Arithmetic and Geometric Sequences

Home , Power of two

TEKS AR.2A Determine the Arithmetic and Geometric patterns that identify the 1.1 Sequences relationship between a function and its common ratio or related finite dif- FOCUSING QUESTION How are arithmetic and geometric sequences alike? ferences as appropriate, How are they different? including linear, quadrat- LEARNING OUTCOMES ic, cubic, and exponential functions. ■■ I can determine patterns that identify a linear function or an exponential function. ■■ I can identify terms of an arithmetic or geometric sequence. (Algebra 1) A.12C Identify terms of ■■ I can write a formula for the nth term of an arithmetic or geometric sequence. arithmetic and geometric (Algebra 1) sequences when the se- ■■ I can use symbols, tables, and language to communicate mathematical ideas. quences are given in function form using recursive processes. ENGAGE A.12D Write a formula for the nth term of Brenda is at the farmer’s market. There are several baskets of tomatoes on a table. Each basket contains 6 tomatoes. What sequence arithmetic and geometric would Brenda create if she listed the number of tomatoes in a set of sequences, given the value baskets (1 basket, 2 baskets, 3 baskets, etc.)? of several of their terms. 6, 12, 18, 24, 30, … MATHEMATICAL PROCESS SPOTLIGHT AR.1D Communi- cate mathematical ideas, reasoning, and their im- EXPLORE plications using multiple The first few terms of two different sequences are shown. representations, including symbols, diagrams, SEQUENCE 1 graphs, and language as appropriate. ELPS 3G Express opin- DRAFTSEQUENCE 2 ions, ideas, and feelings ranging from communicat- ing single words and short FOR phrases to participating in extended discussions Use toothpicks or counters to build the next two terms of each sequence. Record on a variety of social and your information in the table. See margin. grade-appropriate academic topics. 2 CHAPTER 1: ALGEBRAIC PATTERNS VOCABULARY arithmetic sequence, geometric sequence, common MATERIALS SEQUENCE 1: SEQUENCE 2: difference, common ratio, recursive, additive relation- • 50 toothpicks for each ship, multiplicative rela- student group tionship • 65 two-color counters (or other roundPREVIEW objects Term 4 like pennies) for each student group ONLY

Term 5 Term 5 Term 6 16 counters 32 counters

2 CHAPTER 1: ALGEBRAIC PATTERNS 6. The number of counters SEQUENCE 1 SEQUENCE 2 is one power of 2 less than NUMBER OF NUMBER OF TERM NUMBER TERM NUMBER the power of two that TOOTHPICKS COUNTERS corresponds to the term 1 4 1 1 number. 2 7 2 2 7. For the next term, I count- 3 10 3 4 ed the number of counters in the previous term and 4 13 4 8 built a stack that was 5 16 5 16 twice as high. 6 19 6 32 9. In Sequence 1, you add 3 10 31 10 512 to the previous term to get the next term. In Sequence 1. What patterns do you observe in Sequence 1? 2, you multiply the pre- The number of toothpicks increases by 3 (add 3) for the next term. vious term by 2 to get the next term. Both sequences 2. What relationships do you observe between the term number and the number have a recognizable pat- of toothpicks required to build each term in Sequence 1? (Hint: Record the tern, but Sequence 1 is an multiples of 3 next to the number of toothpicks.) additive relationship while See margin Sequence 2 is a multiplicative relationship. Sequence 3. How does the pattern that you mentioned in question 2 appear in the figures 2 grows much faster. that you constructed? See margin INSTRUCTIONAL HINTS Comparing and Contrast- 4. How many toothpicks would you need to build the 10th term of Sequence 1? 31 ing is a high-yield instructional strategy identified 5. What patterns do you observe in Sequence 2? by Robert Marzano and his The number of counters doubles (multiply by 2) for the next term. colleagues (Classroom In- struction That Works, 2001). 6. What relationships do you observe between the term number and the number To support this strategy, of counters required to build each term in Sequence 2? (Hint: Record powers of 2 consider asking students next to the number of counters.) to make a Venn diagram See margin comparing properties of Sequence 1 to properties 7. How does the pattern that you mentioned in question 6 appear in the figures of Sequence 2. This strate- thatDRAFT you constructed? See margin gy helps students identify differences between linear

th and exponential relation- 8. How many counters would you needFOR to build the 10 term of Sequence 2? 512 ships, which they will spend time during this 9. Compare the two sequences. How are they alike? How are they different? chapter studying. See margin ELL STRATEGY 1.1 • ARITHEm ETIC ANd GE omETRIC S E quENCES 3 Expressing ideas and opinions (ELPS: c3G) is an important part of stu- 2. Possible answer: The number of toothpicks was 1 more than 3 times the term number. dent-student discourse about mathematics. Placing 3. Possible answer based on response to previous question: English language learners For the next term, I started with what I had before and then added 3 more toothpicks to place in small groups with their another square onto the end of the term. The first term begins with 1 toothpick on the left edge of peers, as in the activity in the square and 3 toothpicks added to complete the square. PREVIEWthis lesson, provides a safe

environment for them to ONLY sharpen their skills using

Note: additional patterns are possible, such as n sets of 2 horizontal toothpicks and n + 1 erticalv the English language while toothpicks. they are learning about mathematics.

3 REFLECT ANSWERS: REFLECT Sequence 1 has a constant ■■ Which sequence has a constant difference between the numbers of ob- difference because to build or jects for successive terms? How do you know? get to the next term, you add 3 See margin. toothpicks to the previous term ■■ Which sequence has a constant ratio between the numbers of objects every time. for successive terms? How do you know? See margin. Sequence 2 has a constant ratio because to build or get to the next term, you multiply the previous number of counters by EXPLAIN 2 every time. A sequence is a set of numbers that are listed in order and that Watch Explain and follow a particular pattern. Anarithmetic sequence is a sequence You Try It Videos that has a constant difference between consecutive terms.

For example, suppose Bernar- do earns tickets from the local QUESTIONING arcade. He starts out with 11 STRATEGY tickets and earns 4 more tickets or click here In mathematics, difference for each game he plays. Bernar- do can create an arithmetic se- is defined as the result of quence to show the number of subtraction, yet in the “ex- tickets he has after each game. plain” problem, the common difference is 4. Why is an “addend of 4” consid- 11, 15, 19, 23 27, 31, ... ered a difference? +4 +4 +4 +4 +4

Notice that the next term in the sequence can be generated by adding 4 to the previous term. You can use recursive no- A subscript is used to indicate a special case of a tation to generalize an arithmetic sequence like Bernardo’s. variable. a1 indicates the first term of a sequence, a = 11 1 a2 indicates the second an = an – 1 + 4 term of a sequence, and so on. a is read “a-sub DRAFT 1 The addend of 4 is used to generate the next term in this arith- one,” where “sub” indicates a subscript. metic sequence. In general, the addend in an arithmetic se- quenceFOR is called a common difference , since it is also the difference between consecutive terms.

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4 CHAPTER 1: ALGEBRAIC PATTERNS A geometric sequence is a sequence that has a constant ratio between consecutive terms. For example, Kayla earns $0.02 the first week for her allowance, but each week she earns twice as much as she did the week before. She can create a geometric sequence to show the amount of money she earns each week through her allowance.

$0.02 $0.04 $0.08 $0.16 $0.32 $0.64 $1.28 $2.56 $5.12 $10.24 ...

×2 ×2 ×2 ×2 ×2 ×2 ×2 ×2 ×2

The multiplier of 2 is used to generate the next term in this geometric sequence. In general, the multiplier in a geometric sequence is called a common ratio, since it is also the ratio between consecutive terms.

Kayla’s sequence can be represented by a recursive rule.

a = 0.02 1 INSTRUCTIONAL HINTS a = 2a n n – 1 Students process new skills better when they write ARITHMETIC AND GEOMETRIC SEQUENCES about them in their own An arithmetic sequence has a constant addend or words. Take a moment to common difference. It is an additive relationship be- have students write about tween terms of the sequence. the difference between A geometric sequence has a constant multiplier or arithmetic and geometric common ratio. It is a multiplicative relationship between terms of the sequence. sequences. Then have students share their explana- tions with one or two other If you have a sequence, you can use the constant difference or constant multiplier to students. determine subsequent terms of the sequence.

ADDITIONAL EXAMPLE Consider the scenario: EXAMPLE 1 Ericka and her mother bake 120 cookies for a bake Anthony worked all summer to save $1950 for spending money during the school sale. Every 15 minutes they year. He plans to withdraw the same amount from his savings account at the end of sell half of the cookies on each week. Anthony can create an arithmetic sequence that shows the balance of his the table. They can create savings accountDRAFT at the beginning of each week of the school year. a geometric sequence to show their cookie sale. $1950, $1885, $1820 120, 60, 30, 15, … How much money will be in Anthony’s savingsFOR account at the beginning of the fourth week of the school year? How much money will be in Anthony’s savings account at Ask students what the the beginning of the fifth week of the school year? constant multiplier in this sequence would be. Listen for students to say “divide by 2.” Ask them how to 1.1 • ARITHEm ETIC ANd GE omETRIC S E quENCES 5 write “divide by 2” as multiplication. 1 – The multiplier of 2 is used to generate the next term in the sequence. Ask students to write a recursive rule for the se- PREVIEWquence. ONLYa1 = 120 1 – an = 2an – 1 Ask students to continue the sequence. Discuss rea- sonableness of terms 5 and beyond. Would a customer buy half of a cookie?

a5 = 7.5, a6 = 3.75 5 ADDITIONAL EXAMPLES 1. Daniel’s grandmother STEP 1 First, use the existing data to determine the common difference. gave him a $50 gift card for music downloads for TIME SAVINGS BALANCE his birthday. If he down- (WEEKS) (DOLLARS) loads one song per day, he 1 $1950 –$65 = $1885 − $1950 can create an arithmetic 2 $1885 –$65 = $1820 − $1855 sequence that shows the 3 $1820 balance of his gift card each day. The common difference is –$65. $50, $48.55, $47.10, $45.65,... STEP 2 Next, apply the common difference to the balance at the What is the common beginning of the third week to determine the balance at the difference, and what does beginning of the fourth week. it represent? How much $1820 + (–$65) = $1755 money will be left on Dan- iel’s gift card at the end of STEP 3 Then, apply the common difference to determine the balance one week? in Anthony’s savings account at the beginning of the fifth week of school. The common difference is -$1.45. $1755 + (–$65) = $1690

At the end of one week (7 days), The balance in Anthony’s savings account is $1755 at the beginning of the fourth week Daniel will have $39.85 on his of school and $1690 at the beginning of the fifth week of school. gift card.

2. Keniesha went for a hike. She hiked at a steady pace, noting her distance, YOU TRY IT! #1 in miles, every 15 minutes. She can create an arithme- Rachel helps the student council create a paper chain that contains students’ writ- tic sequence to reflect her ten pledges not to bully or tolerate bullying. When Rachel begins stapling, there distance traveled every 15 are 54 inches of paper chain. She measures after adding each link and records the minutes of her hike. results in a table.

NUMBER OF LINKS 0.75, 1.5, 2.25, 3, … 1 2 3 RACHEL ADDED What is the commonDRAFT LENGTH OF PAPER CHAIN 1 1 58— 63 67— difference, and what does (INCHES) 2 2 it represent? How far will Keneisha have hiked if she WhatFOR is the common difference in this situation, and how long will the paper chain be hikes for one and a half after Rachel has added a total of six links to it? hours? See margin. The common difference is 0.75. It represents the distance in miles that Keneisha hiked every 6 CHAPTER 1: ALGEBRAIC PATTERNS 15 minutes. In one and a half hours, Keneisha will have hiked 4.5 miles. YOU TRY IT! #1 ANSWER: 1 – The common difference is 42 and the paper chain will be 81 inches long after Rachel has added six links to it. PREVIEW QUESTIONING STRATEGY FOR YOU TRY IT #1 Some students may expect the table of valuesONLY to start with (1, 54). Ask students why the 1 – dependent variable starts with 582 rather than 54. What column could be added to the table to account for the 54 inches of paper chain that Rachel started with?

