the problem situation is a subset o the nonnegative integers. The unctions graphed on the calculator will have a domain o all real numbers.

1.2

Calculation to Find Width (cm)

0

4

1

4(1.2)

2

4(1.2)(1.2)

3

4(1.2)(1.2)(1.2)

4

4(1.2)(1.2)(1.2)(1.2)

5

4(1.2)(1.2)(1.2)(1.2)(1.2)

10

4(1.2)10

n

4(1.2)n

b. Express regularity in repeated reasoning. Write a unction that CONNECT

TO TECHNOLOGY

expresses the resulting width W ater n magniications o 120%. n

Confirm the reasonableness of your function in Item 2b by using a graphing calculator to make a scatter plot of the data in the table in Item 8 in Lesson 21-1. Then graph the function to see how it compares to the scatter plot.

W(n) = 4(1.2)

c. Use the unction in part b to ind the width o the 11th magniication. W(11)≈ 29.720

Technology Tip Students without a graphing calculator may evaluate the expressions in the Calculation to Find Width column or 0 to 5 magniications to conirm that the unction is reasonable. For additional technology resources, visit SpringBoard Digital.

Developing Math Language Review exponential expressions and compare and contrast them to exponential functions. Discuss the dierence between growth and decay to help students understand the meanings o exponential growth factors and exponential decay factors.

326






The general orm o an exponential unction is f(x) = a(bx), where a and b are constants and a ≠ 0, b > 0, b ≠ 1.

My Notes

3–4 Quickwrite, Create Representations, Debriefing In Item 3, students will explain the meaning o parameters o the equation in terms o the problem situation.

Some students may extend their response in Item 3 to a new resizing situation and simply write the unction using a = 6 cm and b = 1.2. Other students may replicate the work in Item 2 by making a table o values.

3. For the exponential unction written in Item 2b, identiy the value o the parameters a and b. Then explain their meaning in terms o the problem situation. a represents the initial width of 4 cm, and b represents the growth factor of 1.2.

4. Starting with Ramon’s srcinal 4 cm × 6 cm rectangle containing his graphic design, write an exponential unction that expresses the resulting length L ater n magniications o 120%.


L(n) = 6(1.2)n

To supportstudents’ writing eorts, you may want to add a section to your Word Wall on basic sentence structure in English writing (simple sentence, compound sentence, complex sentence, transition words, etc.). Review these structures with students prior to the writing assignment, and provide an opportunity to clariy any questions about language structures.

Ramon decides to print ive dierent reduced drat copies o his srcinal design rectangle. Each one will be reduced to 90% o the previous size.

5. Complete the table below to show the dimensions o the irst ive drat versions. Include all decimal places.

Number of Reductions

W i d t h (c m )

Le n g t h (c m )

0

4

6

1

3.6

5 .4

2

3.24

4 .8 6

3

2 . 9 16

4 .3 74

4

2.62 4 4

3 .9 3 6 6

5

2 . 3 6 19 6

3 .5 4 2 9 4

6. Write the exponential decay actor and the decay rateor the data in the table in Item 5. The exponential decay factor is 0.9 and the decay rate is 10%.

7. Model with mathematics. Use the data in the table in Item 5. a. Write an exponential unction that expresses the width w o a reduction in terms o n, the number o reductions perormed. . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

w(n) = 4(0.9)n

b. Write an exponential unction that expresses the length l o a reduction in terms o n, the number o reductions perormed.

5–6 Create Representations, Vocabulary Organizer, Think-PairShare In Item 5, check to see that students ollow instructions and do not round numbers in t heir computations.

The intent o Item 6 is to distinguish between decay actor and decay rate. The Math Tip can help guide the students.

MATH TIP To compare change in size, you could also use thedecay rate, or percent decrease. This is the percent that is equal to the ratio of the decrease amount to the srcinal amount.

TEACHER to TEACHER In this section o the activity, students investigate exponential decay in the context o the srcinal problem. Instead o repeated magniications o 120%, this second considers repeated reductions o 90%.

l(n) = 6(0.9)n

c. Use the unctions to ind the dimensions o the design i the srcinal design undergoes ten reductions. w(n) ≈ 1.395;l(n) ≈ 2.092

7 Create Representations, ThinkPair-Share, DebriefingStudents may extend their response in Item 3 to a new resizing situation and simply write the unctions using the appropriate values or a and b.


327

ACTIVITY 21 Continued continued

Debrie students’ answers to these items to ensure that they can write an exponential unction and identiy the meaning o a and b in context.

Ans wer s 8. Sample answer: I a = 0, the unction would be f(x) = 0, a constant unction. I b = 1, the unction would be f(x) = 1, also a constant unction. I b < 0, the unction would not be continuous. 9. a. 2 b. Exponential growth; The unction values are increasing. 10. a = 2000; b = 1.05; $2000 is deposited in an account with an annual interest rate o 5%.


11. Exponential; x increases by a constant amount while y increases at a rate o 3 x; y = 3x. 12. Neither; x increases by a constant amount; y increases, but not by a constant amount or a constant multiplier. 13. 2

My Notes

Check Your Understandin g 8. Why is it necessary to place restrictions that a ≠ 0, b > 0, and b ≠ 1 in the general orm o an exponential unction?

9. An exponential unction contains the ordered pairs (3, 6), (4, 12), and (5, 24). a. What is the scale actor or this unction? b. Does the unction represent exponential decay or growth? Explain your reasoning.

10. Make sense o f problems. For the equation y = 2000(1.05)x, identiy the value o the parameters a and b. Then explain their meaning in terms o a savings account in a bank.

   Construct viable arguments. Decide whether each table o data can be modeled by a linear unction, an exponential unction, or neither, and justiy your answers. I the data can be modeled by a linear or exponential unction, give an equation or the unction.

11.

12.

x

0

1

y

1

3

x

0

y

4

2

1

3

2

4

9

27

81

14

22

32

34

8

13. Given that the unction has an exponential decay actor o 0.8, complete the table.


1


ACTIVITY 21


x

0

y

64 51.2 40.96 32.768 26.2144

3

x

01234

y

64

5 1.2

40 .96

4

14. 20% 15. f(x) = 64(0.8)x

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand the dierence between exponential growth and exponential decay. Discuss how they can use a table to determine whether the unction represents growth or decay.

328

3 2 . 76 8

2 6 .2 1 4 4

14. What is the decay rate or the unction in Item 13? 15. Write the unction represented in Item 13.




  Exponential Graphs and Asymptotes


Lesson 21-3 PLAN

My Notes

Learning Targets:


• Determine when an exponential unction is increasing or decreasing. • Describe the end behavior o exponential unctions. • Identiy asymptotes o exponential unctions.

Chunking the Lesson #1–2 #3 #4–7 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Close Reading, Vocabulary Organizer, Think-Pair-Share, Group Presentation

TEACH

1. Graph the unctions y = 6(1.2)x and y = 6(0.9)x on a graphing calculator or other graphing utility. Sketch the results.

y

=

6(0.9)x

2. Determine the domain and range or each unction. Use interval notation. Domain

a. y = 6(1.2)

b. y = 6(0.9)x

(−∞, ∞)

(0,∞)

(−∞, ∞)

(0,∞)

A unction is said to increase i the y-values increase as the x-values increase. A unction is said to decrease i the y-values decrease as thex-values increase.

3. Describe each unction asincreasing ordecreasing. a. y = 6(1.2)x increasing

b. y = 6(0.9)

x

decreasing

As you learned in a previous activity, theend behavior o a graph describes the y-values o the unction as x increases without bound and as x decreases without bound. I the end behavior approaches some constant a, then the graph o the unction has a horizontal asymptote at y = a. When x increases without bound, the values ox approach positive ininity,∞. When x decreases without bound, the values ox approach negative ininity, −∞.

9 8 7 6

y

=

Ask students to ind the domain and range o each unction.

6(1.2)x

5 4 3 2 1

Range x


y

–9 –8 –7 –6 –5 –4 –3 –2 –1

CONNECT

123456789

x

TO AP

Not all functions increase or decrease over the entire domain of the function. Functions may increase, decrease, or remain constant over various intervals of the domain. Functions that either increase or decrease over the entire domain are calledstrictly monotonic.

MATH TERMS If the graph of a relation gets closer and closer to a line, the line is called an asymptoteof the

1. f ( x ) = x2 [domain: all real numbers except 0; range: all real numbers except 0] 2. f(x) = x2 + 1 [domain: all real numbers; range: all real numbers greater than or equal to 1] 3. f(x) = x2 − 2x + 4 [domain: all real numbers; range: all real numbers greater than or equal to 3]

Differentiating Instruction For advanced learners who wish to urther explore the Connect to AP, ask students to identiy the intervals on which each unction shown below is increasing, decreasing, and/or constant. Students should also identiy any unctions that are strictly monotonic. 3

graph.

2 1 –4

4. Describe the end behavior o each unction as x approaches ∞. Write . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

y 4

a.

–3

–2

–1

1

2

3

4

x

–1 –2

the equation or any horizontal asymptotes. a. y = 6(1.2)x

–3 –4

As x goes to in nity, y goes to in nity.

[increasing for all reals; strictly monotonic ] y 5

b.

b. y = 6(0.9)x

4 3

As x goes to in nity, y gets close to 0; there is a horizontal asymptote aty = 0.

2 1 –4

–3

–2

–1

1

2

3

4

x

–1 –2

1–2 Create Representations, Activating Prior Knowledge, DebriefingStudents should use a graphing calculator or other graphing utility or Item 1.

Students should note that the domain o the unctions in the problem situation representing enlargements and reductions are subsets o the counting numbers, whereas the domain o the unction in Items 2a and 2b are all real numbers. Students can explore the range o each unction by using their graphing calculators.

3 Activating Prior Knowledge, DebriefingStudents should have an intuitive knowledge o when a unction is increasing and when it is decreasing. Ask students to describe what the graph o an increasing unction looks like and what the graph o a decreasing unction looks like.

[decreasing for x < 0; constant for 0 < x < 2; increasing for x > 2] y 4

c.

3 2 1 –4

–3

–2

–1

1

2

3

4

x

–1 –2 –3 –4

[decreasing for−2 < x < 2; increasing forx < −2 and x > 2]


329

ACTIVITY 21 Continued 4–7 Create Representations, Think-Pair-Share, Group Presentation, Debriefing Use Items 4–7 to assess student understanding o the concepts o end behavior, ∞, −∞, and asymptotes as related to the graphs o exponential unctions.

Students should be able to answer Item 6 by reerring to the graphs o the unctions. Item 7 is intended to spark a class discussion o the eatures o the graphs o exponential unctions as related to the parameters o the unction. Some students may not think about the possibility o the value o a being negative. Try to ind a group to consider this and have them present their indings to the class.

  Exponential Graphs and Asymptotes


My Notes

5. Describe the end behavior o each unction as x approaches −∞. Write the equation or any horiz ontal asymptotes. a. y = 6(1.2)x As x goes to negative in nity, y gets close to 0; there is a horizontal asymptote aty = 0.

b. y = 6(0.9)x As x goes to negative in nity, y goes to in nity.

6. Identiy any x- or y-intercepts o each unction. a. y = 6(1.2)x x-intercept: none; y-intercept: (0, 6) x

b. yx-intercept: = 6(0.9) none;y-intercept: (0, 6) 7. Reason abstractly. Consider how the parameters a and b aect the graph o the general exponential unction f(x) = a(b)x. Use a graphing


calculator to graph f or various values o a and b.

Debrie students’ answers to these items to ensure that they understand the meaning o increasing and decreasing and can identiy the domain, range, and asymptotes o an exponential unction.

a. When does the unction increase?

Ans wer s 8. domain (both): ( −∞, ∞); range (both): ( −∞, 0) 9. f(x): y approaches −∞; g(x): y approaches 0 10. f(x): y approaches 0; g(x): y approaches −∞


11. Increases because a > 0 and b > 1; y-intercept is (0, 8) because a = 8 gives the value o the y-intercept. 12. Decreases because a > 0 and 0 < b < 1; y-intercept is (0, 0.3) because a = 0.3 gives the value o the y-intercept.

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand when an exponential unction is increasing and when it is decreasing by looking at the unction deinition. Students can graph each unction to conirm their results.

330

for a > 0 and b > 1 or fora < 0 and 0< b < 1

b. When does the unction decrease? for a > 0 and 0< b < 1 or fora < 0 and b > 1

c. What determines the y-intercept o the unction? the value ofa

d. State any horizontal asymptotes o the unction. y=0

Check Your Understandin g Graph the unctions f(x) = −6(1.2)x and g(x) = −6(0.9)x on a graphing calculator or other graphing utility.

8. Determine the domain and range or each unction. 9. Describe the end behavior o each unction as x approaches ∞. 10. Describe the end behavior o each unction as x approaches −∞.

   Make use of structure.For each exponential unction, state whether the unction increases or decreases, and give the y-intercept. Use the general orm o an exponential unction to explain your answers.

11. y = 8(4)x 13. y = −2(3.16)x

12. y = 0.3(0.25)x 14. y = −(0.3)x

15. Construct vi able arguments. What is true about the asymptotes and y-intercepts o the unctions in this lesson? What conclusions can you draw?

13. Decreases because a < 0 and b > 1; y-intercept is (0, −2) because a = −2 gives the value o the y-intercept. 14. Increases because a < 0 and 0 < b < 1; y-intercept is (0, −1) because a = −1 gives the value o the y-intercept. 15. The asymptotes are all y = 0; the yconditions -interceptsare aretrue all xor a. These =all exponential unctions o the orm f(x) = a(bx): when x becomes either very large or very small, a(bx) will approach 0; when x = 0, f(x) = a.



  Transforming Exponential Functions



My Notes

Learning Targets:

PLAN

how changing parameters aects the graph o an exponential • Explore unction. • Graph transormations o exponential unctions. SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations, Quickwrite You can use transormations o the graph o the unction f(x) = bx to graph unctions o the orm g(x) = a(b)x−c + d, where a and b are constants, and a ≠ 0, b > 0, b ≠ 1. Rather than having a single parent graph or all exponential unctions, there is a dierent parent graph or each base b.

Pacing:1 class period Chunking the Lesson

MATH TIP You can draw a quick sketch of the parent graph for any baseb by

(

)

plotting the points −1, 1 , (0, 1), b and (1, b).

1. Graph the parent graph f and the unction g by applying the correct

( )

( )

CONNECT

Exponential functions are important in the study of calculus.

y

f

1234567

x

The graph ofg is a vertical stretch of the graphf by of a factor of 4.

g ( x ) = − 1 (3)x 2 y 7


6 5 4 3 2 1 –7 –6 –5 –4 –3 –2 –1 –1

–4

parent unction f(x) = x2. 1. g(x) = (x + 3)2 −1 [shifted 3 units left and 1 unit down] 2. g(x) = −2x2 [reflection across x-axis and a vertical stretch by a factor of 2] 3. g(x) = 3(x − 1)2 [vertical stretch by a factor of 3 and shifted 1 unit right]

In this lesson, students will explore graphing exponential unctions through transormations. Use the Bell-Ringer Activity as a review o transormations. 1–2 Create Representations, Activating Prior Knowledge, DebriefingStudents should see that the constant a can cause a vertical stretch or shrink o the parent graph.

When a < 0, there is a relection over the x-axis.

f

CONNECT TO AP 1234567

–2 –3


TEACHER to TEACHER

–7 –6 –5 –4 –3 –2 –1 –1

b. f(x) = 3x

TO AP

x

13 12 11 10 9 g 8 7 6 5 4 3 2 1

TEACH Ask students to describe each transormation o the graph o the

vertical stretch, shrink, and/or relection over the x-axis. Write a description o each transormation. x 1 a. f (x ) = 12 g (x ) = 4 2


g

–5 –6 –7

The graph ofg is a vertical shrink of the graphf of by a factor of1 2 and a re ection over the x-axis.

Exponential unctions play an important part in the study o calculus and have some very interesting properties. The instantaneous rate o change (derivative) o any exponential unction in the ormf(x) = bx is a multiple o the unctionf. This makes sense when one considers that the slope o an exponential unction is always increasing at an increasing rate as x increases. A unction that its this description well is an exponential unction. Even more surprising is that the derivative o the exponential unction f(x) = ex—a special exponential unction that is discussed in the next lesson—is the unction itsel. This is the only nonzero unction that has the property o being its own derivative.


331

ACTIVITY 21 Continued continued

My Notes

CONNECT

Technology Tip Discuss asymptotic behavior as it relates to graphs o exponential unctions. Explain that in Item 2a, it looks like the graph touches the x-axis but, in act, it never does. Have students graph the unction using an online calculator or geometry sotware. Have them zoom in to t he x-axis, where they will see that the unction does not cross or intersect it.


ACTIVITY 21

1–2 (continued)Create Representations, Quickwrite, DebriefingStudents should see that or a(b)x−c + d, the constant c causes a horizontal translation o the parent graph, and the constant d causes a vertical translation o the parent graph.

TO TECHNOLOGY

2. Sketch the parent graph f and the graph o g by applying the correct horizontal or vertical translation. Write a description o each transormation and give the equations o any asymptotes. a. f(x) = 2x g(x) = 2(x−3)

You can use a graphing calculator to approximate the range values when the x-coordinates are not integers. Forf(x) = 2x, use a calculator to findf 1 and f 3 . 2

()

1 22

2

y 13 12 11 10 9 8 7 6 5 4 3 2 1

( )

1.414

3≈

≈ 3.322

Then use a graphing calculator to



1

verify that the points  1 , 2 2  and  2  3,2 3 lie on the graph of

(

)

f(x) = 2x.

g f

–7 –6 –5 –4 –3 –2 –1 –1

1234567

x

The graph ofg is a horizontal translation of the graph f to of the right 3 units. Asymptote for f: y = 0; asymptote for g: y = 0 x

() g (x) = ( 1 ) 3

b. f ( x ) = 13

x

−2 y 7 6 5 4 3

g 12 –7 –6 –5 –4 –3 –2 –1 –1

f 1234567

x

–2 –3 –4 –5 –6 –7

The graph ofg is a vertical translation of the graph f down of 2 units. Asymptote for f: y = 0; asymptote for g: y = −2

332






3. Attend to precision. Describe how each unction results rom transorming a parent graph o the orm f(x) = bx. Then sketch the parent graph and the given unction on the same axes. Give the domain and range o each unc tion in interval notation. Give the e quations o any asymptotes. a. g(x) = 3x+4 + 1 y 7 6 5 4

g

f

1 2 3

Have students identiy the domain and range o each unction using its equation. Then have them use the graph to conirm their indings. Review dierent ways to write the domain and range o a unction. They can be written in words, in set notation, or in interval notation. Present students with examples o each to review. For example, or the unction f(x) = 2x, the domain and range can be expressed as:

3 2 1 –7 –6 –5 –4 –3 –2 –1 –1

My Notes

3–4 Create Representations, Quickwrite, DebriefingItem 3 is an opportunity or students to practice transormations o parent graphs o exponential unctions and to assess their understanding.

x

–2

To obtain the graph of g, horizontally translate the graphf of left 4 units and then vertically translate up 1 unit. f g −∞, ∞) Domain: ( (−∞, ∞) Range: (0,∞) (1,∞) Asymptotes:y = 0 y=1

Domain: all real numbers Range: all real numbers greater than 0 Domain: (−∞, ∞) Range: (0, ∞) D = {x | x ∈ ℜ} R = {y | y > 0}

x b. g ( x ) = 2 13 − 4

()

y 7 6 5 4 3 2 1 –7 –6 –5 –4 –3 –2 –11 –

f

x

1234567

–2 –3

g

–4


–5 –6 –7

To obtain the graph of g, vertically stretch the graphf of by a factor of 2 and then vertically translate down 4 units. f g −∞, ∞) Domain: ( (−∞, ∞) Range: (0,∞) (4,∞) Asymptotes:y = 0 y = −4


333



ACTIVITY 21

LESSON 21-4 PRACTICE x

7. The parent graph is f(x) = 4 ; there is a horizontal translation 2 units to the right and a shit down 3 units; domain: (−∞, ∞); range: (−3, ∞); asymptote: y = −3.

continued

My Notes

c. g ( x ) = 12 (4)x−4 − 2 y

y 11 10 9 8 7 6 5 4 3

7 6 5 4 3 2 1

f g

–7 –6 –5 –4 –3 –2 –1 –1

g

x

1234567

–2 –3

2 1 –7 –6 –5 –4 –3 –2 –1 –1

x

1234567

–2 –3

8. The parent graph is f(x) = 2x; there is a vertical shrink o 1 , a horizontal 2 translation 4 units to the r ight, and a vertical shit up 1 unit; domain: (−∞, ∞); range: (1, ∞); asymptote: y = 1. y

To obtain the graph of g, vertically shrink the graph fof by a factor 1 of 2, translate horizontally to the right 4 units, and then translate vertically down 2 units. f g −∞, ∞) Domain: ( (−∞, ∞) Range: (0,∞) (−2, ∞) Asymptotes:y = 0 y = −2

4. Describe how the unction g(x) = − 3(2)x−6 + 5 results rom transorming a parent graph f(x) = 2x. Sketch both graphs on the same axes. Give the domain and range o each unction in interval notation. Give the equations o any asymptotes. Use a graphing calculator to check your work. To obtain the graph of g, re ect the graph of f over thex-axis, vertically stretch the graph of f by a factor of 3, and then vertically translate up 5 units and horizontally translate 6 units to the right.