6 CHAPTER 1: ALGEBRAIC PATTERNS QUESTIONING STRATEGIES EXAMPLE 2 How is 20% written as a decimal? At the beginning of the year, an investor puts $1000 into a fund that pays 20% annu- The fund pays 20% annual- ally. The investor projects how much will be in the fund at the end of each year for ly, yet the common ratio is the next three years. 1.2, not 0.2. What does 1.2 $1200, $1440, $1728 represent?

How much money will be in the investment fund at the end of four years? How much ADDITIONAL money will be in the investment fund at the end of five years? EXAMPLES 1. Jack purchased a new car STEP 1 First, determine the common ratio in this situation. for $25,120. Each year the $1440 ÷ $1200 = 1.2 value of the car will de- $1728 ÷ $1440 = 1.2 crease by 30%. Jack project- The common ratio is 1.2. ed how much his car will be worth at the end of each STEP 2 Next, multiply the third value in the geometric sequence year for the next four years. by the common ratio to determine the fourth value in the geometric sequence. $25,120, $17,584, $12,308.80, $8,616.16 ($1728)(1.2) = $2073.60 What is the common ratio STEP 3 Then, multiply by the common ratio to determine the fifth value in this situation? Is the in the geometric sequence. common ratio what you ex- ($2073.60)(1.2) = $2488.32 pected? Why or why not? The common ratio is 0.7. It There will be $2073.60 in the fund after four years and $2488.32 in the fund after five years. represents 70% of the value of Jack’s car from the previous year. What will the value of Jack’s car be after 8 years? If Jack’s car continues to decrease in value by 30% each year, it will be worth $1,448.12 DRAFT at the end of 8 years. 2. Montrell gets a job icing FOR cupcakes for a local bakery. On his first day on the job, he is proud to notice that the more cupcakes he ices, the faster he gets. Each hour, he takes note of how 1.1 • ARITHEm ETIC ANd GE omETRIC S E quENCES 7 many cupcakes he has iced in that hour. 5, 10, 20, … If Montrell continues icing cupcakes at his current rate, how many cupcakes will he ice in his fifth hour PREVIEWat the bakery? How many ONLYtotal cupcakes will he have iced by the end of his 6-hour shift? Montrell will ice 80 cupcakes in his 5th hour. At the end of his 6-hour shift, he will have iced 315 cupcakes.

7 YOU TRY IT! #2 ANSWER: YOU TRY IT! #2 The common ratio is 0.8 and there are approximately 164 A patient takes 500 milligrams of medicine. A nurse charts the amount of medication milligrams of medicine in in the patient’s system. the patient’s system after five hours. TIME SINCE DOSAGE MEDICINE IN PATIENT’S SYSTEM (HOURS) (MG) QUESTIONING 1 400 STRATEGIES 2 320 Assist students’ thinking 3 256 as they process the difference between recursive and 4 204.8 explicit rules by asking the following questions. What is the common ratio in this situation and approximately how much medicine, to the nearest milligram, will remain in the patient’s system after five hours? • Explain how a recur- See margin. sive rule differs from an explicit rule? • For which type of rule must you know the EXAMPLE 3 A recursive rule shows previous term in order how to determine the nth

to find the next? term, an, using the value of For the sequence shown, write a recursive rule and an the previous term. An ex- • For which type of rule plicit rule, like a function, can you find any term explicit rule. shows how to determine in the sequence without the nth term, an, using the 4, 6.5, 9, 11.5, 14, … needing to continue the term number, n. sequence? STEP 1 First, determine the common difference or ratio in this situation. • Which rule do you pre- fer and why? 4, 6.5, 9, 11.5, 14, ... • How does the explicit rule relate to a function +2.5 +2.5 +2.5 +2.5 rule? The common difference is +2.5.

STEP 2 Next, write the first term in the sequence,a . Use the common DRAFT 1 difference to write a recursive rule relating an to the previous term, an – 1.

a = 4 FOR 1 an = an – 1 + 2.5

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8 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES YOU TRY IT! #2 STEP 3 Use the common difference to work backwards to determine Write the recursive, explic- the value of term 0. it, and function rules for A patient takes 500 milligrams of medicine. A nurse charts the amount of medication 1.5, 4, 6.5, 11.5, 14, ... the additional examples on in the patient’s system. page. 7. +2.5 1. Jack’s car TIME SINCE DOSAGE MEDICINE IN PATIENT’S SYSTEM (HOURS) (MG) STEP 4 Use the common difference and the value of term 0 to write an explicit rule with term 0 as the starting point and the common recursive rule: 1 400 difference as the rate of change. a1 = 25,120, an = 0.70an–1 2 320 explicit rule: a = 1.5 + 2.5n n n 3 256 an = 25,120(0.70) 4 204.8 function rule: f(x) = 25,120(0.70)x What is the common ratio in this situation and approximately how much medicine, to the nearest milligram, will remain in the patient’s system after five hours? 2. Montrell’s job See margin. YOU TRY IT! #3 recursive rule: 1 2 1 — — — Write a recursive rule and an explicit rule for the sequence 6, 7 3 , 8 3 , 10, 11 3 , … a1 = 5, an = 2an–1 See margin. explicit rule: n an = 2.5(2) EXAMPLE 3 A recursive rule shows how to determine the nth function rule: term, a , using the value of n f(x)=2.5(2)x For the sequence shown, write a recursive rule and an the previous term. An ex- PRACTICE/HOMEWORK plicit rule, like a function, explicit rule. shows how to determine For questions 1 – 4 write an explicit rule that describes the number of items used to construct YOU TRY IT! #3 ANSWER: the nth term, a , using the n the pattern in terms of the term number, n. recursive rule: 4, 6.5, 9, 11.5, 14, … term number, n. 1 – 1. a1 = 6, an = an – 1 + 13 STEP 1 First, determine the common difference or ratio in this situation. explicit rule: 2 1 – – 4, 6.5, 9, 11.5, 14, ... an = 43 + 13n an = 2 + 2n

+2.5 +2.5 +2.5 +2.5 2.

The common difference is +2.5.

STEP 2 Next, write the first term in the sequence,a . Use the common 1 DRAFT2 difference to write a recursive rule relating an to the previous an = n term, an – 1. a = 4 1 FOR an = an – 1 + 2.5

8 CHAPTER 1: ALGEBRAIC PATTERNS 1.1 • ARITHEm ETIC ANd GE omETRIC S E quENCES 9 PREVIEWONLY

9 3.

an = 1 + 2n

an = 3 + n

For questions 5 and 6, use the following situation. FINANCE Segway Tours in Corpus Christi charges $12 for an hour to rent a Segway and an additional charge of $4 to rent a required helmet. David can create an arithmetic sequence that shows the cost of renting a Segway.

16, 28, 40, 52, …

5. How much will David spend to rent the Segway with a helmet for 6 hours? $76.00

6. Write a function rule that describes the cost of renting a Segway, f(n), in terms of the number of hours, n, David rents the Segway. f(n) = 4 + 12n

For questions 7 and 8, use the following situation. SCIENCE Roger dropped a ball from a height of 1000 centimeters. The height of the ball is 80% of the previous height after each bounce of the ball. Roger can create a geometric sequence that shows the height of the ball at the end of each bounce.

800, 640, 512, 409.6, …

7. What is the height of the ball after the 5th bounce? 327.68 centimeters

8. Write a function rule that describes the height of the ball, f(n), after the number DRAFTof bounces, n, the ball makes. FORf(n) = 1000(0.8)n

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10 CHAPTER 1: ALGEBRAIC PATTERNS For questions 9 and 10, use the following situation. FINANCE Clayton opens a savings account with $11 he got from his grandmother. Each month after the initial deposit, he adds $15 to the account. Clayton can create an arithmetic sequence that shows the balance of his savings account at the end of each month after he deposits funds in the savings account.

26, 41, 56, …

9. How much money will Clayton have in his account after he deposits money for 12 months? $191.00

10. Write an explicit rule that describes the amount of money in Clayton’s account,

an, in terms of the number of months, n, he deposits money. an = 11 + 15n

For questions 11 – 16 determine whether the sequences shown are arithmetic or geometric sequences. Then, write a recursive rule and an explicit rule. 11. 1, 8, 15, 22, 39, … 12. 2, 6, 18, 54, 162, … See margin. See margin. 11. arithmetic a1 = 1; an = an – 1 + 7 13. −10, −6.5, −3, 0.5, 4, 7.5, … 14. 1.5, 7.5, 37.5, 187.5 … See margin. See margin. an = −6 + 7n 12. geometric 15. 64, 16, 4, 1, 0.25, … 16. 147, 127, 107, 87, 67, … See margin. See margin. a1 = 2; an = 3an – 1 n – 1 an = 2(3) For questions 17 – 20 for each recursive rule and explicit rule given below, write the first 4 terms in the sequence. 13. arithmetic a1 = −10; an = an – 1 + 3.5 17. a = 9.5; a = a + 6.5 18. a = 3; a = 4a 1 n n – 1 1 n n – 1 n n – 1 a = −13.5 + 3.5n an = 3 + 6.5n an = 3(4) 9.5, 16. 22.5, 29 3, 12, 48, 192 14. geometric 1 a1 = 1.5; an = 5an – 1 19. a = 625; a = a ÷ 5 = — a 20. a = 140; a = a – 30 n – 1 1 n n – 1 5 n – 1 1 n n – 1 an = 1.5(5) n – 1 a = 170 – 30n —1 n an = 625( ) 140, 110, 80, 50 15. geometric 5 1 – 625, 125, 25, 5 a1 = 64; an = an – 1 ÷ 4 = 4an – 1 1 – n – 1 an = 64(4) DRAFT 16. arithmetic a1 = 147; an = an – 1 – 20 FOR an = 167 – 20n

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11 TEKS AR.2A Determine the patterns that identify the 1.2 Writing Linear Functions relationship between a function and its common ratio or related finite dif- FOCUSING QUESTION What are the characteristics of a linear function? ferences as appropriate, including linear, quadrat- LEARNING OUTCOMES ic, cubic, and exponential ■■ I can determine patterns that identify a linear function from its related finite functions. differences. ■■ I can determine the linear function from a table using finite differences, includ- AR.2C Determine the ing any restrictions on the domain and range. function that models a giv- ■■ I can analyze patterns to connect the table to a function rule and communicate en table of related values the linear pattern as a function rule. using finite differences and its restricted domain and range. ENGAGE MATHEMATICAL Marcus works in an orchard. Each row in the orchard PROCESS SPOTLIGHT contains 20 trees. How can Marcus use this information to make a table of values to represent the number AR.1F Analyze math- of trees in the orchard? ematical relationships to See margin. connect and communicate mathematical ideas. ELPS 1A Use prior knowl- edge and experiences to EXPLORE understand meanings in Miranda and her family will spend their summer vacation on the beach. They plan English. to rent a beach house that has a fixed cleaning fee and a daily rental fee. The table below shows the rental cost for each of a certain number of days. VOCABULARY finite differences, slope, NUMBER OF DAYS RENTAL COST y-intercept, linear function, 1 $170 restrictions 2 $285 MATERIALS 3 $400 DRAFT 4 $515 • N/A 5 $630 FOR 6 $745 ENGAGE ANSWER: 7 $860 Marcus could place the number of rows in one column and the number of trees in the other column. 12 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

12 CHAPTER 1: ALGEBRAIC PATTERNS 1. What is the difference between the numbers of days in consecutive rows in the table? 1 day 2. What is the difference between the rental cost in consecutive rows in the table? $115 3. Use the pattern in the table to predict the rental cost for 0 days. $55 4. Based on the pattern in the table, what do you think the cleaning fee is? Explain 4. The cleaning fee is $55, how you know. because that is the amount See margin. that you start with on day 5. Based on the pattern in the table, what do you think the daily rental fee is? 0 in the table. Explain how you know. See margin. 5. The daily rental fee is 6. Use the pattern in the table to write an equation that shows the relationship $115, because that is the between n, the number of days the beach house will be rented and r, the total amount that is added to rental cost. the total rental cost for r = 55 + 115n each day that Miranda’s Miranda and her sister have pooled their money for meals. From the initial amount family rents the beach of money they placed in an envelope, they will spend a certain amount each day on house. food. The table below shows the balance of money remaining in the envelope after a certain number of days. QUESTIONING STRATEGIES NUMBER OF DAYS BALANCE As students move through 1 $225 the second scenario, call 2 $190 attention to how they did 3 $155 the first scenario. Ask ques-

4 $120 tions such as:

5 $85 • How does your answer for this situation com- 6 $50 pare to the beach house 7 $15 rental? • Which of the two situ- 7. What is the difference between the numbers of days in consecutive rows in ations is an increasing the table? function? Decreasing 1 dayDRAFT function? 8. What is the difference between the balances in consecutive rows in the table? −$35 9. Use the patterns in the table to predictFOR the balance on day 0. $260

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13 10 Miranda and her sister 10. Based on the patterns in the table, how much money do you think Miranda and initially pooled $260, her sister initially pooled? Explain how you know. because that is the amount See margin. that you start with on Day 11. Based on the patterns in the table, how much money did Miranda and her sister 0 in the table. spend on meals each day? Explain how you know. See margin. 11. Miranda and her sister 12. Use the patterns in the table to write an equation that shows the relationship spent $35 each day on between n, the number of days of the vacation and b, the balance of pooled meals, because that is the money remaining. amount that is subtracted b = 260 – 35n from the balance each day.

REFLECT ANSWERS: REFLECT The differences in values for successive table rows are con- ■■ What do you notice about the differences in values for successive table stant. rows for both the independent and dependent variable? See margin. The ratio of the differences is the same as the rate of change ■■ What relationship exists between the ratio of the differences in the de- in the equation. pendent variable to the differences in the independent variable and the equations that you have written? See margin. ELL STRATEGY Connecting to prior knowl- edge (ELPS: c1A) helps stu- EXPLAIN dents create understanding of new mathematical The differences in values for successive table rows are calledfinite Watch Explain and You Try It Videos topics. Connecting finite differences. When you have a table of data, you can use finite differences to independent differences to determine the type of function the data represents and to write a function representing the relationship between the and dependent variables variables in the table. helps students recall important ideas from Algebra Let’s look more closely at a linear func- 1 and extend them to new The slope, which is the tion. The table on the next page shows rate of change, of a linear or click here content. function connecting two the relationship between x and f(x). In a linear function, f(x) = mx + b, m represents the slope or rate of points, (x1, y1) and (x2, y2) QUESTIONING is found using the slope change, and b represents the y-coordinate of the y-intercept, STRATEGIES formula. or starting point. ∆y y –y DRAFTm = = 2 1 Think back to Miranda and ∆x x2–x1 her family. • Which value(s) repre- FOR sented slope? • Which value(s) represented the y-intercept?

14 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

14 CHAPTER 1: ALGEBRAIC PATTERNS x y = f(x)

0 b ∆x = 1 – 0 = 1 ∆y = (b + m) – b = b + m – b = b – b + m = m 1 b + m ∆y = (b + 2m) – (b + m) = b + 2m – b – m ∆x = 2 – 1 = 1 = b – b + 2m – m = m 2 b + 2m ∆y = (b + 3m) – (b + 2m) = b + 3m – b – 2m ∆x = 3 – 2 = 1 = b – b + 3m – 2m = m 3 b + 3m ∆y = (b + 4m) – (b + 3m) = b + 4m – b – 3m ∆x = 4 – 3 = 1 = b – b + 4m – 3m = m 4 b + 4m ∆y = (b + 5m) – (b + 4m) = b + 5m – b – 4m ∆x = 5 – 4 = 1 = b – b + 5m – 4m = m 5 b + 5m

Notice that in the table, the difference between each pair of x-values, ∆x, is 1 and the difference between each pair ofy -values is m. The finite differences iny -values, ∆y, for a linear function are the same, so we can say that the finite differences are constant.

INTEGRATE FINITE DIFFERENCES AND LINEAR FUNCTIONS TECHNOLOGY In a linear function, the finite differences between suc- Use technology such as cessive y-values, ∆y, are constant if the differences between successive x-values, ∆x, are also constant. a graphing calculator or spreadsheet app on a If the finite differences in a table of values are constant, then the values represent a linear function. display screen to show students how, no matter the numbers present in the You can also use the finite differences to write a linear function describing the relation- linear function, the finite ship between the independent and dependent variables. differences will always be constant. INSTRUCTIONAL HINTS EXAMPLE 1 Remind students of pri- Does the set of data shown below represent a linear function? Justify your answer. or learning in Algebra I. Students learned multiple x y ways to test if data repre- 0 11 sented a linear function. DRAFT1 17 2 23 3 29 4 FOR35

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15 ADDITIONAL EXAMPLES Provide students with STEP 1 Determine the finite differences between successivex -values and successive y-values. the lists of ordered pairs of numbers. Have them x y place the numbers in tables 0 11 and answer the following ∆x = 1 – 0 = 1 ∆y = 17 – 11 = 6 1 17 question. ∆x = 2 – 1 = 1 ∆y = 23 – 17 = 6 2 23 Does the data set represent ∆x = 3 – 2 = 1 ∆y = 29 – 23 = 6 3 29 a linear function? Justify ∆x = 4 – 3 = 1 ∆y = 35 – 29 = 6 your answer. 4 35 1. (2, 13), (3, 6), (4, -1), STEP 2 Determine whether or not the ratios of the differences are (5, -8), (6, -15) constant. Yes, the data set represents The differences inx , ∆x, are all 1, so they are constant. a linear function because the finite differences in y-values The differences iny , ∆y, are all 6, so they are constant. are constant when the finite differences in x-values are also ∆y 6 — = — = 6 for all pairs of ∆x and ∆y. constant. ∆x 1

2. (1, 3), (2, 6), (4, 9), (8, 12), STEP 3 Determine whether or not the set of data represents a linear (16, 15) function. No, data set does not represent Yes, the set of data represents a linear function because the finite differences iny -values a linear function because while are constant when the finite differences in x-values are also constant. the finite differences in y-values are constant the finite differences in x-values are not constant. 3. (-1, -12), (1, -6.5), (3, -1), YOU TRY IT! #1 (5, 4.5), (7, 10) Does the set of data shown below represent a linear function? Justify your answer. Yes, the data set represents a linear function because the x y finite differences in y-values are constant when the finite 0 6.4 differences in x-valuesDRAFT are also 1 7.2 constant. 2 8.0 Note: Often students forget FOR3 8.8 to check the finite differences 4 9.6 between successive x-values. Draw special attention to the See margin. second additional example.

YOU TRY IT! #1 ANSWER: 16 CHAPTER 1: ALGEBRAIC PATTERNS Yes, the set of data represents a linear function, because the finite differences in x-values and the finite differences in y-values are both constant. PREVIEWONLY

16 CHAPTER 1: ALGEBRAIC PATTERNS QUESTIONING STRATEGIES EXAMPLE 2 Example 2 does not represent a linear function. Does the set of data shown below represent a linear function? Justify your answer. How does this data look on a scatterplot? How can x y you tell from looking at ∆x = 43 – 32 = 1 0 17 the graph, that this is not a 1 15.5 linear function? 2 14 3 11.5 4 10

STEP 1 Determine the first differences between successive x-values and successive y-values.

x y 0 17 ∆x = 1 – 0 = 1 ∆y = 15.5 – 17 = -1.5 1 15.5 ∆x = 2 – 1 = 1 ∆y = 14 – 15.5 = -1.5 2 14 ∆x = 3 – 2 = 1 ∆y = 11.5 – 14 = -2.5 3 11.5 ∆x = 4 – 3 = 1 ∆y = 10 – 11.5 = -1.5 4 10

STEP 2 Determine whether or not the differences are constant.

The differences inx , ∆x, are all 1, so they are constant.

The differences iny , ∆y, are not all the same, so they are not constant.

STEP 3 Determine whether or not the set of data represents a linear function.

No, the set of data does not represent a linear function because the finite differences for the y-values are not constant when the finite differences for the x-values are constant.DRAFT FOR

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17 YOU TRY IT! #2 ANSWER: YOU TRY IT! #2 No, the set of data does not represent a linear function, Does the set of data shown below represent a linear function? Justify your answer. because the finite differences in x-values are constant but the x y finite differences in y-values are 0 7.1 not constant. 1 7.5 QUESTIONING 2 8.1 STRATEGIES 3 8.9

After reading Example 3’s 4 9.9 scenario, guide students to plan their steps. See margin. How will the finite differences help you determine the slope? How can you identify the EXAMPLE 3 y-intercept from a table of values? For the data set below, determine if the relationship is a linear function. If so, deter- Do you need to draw a mine a function relating the variables. graph to write a function x y rule? 1 7.5 2 10 3 12.5 4 15 5 17.5

STEP 1 Determine the finite differences between successive x-values and successive y-values.

x y 1 7.5 DRAFT∆x = 2 – 1 = 1 ∆y = 10 – 7.5 = 2.5 2 10 ∆x = 3 – 2 = 1 ∆y = 12.5 – 10 = 2.5 3 12.5 ∆x = 4 – 3 = 1 ∆y = 15 – 12.5 = 2.5 FOR4 15 ∆x = 5 – 4 = 1 ∆y = 17.5 – 15 = 2.5 5 17.5

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18 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES STEP 2 Determine whether or not the relationship is a linear function. For the following data sets, determine if the relation- The differences inx , ∆x, are all 1, so they are constant. The differences iny , ∆y, are all 2.5, so they are constant. ship is a linear function. If Since the finite differences are all constant, the relationship is a so, determine a function linear function. relating the variables.

STEP 3 Determine the slope, or rate of change, of the linear function. 1.

-3 -1 1 3 5 ∆y 2.5 x Slope = — = — = 2.5 ∆x 1 y 0 3 6 9 12

STEP 4 Determine the y-intercept of the linear function. Linear, y = 1.5x + 4.5 Work backwards from x = 1 and y = 7.5. 2. x y x 0 1 2 3 4 0 b 1 – 0 = 1 7.5 – b = 2.5 y -4 -8 -16 -32 -64 1 7.5 2 – 1 = 1 10 – 7.5 = 2.5 2 10 Not Linear 3. (1, 2), (2, 5), (3, 8), (4, 12), 7.5 – b = 2.5 (5, 16) 7.5 – 7.5 – b = 2.5 – 7.5 −b = −5 Not Linear b = 5 4. (1, -5), (2, -13), (3, -21), The y-intercept is (0, 5). (4, -29), (5, -37) STEP 5 Use the slope and the y-coordinate of the y-intercept to write Linear, y = -8x - 13 the function in slope-intercept form.

y = mx + b y = 2.5x + 5 DRAFT FOR

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19 INSTRUCTIONAL HINT Encourage students to write the finite differences YOU TRY IT! #3 on the table of values for For the data set below, determine if the relationship is a linear function. If so, deter- every problem. When mine a function, in slope-intercept form, relating the variables. x = 0, highlight or circle the y-intercept in the table. x y 1 9

2 4

3 -1

4 -6

5 -11

Answer: y = –5x + 14

EXAMPLE 4

For the data set below, determine if the relationship is a linear function. If so, write a function, in slope-intercept form, relating the variables.

x y 2 22 4 21 6 20 8 19 10 18

STEP 1 Determine the finite differences between successive x-values and successive y-values.

x y DRAFT∆x = 4 – 2 = 2 2 22 ∆y = 21 – 22 = –1 ∆x = 6 – 4 = 2 4 21 ∆y = 20 – 21 = –1 FOR∆x = 8 – 6 = 2 6 20 ∆y = 19 – 20 = –1 ∆x = 10 – 8 = 2 8 19 ∆y = 18 – 19 = –1 10 18

20 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

20 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES STEP 2 Determine whether or not the relationship is a linear function. Some students may need additional practice with The differences inx , ∆x, are all 2, so they are constant. The differences iny , ∆y, are all −1, so they are constant. identifying slope and y-in- Since the finite differences are all constant, the relationship is a linear tercept from an equation function. and with writing functions given slope and y-intercept. STEP 3 Determine the slope of the linear function. Identify the slope, m, and ∆y –1 1 y-intercept, b, of the follow- Slope = — = — = – – ∆x 2 2 ing linear functions. STEP 4 Determine the y-intercept of the linear function. 1. y = 0.5x + 2 Work backwards from x = 2 and y = 22. m = 0.5, b = 2 x y 2. y = 3 – 4x 0 b 2 – 0 = 2 22 – b = –1 2 22 m = -4, b = 3 4 – 2 = 2 21 – 22 = –1 2 4 21 – 3. y = - 3x – 5.5 2 m = -–, b = -5.5 22 – b = −1 3 22 – 22 – b = −1 – 22 4. y = 0.6x −b = −23 b = 23 m = 0.6, b = 0

The y-intercept is (0, 23). 5. y = 7 m = 0, b = 7 STEP 5 Use the slope and the y-coordinate of the y-intercept to write the function in slope-intercept form. Write the function given y = mx + b the slope and y-intercept. 1 – y = −2x + 23 1. m = -0.25, b = -3 y = -0.25x – 3 4 – 2. slope = 5, contains the point (0, 2) 4 y = –x + 2 DRAFT 5 1 – 3. slope = 2, contains the point (0, 0) 1 FOR – y = 2x 4. slope = -4, contains the point (2, -3) y = -4x + 5 1.2 • WRITING L INEAR FuNCTIo NS 21 5. slope = -0.7, contains the point (10, -10) y = -0.7x – 3 PREVIEWONLY

21 YOU TRY IT! #4

For the data set below, determine if the relationship is a linear function. If so, determine a function relating the variables.

x y 6 11

9 16

12 21

15 26

18 31

5 — Answer: y = 3x + 1

PRACTICE/HOMEWORK

For questions 1 - 4 determine the equation of the linear function with the given characteristics.