12 11 10 9 8 7 6 5 4 3 2 1 –5 –4 –3 –2 –1 –1

f

y

f

7 6 5 4 3

g

–2

g f

2 1

x

123456789

–7 –6 –5 –4 –3 –2 –1 –1

1234567

x

–2 –3

x

()

9. The parent graph is f (x) = 1 ; 3 there is a relection over the x-axis, a vertical stretch by a actor o 2, and a horizontal translation let 4 units; vertical shit up 1 unit; domain: (−∞, ∞); range: ( −∞, 0); asymptote: y = 0.

–4 –5 –6 –7

f −∞, ∞) Domain: ( −∞, ∞) Range: ( Asymptotes:y = 0

g (−∞, ∞) (−∞, 5) y=5

y

f

9 8 7 6 5 4 3 2 1

–6 –5 –4 –3 –2 –1 –1

g

123456

x

–2 –3 –4 –5

334






My Notes

Check Your Understanding 5. Reason q uantitatively. Explain how to change the equation o a parent graph f(x) = 4x to a translation that is let 6 units and a vertical shrink o 0.5.

6. Write the parent unction f(x) o g(x) = −3(2)(x+2) − 1 and describe how the graph o g(x) is a translation o the parent unction.

   Describe how each unction results rom transorming a parent graph o the orm f(x) = bx. Then sketch the parent graph and the given unction on the same axes. State the domain and range o each unction and give the equations o any asymptotes.

7. g(x) = 4x−2 − 3 8. g ( x) = 12 (2)x − 4 + 1 x +4

( )

9. g ( x) = −2 13

Ans wers 5. Replace a with 0.5, c with −6, and d with 0 in the equation a(b)x−c + d. 6. The parent unction is f(x) = 2x; g(x) is a translation o f(x) in that it is relected over the x-axis, vertically stretched by a actor o 3, horizontally translated let 2 units, and vertically translated up 1 unit.


Make use of structure.Write the equation that indicates ea ch transormation o the parent equation f(x) = 2x. Then use the graph below and draw and label each transormation.


10. For g(x), the y-intercept is at (0, 3).


11. For h(x), the exponential growth actor is 0.5. 12. For k(x), the graph o f(x) is horizontally translated to the right 3 units.

7–9. See page 334.

13. For l(x), the graph o f(x) is vertically translated upward 2 units. y


Check Your Understanding Debrie students’ answers to these items to ensure that they understand how to describe the transormation o an exponential quadratic unction.

13 12 11 10 9 8 7 6 5 4 3 2 1 –7 –6 –5 –4 –3 –2 –1 –1

10. g(x) = 3(2)x 11. h(x) = (0.5)x 12. k(x) = (2)x−3 13. l(x) = (2)x + 2 y g (x ) = 3(2)x

h(x ) = (0.5)x

l(x ) = (2)x + 2

1234567

x

13 12 11 10 9 8 7 6 5 4 3 2 1

–7 –6 –5 –4 –3 –2 –1 –1

k(x ) = (2)x –3 1234567

x

ADA PT Check students’ answers to the Lesson Practice to ensure that they can identiy attributes o a transormation rom an equation. Encourage students to graph the unctions in Items 7–9 to check their answers.


335


  Natural Base Exponential Functions


Lesson 21-5 PLAN

My Notes

Learning Targets:


• Graph the unction f(x) = e . • Graph transormations o f(x) = e . x

Chunking the Lesson

x


SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation, Debrieing

1. Use appropria te tools strateg ically.On a graphing calculator, set

TEACH

(

Ask students to determine whether each expression evaluates to a number that is

≈2.7183. Sample answer: The values of Y a constant value, 2 approach

rational or irrational. 1. π + 2 [irrational]

2.

4+ 3

3. 1 + 4 3 3

This irrational constant is called e and is oten used in exponential unctions. 2. a. On a graphing calculator, enter Y1 = ex. Using the table o values associated with Y1, complete the table b elow.

MATH TIP Exponential functions that describe examples of (continuous) exponential growth or decay usee for the base. You will learn more about the importance of e in Precalculus.

[irrational] [rational]

Technology Tip

Y1 = ex

x

I students are unamiliar with how to use a graphing calculator, walk them through Item 1. Students should consult their manuals i they are using a calculator other than a TI-Nspire. For some students, writing this process in their notes will be helpul, as they can reer to it repeatedly as they work through the course. For additional technology resources, visit SpringBoard Digital.

0

1

1

2.7183

2

7.3891

3

20.086

b. Reason quantitatively. Which row in the table gives the approximate value o e? Explain. The row in whichx = 1; whenx = 1, Y1 = e1 = e.

c. What kind o number does e represent? an irrational number

3. a. Complete the table below.

1–3 Create Representations,

Students Quickwrite, Debriefing should understand that because e is an irrational number,ex is an irrational number when x does not equal 0. In the table or Item 3, have students identiy which values are rational and which are irrational.

x

x−1

x0

x1

x2

2

0.5

1

2

4

e

0 .36 79

1

3

0.3333

1

2 .7 1 8 3

7.3 89 1

x3

8 2 0.0 86

y

3

9

27

5 4

b. Graph the unctions f(x) = ex, g(x) = 2x, and h(x) = 3x on the same

3

coordinate plane.

2 1 6

336

x

)

Y1 = x and Y2 = 1 + 1 . Let x increase by increments o 100. Describe x what happens to the table o values or Y 2 as x increases.


–5 –4 –3 –2 –1 –1 –2

123456

x

c. Compare f(x) with g(x) and h(x). Which eatures are the same? Which are dierent? All three functions have the y-intercept (0, 1) and the horizontal asymptotey = 0. The graph of the function f( x) is between the graphs ofg(x) and h(x), which makes sense because<2e < 3.






4. Graph the parent unction f(x) = ex and the unction g(x) by applying the correct vertical stretch, shrink, relection over the x-axis, or translation. Write a description or the transormation. State the domain and range o each unction. Give the equation o any asymptotes. a. f (x ) = e x g ( x ) = − 1 (e x ) 2 y


5 4 3 2 1 –6 –5 –4 –3 –2 –1 –1

My Notes

4 Create Representations, Quickwrite, DebriefingReer back to transormations o exponential unctions in the previous lesson, and have students predict what the transormations or Items 4a and 4b will look like beore graphing. Discuss the similarities and dierences between quadratic and c ubic transormations and exponential transormations.

123456

x

–2

The graph ofg is a vertical shrink of the graphf by of a factor of1 2 and a re ection over the x-axis. The domain of both functions is (−∞, ∞). The range off is (0,∞); the range ofg is (−∞, 0). The = 0. asymptote of both functionsy is

As students respond to questions or discuss possible solutions to problems, monitor their use o new terms and descriptions o applying math concepts to ensure their understanding and ability to use language correctly and precisely.

Universal Access Have students use an online calculator or a geometry tool to graph the parent unction and its transormation. With these tools, they can drag the unctions to help them answer Items 4–6.

b. f ( x ) = e x x−4 g ( x) = e +1 y 3 2 1 –6 –5 –4 –3 –2 –1 –1

123456

x

–2 –3


The graph ofg is a horizontal translation to the right 4 units and a vertical translation up 1 unitf.ofThe domain of both functions is (−∞, ∞). The range off is (0,∞); the range ofg is (1,∞). The asymptote off is y = 0; the asymptote of g is y = 1.


337

ACTIVITY 21 Continued 5 Create Representations, Quickwrite, DebriefingDebrie students’ answers to Item 5b by having them identiy each transormation and the reason or it. For example, the parent unction f(x) is translated down 2 units by the −2 in g(x).

Discuss how each transormation in Item 5 aects the domain and range o the unction. Ask students why the domain stays the same or each transormation and why the range changes.



My Notes

5. Graph the parent graph f and the unction g by applying the correct transormation. Write a description o each transormation. State the domain and range o each  unction. Give the equation o any asymptotes. a. f(x) = ex g(x) = 2ex − 5 y 8 6 4 2

ELL Support Students may struggle with the vocabulary in this lesson. Items 4–5 require students to describe transormations using math language. Have students reer to the Word Wall or their notes to help them articulate their answers.

–12–10 –8 –6 –4 –2 –2

2 4 6 81 0 1 2

x

–4 –6

The graph ofg is a vertical stretch of the graphf by of a factor of 2 and a vertical translation down 5 units. The domain of both ∞). The range off is (0,∞); the range ofg is functions is −∞, ( (−5, ∞). The asymptote of f is y = 0; the asymptote of g is y = −5.

b. f ( x ) = e x x −4 g ( x ) = 1 (e )−2 2

y 12 10 8 6 4 2 –12–10 –8 –6 –4 –2 –2

2 4 6 81 0 1 2

x

The graph ofg is a vertical shrink of the graphf of by a factor of1, 2 . a horizontal translation right 4 units, and a vertical translation down d e 2 units. The domain of both functions−∞, is ( ∞). The range off is rv e s (0, ∞); the range ofg is (−2, ∞). The asymptote of f is y = 0; the re asymptote ofg is y = −2. s t

h g ri ll A . d r a o B e g e ll o C 5 1 0 2 ©

338





My Notes

6. Explain how the parameters a, c, and d transorm the parent graph f(x) = bx to produce the graph o the unction g(x) = a(b)x−c + d. |a| > 1 ⇒ a vertical stretch by a factoraof 0 < |a| < 1 ⇒ a vertical shrink by a factoraof a < 0 ⇒ a re ection over the x-axis c > 0 ⇒ a horizontal translation right c units c < 0 ⇒ a horizontal translation left c units

Monitor group discussions to ensure that all members o the group are participating and that each member understands the language and terms

d < 0 ⇒ a vertical shift down d units d > 0 ⇒ a vertical shift up d units


used in the discussion.


Match each exponential expression with its graph.

7. f(x) = 3ex

A.

B.

Debrie students’ answers to these items to ensure that they can identiy key attributes o a natural base e xponential unction given an equation.

y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 –1

8. f(x) = −0.4ex

6 Create Representations, ThinkPair-Share, Group Presentation, Debriefing Use Item 6 to assess student understanding o transormations o the graphs o all exp onential unctions. I students are having diiculty completing this item, suggest they substitute numbers or a, c, and d to help them. Have students work in pairs or groups to complete Item 6.

123456

x

y

Ans wers 7. D 8. B 9. A 10. C

1 1 2 3 4 5 6

–6 –5 –4 –3 –2 –1 –1

x

–2 –3 –4 –5 –6

9. f(x) = ex + 2

C.

y 1 1 2 3 4 5 6

–6 –5 –4 –3 –2 –1 –1

x

–2 –3 –4 –5 –6


10. f(x) = −e

x

y

D. 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 –1

123456

x


339



ACTIVITY 21

ASS ESS

continued


My Notes

   Model with mathematics.Describe how each unction results rom transorming a parent graph o the orm f(x) = ex. Then sketch the parent graph and the given unction on the same axes. State the domain and range o each unction and give the equations o any asymptotes. 11. g ( x) = 14 e x + 5 12. g(x) = ex−3 − 4


13. g(x) = −4ex−3 + 3


14. g(x) = 2ex+4 15. Critique the rea soning of others On . Cameron’s math test, he was

11. vertical shrink by a actor o 1 and 4 a translation 5 units up y

x

−2 transormations rom the graph o f(x) = e to the asked to graph o describe g(x) = exthe − 2. Cameron wrote “translation let 2 units and down 2 units.” Do you agree or disagree with Cameron? Explain your reasoning.

20

16. What similarities, i any, are there between the unctions studied in this lesson and the previous lesson?

10 f (x ) = ex –20

domain range asymptote

–10

10

f (−∞, ∞) (0, ∞) y=0

x

g (−∞, ∞) (5, ∞) y=5

12. a translation 4 units down and 3 units to the right y 20 10 f (x ) = ex –20


–10

f (−∞, ∞) (0, ∞) y=0

10

x

g (−∞, ∞) (−4, ∞) y = −4

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand how to graph transormations o natural base exponential unctions. Allow students to use graphing calculators or geometry to check their work.

13. a relection over the x-axis, a vertical stretch by a actor o 4, and a translation 3 units up

14. a vertical stretch by a actor o 2 and a translation 4 units to the let y

y 20

3 10 f (x ) = ex

1

x

x –3


340

–1 –1

1

f (−∞, ∞) (0, ∞) y=0

3

g (−∞, ∞) (−∞, 3) y=3

f (x ) = ex

–20


–10

f (−∞, ∞) (0, ∞) y=0

g (−∞, ∞) (0, ∞) y=0



    Sizing Up the Situation


  

c. Write an equation to express the size o the

Write your answers on notebook paper. Show your work.

smallest violet in terms o the number o violets on the plate. d. I a plate has a total o 10 violets, explain two dierent ways to determine the size o the smallest violet.

Lesson 21-1 1. a. Complete the table so that the unction represented is a linear unction. x f(x)

6.7

10.6

b. What unction is represented in the data? 2. a. How do you use a table o values to determine i the relationship o y = 3x + 2 is a linear relationship?

b. How do you use a graph to determine i the relationship in part a is linear?

unction f(x) = 4(0.75)x ? A. Exponential growth actor is 75%. B. Percent o decrease is 25%. C. The scale actor is 0.75.

identiy the value o the parameters a and b. Then explain their meaning, using the vocabulary rom the lesson.

a.

a linear unction. I so, give an equation or the unction. I not, explain why not.


1 5

x 0 y

b.

y

14 5

12 5

1

1 23 24

18

36

18

x 0 y

3 5

4 12

6

9

4.5

123

0 4 2.25

10. Sixteen teams play in a one-game elimination

5. Which relationship hasthe greatest valueorx = 4? A. y = 5(3)x + 2 B. y = 5(2x + 3) C. y = 5(3x + 2)

match. The winners o the irst round go on to play a second round until only one team remains undeeated and is declared the champion. a. Make a table o values or the number o rounds and the number o teams participating. b. What is the reasonable domain and the range

D. y = 5(2)x+3 6. Ida paints violets onto porcelain plates. She paints

o thisthe unction? Explain. c. Find rate o decay. d. Find the decay actor.

a spiral that is a sequence o violets, the size o each consecutive violet being a raction o the size o the preceding violet. The table below shows the width o the irst three violets in the continuing pattern.

Violet Number Width (cm)

1 4

2 3.2

2.

5.

D. The decay rate is 25%. 8. For the exponential unction f(x) = 3.2(1.5)x,

modeled by a linear unction, an exponential unction, or neither. I the data can be modeled by a linear or exponential unction, give an equation or the unction.

4. Determine i the table o data can be modeled by

01234

x

3 2.56

a. Is Ida’s shrinking violet pattern an example o an exponential unction? Explain.

b. Find the width o the ourth and ith violets in the sequence.

LESSON 21-5 PRACTICE (continued)

15. The correct answer is “translation right 2 units and down 2 units.” Cameron likely made the error o thinking that subtracting 2 in the exponent results in a translation to the let because, on a number line, subtraction is usually with moving to the let. 16. associated Since f(x) = ex is an exponential unction, the unctions in both lessons are exponential unctions and ollow the same rules.

12345

f(x)

3. 4.

9. Decide whether each table o data can be

3. Which relationship is nonlinear? A. (2, 12), (5, 18), (6.5, 21) B. (6, x + 2), (21, x + 7), (−9, x − 3) C. (0.25, 1.25), (1.25, 2.50), (2.50, 5.00) D. ( −5, 20), (−3, 12), (−1, 4)

x

ACT IV IT Y PR ACT ICE 1. a.

Lesson 21-2 7. Which statement is NOT true or the exponential

12345 5.4


5.4

6.7

8.0

9.3

10.6

b. f(x) = 1.3x + 4.1 a. Sample answer: Subtract consecutive y-values to determine i there is a common dierence. b. Sample answer: The graph is a straight line. C yes; y = 2 x + 1 5 5 D

6. a.Yes, it is an example o an exponential unction because as the x-values increase by 1 each time, there is a constant ratio o 0.8 between the y-values. b. 2.56(0.8)= 2.048 cm; 2.048(0.8)= 1.6384 cm c. y = 4(0.8)x−1 or y = 5(0.8)x d. You can substitute 10 into the unction given in part c and have y = 4 (0.8)10−1 = 0.537 cm rounded to three decimal places, or you can use a calculator and repeatedly multiply the starting violet width by 0.8 and ind the same result. 7. A 8. a = 3.2, the initial value; b = 1.5, the growth actor or scale actor 9. a. linear; y = −6x + 24 x b. exponential; y = 36 1 2 10. a.

()

x f(x)

12345 16

8

4

2

1

b. domain: {x: 1, 2, 3, 4, 5}; range: {y: 16, 8, 4, 2, 1} c. 50% d. 0.50 11. A and C 12. B 13. a.( −∞, ∞); (0, ∞); increases; (0, 2) b. ( −∞, ∞); (0, ∞); decreases; (0, 3) c. ( −∞, ∞); (−∞, 0); increases; (0, −1) d. ( −∞, ∞); (−∞, 0); decreases; (0, −3) 14. a.1.37%; 0.84%; 0.362% b. 1.37% − 0.84% + 0.362% = 0.892% c. 100.892% d. P(n) = 313,847,465 (1.00892)n, where P = population and n = years since 2012 e. P(38) ≈ 439,819,438 15. f(x) is increasing when a > 0.


341


    Sizing Up the Situation

ACTIVITY 21 y

16. a. 7 6 5 4 3 2 1

Lesson 21-3 11. Which o the ollowing unctions have the same

f

–7 –6 –5 –4 –3 –2 –1 –1

g

continued

graph? x

1 2 3

B. f(x) = 4x C. f(x) = 4−x D. f(x) = x4

–2 –3 –4

12. Which unction is modeled in the graph below?

Translate the graph of horizontally to the let 3 units and then vertically translate down 4 units. f : domain (−∞, ∞); range (0, ∞); asymptote: y = 0 g: domain (−∞, ∞); range (−4, ∞); asymptote: y = −4 y b. f

4 3 2 1 –2 –1 –1

x

1234567

–2 –3

g

Vertically stretch the graph o f by a actor o 3, relect over the x-axis, and vertically translate up 2 units. f : domain (−∞, ∞); range (0, ∞); asymptotes: y = 0 g: domain (−∞, ∞); range (−∞, 2); asymptote: y = 2 y c. 4 3 2 1 –7 –6 –5 –4 –3 –2 –1 –1

( )

4 (1, 2.2) 3 2 (0, 2) 1 –4 –3 –2 –1

g

transorming a parent graph o the orm f(x) = bx. Then sketch the parent graph and the given unction on the same axes. State the domain and range o each unction and give the equations o any asymptotes. a. g(x) = 2x+3 − 4 x b. g ( x ) = −3 12 + 2

1234

c. g ( x ) = 12 (3) x

A. y = (2)x B. y = 2(1.1)x C. y = (2)1.1x D. y = 2.1x

x +3

−4

17. a. Explain why a change in c or the unction a(b)x−c + d causes a horizontal translation. b. Explain why a change in d or the unction a(b)x−c + d causes a vertical translation. 18. Which transormation maps the graph o x f(x) = 3x to g ( x) = 1 ? 3 A. horizontal translation B. shrink C. relection D. vertical translation

()

13. For each exponential unction, state the domain and range, whether the unction increases or decreases, and the y-intercept. x a. y = 2(4)x b. y = 3 12 c. y = −(0.3)x d. y = −3(5.2)x

14. The World Factbookproduced by the Central Intelligence Agency estimates the July 2012 United States population as 313,847,465. The ollowing rates are also reported as estimates or 2012. Birth rate: 13.7 births/1000 population Death rate: 8.4 deaths/1000 population

Lesson 21-5 19. Is f(x) = ex an increasing or a decreasing unction? Explain your reasoning.