1. slope = 0.4, y-intercept = (0, −3) y = 0.4x – 3

2 1 — — 2. slope = 3, y-intercept = (0, 3 3 ) 2 1 — — y = 3x + 33 2 — 3. slope = − 5, contains the point (10, 3) 2 — y = − 5x + 7 1 — 4. slope = 4, contains the point (−8, 1) 1 — y = 4 x + 3 For questions 5 - 16, determine whether or not the relationship shows a linear function. If the data set represents a linear function, write the equation for the function. DRAFT5. x y 6. x y 7. x y 1 1 1 5.5 1 8 2 4 2 7.5 2 11 3 9 3 9.5 3 14 FOR4 16 4 11.5 4 17 5 25 5 13.5 5 20

not linear linear; y = 2x + 3.5 linear; y = 3x + 5

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22 CHAPTER 1: ALGEBRAIC PATTERNS 8. x y 9. x y 10. x y 1 24 0 1.7 0 2 2 20 1 1.1 2 4 3 16 2 0.5 4 8 4 12 3 –0.1 6 16 5 8 4 –0.7 8 32 linear; y = –4x + 28 linear; y = –0.6x + 1.7 not linear

11. x y 12. x y 13. x y 2 4 2 8 3 2 4 5 4 9 5 10 6 7 6 10 7 18 8 10 8 11 9 26 10 14 10 12 11 34 1 — not linear linear; y = 2x + 7 linear; y = 4x – 10

14. x y 15. x y 16. x y 1 10 1 16 1 120 2 8 2 15 2 60 3 6 3 13 3 40 4 4 4 10 4 30 5 2 5 6 5 24

linear; y = −2x + 12 not linear not linear

For questions 17 - 20, use the information in the problem to create a table of data. Then, use the table to determine if the situation is linear or not. If the situation is linear, then use the table to determine a linear function. SCIENCE 17. The elevation of Lake Sam Rayburn is 164 feet above mean sea level. During the summer, if it does not rain, the elevation of the lake decreases by 0.5 feet each week.

x y ∆x = 1 and ∆y = −0.5, so the situation is linear. 0 164 y = −0.5x + 164 1 163.5 2 163

3 162.5 4DRAFT162 FOR

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23 18. A swimming pool has a capacity of 10,000 gallons of water. The swimming pool was about 20% full when a water hose was turned on to fill the pool at a rate of 75 gallons every 5 minutes.

x y ∆x = 5 and ∆y = 75, so the situation is linear. 0 2,000 y = 15x + 2,000 5 2,075 10 2,150 15 2,225 20 2,300

19. According to a recent county health department survey, there were 750 mosquitos per acre in a county park. After a recent rainstorm, the number of mosquitos doubled every 2 days.

x y ∆x = 2 but ∆y = is not constant, so the situation 0 750 is not linear. 2 1,500

4 3,000 6 6,000 8 12,000

FINANCE 20. Marla has $85 in her savings. She earns $6.50 per hour after taxes and payroll deductions and plans to save half of what she earns each hour.

x y ∆x = 1 and ∆y = 3.25, so the situation is linear. 0 85.00 y = 3.25x + 85 1 88.25 2 91.50

3 94.75 DRAFT4 98.00 FOR

24 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

24 CHAPTER 1: ALGEBRAIC PATTERNS TEKS AR.2D Determine 1.3 Modeling with Linear Functions a function that models real-world data and mathematical contexts using FOCUSING QUESTION How can you use finite differences to construct a finite differences such as linear model for a data set? the age of a tree and its circumference, figurative LEARNING OUTCOMES numbers, average velocity, ■■ I can use finite differences to write a linear function that describes a data set. and average acceleration. ■■ I can apply mathematics to problems that I see in everyday life, in society, and in the workplace. MATHEMATICAL PROCESS SPOTLIGHT AR.1A Apply mathe- ENGAGE matics to problems arising Mariette, a dendrochronologist, observed that in everyday life, society, some tree stumps have rings that are close to- and the workplace. gether while other tree stumps have rings that are farther apart. Why does a tree stump have rings? ELPS What might cause the rings to be closer together 5B Write using or farther apart? See margin. Image credit: Adrian Pingstone, Tree ring, Wikimedia newly acquired basic vo- Commons cabulary and content-based grade-level vocabulary. EXPLORE VOCABULARY

Each year during the growing season, trees grow larger by adding another layer linear function, finite differ- of cells just beneath the bark. This layer is called a tree ring. Because a tree ring is ences added each year, scientists can determine the age of a tree by counting the number of tree rings that are present. MATERIALS • graphing calculator, However, not all tree rings have the same width. Trees grow more when there is spreadsheet, or a plenty of rain and the soil is fertile. Scientists can draw conclusions about temperature and rainfall for a particular year based on the width of the tree ring for graphic application that year.

Mariette measured the width of tree rings from a core sample she took from a post ENGAGE ANSWER: oak tree in the Brazos River valley of central Texas. From the tree ring width, she Possible answer: The rings calculatedDRAFT the radius of the tree. The table below shows her results. are formed because, as the tree grows, each year it adds a new YEAR 2000 2001 2002 2003 2004 2005 2006 layer of cells under the bark. YEAR The amount of rain might cause 0 1 2FOR 3 4 5 6 NUMBER the rings to be closer together or farther apart. RADIUS 2.5 3.1 3.5 3.9 4.4 4.9 5.5 (CM)

1.3 • Modeling With l inear Functions 25 PREVIEWONLY

25 TECHNOLOGY 1. Calculate the finite differences between the year number and the radius. INTEGRATION See margin When modeling with real- world data, a scatterplot 2. Are the first differences in the radius constant? Explain how you know. of the data set and the The first differences in radius are not exactly constant but are very close function model helps in value. students visualize the 3. What is the average finite difference in radius? relationship between the 0.5 cm two variables. Scatterplots can be made using 4. Use the information from the table to write a function rule that models the data. graphing calculators, f(x) = 2.5 + 0.5x, where x represents the year number, or number of years since 2000. spreadsheets, or graphing apps. 5. What do the slope and y-intercept from your function rule mean in the context of this situation? ELL STRATEGY See margin Writing with newly acquired vocabulary 6. Use your model to predict the radius of the tree in 2015. (ELPS: c5B) helps English f(15) = 2.5 + 0.5(15) = 10 centimeters language learners internalize the vocabulary 7. In what year will the radius of the tree be 12.5 centimeters? f(x) = 2.5 + 0.5x = 12.5, x = 20, so the year will be 2020 terms that they have recently learned. Using 8. What would the circumference of the tree be in 2015? vocabulary from current C = 2πr = 2π(10) = 20π ≈ 62.8 centimeters and past learning experiences (e.g., linear 9. Make a scatterplot of your data set and graph the function model over the function, finite differences) scatterplot. How well would you say the function model predicts the actual to explain their thinking values in the data set? Explain your reasoning. and mathematical See margin reasoning reinforces how these terms are consistent through a variety of settings and contexts. REFLECT

■■ How can you determine a linear function model for a data set if the first REFLECT ANSWERS: differences are not exactly the same, but are almost constant? See margin. Use an average value of the first differences as theDRAFT slope of a ■■ Once you have your linear function model, how can you use the model linear function model. to determine a value of the independent variable that generates a particular value of the dependent variable? Use your linear function model See margin. to write an equation where the FOR linear function is equal to a particular value. Then, solve for x.

26 chaP ter 1: algeB raic Patterns

1. +1 +1 +1 +1 +1 +1 5. The slope, 0.5 centimeters 9. The function model con- per year, represents the nects several of the data growth rate of the tree, or values and is very close YEAR 2000 2001 2002 2003 2004 2005 2006 the number of centimeters to the remaining data YEAR 0 1 2 3 4 5 6 that the radius of the tree values. The function model NUMBER PREVIEWgrows each year. appears to closely predict RADIUS the actual data values. 2.5 3.1 3.5 3.9 4.4 4.9 5.5 The y-intercept,ONLY 2.5 cen- (CM) timeters, represents the See page 27. radius of the tree in the +0.6 +0.4 +0.4 +0.5 +0.5 +0.6 year 2000, when the data were first collected.

26 CHAPTER 1: ALGEBRAIC PATTERNS EXPLAIN

A linear function model can be used to represent sets of math- Watch Explain and ematical and real-world data. Dendrochronologists use core You Try It Videos samples, or cylinders that are about 5 millimeters in diameter that are drilled into and extracted from the tree to measure the width of tree rings. Once they have their model, they can use different measures, such as circumference of a tree, to calculate the age of the tree.

You can use Mariette’s data to generate a model relating the cir- or click here cumference of a tree to the age of the tree. For trees that were planted in 2000, the table shows the growth rate.

YEAR 2000 2001 2002 2003 2004 2005 2006

YEAR 0 1 2 3 4 5 6 NUMBER

RADIUS 2.5 3.1 3.5 3.9 4.4 4.9 5.5 (CM)

Add a new row to the table to calculate the circumference. Recall that circumference can be calculated using the formula C = 2πr, where r represents the radius of the circle and C represents the circumference of the circle. Round the circumference to the nearest tenth if necessary.

YEAR 2000 2001 2002 2003 2004 2005 2006

YEAR NUMBER 0 1 2 3 4 5 6

RADIUS (CM) 2.5 3.1 3.5 3.9 4.4 4.9 5.5

CIRCUMFERENCE 15.7 19.5 22.0 24.5 27.6 30.8 34.5 (CM)

Use the rows for year number and circumference to calculate the first finite differences. +1 +1 +1 +1 +1 +1

YEAR NUMBER 0 1 2 3 4 5 6

CIRCUMFERENCE DRAFT15.7 19.5 22.0 24.5 27.6 30.8 34.5 (CM) +3.8 +2.5FOR +2.5 +3.1 +3.2 +3.7

1.3 • Modeling With l inear Functions 27 PREVIEWONLY

27 These first finite differences are not equal, but are all close to +3. Calculate the average finite difference, and use that to determine the slope of the linear function model. 3.8 + 2.5 + 2.5 + 3.1 + 3.2 + 3.7 ∆y = ≈ 3.13 6

∆x = 1

∆y 3.13 = = 3.13 ∆x 1

Using the slope and y-intercept, you can write the function model, f(x) = 15.7 + 3.13x. Once you have a function model, you can use that model to make predictions.

MODELING WITH LINEAR FUNCTIONS

Real-world data rarely follows exact patterns, but you can use patterns in data to look for trends. If the data increases or decreases at about the same rate, then a linear function model may be appropriate for the data set.