20. Which unction has a y-intercept o (0, 0)? A. y = ex + 1 B. y = −ex + 1 C. y = ex − 1 D. y = ex 21. What ordered pair do f(x) = ex and g(x) = 2x have in common?

Net migration rate: 3.62 migrant(s)/1000 population f 1

x

–2 –3 –4

Vertically shrink the graph o f by a actor o 1 , horizontally translate 2 let 3 units, and then vertically translate down 4 units. f :domain ( −∞, ∞); range (0, ∞); asymptote: y = 0 g: domain (−∞, ∞); range (−4, ∞); asymptote: y = −4

ADD ITI ONA L PR ACT ICE I students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital or additional practice problems.

342

y

( )

–4

increasing?

Lesson 21-4 16. Describe how each unction results rom

x

( )

A. f(x) = 14

15. Under what conditions is the unction f(x) = a(3)x

a. Write a percent or each rate listed above. b. Combine the percents rom part a to ind the

MATHEMATICAL PRACTICES

overall growth rate or the United States. c. The exponential growth actor or a population is equal to the growth rate plus 100%. What is the exponential growth rate or the United States? d. Write a unction to express the United States population as a unction o years since 2012. e. Use the unction rom part d to predict the United States population in the year 2050.

22. Explain the dierence between y = x2 and y = 2x.

17. a.Sample answer: By adding to or subtracting rom the x-value o the exponential term, the exponential term is being evaluated or a dierent value than x. b. Sample answer: By adding to or subtracting rom the exponential term containing x, the value o the unction will be increased or decreased.

Att en d t o P re ci sio n

18. C 19. increasing; because e > 1 20. B 21. (0, 1) 22. Sample answer: The irst is a polynomial unction with a constant exponent and a changing base whose graph is a parabola. The second is an exponential unction with a constant base and a changing exponent whose graph is nonlinear but is not a parabola.



ACTIVITY

   

ACTIVITY 22

Earthquakes and Richter Scale Lesson 2 2-1 Exponential Data Learning Targets:

Act iv it y St and ard s F ocu s In Activity 22, students examine logarithmic unctions and their graphs. They begin reviewing exponential unctions. Then they examine the relationship between logarithmic and exponential unctions and write equations using both orms. Students discover and use the properties o logarithms and graph logarithmic unctions.

My Notes

• Complete tables and plot points or exponential data. • Write and graph an exponential unction or a given context. • Find the domain and range o an exponential unction. SUGGESTED LEARNING STRATEGIES: Summarizing, Paraphrasing, Create Representations, Quickwrite, Close Reading, Look or a Pattern In 1935, Charles F. Richter developed the Richter magnitude test sca le to compare the size o earthquakes. The Richter scale is based on the amplitude o the seismic waves recorded on seismographs at various locations ater being adjusted or distance rom the epicenter o the earthquake.

Lesson 22-1

Richter assigned a magnitude o 0 to an earthquake whose amplitude on a seismograph is 1 micron, or 10−4 cm. According to the Richter scale, a magnitude 1.0 earthquake causes 10 times the ground motion o a magnitude 0 earthquake. A magnitude 2.0 earthquake causes 10 times the ground motion o a magnitude 1.0 earthquake. This pattern continues as the magnitude o the earthquake increases.

PLAN Pacing:1 class period Chunking the Lesson #1 #2 #3 Check Your Understanding

1. Reason quantitatively How . does the ground motion caused by earthquakes o these magnitudes compare? a. magnitude 5.0 earthquake compared to magnitude 4.0

#6–7 Check Your Understanding

A magnitude 5.0 earthquake’s ground motion is 10 times that of a magnitude 4.0 earthquake.

Lesson Practice

TEACH

b. magnitude 4.0 earthquake compared to magnitude 1.0 A magnitude 4.0 earthquake’s ground motion is 1000 3or times 10 that of a magnitude 1.0 earthquake.

Bell-Ringer Activity Ask students to simpliy each expression using properties o exponents.  x3  1. x 2 y −3 xy 2    y 

c. magnitude 4.0 earthquake compared to magnitude 0 A magnitude 4.0 earthquake’s ground motion is 10,000 4or times 10 that of a magnitude 0 earthquake.

⋅

2

( )

2. 23

The table below describes the eects o earthquakes o dierent magnitudes.


22

Investigative

5 3. ab a 4b−1 5 4. 3 −2

Typical Effects of Earthquakes of Various Magnitudes 1.0 Very weak, no visible damage 2.0 Not elt by humans 3.0 Oten elt, usually no damage 4.0 Windows rattle, indoor items shake 5.0 Damage to poorly constructed structures, slight damage to well-designed buildings 6.0 Destructive in populated areas 7.0 Serious damage over large geographic areas 8.0 Serious damage across areas o hundreds o miles 9.0 Serious damage across areas o hundreds o miles 10.0 Extremely rare, never recorded

[64]

 b6   3  a  [19,683]

(32 ) Introduction,1 Summarizing, Paraphrasing, Debriefing The table o typical eects will help students understand the physical implications o dierent magnitudes. Lead a class discussion beore starting this lesson to make certain that all students are comortable with earthquake terminology. Use the table on the next page to assess student understanding o magnitude and ground motion caused by an earthquake.

Common Core State Standards for Activity 22

Universal Access

HSF-IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

HSF-IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 

Have students look up all the meanings o the word magnitude to help them understand its meaning with respect to earthquakes. Discuss i and where they have seen or heard the word used beore.

HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Activity 22 • Logarithms and Their Properties

343

ACTIVITY 22 Continued 2 Create Representations, DebriefingThe purpose o this item is to lead students to the idea that there is an exponential relationship between ground motion caused by an earthquake and its magnitude. 3 Create Representations, Quickwrite, DebriefingStudents experience how cumbersome it is to work with speciic amounts o ground motion since the values vary so greatly. They should understand why Richter devised a scale using more manageable numbers. In part d, students should

write an exponential unction. As students work on this item, monitor their writing to ensure that they are using language correctly, including adequate details, and describing mathematical reasoning using precise terms.

  Exponential Data


My Notes

2. Complete the table to show how many times as great the ground motion is when caused by each earthquake as compared to a magnitude 0 earthquake.

Magnitude

Ground Motion Compared to Magnitude 0

1.0 2.0 3.0 4.0

10 100 1000 10,000

5.0 6.0

100,000 1,000,000

7.0

10,000,000

8.0

100,000,000

9.0

1,000,000,000

10.0

10,000,000,000

Technology Tip 3. In parts a–c below, you will graph the Allow students to use a graphing calculator, an online calculator, or a geometry tool to create the graph in Item 3d. With any o these tools, students can easily change the displays to determine an appropriate window or displaying the unction. For additional technology resources, visit SpringBoard Digital.

data rom Item 2. Let the horizontal axis represent the magnitude o the earthquake and the vertical axis represent the amount o ground motion caused by the earthquake as compared to a magnitude 0 earthquake. a. Plot the data using a grid that displays −10 ≤ x ≤ 10 and −10 ≤ y ≤ 10. Explain why this grid is or is not a good choice.

10 8 6 4 2 –10

–8

–6

–4

–2

246

8

10

–2 –4 –6 –8 –10

A [−10, 10]× [−10, 10] window would not be appropriate since only the ordered pair (1, 10) would be plotted on the graph, as shown.

b. Plot the data using a grid that displays −10 ≤ x ≤ 100 and −10 ≤ y ≤ 100.

100 90 80 70 60 A [−10, 100]× [−10, 100] window 50 would not be appropriate since only 40 the ordered pairs (1, 10) and (2, 100) 30 20 would be plotted, as shown. In 10

Explain why this grid is or is not a good choice.

addition, thex-axis does not need to be larger than 10 units.

344


– 10

20

40

60

80

1 00





c. Scales may be easier to choose i only a subset o the data is graphed and i dierent scales are used or the horizontal and vertical axes. Determine an appropriate subset o the data and a scale or the graph. Plot the data and label and scale the axes. Draw a unction that its the plotted data.

My Notes

G(x ) d re a p m o C n ito o M d n u o r G

e10000 k a 9000 u q 8000 h tr 7000 a E 6000 d r 5000 a 4000 d n 3000 a t S 2000 a 1000 to

(4, 10,000)

(1, 10) (2, 100)

Check Your Understanding Debrie students’ answers to these items to ensure that they understand why the upper limit o the domain is 10 in the context o the problem.

Ans wers 4. [0, 10]; Yes, it is continuous. 5. Estimate is (3.5, 3000); actual is (3.5, 3162.278).

(3, 1000) x

12345

Richter Magnitude

One possible answer is to choose the subset {(1, 10), (2, 100), (3, 1000), (4, 10,000)} and plot those points.

d. Write a unction G(x) or the ground motion caused compared to a magnitude 0 earthquake by a magnitude x earthquake. G(x) = 10x

6–7 Close Reading, Create Representations, Debriefing Students should realize that even though they were asked to plot ground motion

as a unction o themagnitude magnitudeisina Item 3d, in reality unction o ground motion. An earthquake’s magnitude is not assigned until an earthquake actually happens, so it does not make sense or the earthquake’s ground motion to be a unction o its magnitude.

Differentiating Instruction Check Your Understanding 4. What is the domain o the unction in Item 3d? Is the graph o the unction continuous?

5. Use the graph rom Item 3c to estimate how many times greater the ground motion o an earthquake o magnitude 3.5 is than a magnitude 0 earthquake. Solve the equation you wrote in Item 3d to check that your estimate is reasonable.

6. Make sense of problems. In Item 3, the data were plotted so that the ground motion caused by the earthquake was a unction o the magnitude o the earthquake. a. Is the ground motion a result o the magnitude o an earthquake, or is the magnitude o an ear thquake the result o ground motion? . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

To supportstudents in reading problem scenarios, careully group students to ensure that all students participate and have an opportunity or meaningul reading and discussion. Suggest that group members each read a sentence and explain what that sentence means to them. Group members can then conirm one another’s understanding o the key inormation provided or the problem.

An earthquake’s magnitude is not assigned until an earthquake actually happens, so an earthquake’s magnitude is a result of ground motion.

b. Based your answer to part a, would you choose ground motion or magnitude as the independent variable o a unction relating the two quantities? What would you choose as the dependent variable? Ground motion should be the independent variable and magnitude should be the dependent variable of a function relating the two quantities.


345

ACTIVITY 22 Continued 6–7 (continued)Close Reading, Create Representations, Debriefing In Item 7a, the intent is that students consider each unction that they graphed to it the data over the entire domain and to begin to realize that the unctions are inverses.

Ater completing Items 7a–c, students should be aware that the two unctions are inverses. It is not expected that students be able to determine an algebraic rule or the unction that is the inverse o y = 10x.



My Notes

c. Make a new graph o the data plotted Item 3c so that the magnitude o the earthquake is a unction o the ground motion caused by the earthquake. Scale the axes and draw a unction that its the plotted data. M(x )

5 e d 4 tu i n g 3 a M r 2 te h ic R 1

(10,000, 4)

(1000, 3) (100, 2) (10, 1) x

2 000

400 0

6 000

80 00 1 0 ,00 0

Ground Motion Compared to a Standard Earthquake

7. Let the unction you graphed in Item 6c be y = M(x), where M is the magnitude o an earthquake or which there is x times as much ground motion as a magnitude 0 earthquake. a. Identiy a reasonable domain and range o the unction y = G(x) rom Item 3d and the unction y = M(x) in this situation. Use interval notation.

Do m a i n

R a n ge

y = G(x)

[ 0,10]

(0 ,10 ,0 0 0,0 0 0 ,0 0 0)

y = M(x)

( 0,10,0 0 0 ,0 0 0,0 0 0 )

[ 0,10]

b. In terms o the problem situation, describe the meaning o an ordered pair on the graphs o y = G(x) and y = M(x). Ground motion compared

y = G(x) Richter magnitude

, to magnitude 0 earthquake

Ground motion compared

y = M(x) to magnitude 0 earthquake , Richter magnitude . d e rv e s e r s t h g ri ll A . d r a o B e g e ll o C 5 1 0 2 ©

346





c. A portion o the graphs o y = G(x) and y = M(x) is shown on the same set o axes. Describe any patterns you observe. 10

G(x )

Ans wers 8. reversed the axes rom the graph

8 6

or Item 3c 9. The unctions G and M are inverse unctions.

4 2 –2

My Notes

Check Your Understanding Debrie students’ answers to these items to ensure that they understand the unctions are inverses. Have them list a ew points rom each unction and reverse the coordinates to help them understand this.

M(x )

2468

10

–2

Sample answer: The two functions are symmetric about the line y = x. The values ofx and y in M(x) are the values of y and x in G(x).

Check Your Understanding 8. How did you choose the scale o the graph you drew in Item 6c? 9. What is the relationship between the unctions G and M?


   How does the ground motion caused by earthquakes o these magnitudes compare?

10. magnitude 5.0 compared to magnitude 2.0 11. magnitude 7.0 compared to magnitude 0 12. magnitude 6.0 compared to magnitude 5.0 13. A 1933 Caliornia earthquake had a Richter scale reading o 6.3. How many times more powerul was the Alaska 1964 earthquake with a reading o 8.3?

14. Critique the rea soning of others. Garrett said that the ground

motion o an earthquake o magn itude 6 is twice the ground motion o an earthquake o magnitude 3. Is Garrett correc t? Explain.


10. Magnitude 5.0 is 1000 times greater than magnitude 2.0. 11. Magnitude 7.0 is 10,000,000 times greater than magnitude 0. 12. Magnitude 6.0 is 10 times greater than magnitude 5.0. 13. 10,000 14. No; the unction does not increase linearly. The ground motion o an earthquake o magnitude 6 is 1000 times the ground motion o an earthquake o magnitude 3.

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand that the ground motion unction is not linear. Have students reer back to the tables in Item 1 to conirm this.


347


  The Common Logarithm Function


Lesson 22-2 PLAN

My Notes

Pacing:1 class period #1 #2 #3–5 Check Your Understanding Lesson Practice

TEACH

SUGGESTED LEARNING STRATEGIES: Close Reading, Vocabulary Organizer, Create Representations, Quickwrite, Think-Pair-Share

Bell-Ringer Activity Ask students to ind the inverse o each unction.

1. f (x) = −2 x + 9 2. f (x) = x 2 − 4 3. f (x ) = x 3 + 1

 f −1( x ) = 9 − x  2    f −1 ( x ) = x + 4     f −1( x ) = 3 x − 1   

1 Quickwrite, Create Representations, DebriefingEncourage students to graph y = log x over various windows. A window o [ −7, 40] with a scale o 10 or x and [−1, 2] with a scale o 1 or y gives a nice but limited view o the unction. In part b, students can use the graph, tables, or the log unction to evaluate M(120,000). In part c, students can ind the intersection o the graphs o M(y) = log x and y = 7.9. Alternatively, students could use the TABLE eature o the calculator to determine the answer. Students may need to appropriately rescale their graphs. Students should consult their graphing calculator manuals. For some students, writing this process in their notes will be helpul, as they can reer to it again and again as they work through the course. For additional technology resources, visit SpringBoard online.

Developing Math Language Students should add the deinition o common logarithmic functionto their math notebooks. Discuss why the use o common is necessary when describing these types o unctions. Students will learn about logarithms in other bases in Activity 23.

Learning Targets:

• Use technology to graph y = log x. • Evaluate a logarithm using technology. exponential equations as their corresponding logarithmic • Rewrite equations. logarithmic equations as their corresponding exponential • Rewrite equations.

Chunking the Lesson

The Richter scale uses a base 10 logarithmic scale. A base 10 logarithmic scale means that when the ground motion is expressed as a power o 10, the

MATH TERMS A logarithmis an exponent to which a base is raised that results in a specified value. A common logarithm is a base 10 logarithm, such as logx log 100 = 2 because 102 = 100.

⋅

magnitudex o the earthquake is the exponent. You have seen this unction G(x) = 10 , where x is the magnitude, in Item 3d o the previous lesson. The unction M is the inverse o an exponential unction G whose base is 10. The algebraic rule or M is a common logarithmic unction. Write this unction as M(x) = log x, where x is the ground motion compared to a magnitude 0 earthquake.

1. Graph M(x) = log x on a

TECHNOLOGY TIP The LOG key on your calculator is for common, or base 10, logarithms.

graphing ca lculator. a. Make a sketch o the calculator graph. Be certain to label and scale each axis. b. Use M to estimate the magnitude o an earthquake that causes 120,000 times the ground motion o a magnitude 0 earthquake. Describe what would happen i this earthquake were centered beneath a large city.

M(x )

5 e d u it n g a M r te h ic R

3 2 1 x

2000 4 000 6000 8 000 10,0 00

Ground Motion

M(120,000)≈ 5.08. According to the information given in the previous lesson, an earthquake of this magnitude could cause damage to buildings.

c. Use M to determine the amount o ground motion caused by the 2002 magnitude 7.9 Denali earthquake compared to a magnitude 0 earthquake. 79,432,823.47 times as much motion as magnitude 0.

MATH TIP You can also write the equation y = log x as y = log10 x. In the equation y = log x, 10 is understood to be the base. Just as exponential functions can have bases other than 10, logarithmic functions can also be expressed with bases other than 10.

2. Complete the tables below to show the relationship between the exponential unction base 10 and its inverse, the common logarithmic unction. x

y = 10x

0

100 = 1

x

1 = 100

y = log x

log 1 = 0

1

101 = 10

10 = 101

log (10)= 1

2

102 = 100

100 = 102

log (100)= 2

3

103 = 1000

1000 = 103

log (1000)= 3

log x

10log x = x

10x

log 10x = x

2 Create Representations, Debriefing Students numerically veriy that y = 10x and y = log x are inverses o each other. Students may struggle with the last row o the t able. Class discussion should reinorce the property o inverse unctions, which states that f ( f −1( x)) = f −1( f ( x)) = x . It is

important that students know that 10log x = x and log 10x = x. Have them add these identities to their math notebooks.

348

4




  The Common Logarithm Function


3. Use the inormation in Item 2 to write a logarithmic statement or each exponential statement.

MATH TIP

b. 10−1 = 1

a. 104 = 10,000

10

log 4 (10,000)=

Recall that two functions are inverses whenf(f −1 (x)) = f −1(f(x)) = x.

( )

1 = −1 10

log

4. Use the inormation in Item 2 to write each logarithmic statement as an exponential statement.

a. log 100,000 = 5

b. log

100,000105 =

(1001 ) = −2

−2

10

= 1 100

5. Evaluate each logarithmic expression without using a calculator. a. log 1000 3

b. log

1 10, 000

−4

Check Your Understanding 6. What unction has a graph that is symmetric to the graph o y = log x about the line y = x? 7. Evaluate log 10x or x = 1, 2, 3, and 4. 8. Let f(x) = 10x and let g(x) = f−1(x). What is the algebraic rule or g(x)?

   9. Evaluate without using a calculator. a. log 106

b. log 1,000,000

c. log 1

100

10. Write an exponential statement or each. a. log 10 = 1


b. log

1 = −6 1, 000, 000

My Notes

c. log a = b

11. Write7a logarithmic statement or0 each. a. 10 = 10,000,000 b. 10 = 1 c. 10m = n 12. Model with m athematics. The number o decibels D o a sound is  I  modeled with the equation D = 10 log  −  10 12 where I is the intensity o the sound measured in watts. Find the number o decibels in each o the ollowing: a. whisper with I = 10−10 b. normal conversation with I = 10−6 c. vacuum cleaner with I = 10−4 d. ront row o a rock concert with I = 10−1 e. military jet takeo with I = 102

The exponent x in the equation y = 10x is the common logarithm of y. This equation can be rewritten as log y = x.

3–5 Create Representations, Think-Pair-Share, Debriefing Students should have noticed the relationship between exponentials and logarithms. Additional guidance is provided in the Math Tip.

I students have trouble with Item 5, remind them that a common logarithm is simply an “exponent” or base 10. To evaluate log 1000, ind the exponent to which 10 should be raised in order to get 1000.

Check Your Understanding Debrie students’ answers to these items to ensure that they understand that the inverse o an exponential unction is a logarithmic unction and the inverse o a logarithmic unction is an exponential unction.