You can also use a scatterplot and a graph to show how the values in the data set are related to the function model. The graph of the function model could also be useful in making predictions from the model.

DRAFT

FORf(x) = 15.7 + 3.13x

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28 CHAPTER 1: ALGEBRAIC PATTERNS EXAMPLE 1

A student takes small steps away from a motion detector at an approximately constant rate. The time, in seconds, for which the student walks and the distance, in meters, the student walks are recorded in the table.

TIME (S) 0 1 2 3 4 DISTANCE (M) 0.25 0.85 1.55 2.2 2.75

Generate a linear function model for this situation. Based on your model, how far away will the student be from the motion detector after 10 seconds?

STEP 1 Calculate the finite differences in the table. +1 +1 +1 +1

TIME (S) 0 1 2 3 4 DISTANCE (M) 0.25 0.85 1.55 2.2 2.75

+0.6 +0.7 +0.65 +0.55 STEP 2 Calculate the average finite difference in the table and use this to determine the slope, or average velocity, for a linear function model.

0.6 + 0.7 + 0.65 + 0.55 2.5 = = 0.625 4 4

STEP 3 Use the slope and y-intercept to write a linear function model.

y = 0.625x + 0.25

STEP 4 Use your linear function model to make a prediction.

y = 0.625(10) + 0.25 y = 6.25 + 0.25 DRAFTy = 6.5 According to the linear function model, the student will be 6.5 meters away from the motion detector after 10 seconds. FOR

1.3 • Modeling With l inear Functions 29

ADDITIONAL EXAMPLE Generate a linear function model for the situation below and answer the questions. A football player gets the ball at the 50 yard line and makes a clean break running for the end zone at the 0 yard line. His distance, in yards, over the time of his run, in seconds, is recorded in the table. If he doesPREVIEW not get tackled, how many seconds will it take him to reach the end zone? Why would the finite differences not beONLY constant in this scenario?

TIME (S) 0 1 2 3 4 DISTANCE (YD) 0 8.2 16.9 25.4 34

He will reach the end zone in between 5 and 6 seconds.

29 YOU TRY IT! #1 ANSWER: YOU TRY IT! #1 y = 8.325x + 0.175; according Tracy, a long distance runner, times herself as she runs a half-marathon, which is 13.1 to the linear function model, miles long. The distance is measured in miles and the time is measured in minutes. Tracy will run the half-marathon in approximately 109 DISTANCE 1 2 3 4 5 minutes. (MI)

TIME (MIN) 8.5 16.2 24.6 33.1 41.8

Generate a linear function model for this situation. Based on your model, how long to the nearest minute will it take Tracy to run the half-marathon? See margin

EXAMPLE 2 Engineers conducting experiments in accident reconstruction want to see how long it would take a vehicle on a highway to coast to a stop if the brakes were inoperable. The first few seconds of the experiment are recorded in the table below. Time is measured in seconds and speed is measured in miles per hour.

TIME (S) 1 2 3 4 5

SPEED (MPH) 65 62 58 56 53

Generate a linear function model for this situation. Based on your model, when will the car come to a stop, to the nearest second?

STEP 1 Calculate the first finite differences in the table. +1 +1 +1 +1

TIME (S) 1 2 3 4 5 DRAFTSPEED (MPH) 65 62 58 56 53 FOR-3 -4 -2 -3

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30 CHAPTER 1: ALGEBRAIC PATTERNS STEP 2 Calculate the average first finite difference in the table and use this to determine the slope, or average acceleration, for a linear function model.

–3 + (–4) + (–2) + (–3) –12 = = –3 4 4

STEP 3 Use the average acceleration and first finite differences in the x-values to determine the y-intercept of a linear function model. +1

TIME 0 1 SPEED b 65

65 – b = – 3 65 – b = – 3 65 – b + 3 = – 3 + 3 68 – b = 0 68 – b + b = 0 + b 68 = b

STEP 4 Write a linear function model.

y = – 3x + 68

STEP 5 Use the linear function model to make a prediction.

0 = –3x + 68 0 – 68 = –3x + 68 – 68 –68 = –3x (–68) ÷ (–3) = (–3x) ÷ (–3) 22.667 ≈ x

According to the linear function model, the car will come to a stop after approximately 23 seconds.DRAFT FOR

1.3 • Modeling With l inear Functions 31

ADDITIONAL EXAMPLE Mrs. Norris started the school year with a large supply of pencils for her students to borrow and use. Each month she noticed that her supply had dwindled. After one month she recorded how many pencils remained. Each month after she counted and recorded again. The data for the firstPREVIEW 4 months of the school year are shown in the table below. ONLY TIME (MONTHS) 1 2 3 4 PENCILS 195 185 173 159

Based on the model, how many months will it take for Mrs. Norris to drop below 100 pencils? Will she have enough pencils to last the school year? How many pencils did Mrs. Norris have at the start of the school year? It would take about 9 months for Mrs. Norris’ pencil supply to drop below 100 pencils. She will have enough pencils to last the school year. She started the year with about 207 pencils.

31 YOU TRY IT! #2 ANSWER: YOU TRY IT! #2 y = –189.75x + 1550; the check- Caleb’s parents set up a checking account for him before college so that he will be able ing account Caleb’s parents set to pay the utilities for his apartment. Caleb keeps track of his spending in the table up will have a balance below below. Time represents the number of months he has been in his apartment and the $100 after 8 months. checking account balance is measured in dollars.

TIME 0 1 2 3 4 (MONTHS)

BALANCE $1550 $1355 $1170 $978 $791

Generate a linear function model for this situation. Based on your model, when will the checking account balance dip below $100? See margin

PRACTICE/HOMEWORK

For the following sets of data, calculate the average finite difference, and use that to determine the slope of a linear function that could model the data.

1. x y 1 15.3 2 25.3 3 35.2 4 45.4 5 55.2 6 65.3

2. x 0 1 2 3 4 5 6 y 50 47.2 44.3 41.5 38.5 35.6 32.5 DRAFT-2.917

3. x 1 2 3 4 5 FORy 14.25 14.05 15.35 16 16.55 0.575

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32 CHAPTER 1: ALGEBRAIC PATTERNS For problems 4 – 6, determine a linear function to model the situation. FINANCE 4. Madeleine has a gift card to her favorite coffee shop. The table below shows how much is remaining on the gift card after each purchase at the coffee shop.

PURCHASES 0 1 2 3 4 5

BALANCE 40 35.68 31.22 26.97 22.65 18.40 (DOLLARS)

y = 40 – 4.32x or y = -4.32x + 40, where x represents the number of purchases and y represents the balance remaining on the gift card. SCIENCE 5. Gus records the mileage on his car, so he can determine his average mileage per month. Below are some of his collected data.

TIME 1 2 3 4 5 6 (MONTHS) MILEAGE 11,540 12,482 13,570 14,670 15,682 16,757 (MILES)

y = 10,496.6 + 1043.4x or y = 1043.4x + 10,496.6, where x represents time and y represents mileage. FINANCE 6. David is purchasing apps for his cell phone. The table below shows how his total cost changes with each app that he selects.

NUMBER OF 1 2 3 4 5 APPS PURCHASED

COST 1.25 2.50 3.75 5.00 6.25 (DOLLARS)

y = 1.25x, where x represents the number of apps purchased and y representsDRAFT total cost. FOR

1.3 • Modeling With l inear Functions 33 PREVIEWONLY

33 Use the following situation to answer problems 7 – 10. SCIENCE Charlie is measuring his little brother’s height throughout the year to see how much he grows. The table below shows how his height changes during the first 5 months.

TIME 0 1 2 3 4 5 (MONTHS)

HEIGHT 54 54.20 54.45 54.85 55.10 55.25 (INCHES)

7. Write a function rule to model the situation. f(x) = 54 + 0.25x or f(x) = 0.25x + 54; where x represents the number of months. 8. The slope, 0.25 inches 8. What do the slope and y-intercept from your function rule mean in the context per month, represents of this situation? See margin the growth rate of Char- lie’s brother (how much 9. Use your model to predict the height of Charlie’s brother after a year. his height increases each f(12) = 54 + 0.25(12) = 57 inches month). The y-intercept, 54 inches, represents the 10. In what month will his height be approximately 56 inches? height of Charlie’s brother f(x) = 54 + 0.25x = 56, x = 8 months when he started tracking Use the following situation to answer problems 11 – 13. his height. CRITICAL THINKING 12. The slope represents a Jeff noticed that the nutrition information on his box of cereal states that there are 14 decrease of 1.12 ounces per servings in the cereal box. He decided to put their claim to the test. He recorded the weight of the remaining cereal after each serving, as shown in the table below. serving (each serving is 1.12 ounces). The y-inter- NUMBER OF 0 1 2 3 4 5 cept, 14 ounces, represents SERVINGS the starting weight of the box of cereal. WEIGHT OF REMAINING 14 12.7 11.6 10.7 9.5 8.4 CEREAL 13. No. (OUNCES) f(x) = 14 – 1.12x = 0; x = 12.5. He will finish 11. Write a function rule that models the situation. the cereal at about 12.5 f(x) = 14 – 1.12x or f(x) = -1.12x + 14, where x represents the number of cereal servings servings, instead of 14 DRAFT12. What do the slope and y-intercept from your function rule mean in the context servings. However, he of this situation? may have used more than See margin the suggested number of FOR ounces per serving, so his 13. Was Jeff able to confirm the claim on the cereal box by eating 14 servings? Ex- results are not conclusive. plain your answer. See margin

34 chaP ter 1: algeB raic Patterns PREVIEWONLY

34 CHAPTER 1: ALGEBRAIC PATTERNS Use the following situation to answer problems 14 – 17. SCIENCE Bob is tracking a hurricane moving toward the coast of Florida. The table below shows its distance from land over time.

TIME 0 1 2 3 4 (HOURS)

DISTANCE 704 684 663.7 644.2 624.4 (MILES)

14. Write a function rule that models the situation. 14. f(x) = 704 – 19.9x or See margin f(x) = -19.9x + 704, where x represents the time (in 15. What do the slope and y-intercept from your function rule mean in the context of this situation? hours) since Bob started See margin tracking the hurricane.

15. The slope, -19.9, means 16. About how far will the hurricane be from land after 24 hours? f(24) = 704 – 19.9(24) = 226.4 miles from land that the distance of the hurricane to land is 17. Approximately when will the hurricane make landfall? decreasing by 19.9 miles f(x) = 704 – 19.9x = 0; x = 35.38; in a little over 35 hours each hour. The y-inter- Use the following situation to answer problems 18 – 19. cept, 704, gives us the CRITICAL THINKING original distance from land when Bob started tracking Maddie is running a 10-K (10 kilometer) race. She wears an electronic chip that tracks her progress throughout the race. She runs at a fairly steady pace throughout the race, the hurricane. as shown in her chip data below. 19. No.

DISTANCE f(10) = 3.8(10) = 38. She 0 1 2 3 4 5 (KILOMETERS) will finish the race in about 38 minutes, which is TIME 0 4.1 7.9 11.8 15.5 19 (MINUTES) a half minute slower than her previous best time.

18. Write a function rule that models the situation. f(x) = 3.8x, where x represents the number of kilometers Maddie has run.

19. If Maddie continues at this rate, will she beat her previous best time of 37.5 minutes?DRAFT Explain your answer. See margin FOR

1.3 • Modeling With l inear Functions 35 PREVIEWONLY

35 Use the following situation to answer problems 20 – 21. FINANCE Nikki has a job as a waitress where she gets an hourly wage plus tips. The table below shows her total earnings for working one weekend.

TIME WORKED 1 2 3 4 5 (HOURS)

TOTAL EARNED 9.3 19.3 30.73 43.23 56.33 (DOLLARS)

Nikki calculates that the function f(x) = 11.8x – 2.5 models her earnings over time. She understands that the slope of 11.8 means she earned an average of about $11.80 per hour. However, she is uncertain about why she has a negative y-intercept in her function equation, since she didn’t earn -$2.50 for working 0 hours.