Ans wers 6. y = 10x 7. 1, 2, 3, 4 8. g(x) = log x


9. a.6 b. 6 c. −2 10. a.10 1 = 10

1 1, 000, 000 c. 10 b = a 11. a.lo g 10,000,000 = 7 b. log 1 = 0 c. log n = m 12. a. 20 b. 60 c. 80 d. 110 e. 140

b. 10−6 =

ADA PT Check students’ answers to the Lesson Practice to ensure that they can switch between exponential and logarithmic orm. Allow students to conirm their answers to Items 9 and 12 using a scientiic calculator.


349


  Properties of Logarithms


PLAN My Notes


Learning Targets:

• Make conjectures about properties o logarithms. and apply the Product Property and Quotient Property o • Write Logarithms. • Rewrite logarithmic expressions by using properties.

Chunking the Lesson #1–2 #3–4 #5–6 Check Your Understanding #9–11 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look or a Pattern, Quickwrite, Guess and Che ck You have already learned the properties o exponents. Logarithms also have properties.

TEACH

1. Complete these three properties o exponents.


m+n

Ask students to evaluate each expression.

am an =

1. 23 2−1

am = an

⋅

−2

[4]

( )

[81]

3

[16]

2. 3−2 5 3. 4

( 22 )

⋅

(am)n =

TEACHER to TEACHER

a

am−n

am⋅n

2. Use appropriate tools

strategically.Use a calculator to complete the tables below. Round each answer to the nearest thousandth.

In this lesson and the next, students will discover that the product rule, the quotient rule, and the power rule hold true or common logarithms. In Activity 23, students will learn that these properties extend to other logarithms o other bases as well.


x

y = log x

x

y = log x

1 2 3 4 5

0

6 7 8 9 10

0.778

0.301 0.477 0.602 0.699

0.845 0.903 0.954 1

3. Add the logarithms rom the tables in Item 2 to see i you can develop a

Students need to be amiliar with the

property. Find each sum and round each answer to the nearest thousandth.

properties exponents beore continuingo with this activity. Review these properties with students.

log 2 + log 3 = log 2 + log 4 =

1–2 Create Representations, Activating Prior Knowledge, DebriefingItem 1 prompts students or inormation that they should have learned previously. The inormation, in turn, will lead them to make conjectures about the connections between these properties and the properties o logarithms.

Students will use data rom the tables in Item 2 in items that ollow. It is important to see that students have correct answers or Items 1 and 2 beore they proceed to the next items. 3–4 Create RepresentationsI students do not recognize a property in Item 3, the next items will guide them to the property.

350

MINI-LESSON:

0.778 = log 6 0.903 = log 8

log 2 + log 5 =

1 = log 10

log 3 + log 3 =

0.954 = log 9

Review of Exponent Properties

Students need to be familiar with the properties of exponents in this lesson. If students need to review these properties, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.






4. Compare the answers in Item 3 to the tables o data in Item 2. a. Express regularity in repeated reasoning. Is there a pattern or

My Notes

5–6 Quickwrite, Create Representations, Debriefing Students may struggle with making the connection in Item 5. When ex ponential expressions with like bases are multiplied, the exponents are added. When logarithms with like bases are added, the result is the logarithm o the product o their inputs. Help students to

property when these logarithms are added? I yes, explain the pattern that you have ound. When two logarithms of like bases are added together, the result is the logarithm of the product of the input values.

b. State the property o logarithms that you ound by completing the ollowing statement. log m + log n =

log (mn)

5. Explain the connection between the property o logarithms stated in

see that logarithms are exponents that have been expressed using dierent notation.

Item 4 and the corresponding property o exponents in Item 1.

When exponential expressions with like bases are multiplied, the exponents are added. When logarithms with like bases are added, the result is the logarithm of the product of their inputs.

6. Graph y1 = log 2 + log x and y2 = log 2x on a graphing calculator. What do you observe? Explain. The two functions are identical because y2 = log (2x) = log (2) + log (x) = y1 by the property in Item 4b.

Check Your Understanding Identiy each statement as true or alse. Justiy your answers.

7. log mn = (log m)(log n) 8. log xy = log x + log y

9. Make a conjecture about the property o logarithms that relates to the property o exponential equations that states the ollowing: am = a m−n. an

( )

The c o nj ec ture is lo g(m )− lo g(n )= log m n .

10. Use the inormation rom the tables in Item 2 to provide examples that . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

support your conjecture in Item 9. Sample answers:

() ()

lo g ()4 − log () 2 = 0 .3 0 1= log 4 = lo g () 2 2 lo g ()6 − log () 2 = 0 .47 7= log 6 = log () 3 2

11. Graph y1 = log x − log 2 and y 2 = log x on a graphing calculator. What do you observe?

2

The graphs of the two functions are identical.

3–4 (continued)Look for a Pattern, DebriefingStudents should realize that when two logarithms with like bases are added, the result is the logarithm o the product o the input values.

TECHNOLOGY TIP When using the LOG key on a graphing calculator, a leading parenthesis is automatically inserted. The closing parenthesis for logarithmic expressions must be entered manually. So entering log 2 + log x without closing the parentheses will NOT give the correct result.

Students have looked at the property in Item 6 both numerically and analytically in terms o the corresponding property o exponents. Now they will validate the property graphically. A suggested graphing window would be [0, 10] or the x-axis and [−2, 2] or the y-axis.

Check Your Understanding Debrie students’ answers to these items to ensure that they understand the properties o logarithms presented in this lesson.

Ans wers 7. False; it is the sum o the logs: log m + log n. 8. true by the property in Item 4b 9–11 Guess and Check, Create

Representations, for aa Pattern, DebriefingStudents Look willmake conjecture or guess in Item 9 and then validate or check it in the items that ollow. Students who struggle with Item 9 should be directed to take a similar approach to what they did in Item 3 and Item 4. I students continue to have diiculty with this concept, have them answer Item 11 and then return to Item 9. In Items 10 and 11, students validate their conjecture both numerically and graphically. A suggested graphing window would be [0, 10] or the x-axis and [−2, 2] or the y-axis.


351



ACTIVITY 22


continued

Debrie students’ answers to these items to ensure that they can use the Product and Quotient Properties to rewrite logarithms.

Ans wers 12. Sample answers: log (9 4) = log 9 + log 4 = 0.954 + 0.602 = 1.556; log (6 6) = log 6 + log 6 = 0.778 + 0.778 = 1.556 13. Sample answers: log 10 =

⋅

⋅

( )

My Notes

Check Your Understandin g Use the inormation rom the tables in Item 2 and the properties in Items 4b and 9.

12. Write two dierent logarithmic expressions to ind a value or log 36. 13. Write a logarithmic expression that contains a quotient and simpliies to 0.301.

14. Construct viable arguments. Show that log (3 + 4) ≠ log 3 + log 4.

5 log 10 − log 5 = 1 − 0.699 = 0.301

14. log 7 = 0.845; log 3 = 0.477; log 4 = 0.602; 0.477 + 0.602 = 1.079, not 0.845.

  

ASS ESS

15. a. log 8

Students’ answers to Lesson Practice problems will provide you with a ormative assessment o their understanding o the lesson concepts and their ability to apply their learning. See the Activity Practice or additional problems or this lesson. You may assign the problems here or use them as a culmination or the ac tivity. LESSON 22-3 PRACTICE

()

15. a.log 8 = log 8 − log 3 3 = 0.903 − 0.477 = 0.426 b. log 24 = log (8 3) = log 8 + log 3 = 0.903 + 0.477 = 1.38 c. log 64 = log (8 8) = log 8 +

⋅ ⋅ log 827==0.903 d. log log (3+ 0.903 ⋅ 3 ⋅ 3)= 1.806 = log 3 + log 3 + log 3

Use the table o logarithmic values at the beginning o the lesson to evaluate the logarithms in Items 15 and 16. Do not use a calculator.

(3)

b. log 24 c. log 64 d. log 27

(9)

16. a. log 4

b. log 2.25 c. log 144 d. log 81 17. Rewrite log 7 + log x − (log 3 + log y) as a single logarithm.

( 9n )

18. Rewrite log 8m as a sum o our logarithmic terms. 19. Make use o f structure. Rewrite log 8 + log 2 − log 4 as a single logarithm and evaluate the result using the table at the beginning o the lesson.

= 0.477 + 0.477 + 0.477 = 1.431 16. a.log 4 = log 4 − log 9 9 = 0.602 − 0.954 = −0.352 b. log 2.25 = log 9 = log 9 − log 4 4 = 0.954 − 0.602 = 0.352 c. log 144 = log (4 4 9) = log 4 + log 4 + log 9 = 0.602 + 0.602 + 0.954 = 2.158 d. log 81= log (9 9) = log 9 + log 9 = 0.954 + 0.954 = 1.908   17. log  7 x  3y  18. log 8 + log m − (log 9 + log n) 19. log 8 2 = log 4 = 0.602 4

()

⋅ ⋅

⋅

(⋅)

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand how to evaluate a logarithmic expression and rewrite an expression as a single logarithm. As an additional activity, use index cards to create a game or students to match expressions. For example, log 4 9 would match the expression log 4− log 9.

()

352




  More Properties of Logarithms


Lesson 22-4 PLAN

My Notes

Learning Targets:

• Make conjectures about properties o logarithms. • Write and apply the Power Property o Logarithms. • Rewrite logarithmic expressions by using their properties. SUGGESTED LEARNING STRATEGIES: Think-Pair-Share, Create Representations

1. Make a conjecture about the property o logarithms that relates to the property o exponents that states the ollowing: (am)n = amn. log (mn) = n log (m)

Pacing:1 class period Chunking the Lesson #1–3 Check Your Understanding #6–9 Check Your Understanding Lesson Practice

TEACH Bell-Ringer Activity

2. Use the inormation rom the tables in Item 2 in the previous lesson and the properties developed in Items 4 and 9 in the previous lesson to support your conjecture in Item 1. Possible answers are given below.

⋅ ⋅ log (2 ) = log (2⋅ 2 ⋅ 2) = log (2)+ log (2)+ log (2)= 3 log (2) log (32) = log (3 3) = log (3) + log (3) = 2 log (3) log (42) = log (4 4) = log (4) + log (4) = 2 log (4) 3

3. Use appropriat e tools strategi cally.Graph y1 = 2 log x and y2 = log x2 on a graphing calculator. What do you observe? The graphs of the two functions are identical.

Check Your Understanding Identiy each statement as true or alse. Justiy your answer.

4. 2 log m = log m 5. log 10 2 = log 210

6. Express regularity i n repeated reasoning.The logarithmic properties that you conjectured and then veriied in this lesson and the previous lesson are listed below. State each property. Product Property: . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

Quotient Property:

Power Property:

log m + log n = log (mn)

( )

log (m )− lo g (n )= log m n

log (mn) = n log (m)

Ask students to identiy each st atement as true or alse. I the statement is alse, have them correct it.

1. log 3 + log 5 =log 8 [ false; log 3 + log 5 = log 15] 2. log 10 − log 5 = log 2 [true] 3. log 1000 = 3 [true] 1–3 Guess and Check, Create Representations, Look for a Pattern, DebriefingStudents may need coaching to develop the property in Item 1. There are two ways to approach this problem, ollowing the patterns rom Items 2–11. Students could try either log (u)v or log (uv). They may struggle with log (u)v as there is no pattern to be ound. However, i students try log (uv), they will have more success i they recognize exponentiation as repeated multiplication and apply the Product Property o Logarithms:

log(uv ) = log(u⋅ ⋅ u ⋅ u⋅ . . . u) = log(+ u) log( + u+) . . . = v log(u)

log( u)

In Item 2, students should apply the Product Property o Logarithms to veriy their conjecture numerically. For example,

⋅

log(32 ) = log(3 3) = log 3 + log 3 = 2 log 3 In Item 3, students veriy their conjecture graphically. A suggested graphing window would be [0, 10] or the x-axis and [−2, 2] or the y-axis.

6–9 Think-Pair-Share, Create Representations, Debriefing Item 4 gives students an opportunity to relect on the properties derived in the activity. Ater sharing answers with the entire group to make sure all responses are correct, students can record the inormation in

their math notebooks. Monitor group discussions to ensure that all members are participating. Pair or group students careully to acilitate discussions and understanding o both routine language and mathematical terms.

Check Your Understanding Debrie students’ answers to these items to ensure that they can correctly use properties o logarithms to justiy their answers.

Ans wers 4. True; by the property in Item 1, 2

2 log m = log m = log m . 5. False; log 102 = 2


353

ACTIVITY 22 Continued 6–9 (continued)Items 7–9 give students an opportunity to apply the logarithm properties. They also provide an opportunity to assess student understanding.

  More Properties of Logarithms


My Notes

7. Use the properties rom Item 6 to rewrite each expression as a single logarithm. Assume all variables are positive. a. log x − log 7


log x 7

Debrie students’ answers to these items to ensure that they understand that the log o a negative number is not a real number.

Ans wer s 10. The log o a sum does not equal the

b. 2 log x + log y log (x2y)

8. Use the properties rom Item 6 to expand each expression. Assume all variables are positive. a. log 5xy4

sum o the logs; only the product the sum o deined the logs.since it 11. equals log (−100) is not represents the exponent you would raise 10 to in order to get −100. A base o 10 will never give a negative value.

log 5 + log x + 4 log y

b. log x3 y

log x − 3 log y

9. Rewrite each expression as a single logarithm. Then evaluate. a. log 2 + log 5 log 10= 1

ASS ESS

b. log 5000 − log 5 log 1000= 3

Students’ answers to Lesson Practice problems will provide you with a ormative assessment o their understanding o the lesson concepts and their ability to apply their learning. See the Activity Practice or additional problems or this lesson. You may assign the problems here or use them as a culmination or the activity.

c. 2 log 5 + log 4 log 100 = 2

Check Your Understandin g 10. Explain why log (a + 10) does not equal log a + 1. 11. Explain why log (−100) is not deined.


12. log 100 = 2 13. log 1 = −1 10 14. log 10,000 = 4 15. log 1 = −2 100 16. log 100 1 = log 1 = 0 100 17. log b + 3 log c + 2 log d

( )

    Attend to precision.Rewrite each expression as a single logarithm. Then evaluate the expression without using a calculator.

12. log 5 + log 20 13. log 3 − log 30 14. 2 log 400 − log 16 og 1 + 2 log2 400

ADA PT

15.l

Check students’ answers to the Lesson Practice to ensure that they can both rewrite logarithmic expressions as a single logarithm and expand a single logarithm into an expression. Students should easily and luently go back and orth between the two orms. Allow students to use a calculator to check their work.

16. log 100 + log

354

1 . (100 )

17. Expand the expression log bc3d2.




    Earthquakes and Richter Scale


7. Write an exponential statement or each

   Write your answers on notebook paper. Show your work.

Lesson 22-1 M(x )

logarithmic statement below. a. log 10,000 = 4 1 9 b. log 1, 000, 000 , 000 = − c. log a = 6

8. Write a logarithmic statement or each

5

exponential statement below. a. 10 −2 = 1 100 b. 101 = 10 c. 104 = n

e d 4 tiu n g 3 a M r 2 e t h ic R 1

x

2 00 0 4 0 0 0 60 0 0 80 0 0 1 0 , 00 0

Ground Motion

1. What is the y-intercept o the graph? 2. What is the x-intercept o the graph? 3. Is M(x) an increasing or decreasing unction? 4. Which o these statements are NOT true regarding the graph above? A. The graph contains the point (1, 0). B. The graph contains the point (10, 1). C. The domain is x > 0. D. The x-axis is an asymptote.

Lesson 22-2 5. Use a calculator to ind a decimal approximation rounded to three decimal places. a. log 47 b. log 32.013 c. log 5 7 d. log −20

9. Evaluate 5without using a calculator. a. log 10 b. log 100 1 c. log 100, 000

10. I log a = x, and 10 < a < 100, what values are acceptable or x? A. 0 < x < 1 B. 1 < x < 2 C. 2 < x < 3 D. 10 < x < 100

ACT IV IT Y PR ACT ICE 1. There is no y-intercept. 2. (1, 0) 3. M(x) is increasing. 4. D 5. a.1.672 b. 1.505 c. −0.146 d. not a real number 6. C 7. a.10 4 = 10,000 1 b. 10−9 = 1, 000, 000, 000 c. 10 6 = a 8. a. log 1 = −2 100= 1 b. log 10 c. log n = 4 9. a.5 b. 2 c. −5 10. B 11. a.0.301 + 0.447 = 0.748 b. 0.301 − 0.447 = −0.146 c. 0.447 − 0.301 = 0.146 d. 0.301 + 0.447 + 0.447 = 1.195

Lesson 22-3 11. I log 2 = 0.301 and log 3 = 0.447, ind each o the ollowing using only these values and the properties o logarithms. Show your work. a. log 6 b. log 2 3 c. log 1.5 d. log 18

()

()


6. A logarithm is a(n) A. variable. B. constant. C. exponent. D. coeicient.


355

ACTIVITY 22 Continued 12. B 13. Sample answer: Simpliy log 103 + log 105 = log 108 to 3 log 10 + 5 log 10 = 8 log 10. Since log 10 = 1, the log equation becomes 3 + 5 = 8, which is precisely the same operation used in the exponent product. 14. a.log 2x 3y b. log 5 7 c. log 24 ⋅ 12 = log 48 6 15. a.log 3 + log x − (log 8 + log y) b. log (m + v) − log 3 c. log 4 − log (9 − u) 16. a.0.602 b. 1.431 c. 0.151 d. 0.540 17. a.log uv = log u + log v b.l ogu = log u ( )− log () v = v c. log uv = v log (u) 18. a.log 1000 = 3 b. log 1 = 0 c. log 10 = 1 19. a.log x + 2 log y b. log x + log y − log z c. 3 log a + 2 log b 20. a.3.816 b. 2.709 c. 1.857 21. a.2 m log n b. 0 c. 7 log 2 22. D 23. log 10x − log 104 = x log 10 − 4 log 10 = (x − 4) log 10 = (x − 4)(1) =x−4 log 10π − log 104 = 3.14 −4 = −0.86

    Earthquakes and Richter Scale


12. Which expression does NOT equal 3? A. log 103 B.

log 105 log 102

 7 C. log  104   10  D. log 104 − log 10

⋅

15. Expand each expression.   a. log  3x   8 y  b. log m + v

log x A. log xy =

log y

the ollowing using the properties o logarithms. a. log 4 b. log 27 c. l og 2 d. log 12

Lesson 22-4 17. Complete each statement to illustrate a property

Power Property c.

the ollowing using the properties o logarithms. a. log 38 b. log (23)3 2 c. log 8(3 )

22. Which o the ollowing statements is TRUE?

16. I log 2 = 0.301 and log 3 = 0.477, ind each o

b.Quotient Property

xy

b. log z c. log a3b2 20. I log 8 = 0.903 and log 3 = 0.477, ind each o

21. Write each expression without using exponents. a. m log n + log nm b. log (mn)0 c. log 24 + log 23

( 3 ) ( 9 −4 u )

or logarithms. a.Product Property

Then evaluate without using a calculator. a. log 500 + log 2 b. 2 log 3 + log 1 9 c. log 80 − 3 log 2

19. Expand each expression. a. log xy2

13. Explain the connection between the exponential equation (103 105 = 108) and the logarithmic equation (log 103 + log 105 = log 108). 14. Rewrite each expression as a single logarithm. a. log 2 + log x − (log 3 + log y) b. log 5 − log 7 c. (log 24 + log 12) − log 6

c. log

18. Rewrite each expression as a single logarithm.

uv = ? log u = ? v log uv = ? log

B.l ogxy = y logx C. log (x + y) = log x + log y D.l

og x = 1 logx 2

MATHEMATICAL PRACTICES Reason Abstractly and

Quantitatively 23. Veriy using the properties o logarithms that log 10x − log 104 = x − 4. Then evaluate or x = π, using 3.14 or π.


356





Embedded Assessment 2 Use after Activity 22

1. Reason quantitatively. Tell whether or not each table contains data that can be modeled by an exponential unction. Provide an equation to show the relationship between x and y or the sets o data that are exponential.

a.

b.

c.

x

0

21

y

3

6

x

0123

y

2468

x y

3 12

01 2 3 108 36 12

24

Ans wer Key 1. a. exponential; y = 3(2)x b. not exponential x c. exponential; y = 108 1 3 2. a. decreasing; domain: ( −∞, ∞); range: (0, ∞); y-intercept: 4 b. decreasing; domain: ( −∞, ∞); range: (−∞, 0); y-intercept: −3 3. a. To obtain the graph og, vertically stretch the graph o f by a actor o 2, horizontally translate to the lef 3 units, and then vertically translate down 5 units. y b.

()

4

2. Tell whether or not each unction is increasing. State increasing or decreasing, and give the domain, range, and y-intercept o the unction. x a. y = 4 2 b. y = −3(4)x 3 x+3 3. Let g(x) = 2(4) − 5. a. Describe the unction as a transormation o f(x) = 4x. b. Graph the unction using your knowledge o transormations. c. What is the horizontal asymptote o the graph o g?

()

4. Rewrite each exponential equation as a common logarithmic equation. a. 103 = 1000

−4 b. 10 =

Ass ess ment Foc us • Examining exponential patterns and unctions • Identiying and analyzing exponential graphs • Transorming exponential unctions • Graphing and transorming natural base exponential unctions • Examining common logarithmic unctions • Understanding properties o logarithms

WHETHER OR NOT

1 10, 000

c. 107 = 10,000,000

5. Make use of structure. Rewrite each common logarithmic equation

g

as an exponential equation.

a. log 100 = 2

b. log 100,000 = 5

c. log

1 = −5 100, 000

6. Evaluate each expression without using a calculator. a. log 1000 b. log 1 c. log 2 + log 50

–9 –8 –7 –6 –5 –4 –3 –2 –1 –1

7. Evaluate using a calculator. Then rewrite each expression as a single

–3

f

1 2

x

–2

logarithm without exponents and evaluate again as a check. a. log 5 + log 3 b. log 3 4 c. log 3 − log 9


5 4 3 2 1

–4 –5

c. y = −5 4. a. log 1000 = 3 b. log 1 = −4 10,000 c. log 10,000,000 = 7

Common Core State Standards for Embedded Assessment2 HSF-IF.B.5

HSF-IF.B.6

5. a. 102 =100 b. 105 = 10,000 c. 10−5 = 1 10,000 6. a. 3 b. 0 c. 2 7. a. log 15; 1.76091259 b. 4 log 3; 1.908485019 c. log 1 ; −0.4771212547 3

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Calculate and interpret the average rate of change of a function (presented symbolically or

as a table) over a specified interval. Estimate the rate of change from a graph. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.  HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. HSF-IF.C.7


357



Embedded Assessment 2 T EACHER

to TEACHER


Unpacking Embedded Ass essm ent 3


WHETHER OR NOT Scoring Guide

Exe m p l a r y

Pr o f i ci e n t

Mathematics Knowledge and Thinking



help students understand the concepts they will need to know to be successul on Embedded Assessment 3.

•

Problem Solving

I n c o m pl e t e

Mathematical Modeling / Representations (Items 1, 3b)

• Largely correct

understanding of the features of exponential functions and their graphs including domain and range• Little difficulty when evaluating and rewriting exponential and logarithmic equations and expressions

Little or no understanding how to determine whether of how to determine a table of data represents an whether a table of data exponential function represents an exponential Partial understanding of the function features of exponential • Inaccurate or incomplete functions and their graphs understanding of the including domain and range features of exponential functions and their graphs Some difficulty when including domain and range evaluating and rewriting logarithmic and exponential• Significant difficulty when equations and expressions evaluating and rewriting logarithmic and exponential equations and expressions

• An appropriate and efficient• A strategy that may include• A strategy that results in • • Fluency in recognizing exponential data and modeling it with an equation

unnecessary steps but results in a correct answer recognizing exponential data and modeling it with an equation

how to graph an understanding of how to exponential function using graph an exponential transformations function using transformations

• Clear and accurate justification of whether or not data represented an exponential model

some incorrect answers

• Little difficulty in accurately• Some difficulty with

• Effective understanding of • Largely correct

Reasoning and Communication

• Partial understanding of •

understanding of how to determine whether a table of data represents an exponential function •

understanding of the features of exponential functions and their graphs including domain and range Fluency in evaluating and • rewriting exponential and logarithmic equations and expressions strategy that results in a correct answer

(Item 1)

• Largely correct

understanding of how to determine whether a table of data represents an exponential function

(Items 1, 2, 3c, 4–7)

Once students have completed this Embedded Assessment, turn to Embedded Assessment 3 and unpack it with them. Use a graphic organizer to

(Items 1a, 3a)

Em e r g i n g


•

recognizing exponential data and modeling it with an equation

No clear strategy when solving problems Significant difficulty with recognizing exponential data and model it with an equation

• Partial understanding of •

Mostly inaccurate or how to graph an incomplete understanding exponential function using of how to graph an transformations exponential function using transformations

• Adequate justification of • Misleading or confusing • whether or not data justification of whether or represented an exponential not data represented an model exponential model

Incomplete or inadequate justification of whether or not data represented an exponential model

• Precise use of appropriate • Adequate and correct math terms and language to describe a function as a transformation of another function

• Misleading or confusing • Incomplete or mostly description of a function as description of a function as inaccurate description of a a transformation of another function a transformation of another of function as function a transformation function another

Common Core State Standards for Embedded Assessment2 (cont.) HSF-BF.A.1 Write a function that describes a relationship between two quantities.  HSF-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

358



ACTIVITY

     

ACTIVITY 23

Undoing It All Lesson 23-1 Logarithms i n Other Bases

23

Investigative Act iv it y St and ard s F ocu s In Activity 23, students extend the concept o logarithms to bases other than 10. They also extend their knowledge o inverse unctions to include the inverse relationship between y = bx and y = logb x. Students will discover and apply properties o logarithms and apply the concept o graphing by transormations to logarithmic unctions.

My Notes

Learning Targets:

• Use composition to veriy two unctions as inverse. • Deine the logarithm o y with base b. • Write the Inverse Properties or logarithms. SUGGESTED LEARNING STRATEGIES: Close Reading, Create Representations In the irst unit, you studied inverses o linear unctions. Recall that two unctions f and g are inverses o each other i and only i f(g(x)) = x or all x in the domain o g, and g(f(x)) = x or all x in the domain o f.

1. Find the inverse unction g(x) o the unction f(x) = 2x + 1. Show your work. y = 2x + 1

MATH TIP To find the inverse of a function algebraically, interchange the x and

Lesson 23-1

y variables and then solve for y.

PLAN

x = 2y + 1 y = x−1 2 g( x) = x − 1 2

Pacing: 1 class period Chunking the Lesson #1–3 #4 #5–6 #7–8 Check Your Understanding Lesson Practice

2. Use the deinition o inverse unctions to prove that f(x) = 2x + 1 and the g(x) unction you ound in Item 1 are inverse unctions.

(

)

f ( g( x)) = 2 x − 1 + 1= −x+ = 1 2

1

x

(2 x + 1) − 1 2 x = =x g((f x)) = 2 2


3. Graph f(x) = 2x + 1 and its inverse g(x) on the grid below. What is the

Present the ollowing graph o f(x), and ask students to sketch the graphs o f(x) and g(x), the relection o f(x), over the line y = x. Have students use the graph to ind g(−1), g(2), and g(5).

line o symmetry between the graphs? The line of symmetry yis= x. 10 8

y

f

6

10

4

f (x )

g (x )

g

2

8 6 4

–10

–8

–6

–4

–2

246

8

2

10

–2 –10 – 8 –4


–6

–4

–2

–2

24

68

10

x

–4

–6

–6 –8

–8

–10

–10

[g(−1) = 4, g(2) = 2, g(5) = 0]

In a previous activity, you investigated exponential unctions with a base o 10 and their inverse unctions, the common logarithmic unctions. Recall in the Richter scale situation that G(x) = 10x, where x is the magnitude o an earthquake. The inverse unction is M(x) = log x, where x is the ground motion compared to a magnitude 0 earthquake.

Common Core State Standards for Activity 23 HSF-BF.A.1




1–3 Activating Prior Knowledge, Create Representations These items review prior work with inding the inverse o a linear unction and then graphing a unction and its inverse. This then becomes the oundation or working with inverses o logarithmic unctions.

HSF-BF.A.1c (+) Compose functions. HSF-BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment cases and illustrate explanation of thefrom effects ongraphs the graph technology. with Include recognizing evenanand odd functions their and using algebraic expressions for them.

HSF-BF.B.4

Find inverse functions.

HSF-BF.B.4b (+) Verify by composition that one function is the inverse of another.

Activity 23 • Inverse Functions: Exponential and Logarithmic Functions

359

ACTIVITY 23 Continued 4 Activating Prior Knowledge, Create RepresentationsThis item reviews the concept o inverses that students investigated in Activity 22. Students see the equations and graphs o the common logarithmic unction and its inverse, the general exponential unction. This item provides the background as students begin to study logarithms with bases other than 10.

  Logarithms in Other Bases


My Notes

4. A part o each o the graphs o y = G(x) and y = M(x) is shown below. What is the line o symmetry between the graphs? How does that line compare with the line o symmetr y in Item 3? The line of symmetry is y = x. It is the same as the line of symmetry in Item 3. 10

G(x )

8


6

Here students are ormally introduced to the logarithm o y base b and to logarithms with base e, or natural logarithms. Have students add the deinitions and notations to their math noteooks. Point out that the base in a logarithmic unction is the same as the base in the inverse exponential unction. 5–6 Create Representations Students realize that exponential unctions and logarithmic unctions with the same base are inverses. Be aware that students may correctly answer Item 5 without ully understanding what it means or y = bx and y = logb x to be inverse unctions. Item 6 will help students with that understanding.

4 2 –2

M(x )

2468

10

–2

Logarithms with bases other than 10 have the s ame properties as common logarithms. The logarithm o y with base b, where y > 0, b > 0, b ≠ 1, is deined as: logb y = x i and only i y = bx. The exponential unction y = bx and the logarithmic unction y = logb x, where b > 0 and b ≠ 1, are inverse unctions.

5. Let g(x) = f −1(x), the inverse o unction f. Write the rule or g or each

MATH TIP

unction f given below. −1

The notation f is used to indicate the inverse of the function f.

a. f(x) = 5x

b. f(x) = log4 x

c. f(x) = loge x

g(x) = 4x

g( x) = ex

g(x) = log5 x

Logarithms with base e are called natural logarithms , and “loge” is written ln. So, loge x is written ln x.

6. Use the unctions rom Item 5. Complete the expression or each composition.

a. f(x) = 5x log x

5 5 log5 5x

=x =x

f(g(x)) =

log4 4x

=x

g(f(x)) =

4log4x

=x

f(g(x)) =

eln x

=x

g(f(x)) =

ln ex

=x

f(g(x)) = g(f(x)) =

b. f(x) = log4 x

c. f(x) = ex

Common Core State Standards for Activity 23 (continued)

360

HSF-IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 

HSF-IF.C.7e

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.




  Logarithms in Other Bases


7. Use what you learned in Item 6 to complete these Inverse Properties of Logarithms. Assume b > 0 and b ≠ 1. a. blogb x =

x

b. logb bx =

x

8. Simpliy each expression. a. 6log6 x x c.

7log7 x

x

e. ln ex x

My Notes

7–8 Create Representations, Note Taking, DebriefingItem 7 generalizes the results o Item 6. Tell students that these statements are considered identities or logarithms because the result is the same as the input. Have students add these to their math journals.

In Item 8, students apply the identities rom Item 7. In part d, some students may need to be reminded t hat or common logarithms, it is not necessary to write the base o 10 as a subscript; it is assumed to be base 10.

b. log3 3x x x

d. log 10 x f. eln x x


Check Your Understandin g 9. Describe the process you use to ind the inverse unction g(x) i f(x) = 7x + 8. 10. Construct vi able arguments. Look at the graphs in Items 3 and 4. What can you conclude about the line o symmetry or a unction and its inverse?

11. Answer each o the ollowing as true or alse. I alse, explain your reasoning. 1 a. The “−1” in unction notation f−1 means f . b. Exponential unctions are the inverse o logarithmic unctions. c. I the inverse is a unction, then the srcinal must be a unction.

   Let g(x) = f −1(x), the inverse o unction f. Write the rule or g or each

By now, students are amiliar with many mathematical concepts related to inverses, including inverse unctions and inverse operations. The Inverse Properties o Logarithms are similar— they represent an “undoing” process.

Check Your Understanding Debrie students’ answers to these items to ensure that they understand the relationship between a unction and its inverse. Students should be able to describe the relationship between the graph o a unction and its inverse; they should also understand the composition o a unction and its inverse.

unction f given below.


12. f(x) = 3x − 8

13. f (x ) = 12 x + 5

14. f(x) = 5x − 6

15. f(x) = −x + 7

16. f(x) = 7x

17. f(x) = ex

18. f(x) = log12 x

19. f(x) = ln x

Simpliy each expression.

20. log9 9x

21. 15log15 x

22. ln ex

23. 8log8 x


Ans wers 9. Interchange x and y in the srcinal unction, and then solve or y. x = 7y + 8, x − 8 = 7y, y = x − 8 or 7 f −1 ( x ) = x − 8 7 10. The line o symmetry or a unction and its inverse is x = y. 11 a. False; it means the inverse o unction f. b. true c. False; the srcinal is not necessarily a unction.

12. g ( x ) = x + 8 3 14. g ( x ) = x + 6 5 16. g(x) = log7 x 18. g(x) = 12x 20. x 22. x

13. g(x) = 2x − 10 15. g(x) = −x + 7 17. g(x) = ln x 19. g(x) = ex 21. x 23. x

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand the inverse relationship between logarithmic and exponential unctions. Students should be able to write the inverse o a linear, exponential, or logarithmic unction. As needed, provide students with extra practice in the orm o a table or puzzle requiring students to match unctions and their inverses.


361


  Properties of Logarithms and the Change of Base Formula


Lesson 23-2 PLAN

My Notes

Pacing: 1 class period #1–3 Check Your Understanding #6 #7–8 Check Your Understanding #11 #12 #13 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Create Representations, Close Reading When rewriting expressions in exponential and logarithmic orm, it is helpul to remember that a logarithm is an exponent. The exponential statement 23 = 8 is equivalent to the logarithmic statement log2 8 = 3. Notice that the logarithmic expression is equal to 3, w hich is the exponent in the exponential expression.

1. Express each exponential statement as a logarithmic statement. a. 34 = 81 b. 6−2 = 1 c. e0 = 1


5

[m ] [m50]

1

[m5]

Universal Access Some students may need review work with exponents beore working on Items 1–3. While they are likely to remember how to work with p ositive integer exponents, they may need to be reminded that when b ≠ 0, b0 = 1 and b−n = 1n . b Ask students to evaluate each expression. a. 43 [64] −3

( )

c. 14

42 = 16

[m5]

4. (m10 ) 2

[64]

b. 4−3

1  64   

d. 40

[1]

ln 1 = 0

( )

log6 1 = −2 36

2. Express each logarithmic statement as an exponential statement. a. log4 16 = 2 b. log5 125 = 3 c. ln 1 = 0

15

1. m (m ) 2. (m10)5 10 3. m 5 m

36

log3 81 = 4

Ask students to simpliy each expression. 10

Learning Targets:

• Apply the properties o logarithms in any base. • Compare and expand logarithmic expressions. • Use the Change o Base Formula.

Chunking the Lesson

MATH TIP Remember that a logarithm is an exponent. To evaluate the expression log6 36, find the exponent for 6 that gives the value 36. 62 = 36. Therefore, log6 36 = 2.

e0 = 1

53 = 125

3. Evaluate each expression without using a calculator.

( 64 )

a. log2 32 5

b. log 4 1

c. log3 27 3

d. log12 1 0

Check Your Understandin g 4. Why is the value o log−2 16 undeined? 5. Critique the rea soning of others Mike . said that the log 3 o 19 is

undeined, because 3−2 = 1 , and a log cannot have a negative value. Is 9 Mike right? Why or why not?

The Product, Quotient, and Power Properties o common logarithms also extend to bases other than base 10.

6. Use the given property to rewrite each expression as a single logarithm. Then evaluate each logarithm in the equation to see that both sides o the equation are equal. log2 32

1–3 Create Representations, Guess and Check, DebriefingThe ocus o Items 1 and 2 is or students to write equivalent orms o exponential and logarithmic statements.

a. Product Property:log2 4 + log2 8 =

For Item 3, suggest that they set each expression equal to x and then rewrite the equations in exponential orm. They can then see that they are looking or a missing exponent. For example: log2 32 = x → 2x = 32, and so x = 5. They can guess and check to ind the correct exponent.

c. Power Property: 2 log 5 25 = log5 625

Check Your Understanding Debrie students’ answers to these items to ensure that they understand that a

2

3

+

=

3

2

⋅

−

1

2

Ans wer s 4. The value o b must be greater than 0, so log−2 16 is undeined. 5. Mike is mistaken; a logarithm can have a negative value, but it cannot have a negative base.

6 DebriefingStudents validate the Product, Quotient, and Power Properties o l ogarithms numerically.


5

log3 9

b. Quotient Property:log3 27 − log3 3 =

logarithm is an exponent. Discuss why a logarithm may not have a negative base.

362

−3

2 = ______

=

4



(y )

b. log4 x2y 2 log4 x + log4 y c. ln  x 3  2 ln x − 3 ln y y 2



8. Assume that x is any real number, and decide whether the statement is always true, sometimes true, or never true. I the statement is sometimes true, give the conditions or which it is true. log 7 a. log 7 − log 5 = never true log 5

b. log5 5x = x always true 2

c. 2log2 x = x2 sometimes true, when x≠ 0 d. log4 3 + log4 5 − log4 x = log4 15 sometimes true, when x= 1 e. 2 ln x = ln x + ln x sometimes true, when x> 0

Check Your Understandin g 9. Attend to precision. Why is it important to speciy the value o the variables as positive when using the Product, Quotient, and Power Properties o logarithms? Use Item 7 to state an example.

10. Simpliy the ollowing expression: log 7 − log 5

11. Use the common logarithm unction on a calculator to ind the numerical value o each expression. Write the value in the irst column o the table. Then write the numerical value using logarithms in base 2 in the second column.

Nu m er ica l Va l u e

l og2 a

log 2 log 2

1

log2 2

log 4 log 2

2

lo g2 4

log 8 log 2

3

lo g2 8

log 16 log 2

4

lo g2 16

log N log 2

ACTIVITY 23 Continued 7–8 Quickwrite, DebriefingIn Item 7, students apply the Product, Quotient, and Power Properties to expand logarithmic expressions.

In Item 8a, point out that log7 log 7 − log 5 = log 7 ≠ . Students 5 log5 may mistakenly believe that this statement is always true, rather than never true. Item 8b is an identity that holds true or all x. Item 8c is sometimes true. It is true or all real numbers except 0. When x = 0, the let side o the equation is undeined and the right side is equal to 0. Item 8d is sometimes true. It is only true when x = 1, and t hereore log4 x = 0. Part e is sometimes true. It is only true when x is greater than 0.

Check Your Understanding Debrie students’ answers to these items to ensure that they understand the Product, Quotient, and Power Properties or logarithms with bases other than 10. Ask students to write these properties using natural logarithms.

Ans wers 9. Suppose y < 0, in log4 x2y. Then x2y

Sometimes it is useul to change the base o a logarithmic expression. For example, the log key on a calculator is or common, or base 10, logs. Changing the base o a logarithm to 10 makesit easier to work with logarithms on a calculator.


continued

My Notes

7. Expand each expression. Assume all variables are positive. a. log 7 x3 log7 x − 3 log7 y



ACTIVITY 23

would be negative, and the logarithm o a negative number is undeined because b > 0. 10. log 7 5 11 Create Representations, Note Taking, Look for a Pattern Students can use a calculator to evaluate common logarithmic expressions o the orm log( N ) and then look or a pattern to log (2) ind an equivalent expression o the orm log2 N.

log2 N


363



ACTIVITY 23 12 Create Representations, Note Taking, Look for a Pattern Students generalize the pattern rom base 2 to base b. Students should add to their math journals the Change o Base log x Formula: logb x = , where b, x > 0, log b b ≠ 1.

continued

My Notes

12. The patterns observed in the table in Item 11 illustrate the Change of Base Formula . Make a c onjecture about the Change o Base Formula o logarithms. log x logb x = log b 13. Consider the expression log2 12. a. The value o log 2 12 lies between which two integers? 3 and 4

13 DebriefingStudents use the Change o Base Formula to approximate log2 12 on a calculator, irst mentally approximating the value and then checking with a calculator to see t hat their answer is reasonable.

b. Write an equivalent common logarithm expression or log2 12, using the Change o Base Formula. lo g (12 ) lo g( 2 )

c. Use a calculator to ind the value o log2 12 to three decimal places. Compare the value to your answer rom part a.