20. Yes. Her equation models 20. Is her equation correct? Explain why or why not. the data closely. Not every See margin point contained in the function will fit the actual 21. Since she earned $0 for working zero hours, she now decides to include the point (0, 0) in her data set. How will this affect her function equation to model data; it is only an approx- the situation? imation of the relationship See margin between the data 21. Her new equation will be approximately f(x) = 11.3x. This equation also models the data closely, but now the function goes through the point (0, 0) specifically. DRAFT FOR

36 chaP ter 1: algeB raic Patterns PREVIEWONLY

36 CHAPTER 1: ALGEBRAIC PATTERNS TEKS AR.2A Determine the 1.4 Writing Exponential Functions patterns that identify the relationship between a function and its common FOCUSING QUESTION What are the characteristics of an exponential ratio or related finite dif- function? ferences as appropriate, including linear, quadrat- LEARNING OUTCOMES ic, cubic, and exponential ■■ I can determine patterns that identify an exponential function from its related functions. common ratios. ■■ I can classify a function as linear or exponential when I am given a table. AR.2B Classify a func- ■■ I can determine the exponential function from a table using common ratios, tion as linear, quadratic, including any restrictions on the domain and range. cubic, and exponential ■■ I can analyze patterns to connect the table to a function rule and communicate when a function is repre- the exponential pattern as a function rule. sented tabularly using finite differences or common ratios as appropriate ENGAGE AR.2C Determine the Miranda shared a cookie recipe on social function that models a giv- media with three friends. Each of Mi- en table of related values randa’s friends shared the cookie recipe with three of their friends. If this trend using finite differences and continues, how many people will receive its restricted domain and a cookie recipe in the fifth round? range. 243 people. MATHEMATICAL PROCESS SPOTLIGHT EXPLORE AR.1F Analyze mathematical relationships to Begin with a sheet of paper. Fold it in half and record the number of layers of paper connect and communicate after the fold in a table like the one shown. mathematical ideas. NUMBER OF ELPS NUMBER OF FOLDS LAYERS 5F Write using a va- 0 1 riety of grade-appropriate DRAFT1 2 sentence lengths, patterns, 2 4 and connecting words to combine phrases, clauses, 3 8 and sentences in increas- FOR4 16 ingly accurate ways as 5 32 more English is acquired. 6 64 VOCABULARY common ratio, base, expo- 1.4 • Writing ExponE ntial Functions 37 nent, multiplier, exponential relationship MATERIALS • 2 sheets of paper for each student PREVIEWONLY

37 1. What is the difference between the numbers of folds in consecutive rows in the table? 1 fold

2. What is the difference between the numbers of layers in consecutive rows in the table?

NUMBER OF NUMBER OF FOLDS LAYERS

0 1 ∆y = 2 – 1 = 1 1 2 ∆y = 4 – 2 = 2 2 4 ∆y = 8 – 4 = 4 3 8 ∆y = 16 – 8 = 8 4 16 ∆y = 32 – 16 = 16 5 32 ∆y = 64 – 32 = 32 6 64

3. Are the finite differences between the number of layers and the number of folds constant? How can you tell? No, because while ∆x = 1 and is constant for every successive row, ∆y is not constant. 4. Possible response: The 4. What patterns do you observe in the differences in the table? finite differences in the See margin. number of layers is the 5. Is this a linear relationship? How do you know? same as the number of lay- No, because the finite differences are not constant for the same ∆x. ers shifted down one row. For each fold, the number 6. What is the ratio between successive numbers of layers? 2 4 8 16 32 64 of layers doubles. – – – — — — 1 = 2 2 = 2 4 = 2 8 = 2 16 = 2 32 = 2 8. Possible response: An th th exponential function best 7. How many layers would there be after the 7 fold? 10 fold? describes this relationship 7th fold: 128, 10th fold: 1024 because each fold doubles 8. What type of function best describes this relationship? Explain your reasoning. the number of layers, so the number of layersDRAFT is See margin. multiplied by the same 9. Write an equation that could be used to determine y, the number of layers, if number, 2, for every fold you know x, the number of folds. you make. FORy = 2x

38 c H aptE r 1: algEBraic pattE rns

QUESTIONING STRATEGIES As students move through the second scenario, call attention to how they did the first scenario. Ask questions such as: • How does your data for the area of the folded region compare to the data for the number ofPREVIEW layers? • Which of the two situations is an increasingONLY function? Decreasing function?

38 CHAPTER 1: ALGEBRAIC PATTERNS Begin with a new sheet of paper. INSTRUCTIONAL HINTS Suggest that students write 10. What is area of the sheet of paper without any folds? measurements along the Possible response: If the paper is 8.5 × 11 inches, then the area is 93.5 square inches. edges of the piece of paper 11. Fold the paper in half and record the area of the region showing after the each time they fold it. fold in a table like the one shown. If necessary, round to the nearest tenth of a Create a need for an equa- square inch. tion rather than continuing Answers shown are for a regular sheet of 8.5 × 11 inch sheet of paper. If students use different size paper, then their answers will vary. the table. Ask students why an equation in helpful in NUMBER OF NUMBER OF both scenarios. FOLDS LAYERS 0 93.5

1 46.8

2 23.4

3 11.7

4 5.8

5 2.9

6 1.5

12. What is the difference between the numbers of folds in consecutive rows in the table? 1 fold

13. What is the difference between the numbers of layers in consecutive rows in the table?

NUMBER OF NUMBER OF FOLDS LAYERS 0 93.5 ∆y = 46.8 – 93.5 = −46.7 1 46.8 ∆y = 23.4 – 46.8 = −23.4 2 23.4 ∆y = 11.7 – 23.4 = −11.7 3 11.7 ∆y = 5.8 – 11.7 = −5.9 DRAFT4 5.8 ∆y = 2.9 – 5.8 = −2.9 5 2.9 ∆y = 1.5 – 2.9 = −1.4 6 FOR1.5

1.4 • Writing ExponE ntial Functions 39 PREVIEWONLY

39 14. Are the finite differences between the area of the regions and the number of folds constant? How can you tell? No, because while ∆x = 1 and is constant for every successive row, ∆y is not constant. 15. Possible response: The 15. What patterns do you observe in the differences in the table? finite differences in the See margin. number of layers is the 16. Is this a linear relationship? How do you know? same as the number of No, because the finite differences are not constant for the same ∆x. layers shifted down one row. For each fold, the 17. What is the ratio between successive areas of regions? area of the new region is 46.8 23.4 11.7 5.8 2.9 1.5 ≈ 0.5 = 0.5 = 0.5 ≈ 0.5 = 0.5 ≈ 0.5 about half of the area of 93.5 46.8 23.4 11.7 5.8 2.9 the previous region. 18. What would be the area of the region present after the 7th fold? 19. Possible response: An 7th fold: 0.8 exponential function best describes this relationship 19. What type of function best describes this relationship? Explain your reasoning. because each fold gener- See margin. ates a region that has half 20. Write an equation that could be used to determine y, the area of the region, if the area of the previous you know x, the number of folds. region. 1 x y = 93.5 – (2)

REFLECT ANSWERS: When the difference in x-values REFLECT is 1, the successive ratios are ■■ What do you notice about the successive ratios in each relationship? constant. See margin. The successive ratio is the same ■■ What relationship exists between the successive ratios in the depen- as the base in the equation. dent variable and the equations that you have written? See margin.

EXPLAIN

When you found finite differences that were constant for the de- Watch Explain and pendent variable, y, and the differences between values of the You Try It Videos DRAFTindependent variable, x, were the same, the relationship was linear. But as you have seen, this is not true for every functional relationship.FOR If the finite differences are not constant, look at the ratios between successive rows in the table. If these ratios are constant, then the constant ratio is called a common ratio and the relation- or click here ship is an exponential function. An exponential relationship is

40 c H aptE r 1: algEBraic pattE rns

HIGHER-ORDER THINKING STRATEGY Use a graphic organizer such as a Venn diagram to help students compare and contrast linear relationships and exponential relationships. The Venn diagram could be constructed in an interactive math notebook or math journal and students could add more information as theyPREVIEW learn more about linear and exponentialONLY relationships in this chapter.

40 CHAPTER 1: ALGEBRAIC PATTERNS one in which there is repeated multiplication. For example, a geometric sequence has a constant multiplier, so the sequence In an exponential function, if the base can be represented as an exponential function with a domain is greater than 1, then of whole numbers. the function represents exponential growth. If Let’s look more closely at an exponential function. The table the base is between 0 below shows the relationship between x and f(x). In an and 1, then the function exponential function, f(x) = abx, b represents the base of the expo- represents exponential decay. nential function, which is also a common ratio or constant multiplier. The parameter a represents an initial value or y-intercept.

x y = f(x)

0 a ∆x = 1 – 0 = 1 ∆y = ab – a = a(b – 1) 1 ab ∆x = 2 – 1 = 1 ∆y = ab(b) – ab = ab(b – 1) 2 ab(b) ∆x = 3 – 2 = 1 ∆y = ab(b)(b) – ab(b) = ab2(b – 1) 3 ab(b)(b) ∆x = 4 – 3 = 1 ∆y = ab(b)(b)(b) – ab(b)(b) = ab3(b – 1) 4 ab(b)(b)(b) ∆x = 5 – 4 = 1 ∆y = ab(b)(b)(b)(b) – ab(b)(b)(b) = ab4(b – 1) 5 ab(b)(b)(b)(b)

Unlike a linear function, the finite differences for an exponential function, f(x) = abx, are not constant. Instead, the multiplicative pattern that is present in the original data repeats in the finite differences.

Instead of looking at the finite differences, for an exponential function, take a closer look at the successive ratios.

x y = f(x)

0 a yn ab ∆x = 1 – 0 = 1 y = a = b n-1 1 ab yn a(b)(b) ∆x = 2 – 1 = 1 y = a(b) = b n-1 2 ab(b) yn a(b)(b)(b) ∆x = 3 – 2 = 1 y = a(b)(b) = b n-1 DRAFT3 ab(b)(b) yn a(b)(b)(b)(b) ∆x = 4 – 3 = 1 y = a(b)(b)(b) = b n-1 4 ab(b)(b)(b) yn a(b)(b)(b)(b)(b) ∆x = 5 – 4 = 1 y = a(b)(b)(b)(b) = b FORn-1 5 ab(b)(b)(b)(b)

1.4 • Writing ExponE ntial Functions 41 PREVIEWONLY

41 Notice that in the table, ∆x = 1. Knowing that, the common ratio for successive rows is equivalent to b, which is the base of the exponential relationship.

COMMON RATIOS AND EXPONENTIAL FUNCTIONS

In an exponential function, the ratios between succes- y sive y-values, n , are constant if the differences between yn-1 successive x-values, ∆x, are also constant.

If the ratios of successive values of the dependent variable in a table of values are constant, then the values represent an exponential function.

You can also use the common ratio to write an exponential function describing the relationship between the independent and dependent variables.

INTEGRATE TECHNOLOGY EXAMPLE 1 Use technology such as a graphing calculator Does the set of data shown below represent an exponential function? Justify your or spreadsheet app on a answer. display screen to show students how geometric x y sequences are related to 0 1 exponential functions. If the exponential function’s 1 5 domain is restricted to whole numbers, then the 2 25 resulting points represent a 3 125 geometric sequence. DRAFT 4 625 FOR

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42 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES STEP 1 Determine the finite differences between successivex -values Determine if each of the and the ratios between successive y-values. data sets below represent an exponential function. x y Justify your answer.