3.585; The value lies between 3 and 4.

Extendstudents’ knowledge o the Change o Base Formula. Item 12 shows the Change o Base Formula written in terms o the common logarithm because students can evaluate common logarithms on a calculator. However, this property o logarithms can be generalized to any logb a base b: logc a = , where a, b, logb c c > 0 and b, c ≠ 0. Challenge students to use the Change o Base Formula with a base other than 10 to evaluate the ollowing logarithms without a calculator. a. log16 8 [ 3 ; change to base 2] 4 b. log9 27 [ 3 ; change to base 3] 2

Check Your Understanding 14. Change each expression to a logarithmic expression in base 10. Use a calculator to ind the value to three decimal places. a. log5 32 b. log3 104 15. In Item 13, how do you ind out which values the value o log 2 12 lies between?

   Write a logarithmic statement or each exponential statement.

16. 73 = 343 17. 3−2 = 19 18. em = u Write an exponential statement or each logarithmic statement.

19. log6 1296 = 4


20. log 12 4 = −2 21. ln x = t

Debrie students’ answers to these items to ensure that they understand the Change o Base Formula. Ask students to rewrite the Change o Base Formula using natural logarithms.

Evaluate each expression without using a calculator.

22. log4 64 23. log 2 1 32

( )


ADA PT Check students’ answers to the Lesson Practice to ensure that they understand how to rewrite exponential and logarithmic expressions in an equivalent orm. As needed, have students practice creating equivalent expressions by writing each o the elements o an expression on a dierent index card or piece o paper. Then have students reposition the cards to orm equivalent expressions.

364

Change each expression to a logarithmic expression in base 10. Use a calculator to ind the value to three decimal places.

24. log3 7 25. log2 18 26. log25 4

Ans wer s


log 32 ≈ 2.153 14. a. log 5 log 104 ≈ 4.228 b. log 3 15. Sample answer: 12 lies between two powers o 2, 8 and 16. log 2 8 = 3;

16. log7 343 = 3 17. log3 1 = −2 9 18. ln u = m 19. 64 =1296 −2 20. 1 =4 2 21. et = x 22. 3 23. −5

log2 16 = 4. Thereore, log 2 12 lies between 3 and 4.

()

()

log7 ≈ 1.771 log3 log18 25. ≈ 4.170 log2 log4 26. ≈ 0.43 log25

24.




  Graphs of Logarithmic Functions


Lesson 23-3 PLAN

My Notes

Learning Targets:

• Find intercepts and asymptotes o logarithmic unctions. • Determine the domain and range o a logarithmic unction. • Write and graph transormations o logarithmic unctions.

Pacing: 1 class period Chunking the Lesson

SUGGESTED LEARNING STRATEGIES: Create Representations, Look or a Pattern, Close Reading, Quickwrite

1. Examine the unction f(x) = 2x and its inverse, g(x) = log2 x. a. Complete the table o data or f(x) = 2x. Then use that data to complete a table o values or g(x) = log2 x.

#1 #2 Check Your Understanding #5 #6 #7 Check Your Understanding Lesson Practice


x

f(x) = 2x

x

g(x) = log2 x

−2

1 4

1 4

−2

−1

1 2

1 2

−1

0

1

1

0

1

2

2

1

2

4

4

2

b. Graph both f(x) = 2x and g(x) = log2 x on the same grid. c. What are the x- and y-intercepts or f(x) = 2x and g(x) = log2 (x)? For f( x) = 2x, the y-intercept is 1, and there is xno -intercept. For g(x) = log2 (x), thex-intercept is 1, and there is yno -intercept.

d. What is the line o symmetry between the graphs o f(x) = 2x and g(x) = log2 x? y = x e. State the domain and range o each unction using interval notation. Domain f(x) = 2x g(x) = log2 x

Range (−∞, ∞) (0, ∞)

(0,∞) (−∞, ∞) x

f. What is the end behavior o the graph o f(x) = 2 ?

As x approaches−∞, y approaches 0. As x approaches∞, y approaches∞.


g. What is the end behavior o the graph o g(x) = log2 (x)? As x approaches 0,y approaches−∞. As x approaches∞, y approaches∞. y 5 4 3 2 1 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5

f (x ) = 2x g (x ) = log2x 1234567

8 9

x

Ask students to ind any x- and y-intercepts o each o the ollowing unctions without graphing.

1. f(x) = x2 − 4 [x-intercepts: 2 and −2; y-intercept: −4] 2. f(x) = 3x − 12 [x-intercepts: 2 and −2; y-intercept: −4] 3. f(x) = (x − 1)2 + 2 [no x-intercepts; y-intercept: 2] 4. f(x) = 10x [no x-intercept; y-intercept: 1]

TEACHER to TEACHER Lesson 23-3 investigates the graph o g(x) = log2 x by relating it to the graph o the inverse unction,f(x) = 2x. Then transormations are used to graph unctions o the orm g(x) = a logb (x − c) + d in Item 5. 1 Create Representations, Look for a

Pattern, DebriefingAter completing a table o values or f(x) = 2x, students should realize they can interchange the x and y columns o the table to get some values or the inverse, g(x) = log2 x. Remind students o this relationship i necessary. The Math Tip on page 367 will reinorce this concept. Students then observe the symmetry o the graphs o f and g over the line y = x. They recognize that the domains and ranges o the two unctions interchange, and that while one unction has a horizontal asymptote at y = 0, the other has a vertical asymptote at x = 0.


365

ACTIVITY 23 Continued 2 Look for a Pattern, Think-PairShare, Debriefing Students perorm an investigation similar to the work they did in Item 1, with basee instead o base 2. Have students discuss the similarities and dierences in the answers to Items 1 and 2.



My Notes

f(x) = 2x

Technology Tip I students do not have a natural logarithm key on their calculators, log x remind them that ln x = . log e Students can use the common logarithm key and a decimal approximation oe as an alternate method o evaluating natural logarithms.

y=0 x= 0

g(x) = log2 x

ELL Support To support students’ language acquisition, monitor their listening skills and understanding as they compare Items 1 and 2. Careully pair students to ensure that all students participate in and learn rom the discussion.

h. Write the equation o any asymptotes o each unction.

TECHNOLOGY TIP The LN key on your calculator is the natural logarithm key.

2. Examine the unction f(x) = ex and its inverse, g(x) = ln x. a. Complete the table o data or f(x) = ex. Then use those data to complete a table o values or g(x) = ln x. x

f(x) = ex

x

g(x) = ln x

−2

0.135

0.135

−2

−1

0.368

0.368

−1

0

1

1

0

1

2.718

2.718

1

2

7.390

7.390

2

b. Graph both f(x) = ex and g(x) = ln x on the same grid. y 5 4 3 2 1 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1

f (x ) = ex g (x ) = ln x

123456789

x

–2

For additional technology resources, visit SpringBoard Digital.

–3 –4 –5

c. What are the x- and y-intercepts or f(x) = ex and g(x) = ln x? For f(x) = ex, the y-intercept is 1, and there is xno -intercept. For g( x) = ln x, the x-intercept is 1, and there is yno -intercept.

d. What is the line o symmetry between the graphs o f(x) = ex and g(x) = ln x? y = x e. State the domain and range o each unction using interval notation. Domain f( x) = ex g( x) = In x

Range (−∞, ∞) (0, ∞)

(0,∞) (−∞, ∞)

f. What is the end behavior o the graph o f(x) = ex? As x approaches−∞, y approaches 0. As x approaches∞, y approaches∞.

g. What is the end behavior o the graph o g(x) = ln x? As x approaches 0, y approaches−∞. As x approaches∞, y approaches∞.

h. Write the equation o any asymptotes o each unction. y=0 f(x) = ex x= 0 g(x) = ln x

366





My Notes

Check Your Understanding 3. Make sense o f problems. From the graphs you drew or Items 1 and 2, draw conclusions about the behavior o inverse unctions with respect to: a. the intercepts b. the end behavior c. the asymptotes intercept or the inverse unction?

Transormations o the graph o the unction f(x) = logb x can be used to graph unctions o the orm g(x) = a logb (x − c) + d, where b > 0, b ≠ 1. You can draw a quick sketch o each parent graph, f(x) = logb x, by plotting the points 1 , −1 , (1, 0), and (b, 1). b

)

5. Sketch the parent graph f(x) = log2 x on the axes below. Then, or each transormation o f, provide a verbal descr iption and sketch the graph, including asymptotes. a. g(x) = 3 log2 x vertically stretch the graph fof by a factor of 3; asymptote x= 0

b. h(x) = 3 log2 (x + 4) horizontally translate the graphgof left 4 units; asymptote x = −4

c. j(x) = 3 log2 (x + 4) − 2 vertically translate the graphhofdown 2 units; asymptote x = −4

d. k(x) = log2 (8x) k(x) = log2 8 + log2 x = 3 + log2 x, so the graph

h(x ) = 3 log2 (x

+

4)

6 5 4 3 2

j (x ) = 3 log2 (x

MATH TIP Recall that a graph of the exponential function f(x) = bx can be drawn by plotting the points −1, 1 , (0, 1), and (1, b). b

k(x ) = 3 + log2 x g (x ) = 3 log2 x

(

f (x ) = log2 x

1 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1

+ 4) – 2

123456789

x

–2


x

–3

= –4

–4

5 Activating Prior Knowledge, Create Representations, Quickwrite, DebriefingThe Math Tip shows students why they can get a quick sketch o the graph o g(x) = log2 x by plotting the points 1 , −1 , (1, 0), and (2, 1). 2 Students are expected to use their prior knowledge o graphing other unctions by transormations to sketch transormations o the logarithmic unction.

(

of log2 x shifts 3 units up; asymptote x= 0 y

Check Your Understanding Debrie students’ answers to these items to ensure that they understand the behavior o the graphs o inverse exponential and logarithmic unctions. Ask students to explain the relationship between the domains and ranges o inverse unctions and how this relationship supports the answers to Item 3.

Ans wers 3. a. I the unction hasa y-intercept o 1, then the inverse unction has an x-intercept o 1. b. I the unction approaches 0 onthe y-axis asx approaches in nity, then in the inverse unction, y approaches in nity asx approaches 0. c. I the unction has an asymptote at x = 0, then the inverse has an asymptote aty = 0. 4. (0, 0)

4. I a unction has an intercept o (0, 0), what point, i any, will be an

(


)

Switching the x- and y-coordinates of these points gives you three points on the graph of the inverse of f(x) = bx, which is f(x) = logb x.

)

6 Quickwrite Students can use the

Change o Base Formula in order to graph the unction on a calculator.

–5

x

–6 =0

6. Explain how the unction j (x) = 3 log2 (x + 4) − 2 can be entered on a graphing calculator using the common logarithm key. Then graph the unction on a calculator and compare the graph to your answer in j(x) = Item 5c. Use the Change of Base Formula and rewrite 3 log2 (x + 4) − 2 as j( x) =

3 l o g (x + 4 ) − 2. log( 2 )


367

ACTIVITY 23 Continued 7 Group Presentation, Debriefing, ParaphrasingMonitor group presentations on the eects o parameters to ensure that students are communicating clearly and that they are using terms such as stretch, shrink, reflection, and translation correctly. Ater group presentations and a debrieing, have students summarize the inormation about transormations in their math journals.



My Notes

7. Explain how the parameters a, c, and d transorm the parent graph f(x) = logb x to produce a graph o the unction g(x) = a logb (x − c) + d. Transformations of the parent graph f(x) = logb x that produce the graph ofg(x) = a logb (x − c) + d are described below. |a| > 1 ⇒ a vertical stretch by a factora of 0 < |a| < 1 ⇒ a vertical shrink by a factoraof a < 0 ⇒ a re ection over the x-axis c > 0 ⇒ a horizontal translation right c units


c < 0 ⇒ a horizontal translation left c units

Debrie students’ answers to this item to ensure that they connect

d < 0 ⇒ a vertical shift down d units d > 0 ⇒ a vertical shift up d units

transormations o logarithmic unctions with their prior knowledge o transormations o quadratic unctions. I students have diiculty generalizing the eects o a and c, illustrate with speciic examples o quadratic and logarithmic unctions.

Check Your Understandin g 8. Look for and make use of structure. a. Compare the eect o a in a logarithmic unction a logb x to a in a quadratic unction ax2 (assume a is positive).

b. Compare the eect o c in a logarithmic unction logb (x − c) to c in a quadratic unction (x − c)2.

Ans wer s 8. a. Both cause a vertical stretch or shrink by a actor o a. b. Both cause a horizontal translation c units to the right.

    9. Given an exponential unction that has a y-intercept o 1 and no

ASS ESS

x-intercept, what is true about the intercepts o the unction’s inverse?

10. Make sense of problems. The inverse o a unction has a domain o (−∞, ∞) and a range o (0, ∞). What is true about the srcinal


unction’s domain and range? each unction, using a parent graph Model with mathematics Graph . and the appropriate transormations. Describe the transormations.

See the Activity Practice or additional problems or this lesson. You may assign

11. f(x) = 2 log2 (x) − 6 12. f(x) = log (x − 5) + 1 3 13. f ()x = 12 log 4 x

the problems here or use them as a culmination or the activity.

14. f(x) = log2 (x + 4) − 3



9. The x-intercept is 1, and there is no y-intercept. 10. The domain is (0, ∞), and the range is (−∞, ∞).

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand graphing logarithmic and exponential unctions. As needed, provide students with additional practice using transormations to graph unctions. Students may beneit rom additional practice graphing quadratic unctions or square root unctions to better understand the eect o the parameters on the graph o t he parent unction.

y

11.

y

12.

8

8

6 4

4

f (x ) = log2 x

f (x ) = log3 x

2 –8

–6

–4

–2

–2

2

4

6

8

x

f (x ) = 2log2 (x )

48 –4

–4 –6

–4

6

x

12 f (x ) = log (x

–8

5) + 1

3

–8

vertically stretch the parent graph by a actor o 2, and then vertically translate down 6 units

368

horizontally translate the parent graph right 5 units, and then vertically translate up 1 unit



      Undoing It All


3

  

Lesson 23-2

Write your answers on notebook paper. Show your work.

Express each exponential statement as a logarithmic statement.

Lesson 23-1

13. 122 = 144

Let g(x) = f −1(x), the inverse o unction f. Write the rule or g or each unction f given below.

14. 2−3 = 1 8 15. en = m 16. e3x = 2

1. f(x) = 7x − 9 x 2. f (x ) = 13

( )

3. f(x) = 2x − 8

17. 102 =100 18. e0 = 1

4. f(x) = −xx + 3 5. f(x) = 5

Express each logarithmic statement as an exponential statement.

6. f(x) = ex

19. log3 9 = 2

7. f(x) = log20 x 8. f(x) = ln x

20. log2 64 = 6 21. ln 1 = 0 22. ln x = 6

Simpliy each expression.

23. log2 64 = 6

9. log3 3x 10.

24. ln e = 1

12log12x

Expand each expression. Assume all variables are positive.

x

11. ln e 12. 7log7x

25. log2 x2y5  8 26. log 4  x  5 27. ln ex 28. ln ( 1x ) 29. Which is an equivalent orm o the expression ln 5 + 2 ln x ? A. 5 ln x2 B. ln 2x5 C. ln 5x2 D. 2 ln x5


ACT IV IT Y PR ACT ICE 1. g ( x ) = x + 9 7 2. g ( x ) = log(1 x )

3. g ( x ) = x + 8 2 4. g(x) = 3 − x 5. g(x) = log5 x 6. g(x) = ln x 7. g(x) = 20x 8. g(x) = ex 9. x 10. x 11. x 12. x 13. log12 144 = 2 14. log 2 1 = −3 8 15. ln m = n 16. ln 2 = 3x 17. log 100 = 2 18. ln 1 = 0 19. 32 = 9 20. 26 = 64 21. e0 = 1 22. e6 = x 23. 26 = 64 24. e1 = e 25. 2 log2 x + 5 log2 y 26. 8 log4 x + log4 5 27. ln e + ln x 28. ln 1 − ln x 29. C 30. log2 64 2 31. log 3 x y 32. ln 2x 33. ln x3

()

34. 35. 36. 37. 38. 39. 40. 41. 42.

1 3 2 4 log20 log4 log4 log20 log45 log5 log18 log3 D

≈ 2.161 ≈ 0.463 ≈ 2.365 ≈ 2.631

LESSON 23-3 PRACTICE (continued)

y

13.

y

14.

4 2 –2

8 f (x ) = log4 (x )

2 –2

4

4 f (x ) = log2 x 6

8

f (x ) = 12 log4 (x )

–4

vertically shrink the parent graph by a actor o 1 2

x

–8

–4

4 –4

8

f (x ) = log2 (x

x + 4)

3

–8

horizontally translate the parent graph let 4 units, and t hen vertically translate down 3 units


369


      Undoing It All

ACTIVITY 23

y

43.

continued

8 f (x ) = log2 x

4 –8

–4

4 –4

Rewrite each expression as a single, simpliied logarithmic term. Assume all variables are positive.

x

8

f (x ) = 3log2 x

30. log2 32 + log2 2

and the range is (−∞, ∞), what are the domain and range o the inverse o the unction? A. domain: (−∞, ∞), range: (−∞, ∞) B. domain: (0, ∞), range: (−∞, ∞) C. domain: (−∞, ∞), range: (−∞, ∞) D. domain: (−∞, ∞), range: (0, ∞)

31. log3 x2 − log3 y 32. ln x + ln 2

1

–8

33. 3 ln x Evaluate each expression without using a calculator.

vertically stretch parent graph by a actor o 3, and then translate vertically down 1 unit

8

–4

48 –4

Graph each unction, using a parent graph and the appropriate transormations. Describe the transormations.

36. log7 49 37. log3 81

43. f(x) = 3 log2 (x) − 1 44. f(x) = log3 (x − 4) + 2

Change each expression to a logarithmic expression in base 10. Use a calculator to ind the value to three decimal places.

f (x ) = log3 x

4

35. log7 343

34. log12 12

y

44.

x

12

f (x ) = log3 (x

MATHEMATICAL PRACTICES

39. log20 4

Model with Mathematics

47. Given the unction f(x) = 2x + 1 a. Give the domain, range, y-intercept, and any

41. log3 18

–8

45. f (x ) = 1 log 4 x 4 46. f(x) = log2 (x + 3) − 4

38. log4 20 40. log5 45

4) + 2

Lesson 23-3 42. I the domain o a logarithmic unction is (0, ∞)

asymptotes or f(x). Explain.

b. Draw a sketch o the graph o the unction on a grid. Describe the behavior o the unction as x approaches ∞ and as x approaches −∞.

horizontally translate the parent graph right 4 units, and then vertically translate up 2 units

y

45. 4 2

f (x ) = log4 x

x –2

2

–2

4

6

8

f (x ) = 1 log4 x 4

–4

vertically shrink the parent graph by a actor o 1 4

y

46. 8

4 f (x ) = log2 x –8

–4

4 –4

x

8

f (x ) = log2 (x

+

3)

4

–8

47. a.domain: (−∞, ∞); range: (1, ∞); y-intercept: 2; asymptote: y = 1 b. As x approaches ∞, f(x) approaches ∞; as x approaches −∞, f(x) approaches 1. y 10

horizontally translate the parent graph 3 let, and then vertically translate down 4 units


370

8 6 4 2 –4

–2

2

4

x



      College Costs Lesson 24-1 Exponential Equations

ACTIVITY 24

In this activity, students explore exponential and logarithmic equations and solve them using properties o exponents and logarithms. They will also use technology to approximate the solutions o exponential and logarithmic equations using tables o values and graphing. Students will also investigate and learn how to solve exponential and logarithmic inequalities.

• Write exponential equations to representsituations. • Solve exponential equations. Summarizing, Paraphrasing, SUGGESTED LEARNING STRATEGIES: Create Representations, Vocabulary Organizer, Note Taking, Group Presentation Wesley is researching college costs. He is considering two schools: a our-year private college where tuition and ees or the current year cost about $24,000, and a our-year public university where tuition and ees or the current year cost about $10,000. Wesley learned that over the last decade, tuition and ees have increased an average o 5.6% per year in our-year private colleges and an average o 7.1% per year in our-year public colleges.

Lesson 24-1

To answer Items 1–4, assume that tuition and ees continue to increase at the same average rate per year as in the last decade.

PLAN Pacing: 1 class period

1. Complete the table o values to show the estimated tuition or the next our years. Years from Present

Private College Tuition and Fees

Chunking the Lesson #1–3 #4 Example A Example B Check Your Understanding Lesson Practice

Public College Tuition and Fees

0

$24,000

$10,000

1

$ 25 , 3 44

$10, 7 10

2

$ 2 6 , 76 3 . 2 6

$11, 4 7 0. 4 1

3

$ 28 , 2 62 . 01

$12, 2 84 . 8 1

4

$ 29 , 8 44 . 6 8

$13, 15 7. 0 3

TEACH Bell-Ringer Activity Provide students with a list o numbers that can each be expressed in exponential orm with base 2. Have students rewrite these numbers in exponential orm.

2. Express regularity in repeated reasoning.Write two unctions to model the data in the table above. Let R(t) represent the private tuition and ees andU(t) represent the public tuition and ees, where t is the number o years rom the present. t R(t) = 24,00 0(1.056) t U(t) = 10,00 0(1.071)

3. Wesley plans to be a senior in college six years rom now. se U the models above to ind tuition and ees at both the private and public colleges orthe hisestimated senior year in college. R(6) = $33,280.88 U(6) = $15,091.65 . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

4. Use appropr iate tools strategic ally. Write an equation that can be solved to predict the number o years that it will take or the public college tuition and ees to reach the current private tuition and ees o $24,000. Find the solution using both the graphing and table eatures o a calculator. 10,000(1.071)t = 24,000; graphing each side of the

MATH TIP

To solve an equation graphically on a calculator, enter each side of equation gives the intersection point (12.763, 24,000) and a table shows the equation as a separate the values below. Therefore, it will take 13 years. function and find the intersection point of the two functions. t U 12

2 2 , 7 76

13

2 4, 3 93

24

Directed Act iv it y St and ard s F ocu s

My Notes

Learning Targets:

ACTIVITY

Activity 24• Logarithmic and Exponential Equations and Inequalities

371

Common Core State Standards for Activity 24 HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V= IR to highlight resistance R.

1–3 Create Representations, Debriefing Tell students to keep all decimal places in their calculations, but to record answers to two decimal places. Some students may irst ind the amount o increase each year and then add it to the previous year’s amount. Other students may recognize that this situation represents exponential growth and use the growth actor to complete the table. Have students share the dierent methods they use to complete the table. Students should realize that the multiplicative pattern indicates an exponential unction or the model.

Some students may evaluate the unctions att = 6 rom Item 2 to answer Item 3. Other students may extend the pattern in the table. 4 Create Representations, Debriefing Students will write an exponential equation and solve it graphically and numerically. To solve an equation graphically on the ca lculator, students can enter the let side o their equation as one unction and the right side as a

second anpoint appropriate window,unction, and thenchoose ind the o intersection o thetwo unctions. To solve an equation numerically, students can enter the unction U into a graphing calculator and then scroll down a table to ind when the value oU irst exceeds $24,000.

Activity 24 • Logarithmic and Exponential Equations and Inequalities

371


  Exponential Equations

ACTIVITY 24 Example A, Example B Note Taking, Group Presentation Debriefing One method o solving exponential equations is shown in these examples. The equation can be rewritten so that there are like bases on each side o the equation. To use t his method, students will need to understand that bm = bn i and only i m = n.

continued

My Notes

bm = bn i and only i m = n

Have the students work together in groups to solve the Try These items. Encourage students to check their answers by substituting the solution into the srcinal equation. Invite students to

Example A

⋅ ⋅

Solve 6 4x = 96. 6 4x = 96 Step 1: 4x = 16 Step 2: 4x = 42 Step 3:2 x =

share theirinstrategies or orm. rewriting numbers exponential

Check Your Understanding Debrie students’ answers to these items to ensure that they understand how to solve exponential equations using the property that bm = bn i and only im = n.


Divide both sides by 6. Write both sides in terms o base 4. bm = bn, then m = n.

Solve 54x = 125x − 1. 54x = 125x − 1 Step 1: 54x = (53)x−1 Step 2: 54x = 53x−3 Step 3: 4x = 3x − 3 Step 4: x = −3

Write both sides in terms o base 5 . Power o a Power Property: ( am)n = amn I bm = bn, then m = n. Solve or x.

Try These A–B

MATH TIP Check your work by substituting your solutions into the srcinal problem and verifying the equation is true.

Solve or x. Show your work. a. 3x − 1 = 80 b. 2x = 1 −5 32 4

c. 63x−4 = 36x+1 6

x

() ( )

d. 1 = 1 7 2 49

Check Your Understanding 5. When writing both sides o an equation in terms o the same base, how do you determine the base to use? 6. How could you check your solution to an exponential equation? Show how to check your answers to Try These part a.

See the Activity Practice or additional problems or this lesson. You may assign the problems here or use them as a culmination or the activity. 7. 1 2 8. 2 9. 3 10. −2 11. 2 12. 7 13. 2 14. 6 15. No; 2 and 27 cannot be written using the same base.

ADA PT

I

Example B

Ans wer s 5. Look at the bases on each side o the equation and see i you can write one as a power o the other. 6. Substitute your solution into the srcinal equation; 3 4 − 1 = 80; 81 − 1 = 80; 80 = 80.

Solving a problem like the one in Item 4 involves solving an exponential equation. An exponential equation is an equation in which the variable is in the exponent. Sometimes you can solve an exponential equation by writing both sides o the equation in terms o the same base. Then use the act that when the bases are the same, the exponents must be equal:

   Make use of structure.Solve or x by writing both sides o the equation in terms o the same base. 7. 210x = 32 8. 4x − 5 = 11 x 1 9. 24x−2 = 4x+2 10. 8 = 64 11. 4 5x = 100 12. 3 2x = 384 2x 4− x 2x 10−x 13. 13 = 19 14. 12 = 18 15. Can you apply the method used in this lesson to solve the equation 24x = 27? Explain why or why not.

⋅

()

⋅

( )

( )

( )

Common Core State Standards for Activity 24 (continued)

Check students’ answers to the Lesson Practice to ensure that they understand how to solve exponential equations when the equation can be rewritten so that there is the same base on both sides o

HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

the equation. Guide students to practice rewriting numbers in exponential orm, and reinorce the rule:bm = bn i and only i m = n. Remind students that these types o exponential equations are a special case and that this method will not solve all exponential equations.

HSF-LE.A.4

372

For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.



  Solving Equations by Using Logarithms


Pacing: 1 class period

• Solve exponential equations using logarithms. • Estimate the solution to an exponential equation. • Apply the compounded interestormula.

Chunking the Lesson Example A Example B #1 Example C Example D Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Note Taking, Group Presentation, Create Representations, Close Reading, Vocabulary Organizer

For many exponential equations, it is not p ossible to rewrite the equation in terms o the same base. In this case, use the concept o inverses to solve the equation symbolically.

Example A Estimate the solution o 3x = 32. Then solve to three decimal places. Estimate thatx is between 3 and 4, because 33 = 27 and 34 = 81. x

Step 1: Step 2: Step 3: Step 4:

3 = 32 log3 3x = log332 x = log332 let side. log 32 x= log 3 x ≈ 3.155

Take the log base 3 o both sides. Use the Inverse Property to simpliy the

logb bx = x and

blogb x = x

Use the Change o Base Formula. Use a calculator to simpliy.

between 1 and 2; between 3 and 4; between−1 and 0; between 4 and 5 log 12 log 610 lo g 0 .28 ≈ 3.98 5 x = x= ≈ 1. 38 7 x = ≈ −0. 91 8 x = ln 91 ≈ 4.512 log 5 log 6 log 4

Example B Find the solution o 4x−2 = 35.6 to three decimal places. 4x−2 = 35.6 log4 4x−2 = log435.6 Take the log base 4 o both sides. Step 1: x − 2 = log435.6 Use the Inverse Property to simpliy the let side. Step 2: x = log4 35.6 +2 Solve or x. log 35.6 Step 3: +2 x= Use the Change o Base Formula. log 4 Step 4: x ≈ 4.577 Use a calculator to simpliy.

Try These B Find each solution to three decimal places. Show your work. a. 12x+3 = 240 b. 4.2x+4 + 0.8 = 5.7 c. e2x−4 = 148 log 240 − 3 ≈ −0. 7 9 4 log 12


Recall that theInverse Properties of logarithms state that for b > 0,b ≠ 1:

Estimate each solution. Then solve to three decimal places. Show your work. a. 6x = 12 b. 5x = 610 c. 4x = 0.28 d. ex = 91

x=

TEACH

MATH TIP

Try These A


x=

lo g 4 .9 − 4 ≈ −2. 8 9 3 lo g 4 .2

Lesson 24-2 PLAN

My Notes

Learning Targets:


x = ln 1 4 8 + 4 ≈ 4. 49 9 2

Continue the Bell-Ringer Activity rom Lesson 24-1 by providing students with a new list o numbers. This time, include numbers that have dierent bases when written in exponential orm. For example, your list might include 32, 81, 100, 150, 225, 243, and 400.

Example A, Example B Note Taking, Group PresentationThese examples deal with exponential equations that cannot be written in terms o the same base. In this case, students will use logarithms to solve the exponential equations. Students are oten taught to simply rewrite an exponential equation in logarithmic orm in order to solve the equation. When doing this, the concept o an exponential unction base b and the logarithmic unction base b as inverses is oten lost. This is ansuch important concept in later mathematics, as solving some separable dierential equations in calculus. Thereore, it is a good idea to include Step 1 o the solution in each example. As you model Example A, have students approximate logarithms by deciding which two integers the value lies between. Then, when students evaluate the logarithms on their calculators, they will know i their answers are reasonable. As students work through the Try These items, circulate rom group to group, asking and answering questions in order to assist students as needed. Then have them present their work to the entire class.


373

ACTIVITY 24 Continued 1 Create Representations, Debriefing Students return to Item 4 in Lesson 24-1 to solve the exponential equation analytically. Their work is validated when they see the solution is the same as their graphical and numerical solutions.

Developing Math Language As students respond to questions or discuss possible solutions to problems, monitor their use o the vocabulary compound interestand continuously compounded interestto ensure their understanding and ability to use language correctly and precisely.

Example C Create Representations, DebriefingCompound interest problems provide students with an example o an application o exponential growth.

Differentiating Instruction Some students may need examples in order to apply the compound interest ormula correctly. Use the items below or that purpose.

a. I you deposit $500 in an account paying 3.5% annual interest compounded semiannually, how much money will be in the account ater 5 years? 2.5   ≈ $594.72   A = 500 1 + 0.035 2   b. How long will it take or an investment o $2500 to earn $500 interest in an account that pays 4% annual interest compounded quarterly?

(



My Notes

1. Rewrite the equation you wrote in Item 4 o Lesson 24-1. Then show how to solve the equation using the Inverse Proper ty. t = 24,000 10,000 (1.071)

MATH TERMS

1.071t = 2.4

Compound interest is interest that is earned or paid not only on the principal but also on previously accumulated interest. At specific periods of time, such as daily or annually, the interest earned is added to the principal and then earns additional interest during

log1.071 1.071t = log1.071 2.4 log 2 .4 t= log 1 .07 1 t ≈ 12.763

the next period.

Wesley’s grandather gave him a birthday git o $3000 to use or college. Wesley plans to deposit the money in ormula a savingsbelow account. Most banks pay o compound interest , so he can use the to ind the amount money in his savings account ater a given period o time.

MATH TIP When interest is compounded annually, it is paid once a year. Other common compounding times are shown below.

Compound Interest Formula A = amount in account P = principal invested nt r = annual interest rate as a decimal A = P 1+ r n n = number o times per year that interest is compounded t = number o years

(

)

Times per Year Semiannually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365

Example C I Wesley deposits the git rom his grandather into an account that pays 4% annual interest compounded quarterly, how much money will Wesley have in the account ater three years? Substitute into the compound interest ormula. Use a calculator to simpliy. nt

4 ( 3) A = P 1+ r = 3000 1 + 0.04 ≈ $3380.48 4 n Solution: Wesley will have $3380.48 in the account ater three years.

(

)

)

(

)

Try These C How long would it take an investment o $5000 to earn $1000 interest i it is invested in a savings account that pays 3.75% annual interest compounded monthly?

4.869 years⇒ 4 years 11 months

[ A = 2500 + 500 = 3000

(

3000 = 2500 1 + 0.04 4 1.2 = 10. 14t

( 4t )

)

log 101 1.2 = llog 101 1014t log 101 1.2 = 4t

 log1.2  t = 1  ≈ 4.581 4  log101 

( )

4 years and 9 months]

MINI-LESSON:

Compounding Periods and Finding

Time

If students need additional help calculating compound interest, a minilesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

374






Wesley’s grandather recommends that Wesley deposit his git into an account that earns interest compounded continuously, instead o at a ixe d number o times per year. Continuously Compounded Interest Formula A = amount in account P = principal invested A = Pert r = annual interest rate as a decimal t = number o years

Example D I Wesley deposits the git rom his grandather into an account that pays 4% annual interest compounded continuously, how much money will Wesley have in the account ater three years? Substitute into the continuously compounded interest ormula. Use a calculator to simpliy.

My Notes

Example D Note Taking, Group Presentation, DebriefingInvite students to discuss what it means to compound interest continuously. Students can calculate compound interest over shorter and shorter time periods to investigate continuously compounded interest. The idea that unctions have limiting values is oundational in calculus. Exponential growth, like the growth o an investment through compound interest, is a key application o exponential unctions. Ater completing the Try These items, invite students to suggest other examples o exponential growth.


A = Pert = 3000e0.04(3) ≈ $3382.49 Solution: Wesley will have $3382.49 in the account ater three years.

Try These D How long would it take an investment o $5000 to earn $1000 interest i it is invested in a savings account that pays 3.75% annual interest compounded continuously? 4.862 years

Check Your Understanding 2. How is solving exponential and logarithmic equations similar to other equations that you have solved?

3. Attend to precision. In Examples C and D, why are the answers rounded to two decimal places?

4. A bank advertises an account that pays a monthly interest rate o 0.3% compounded continuously. What value do you use or r in the continuously compounded interest ormula? Explain.

Debrie students’ answers to these items to ensure that they understand how to solve exponential equations. Students should also be able to apply the appropriate ormulas or compound interest.

Ans wers 2. Similar to other e quations, solving exponential and logarithmic equations requires perorming inverse operations on both sides o the equation. 3. The answers represent amounts o money, which are always rounded to the nearest cent. 4. 3.6% or 0.036; The monthly interest rate must be multiplied by 12 to get an annual rate to use in the ormula.

    . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

ASS ESS

Solve or x to three decimal places.

5. 8x = 100

6. 3x−4 = 85

7. 3ex+2 = 87

8. 23x−2 + 7 = 25

⋅

9. 2 43x − 3 = 27 10. e2x − 1.5 = 6.7 11. Make sense o f problems. A deposit o $4000 is made into a savings account that pays 2.48% annual interest c ompounded quarterly. a. How much money will be in the account ater three years? b. How long will it take or the account to earn $500 interest? c. How much more money will be in the account ater three years i the interest is compounded continuously?



ADA PT

5. 2.215 6. 8.044 7. 1.367 8. 2.057 9. 0.651 10. 1.052

Check students’ answers to the Lesson Practice to ensure that they understand how to solve exponential equations using logarithms. Monitor students’ work to ensure that they are applying the Change o Base Formula or

$4307.96 11. b. a.4.764 years

logarithms Encourage students to correctly. estimate each answer beore using technology.

c. $0.99


375


  Logarithmic Equations

ACTIVITY 24

Lesson 24-3

continued

PLAN

My Notes

Pacing: 1 class period Example A Example B Example C Example D Check Your Understanding Lesson Practice

TEACH Bell-Ringer Activity Provide students with several expressions that can be simpliied using the laws o logarithms. For example: 2 log x + 3 log y. Students should rewrite each expression as a single logarithm.


Learning Targets:

• Solve logarithmic equations. • Identiy extraneous solutions to logarithmic equations. • Use properties o logarithms to rewrite logarithmic expressions.

Chunking the Lesson

SUGGESTED LEARNING STRATEGIES: Create Representations, Vocabulary Organizer, Note Taking, Group Presentation

MATH TERMS An extraneous solution is a solution that arises from a simplified form of the equation that does not make the srcinal equation true.

Equations that involve logarithms o variable expressions are called logarithmic equations . You can solve some logarithmic equations symbolically by using the concept o unctions and their inverses. Since the domain o logarithmic unctions is restricted to the positive real numbers, it is necessary to check or extraneous solutions when solving logarithmic equations.

Example A Solve log4 (3x − 1) = 2. log4 (3x − 1) = 2 Step 1: 4log4 (3x−1) = 42 3x − 1 =16

Students should add the deinitions o logarithmic equationand extraneous solution to their math journals. Students can use their prior knowledge to understand the meanings o these terms.

Step 2:

Example A Group Presentation, Note Taking, DebriefingThe next set o examples cover equations with one logarithmic expression and equations with two logarithmic expressions in the same base set equal to each other. Focus on Step 1 in Example A, again stressing the concept that the exponential unction base b and the logarithmic unction base b are inverses. Some students may see that you also can get to Step 2 by writing the srcinal e quation in exponential orm.

Try These A

Write in exponential orm using 4 as the base. Use the Inverse Property to simpliy the let side. Solve or x.

17 x = 3 Check: log4 (3 17 − 1)l= og 4 16 = 2 3

Step 3:

⋅

Solve or x. Show your work. a. log3 (x − 1) = 5 b. log2 (2x − 3) = 3 2log2(2x−1) = 23

eln(3x) = e2s

x − 1 = 243

2x − 3 = 8

x = 244

x = 5.5

3x = e2 2 x= e 3

To solve other logarithmic equations, use the act that when the bases are the same, m > 0, n > 0, and b ≠ 1, the logarithmic values must be equal: logb m = logb n i and only i m = n

Ater reviewing the Try These items, emphasize the importance o careul, correct notation. Encourage students to show their work with every step, reminding them that writing the steps is oten more eicient than trying to keep track o too many steps mentally.

376

c. 4 ln (3x) = 8

3log3(x−1) = 35






My Notes

Example B Solve log3 (2x − 3) = log3 (x + 4). log3 (2x − 3) = log3 (x + 4) Step 1: 2x − 3 = x4+ I log b m = logb n, then m = n. x. Step 2: 7x = Solve or Check: log3 (2

⋅ 7 − 3) = log (7 + 4)

Have students work in groups on the Try These B items. Remind students that they must check solutions to logarithmic equations to identiy any that may be extraneous.

?

3

Try These B

a. 3x + 4 = 61 3x = 2 x= 2 3

Solve or x. Check or extraneous solutions. Show your work. a. log6 (3x + 4) = 1 b. log5 (7x − 2) = log5 (3x + 6) c. ln 10 − ln (4x − 6) = 0

b. 7x − 2 = 3x + 6 4x = 8 x= 2

log3 11 = log3 11

Sometimes it is necessary to use properties o logarithms to simpliy one side o a logarithmic equation beore solving the equation.

Example B Note Taking, Group PresentationIn order to understand Example B, students will need to grasp the act that logb m = logb n i and only i m = n.

Example C Note TakingFocus students on using logarithmic properties to simpliy terms in the logarithmic equation beore solving. This example also provides a problem with an extraneous solution.

c. 10= 4x − 6 16 = 4x x= 4

Example C Solve log2 x + log2 (x + 2) = 3. log2 x + log2 (x + 2) = 3 Step 1: log2 [x(x + 2)] =3 Step 2: 2log2 [x(x+2)] = 23 Step 3: Step 4: Step 5: Step 6:

Product Property o Logarithms Write in exponential orm using 2 as the base. Use the Inverse Property to simpliy. Write as a quadratic equation. Solve the quadratic equation. Check or extraneous solutions.

x(x + 2) =8 x2 + 2x − 8 =0 (x + 4)(x − 2) =0 x = −4 or x =2 ?

Check: log2 (−4) + log (−4 + 2) 3= ? log2 (−4) + log (3−2) =

log log

?

2

2 + log (2 + 2) = 3 ? 2 2 + log 4 = 3 ?


log2 8 = 3 3=3 Because log2 (−4) and log ( −2) are not deined, −4 is not a solution o the srcinal equation; thus it is extraneous. The solution is x = 2.