0 1 y n = 5 = 5 1. ∆x = 1 – 0 = 1 yn-1 1 1 5 y 0 1 2 3 4 n = 25 = 5 x ∆x = 2 – 1 = 1 yn-1 5 2 6 18 54 162 2 25 y y n = 125 = 5 ∆x = 3 – 2 = 1 yn-1 25 3 125 y Exponential n = 625 = 5 ∆x = 4 – 3 = 1 yn-1 125 4 625 2. x -1 0 1 2 3 STEP 2 Determine whether or not the differences between successive y 8 4 2 1 0.5 x-values and ratios between successive y-values are constant. Exponential The differences between successive values of x, ∆x, are all 1, so they are constant. 3. y The ratios between successive values of y, n , are all 5, so they are 3 6 9 12 15 yn-1 x constant. y 12 17 22 27 32

y n y y Not Exponential n-1 = 5 = 5 for all pairs of ∆x and n . ∆x 1 yn-1 Note: Pay special attention to STEP 3 Determine whether or not the set of data represents an students’ justification for the exponential function. third additional example. Be Yes, the set of data represents an exponential function because the differences between mindful that some students successive x-values and the ratios between successive y-values are constant. may say “no, because the differences between successive values of x are not 1.” The differences are constant, but the ratio between successive DRAFT values of y are not constant. FOR

1.4 • Writing ExponE ntial Functions 43 PREVIEWONLY

43 YOU TRY IT! #1 ANSWER: YOU TRY IT! #1 Yes, the set of data represents an exponential function, Does the set of data shown below represent an exponential function? Justify your answer. because the finite differences in x and the successive ratios in y are both constant. x y 0 1.2

1 1.44

2 1.728

3 2.0736

See margin.

EXAMPLE 2 Does the set of data shown below represent an exponential function? Justify your answer.

x y

0 8

1 4

2 2

3 0.8

4 0.4 DRAFT FOR

44 c H aptE r 1: algEBraic pattE rns PREVIEWONLY

44 CHAPTER 1: ALGEBRAIC PATTERNS STEP 1 Determine the finite differences between successive x-values and ratios between successive y-values.

x y

0 8 y n = 4 = 1 ∆x = 1 – 0 = 1 yn-1 8 2 1 4 y n = 2 = 1 ∆x = 2 – 1 = 1 yn-1 4 2 2 2 y n = 0.8 = 2 ∆x = 3 – 2 = 1 yn-1 2 5 3 0.8 y n = 0.4 = 1 ∆x = 4 – 3 = 1 yn-1 0.8 2 4 0.4

STEP 2 Determine whether or not the differences are constant.

The differences inx , ∆x, are all 1, so they are constant.

yn The ratios of successive values of y, y , are not all the same, so they are not constant. n-1

STEP 3 Determine whether or not the set of data represents an exponential function.

No, the set of data does not represent an exponential function because even though the differences in successive values of x are constant, the ratios between successive values of y are not constant.

YOU TRY IT! #2 YOU TRY IT! #2 ANSWER: Does the set of data shown below represent an exponential function? Justify your No, the set of data does not rep- answer. resent an exponential function, because although the first dif- x y ferences in x are constant, the ratios between successive values DRAFT0 9.2 of y are not constant. 1 18.4 2 FOR46 3 92

4 230

See margin. 1.4 • Writing ExponE ntial Functions 45 PREVIEWONLY

45 ADDITIONAL EXAMPLES Determine if each of the EXAMPLE 3 data sets below represent For the data set below, determine if the relationship is an exponential function. If so, an exponential function. If write a function relating the variables. so, write a function relating the variables. Justify your x y answer. 1 4 1. 2 10 x 0 1 2 3 4 y -1 -2 -4 -8 -16 3 25

Exponential, y = -2x 4 62.5

2. 5 156.25

x 18 15 12 9 STEP 1 Determine the finite differences between successive x-values y 24 22 20 18 and the ratios between successive y-values. Not Exponential x y 3. 1 4 y 10 2 3 5 8 12 n = = 2.5 x ∆x = 2 – 1 = 1 yn-1 4 2 2 10 — 1 -1.5 2.25 -3.375 y 25 y -3 n = = 2.5 ∆x = 3 – 2 = 1 yn-1 10 3 25 y Not Exponential n = 62.5 = 2.5 ∆x = 4 – 3 = 1 yn-1 25 4 62.5 y 4. n = 156.25 = 2.5 ∆x = 5 – 4 = 1 yn-1 62.5 5 156.25 x 1 2 3 4 y 4.5 6.75 10.125 15.1875 Exponential, y = 3(1.5)x STEP 2 Determine whether or not the relationship is an exponential function.

The differences inx , ∆x, are all 1, so they are constant.

yn DRAFTThe ratios between successive values of y, y , are all 2.5, so they are n-1 constant. FORSince the first differences in x and the successive ratios in y are all constant, the relationship is an exponential function.

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46 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES STEP 3 Determine the y-intercept of the exponential function. After completing the additional examples on pg. 46, Work backwards from x = 1 and y = 4. draw students’ attention back to examples 2 and 3. x y Based on prior lessons, ask if they can write a function 0 a 4 = 2.5 1 – 0 = 1 a rule for either of those sets

1 4 10 of data? = 2.5 2 – 1 = 1 4

2 10 Yes, Additional Example #2 is linear. The function rule is 2 4 y = –x -12. a = 2.5 3

4 a(a ) = 2.5(a)

4 = 2.5a 4 2.5a — — 2.5 = 2.5 1.6 = a

The y-intercept is (0, 1.6).

STEP 4 Use the y-coordinate of the y-intercept and the common ratio to write the exponential function.

y = 1.6(2.5)x

YOU TRY IT! #3

For the data set below, determine if the relationship is an exponential function. If so, write a function relating the variables.

x y DRAFT1 14 2 98 3 FOR686 4 4802

Answer: y = 2(7)x

1.4 • Writing ExponE ntial Functions 47 PREVIEWONLY

47 QUESTIONING STRATEGIES Assist students in process- EXAMPLE 4 ing their learning by hav- For the data set below, determine if the relationship is an exponential function. If so, ing students write the steps write a function relating the variables. in their own words.

• How would you ex- x y plain to a friend how you know if a set of 1 1000 data represents an ex- 2 400 ponential function? • Explain how to write 3 160 an exponential func- 4 64 tion based on a table of values to someone 5 25.6 who has never seen this math lesson. STEP 1 Determine the first differences between successivex -values and the ratios between successive y-values.

x y

1 1000 y n = 400 = 0.4 ∆x = 2 – 1 = 1 yn-1 1000 2 400 y n = 160 = 0.4 ∆x = 3 – 2 = 1 yn-1 400 3 160 y n = 64 = 0.4 ∆x = 4 – 3 = 1 yn-1 160 4 64 y n = 25.6 = 0.4 ∆x = 5 – 4 = 1 yn-1 64 5 25.6

STEP 2 Determine whether or not the relationship is an exponential function.

The differences inx , ∆x, are all 1, so they are constant.

yn DRAFTThe ratios between successive values of y, y , are all 0.4, so they are n-1 constant. FORSince the first differences in x and the successive ratios in y are all constant, the relationship is an exponential function.

48 c H aptE r 1: algEBraic pattE rns PREVIEWONLY

48 CHAPTER 1: ALGEBRAIC PATTERNS STEP 3 Determine the y-intercept of the exponential function.

Work backwards from x = 1 and y = 1000.

x y

0 a 1000 1 – 0 = 1 a = 0.4

1 1000 400 = 0.4 2 – 1 = 1 1000 2 400

1000 a = 0.4 1000 a( a ) = 0.4(a) 1000 = 0.4a 1000 0.4a 0.4 = 0.4 2500 = a

The y-intercept is (0, 2500).

STEP 4 Use the y-coordinate of the y-intercept and the common ratio to write the exponential function.

y = 2500(0.4)x

YOU TRY IT! #4

For the data set below, determine if the relationship is an exponential function. If so, write a function relating the variables.

x y DRAFT1 81 2 27 3 FOR9 4 3

5 1

1 x — Answer: y = 243(3) 1.4 • Writing ExponE ntial Functions 49 PREVIEWONLY

49 PRACTICE/HOMEWORK

For questions 1-4 use finite differences to determine if each table represents an exponential function.

1. x y 2. x y 0 2 0 3 1 6 1 4 2 18 2 7 3 54 3 12 Yes No

3. x y 4. x y 0 0 0 3 1 1 1 6 2 8 2 12 3 27 3 24 No Yes

For questions 5-8 identify if each table represents an exponential function or not. If the table represents an exponential function, identify the common ratio.

5. x y 6. x y 1 2 1 2 2 4 2 4 3 6 3 8 4 8 4 16 Exponential Function? No Exponential Function? Yes Common Ratio: None Common Ratio: 2

7. x y 8. x y 1 3 1 4 2 4.5 2 1 3 6.75 3 0.25 4 10.125 4 0.0625 Exponential Function? Yes Exponential Function? Yes Common Ratio: 1.5 Common Ratio: 0.25

DRAFT FOR

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50 CHAPTER 1: ALGEBRAIC PATTERNS For questions 9-12 use the situation below. CRITICAL THINKING A sheet of paper is 0.1 mm thick. When the paper is folded in half, the total thickness of the layers of paper is 0.2 mm. When the paper is folded in half again, the total thickness of the layers of paper is 0.4 mm.

9. Complete the table below to represent the situation.

NUMBER TOTAL THICKNESS OF FOLDS OF LAYERS x y 0 0.1

1 0.2

2 0.4 3 0.8 4 1.6

10. Does the situation represent a linear function or an exponential function? Justi- 10. Exponential function fy your answer. See margin. While ∆x is constant, ∆y is not constant. The situ- 11. Which of the following represents the function that models this situation? ation does have a common A. y = x + 0.1 C. y = 0.1 • 2x B. y = 2 • 0.1x D. y = 2x + 0.1 ratio of 2.

12. Which of the following statements are true about the situation? • ∆x = 1 • The function is linear. • The situation is an example • The function is decreasing. of exponential decay. • ∆y = 0.1 • The function is increasing. • The common ratio is 0.2. • The common ratio is 2. • The situation is an example of ex- • The y-intercept is (0, 0.1). ponential growth.

For questions 13-18 identify if each table represents an exponential function or not. If the table represents an exponential function, write the function relating the variables.

13. x y 14. x y 0 0 0 40 1DRAFT 4 1 8 2 32 2 1.6 3 108 3 0.32 Exponential Function? No FOR Exponential Function? Yes Function: N/A Function: y = 40(0.2)x

1.4 • Writing ExponE ntial Functions 51 PREVIEWONLY

51 15. x y 16. x y 0 50 1 300 1 25 2 150 2 12.5 3 100 3 6.25 4 75 Exponential Function? Yes Exponential Function? No Function: y = 50(0.5)x Function: N/A 17. x y 18. x y 1 4500 1 14 2 6750 2 56 3 10,125 3 224

4 15,187.5 4 896 Exponential Function? Yes Exponential Function? Yes Function: y = 3,000(1.5)x Function: y = 3.5(4)x

For questions 19-20 use the situation below. CRITICAL THINKING A sheet of paper has an area of 100 square inches. When the paper is cut in half, the area of one piece is 50 square inches. When that piece is cut in half, the area of one piece is 25 square inches.