Try These C Solve or x, rounding to three decimal places i necessary. Check or extraneous solutions. a. log4 (x + 6) − log4 x = 2 2 5

b. ln (2 x + 2) + ln 5 = 2 −0.261 c. log2 2x + log2 (x − 3) = 3

4; −1 is extraneous


377

ACTIVITY 24 Continued Example D Note Taking, Think-PairShare Explain that, in Example D, you are introducing the variable y. Use the transitive property o equality to state −x = y and y = log x. Guide students to understand that the solution o this set o equations is the solution o the srcinal equation.



My Notes

Some logarithmic equations cannot be solved symbolically using the previous methods. A graphing calculator can be used to solve these equations.

Example D

Technology Tip

Solve −x = log x using a graphing calculator. −x = log x Step 1: Enter −x or Y1. Step 2: Enter log x or Y2.

The Try These D equations provide students with an opportunity to explore the use o graphs and tables to approximate the solution o logarithmic equations. For additional technology resources, visit SpringBoard Digital.

Step 3: Graph both unctions. Step 4: Find the x-coordinate o the point o intersection: x ≈ 0.399 Solution: x ≈ 0.399


X=0.399 Y=0.399

Try These D

Debrie students’ answers to these items to ensure that they understand the concept o extraneous solutions or logarithmic equations. I students struggle with the concept o extraneous solutions or logarithmic equations, demonstrate the parallel with extraneous solutions o radical equations.

Solve or x. a. x log x = 3 x ≈ 4.556

b. ln x = −x2 − 1 x ≈ 0.330

c. ln (2 x + 4) = x2 x ≈ −0.89 or 1.38

Check Your Understanding 1. Explain how it is possible to have more than one solution to a simpliied logarithmic equation, only one o which is valid.

2. Critique the rea soning of others Than solves a logarithmic . equation and gets two possible solutions, −2 and 4. Than immediately decides that −2 is an extraneous solution, because it is negative. Do


you agree with his decision? Explain your reasoning.

  


Solve or x, rounding to three decimal places i necessary. Check or extraneous solutions.

3. log5 (3x + 4) = 2 4. log3 (4x + 1) = 4

ADA PT

5. log12 (4x − 2) = log12 (x + 10)

Check students’ answers to the Lesson Practice to ensure that they understand how to solve logarithmic equations. Students who need additional practice approximating solutions can apply graphing techniques to solve linear or quadratic equations. Provide students with examples that they can solve algebraically. Guide them to approximate the solutions graphically and then to conirm the approximation by inding the solution algebraically.

6. log2 3 + log2 (x − 4) = 4 7. ln (x + 4) − ln (x − 4) = 4 8. Construct via ble arguments. You saw in this lesson that logarithmic equations may have extraneous solutions. Do exponential equations ever have extraneous solutions? Justiy your answer.

Ans wer s 1. Sample answer: Solving a logarithmic equation can require simpliying expressions to a quadratic equation having two solutions. Because the input o a logarithm must be a positive real number, one solution may not satisy the srcinal equation. 2. No; Than needs to check both solutions. A solution is extraneous only i it causes a logarithm to have an input that is not positive.

378

Intersection


3. 5. 7. 8.

7 4. 20 4 6. 28 ≈ 9.333 3 4.149 No; logarithmic equations sometimes have extraneous solutions because the domain o a logarithmic unction is limited to positive numbers, so you may not be able to substitute a solution back into the srcinal equation i it results in taking the logarithm o a negative number. However, exp onential unctions have a domain o all real numbers, so no solutions will be extraneous.




  Exponential and Logarithmic Inequalities


Lesson 24-4 PLAN

My Notes

Learning Targets:

Pacing: 1 class period

• Solve exponential inequalities. • Solve logarithmic inequalities.

Chunking the Lesson Example A Example B Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Note Taking, Group Presentation, Create Representations You can use a graphing calculator to solve exponential and logarithmic inequalities.

TEACH Example A


Use a graphing calculator to solve the inequality 4.2x+3 > 9. Step 1: Enter 4.2 x+3 or Y1 and 9 or Y2. Step 2: Find the x-coordinate o the point o intersection: x ≈ −1.469 Step 3: The graph o y = 4.2x+3 is above the graph o y = 9 when x > −1.469. Solution: x > −1.469

Have students solve the ollowing system o inequalities. x2 + 3x + 1 > y 4x + 5y > 20 Intersection

Use a graphing calculator to solve each inequality. a. 3 5.11−x < 75 b. log 10 x ≥ 1.5 c. 7.2 ln x + 3.9 ≤ 12 3.162 x≥

0

< x ≤ 3.080

Example B Scientists have ound a relationship between atmospheric pressure and altitudes up toa50 miles above sea level that can be modeled by P = 147 .(.)05 3a.6 . P is the atmospheric pressure in lb/in.2 Solve the equation P = 147.(.)05 3.6 or a. Use this equation to ind the atmospheric pressure when the altitude is greater than 2 mi les. Step 1: Solve the equation or a. a P = 0.5 3.6 Divide both sides by 14.7. 14.7  a  P  3 . 6 log 0.5 = log 0.5 0.5  Take the log base 0.5 o each side.  14.7  . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

Invite students to share their strategies or solution.

TEACHER to TEACHER

Try These A

⋅ x > −0.978

X=–1.469 Y=9

( ) log ( P ) = a 14.7 3.6 3.6 log ( P ) = a 14.7 3.6 log ( P ) 14.7 = a 0 .5

0 .5

log 0.5

Universal Access Students who ind it challenging to solve exponential or logarithmic unctions algebraically will beneit rom the technology tools discussed in this lesson. These tools allow students concentrate onconused guiding principlestowithout getting by exponential or logarithmic notation. Guide these students to connect the algebraic solutions to those ound using graphs and tables.

Simpliy. Multiply both sides by 3.6.

The items in this lesson give students the opportunity to work with exponential and logarithmic inequalities. Example A Create Representations, Think-Pair-Share, Group Presentation Demonstrate how to use the t able eature o the graphing calculator to solve the exponential inequality. Adjust the value o Δy to ind more precise approximations.

Have students work in their groups to complete the Try These A items. Within each have some students use tablesgroup, o values and other students approximate solutions graphically. Students then compare their methods and their solutions. Invite students to share their methods and strategies with the class.

Technology Tip Use the Change o Base Formula.

In order to use the graphing eature o their calculators to approximate equation solutions, students must deine each unction that the calculator will graph. These unctions are then available or use with the table eature. For additional technology resources, visit SpringBoard Digital.

Example B Create Representations Students solve the inequality x 14.70( .5 )3.6 < 5 by setting each side o the equation equal toy andinding then the graphing these equations, intersection, and selecting the interval where the exponential unction is below the linear unction. There is also a restriction that the model holds or x < 50, so the solution is 5.601< x < 50.


379

ACTIVITY 24 Continued Example B(continued)In Try These B Item a, students should solve the inequality 20 < −400 + 180 log x < 30. They can graph y = 20, y = 30, and y = −400 + 180 log x and ind where the logarithmic unction occurs between the two linear unctions. Students will need to ind an appropriate viewing window. One po ssible window is 0 < x < 300 with a scale o 25 and 0 < y < 40 with a scale o 10.

My Notes

Debrie students’ answers to these items to ensure that they understand the methods used to solve exponential and logarithmic inequalities. Invite students to share their responses and discuss their avorite strategies or solving inequalities.

Ans wer s 2. Sample answer: The answer is a range o values instead o a single value. 3. Sample answer: Graph both sides o the inequality. When one graph is above the other graph, then that side o the inequality is greater than the other side. This will occur on one side o the intersection point. Find the x-value o the intersection point and use the appropriate inequality symbol(s) to write the solution.

ASS ESS

Step 2:

Use your graphing calculator to solve the inequality

( )

P 14.7 > 2. log 0.5

3.6 log

X=10.002 Y=2

( )

x 14.7 is above the graph o y = 2 when log 0.5

3.6 log The graph o y = Intersection


  Exponential and Logarithmic Inequalities


0 < x < 10.002. Solution: When the altitude is greater than 2, the atmospheric pressure is between 0 and 10.002 lb/in.2.

Try These Suppose that B the relationship between C, the number o digital cameras supplied, and the price x per camera in dollars is modeled by the unction C = −400 + 180 log x. a. Find the range in the price predicted by the model i there are between 20 and 30 cameras supplied.

⋅

from $215.44 to $244.84

b. Solve the equation or x. Use this equation to ind the number o cameras supplied when the price per camera is more than $300. C + 400

x = 10

180

; more than 45 cameras

Check Your Understanding 2. How are exponential and logarithmic inequalities dierent rom exponential and logarithmic equations?

3. Describe how to ind the solution o an exponential or logarithmic inequality rom a graph. What is the importance o the intersection point in this process?


   Use a graphing calculator to solve each inequality.

4. 16.4(0.87)x−1.5 ≥ 10 5. 30 < 25 log (3.5x − 4) + 12.6 < 50 6. 4.5ex ≤ 2 7. ln (x − 7.2) > 1.35


4. 5. 6. 7.

x ≤ 5.052 2.562 < x < 10.095 x ≤ −0.812 x > 11.057

ADA PT Check students’ answers to the Lesson Practice to ensure that they understand how to solve exponential and logarithmic inequalities. Students who need additional practice may beneit rom applying the techniques o this lesson to review solving linear or quadratic inequalities.

380




      College Costs    Write your answers on notebook paper. Show your work.

Lesson 24-1 1. Which exponential equation can be solved by rewriting both sides in terms o the same base? A. 4 x = 12 B. 6 2x−3 = 256 C. 3x+2 − 5 = 22 D. ex = 58

⋅

2. Solve or x. a. 16x = 32x−1 b. 8 3x = 216 c. 5x = 1

⋅

d. e. f. g. h.

625 72x = 343x−4 4x + 8 = 72 ex = 3 e3 x = 2 3e5x = 42

Lesson 24-2 3. Solve or x to three decimal places. a. 7x = 300 b. 5x−4 = 135 c. 32x+1 − 5 = 80 d. 3 63x = 0.01 e. 5x = 212 f. 3(2x+4) = 350 4. A deposit o $1000 is made into a savings account

⋅

that pays 4% annual interest compounded monthly. a. How much money will be in the account ater 6 years? b. How long will it take or the $1000 to double?



5. June invests $7500 at 12% interest or one year. a. How much would she have i the interest is compounded yearly?

b. How much would she have i the interest is compounded daily?

6. I $4000 is invested at 7% interest per year compounded continuously, how long will it take to double the srcinal investment?

7. At what annual interest rate, compounded continuously, will money triple in nine years? A. 1.3% B. 7.3% 8.1% C. D. 12.2%

Lesson 24-3 8. Compare the methods o solving equations in the orm o log = log (such as log3 (2x − 3) = log3 (x + 4)) and log = number (such as log4 (3x − 1) = 2).

9. Solve or x. Check or extraneous solutions. a. log2 (5x − 2) = 3 b. log4 (2x − 3) = 2 c. log7 (5x + 3) = log7 (3x + 11) d. log6 4 + log6 (x + 2) = 1 e. log3 (x + 8) = 2 − log3 (x) f. log2 (x + 6) − log2(x) = 3 g. log2 x − log2 5 = log2 10 h. 5 ln 3x = 40 i. ln 4x = 30

ACT IV IT Y PR ACT ICE 1. C 2. a. 5 b. 3 c. −4 d. 12 e. 3 f. ln 3 g. 1 ln 2 3 h. 1 ln14 5 3. a. 2.931 b. 7.048 c. 1.522 − 1.061 d. e. 3.328 f. 2.866 4. a. $1270.74 b. 17.358 years 5. a. $8400.00 b. $8456.06 6. 9.9 years 7. D 8. To ind a solution o alog = log equation, use the act that when the bases are the same and b, m, n > 0, b ≠ 1, the logarithmic values must be equal: logb m = logb n i and only i m = n. For log = number, you can write in exponential orm and solve. 9. a. 2 b. 19 2 c. 4 d. − 12 e. 1 f. 6 7 g. 50 h. 1 e 8 3 i. 1 e 30 4


381

ACTIVITY 24 Continued 10. a.The solutions or log(x − 2) must be greater than 2, because any solutions equal to or less than 2 will result in the log o a negative number. b. The solutions or log (x + 3) must be greater than −3, because any solutions equal to or less than −3 will result in the log o a negative number. 11. a.1.939 b. 0.788; 5.256 12. a.0.611< x <1.473 b. x ≥ 0.753 c. x ≥ 6 d. 0 < x ≤ 1 e. x ≤ −2.24 13. a.y1(1000) ≈ 2.71692; y1(10,000) ≈ 2.71815; y1(1,000,000) ≈ 2.71828 b. The value o the expression gets closer and closer to e. c. Since m = n , r = 1 and r n m n = mr. Thereore, nt mrt A=+ P 1= +r P 1 1 n m m  rt  = P  1+ 1  . m  

( ) () (

      College Costs


10. I an equation contains a. log (x − 2), how do you know the solutions must be greater than 2? b. log (x + 3), how do you know solutions must be greater than −3?

11. Solve or x to three decimal places using a graphing ca lculator. a. ln 3x = x2 − 2 b. log (x + 7) = x2 − 6x + 5

Lesson 24-4 12. Use a graphing calculator to solve each inequality. 12x

a. 2000 < 1500(1.04) < 3000 b. 4.5 log (2x) + 8.4 ≥ 9.2 c. log3 (3x − 5) ≥ log3 (x + 7) d. log2 2x ≤ log4 (x + 3) e. 5x+3 ≤ 2x+4

MATHEMATICAL PRACTICES Look For and Make Use of Structure

13. Explore how the compounded interest ormula is related to the continuously compounded interest ormula. m a. Consider the expression 1 + 1 , where m is m a positive integer. Enter the expression in your calculator as y1. Then ind the value o y1(1000), y1(10,000), and y1(1,000,000). b. As m increases, what happens to the value o the expression? c. The compounded interest ormula is nt A = P 1 + r Let . m = n . Explain r n why the ormula may be written as rt m   A = P  1+ 1  .  m   d. As the number o compounding periods, n, increases, so does the value o m. Explain how your results rom parts b and c show the connection between the compounded interest ormula and the continuously compounded interest ormula.

(

(

(

)

)

)

)

d. Sample answer: The compounded interest ormula may be written as m  rt  A = P  1 + 1  , but as the m   number o compounding periods (n) increases, m also increases, and the expression in square brackets gets closer and closer to e. So it makes sense that the

(

)

ormula or continuously compounded interest is A = Pert.


382



Exponential and Logarithmic Equations

Embedded Assessment3 Embedded Assessment 3 Use after Activity 24

1. Make use of structure. Express each exponential statement as a logarithmic statement. a. 5−3 = 1 125 c. 202 = 400

b. 72 = 49

Ans wer Key

d. 36 = 729

( )

1 = −3 729 d. log11 14,641 = 4

b. log 9

c. log2 64 = 6

3. Evaluate each expression without using a calculator. a. 25log 25 x

b. log3 3x

c. log3 27

d. log8 1

e. log2 40 − log2 5

f. log 25 log 5

4. Solve each equation symbolically. Give approximate answers rounded to three decimal places. Check your solutions. Show your work.

a. 42x−1 = 64

b. 5x = 38

c. 3x+2 = 98.7

d. 23x−4 + 7.5 = 23.6

e. log3 (2x + 1) = 4

f. log8 (3x − 2) = log8 (x + 1)

g. log2 (3x − 2) + log2 8 = 5

h. log6 (x − 5) + log6 x = 2

( )

1 = −3 b. log 49 = 2 7 125 c. log20 400 = 2 d. log3 729 = 6 2. a. 83 = 512 b. 9−3 = 1 729 c. 26 = 64 d. 114 = 14,641 3. a. x b. x c. 3 d. 0 e. 3 f. 2 4. a. 2 b. 2.260 c. 2.180 d. 2.670 e. 40 f. 1.5 g. 2 h. 9 5. a.

1. a. log5

2. Express each logarithmic statement as an exponential statement. a. log8 512 = 3

Ass ess ment Foc us • Solving exponential equations • Solving logarithmic equations • Solving real-world applications o exponential and logarithmic unctions

EVALUATING YOUR INTEREST

y 9 8 7 6 5 4 3 2 1

5. Let f(x) = log2 (x − 1) + 3. a. Sketch a parent graph and a series o transormations that result in

g (x ) = log2(x )

the graph o f.

–1 –2

b. Give the equation o the vertical asymptote o the graph o f.

f (x ) = log2(x

–

1) + 3

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8

h(x ) = log2(x

–

9

x

1)

–3 –4

6. Make sense o f problems. Katie deposits $10,000 in a savings

–5 –6

account that pays 8.5% interest p er year, compounded quarterly. She does not deposit more mone y and does not w ithdraw any money. a. Write the ormula to ind the amount in the account ater 3 years. b. Find the total amount she will have in the account ater 3 years.

–7 –8 –9

b. x = 1 ( 4 )(3) 6. a. A = 10, 000 1 + 0.085 4 b. $12,870.19 7. 4.256 years; Check students’ graphs.

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7. How long would it take an investment o $6500 to earn $1200 interest i it is invested in a savings ac count that pays 4% annual interest compounded quarterly? Show the solution both graphically and symbolically.

)

Solution should appear as 4x point intersection o f(x ) = 1.01 and o f(x) = 1.1846. . d e v r e s re s t h ig r ll A . rd a o B e g e ll o C 5 1 0 2 ©

Common Core State Standards for Embedded Assessment3 HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.  HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. HSF-LE.A.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.


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Embedded Assessment3 Embedded Assessment 3 T EACHER

to TEACHER



Exponential and Logarithmic Equations EVALUATING YOUR INTEREST

Scoring Guide Mathematics Knowledge and Thinking

Exe m p l a r y

Pr o f i ci e n t

Em e r g i n g

• Fluency and accuracy in

• Largely correct work when • evaluating and rewriting evaluating and rewriting exponential and logarithmic exponential and logarithmic equations and expressions equations and expressions

(Items 1–7)

Difficulty when evaluating • and rewriting logarithmic and exponential equations and expressions

• Effective understanding of • Adequate understanding of• Partial understanding of and accuracy in solving logarithmic and exponential equations algebraically and graphically

• Effective understanding of

how to solve logarithmic and exponential equations algebraically and graphically leading to solutions that are usually • correct

logarithmic functions and their key features as • Adequate understanding of transformations of a parent logarithmic functions and graph their key features as transformations of a parent graph

Problem Solving

strategy that results in a correct answer

(Items 6, 7)

unnecessary steps but results in a correct answer

• Little difficulty in accurately• real-world scenario with an modeling a real-world exponential equation or scenario with an graph exponential equation or Effective understanding of graph •

Mostly inaccurate or incomplete work when evaluating and rewriting logarithmic and exponential equations and expressions Inaccurate or incomplete understanding of how to solve exponential and logarithmic equations algebraically and graphically

Partial understanding of logarithmic functions and their key features as • Little or no understanding transformations of a parent of logarithmic functions and graph their key features as transformations of a parent graph some incorrect answers

No clear strategy when solving problems

• Fluency in modeling a

Some difficulty in modeling• a real-world scenario with an exponential equation or graph

•

Partial understanding of how to graph a logarithmic • Mostly inaccurate or function using incomplete understanding transformations of how to graph a logarithmic function using transformations

(Items 5–7)

Reasoning and Communication

how to solve logarithmic • and exponential equations algebraically and graphically

• An appropriate and efficient• A strategy that may include• A strategy that results in •

(Items 6, 7)

Mathematical Modeling / Representations

I n c o m pl e t e


how to graph a logarithmic • Largely correct function using understanding of how to transformations graph a logarithmic function using transformations

Significant difficulty with modeling a real-world scenario with an exponential equation or graph

• Clear and accurate use of • Correct use of mathematical• Partially correct justification• Incorrect or incomplete mathematical work to justify an answer

work to justify an answer

of an answer using mathematical work

justification of an answer using mathematical work

Common Core State Standards for Embedded Assessment3 (cont.) HSF-BF.A.1 Write a function that describes a relationship between two quantities.  HSF-BF.A.1c (+) Compose functions. HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. HSF-BF.B.4 Find inverse functions. HSF-BF.B.4b (+) Verify by composition that one function is the inverse of another.

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Unit 4 TE Exponential and Log

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