NUMBER OF AREA OF CUTS ONE PIECE x y 0 100

1 50

2 25

19. What would be the area of one piece after 5 cuts? 3.125 square inches

20. Write the function relating the variables. DRAFTy = 100(0.5)x FOR

52 c H aptE r 1: algEBraic pattE rns PREVIEWONLY

52 CHAPTER 1: ALGEBRAIC PATTERNS TEKS Modeling with Exponential AR.2B Classify a func- 1.5 Functions tion as linear, quadratic, cubic, and exponential when a function is repre- FOCUSING QUESTION How can you use common ratios to construct an ex- sented tabularly using fi- ponential model for a data set? nite differences or common ratios as appropriate. LEARNING OUTCOMES ■■ I can use finite differences or common ratios to classify a function as either lin- AR.2D Determine ear or exponential when I am given a table of values. a function that models ■■ I can use common ratios to write an exponential function that describes a data real-world data and math- set. ematical contexts using ■■ I can apply mathematics to problems that I see in everyday life, in society, and finite differences such as in the workplace. the age of a tree and its circumference, figurative numbers, average velocity, ENGAGE and average acceleration. DeAnna dropped a basketball and let it bounce several MATHEMATICAL times. What would a graph of the height of the basketball versus time look like? PROCESS SPOTLIGHT See margin. AR.1A Apply mathematics to problems arising in everyday life, society, EXPLORE and the workplace. ELPS According to international basketball guidelines, a basketball should be inflated such that when dropped from a height of 1.8 meters, the ball should bounce and 5B Write using rebound to a height of at least 1.2 meters but no more than 1.4 meters. newly acquired basic vocabulary and content-based Bounce height is an important aspect to making sure that a grade-level vocabulary. ball used in any sport is properly inflated or is not worn out. VOCABULARY DIRECTIONS ■■ Tape two meter sticks to the wall and use them to mea- exponential function, com- sure the bounce height, or the height to which the ball mon ratios, exponential bounces, when dropped from a given height. decay, exponential growth ■■ BeginDRAFT with 180 centimeters. Select a spot on the ball, such as the highest point on the top of the ball, to use as a MATERIALS consistent reference point. Drop the ball from this height • 1 ball for each student and record the height of the first bounce.FOR group (variety of balls, ■■ Repeat for two more trials and calculate the average such as soccer balls, bounce height for a drop from 180 centimeters. basketballs, tennis balls, and racquetballs, keeping in mind that not all balls will bounce 1.5 • Modeling with e xponential Functions 53 on carpet) • 2 meter sticks per student group ENAGE ANSWER: • masking tape Possible sketch PREVIEWONLY

53 STRATEGIES FOR ■■ Use the average bounce height as the starting point for the second bounce. Record SUCCESS the bounce height when the ball is released from this drop height for three trials. It is always best to have Calculate the average bounce height. students collect data on ■■ Repeat for a total of 6 bounces. their own. However, if it is ■■ Record your information in a table like the one shown. not possible to collect data Sample data: for this activity, provide DROP BOUNCE BOUNCE BOUNCE AVERAGE students with the sample BOUNCE HEIGHT HEIGHT 1 HEIGHT 2 HEIGHT 3 HEIGHT data so that they can use (CM) (CM) (CM) (CM) (CM) it to answer the questions 180 135 130 140 135 with this activity. 135 95 105 100 100 Provide different student 100 70 70 85 75 groups with different balls; e.g., basketballs, racquet- 75 50 55 60 55 balls, tennis balls (only if 55 45 35 40 40 used on tile floors), golf 40 30 25 35 30 balls, etc. Each type of ball will generate a different 1. If you were to drop the ball once and let it continue bouncing until it stopped, model. After the activity, a height versus time graph of the ball might look like the figure shown. Use an discuss with students why ordered pair (bounce number, height of bounce) to label each point shown on the models are different the graph. and how the parameters a HT(CM) and b change with different (0, 180) types of balls. (1, 135) (2, 100) TECHNOLOGY (3, 75) (4, 55) INTEGRATION (5, 40)(6, 30) Motion detectors and calculators can be used to capture and graph the height of a ball as it bounces in real time. You can use 2. Calculate the finite differences between the bounce number and the average a motion detector to record bounce height. Do the data appear to be linear? How do you know? the height versus time of a See margin. series of bounces, and then 3. Calculate the ratios between the average bounce heights for successive bounces. use the graph and DRAFTtrace Do the data appear to be exponential? How do you know? features of the calculator See margin. to determine the height of each of the first 5 or 6 4.FOR What is the starting point, or y-intercept, of the data? The starting point is 180 centimeters. bounces and use that data to complete the activity.

54 chapteR 1: algeBRaic patteR ns

2. +1 +1 +1 +1 +1 +1

BOUNCE 0 1 2 3 4 5 6 NUMBER AVERAGE PREVIEW BOUNCE 180 135 100 75 55 40 30 HEIGHT ONLY (CM)

-45 -35 -25 -20 -15 -10

The data do not appear to be linear since the finite differences are not constant.

54 CHAPTER 1: ALGEBRAIC PATTERNS 5. Use either the finite differences or common ratios to determine a function that best models the relationship between the bounce number, x, and the average bounce height, f(x). If it is a linear function, use slope-intercept form. If it is an exponential function, use the form f(x) = abx. f(x) = 180(0.75)x

6. What do the y-intercept and either rate of change or base from your function 6. The y-intercept, 180 rule mean in the context of this situation? See margin. centimeters, represents the height of the ball when it 7. Use your model to predict the height of the 8th bounce. was first dropped. Answer using sample data: f(8) = 180(0.75)8 ≈ 18 centimeters The base, 0.75 or 75%, 8. What is the diameter of the ball that you used? (You may need to use the for- means that the ball will mula C = πd to calculate the diameter of the ball.) bounce back 75% of the Answers may vary. A basketball has an approximate diameter of about height of the previous 24 centimeters. bounce. 9. Thinking about the diameter of the ball, what will be the last bounce observed before the ball bounces to a height that is less than the diameter? 9. Answers may vary de- See margin. pending on the ball used and the data collected. For a basketball with the REFLECT sample data:

7 ■■ How can you determine an exponential function model for a data set if f(7) = 180(0.75) ≈ 24 cen- the common ratios are not exactly the same, but are very close to each timeters, so the 7th bounce other? will be the same height as See margin. the diameter of the ball, ■■ Once you have determined your exponential function model, how can and should be the last you use the model to determine a value of the independent variable that generates a particular value of the dependent variable? “bounce” that is observed. See margin.

REFLECT ANSWERS: EXPLAIN Use an average value of the common ratios as the base of an Exponential function models can be used to represent sets of math- Watch Explain and You Try It Videos exponential function model. ematical and real-world data. If an exponential function is decreasing, then the function is called an exponential decay function, Make a graph of the exponential since the values of the dependent variable, f(x), decay, or become function and then locate the smaller, asDRAFT the values of the independent variable, x, increase. The ordered pair along the curve relationship between bounce height and drop height is an exponential decay relationship. that contains the value of the dependent variable, usually y. Other exponential functions are increasingFOR and are called expo- or click here The x-coordinate of this ordered nential growth functions. The values of the dependent variable, f(x), grow, or become pair is the value of the inde- larger, as the values of the independent variable, x, increase. Population growth is pendent variable that generates sometimes exponential when the population of a city or county grows at the same that y-value. percent each year.

1.5 • Modeling with e xponential Functions 55

3. +1 +1 +1 +1 +1 +1

BOUNCE 0 1 2 3 4 5 6 NUMBER AVERAGE PREVIEW BOUNCE 180 135 100 75 55 40 30 HEIGHT ONLY (CM)

135 100 75 55 40 30 = 0.75 ≈ 0.74 = 0.75 ≈ 0.73 ≈ 0.73 = 0.75 180 135 100 75 55 40

The data appear to be exponential since the successive ratios are very close to a constant of 0.75.

55 For example, the table below shows the population of Hays County, Texas, for certain years since 1985.

Instead of looking at the finite differences, for an exponential function, take a closer look at the successive ratios.

5-YEAR YEAR POPULATION, INTERVAL, x f(x)

0 1985 56,225 y 65,767 n = ≈ 1.17 ∆x = 1 – 0 = 1 yn-1 56,225 1 1990 65,767 y 78,956 n = ≈ 1.20 ∆x = 2 – 1 = 1 yn-1 65,767 2 1995 78,956 y 99,070 n = ≈ 1.25 ∆x = 3 – 2 = 1 yn-1 78,956 3 2000 99,070 y 126,470 n = ≈ 1.28 ∆x = 4 – 3 = 1 yn-1 99,070 4 2005 126,470 y 158,289 n = ≈ 1.25 ∆x = 5 – 4 = 1 yn-1 126,470 5 2010 158,289 y 197,298 n = ≈ 1.25 ∆x = 6 – 5 = 1 yn-1 158,289 6 2015 197,298

Data Source: U.S. Census Bureau and Texas Department of State Health Services

The successive ratios are not equal, but are all close to the same value, 1.25. Calculate the average ratio and use that as the base, b, for the exponential function model.

1.17 + 1.20 + 1.25 + 1.28 + 1.25 + 1.25 ≈ 1.23 6

Use the initial population for 5-Year Interval 0, which is 56,225, as the starting point, a, to write the function, f(x) = 56,225(1.23)x. Once you have a function model, you can use that model to make predictions.

STARTING POINT DRAFT f(x) = 56,225(1.23)x FOR COMMON RATIO

56 chapteR 1: algeBRaic patteR ns PREVIEWONLY

56 CHAPTER 1: ALGEBRAIC PATTERNS Make a scatterplot of the data and graph the function rule over the scatterplot. QUESTIONING STRATEGIES Notice that not all of the data points The scatterplot of data and lie on the curve representing the POPULATION OF HAYS COUNTY, TEXAS graphed function rule are function model. That is because shown to the left. the successive ratios are not exactly equal, but are close to 1.23. • When looking at the two together, how Community leaders need to know how many people will live in the do you know that the community in order to decide graph is a good repre- how many fire stations, schools, or sentation of the data? restaurants to build. Demographers, • Which point(s) from or people who study population the data are included in trends, use population models like the graph? (Listen for this one to make predictions about how many people will live in a com- POPULATION students to mention the munity in a particular year. Com- y-intercept.) munity and business leaders rely on • Why is the y-intercept demographers in order to help them an exact point both make better decisions for people liv- from the data and func- ing in a particular community. tion? A graph helps you to visualize the 5-YEAR INTERVALS SINCE 1985 data in order to make predictions. For example, interval 7 represents seven 5-year intervals, or 35 years, since 1985. 1985 + 35 = 2020. The population model contains the point (7, 240,000) which means that at interval 7, or the year 2020, the population of Hays County, Texas, could be 240,000.

MODELING WITH EXPONENTIAL FUNCTIONS

Real-world data rarely follows exact patterns, but you can use patterns in data to look for trends. If the data increases or decreases with approximately the same ratio, then an exponential function model may be appropriate for the data set.

Of course, not all exponential relationships involve growth. Automobile depreciation is a loss in the value of an automobile over time. If the value of an automobile loses a percent ofDRAFT its value each year, then the depreciation is an exponential decay. FOR

1.5 • Modeling with e xponential Functions 57 PREVIEWONLY

57 EXAMPLE 1 Approximate radiation levels, in millirads per hour, near the Fukushima nuclear power plant near Naraha, Japan are shown in the table below.

4-DAY DATE RADIATION INTERVAL, x LEVEL, f(x)

0 MARCH 22, 2011 71.2

1 MARCH 26, 2011 39.8

2 MARCH 30, 2011 22.3

3 APRIL 3, 2011 12.5

4 APRIL 7, 2011 7.1

5 APRIL 11, 2011 3.9

6 APRIL 15, 2011 2.2

Data Source: U.S. Department of Energy

Use the data set to determine if the relationship is linear or exponential.

STEP 1 Determine the finite differences in values of x and values of f(x).

4-DAY DATE RADIATION INTERVAL, x LEVEL, f(x)

0 MARCH 22, 2011 71.2 ∆x = 1 – 0 = 1 ∆f(x) = 39.8 – 71.2 = –31.4 1 MARCH 26, 2011 39.8 ∆x = 2 – 1 = 1 ∆f(x) = 22.3 – 39.8 = –17.5 2 MARCH 30, 2011 22.3 ∆x = 3 – 2 = 1 ∆f(x) = 12.5 – 22.3 = –9.8 3 APRIL 3, 2011 12.5 ∆x = 4 – 3 = 1 ∆f(x) = 7.1 – 12.5 = –5.4 DRAFT4 APRIL 7, 2011 7.1 ∆x = 5 – 4 = 1 ∆f(x) = 3.9 – 7.1 = –3.2 5 APRIL 11, 2011 3.9 ∆x = 6 – 5 = 1 ∆f(x) = 2.2 – 3.9 = –1.7 FOR6 APRIL 15, 2011 2.2

58 chapteR 1: algeBRaic patteR ns

ADDITIONAL EXAMPLE SALARY 5-YEAR YEAR (MILLIONS OF The average salary, in millions of dollars, for INTERVAL, x professional players on a national sports team is DOLLARS), f(x) shown in the table below. Use the data set to determine if the PREVIEWrelationship is linear or exponential. 0 1995 1.89 Exponential ONLY1 2000 2.4 Encourage students to follow the steps from Ex- ample 1 to complete the additional example. 2 2005 2.88

3 2010 3.8

4 2015 4.38

58 CHAPTER 1: ALGEBRAIC PATTERNS STEP 2 Determine the ratios between successive values of f(x).

4-DAY DATE RADIATION INTERVAL, x LEVEL, f(x)

0 MARCH 22, 2011 71.2 yn 39.8 ∆x = 1 – 0 = 1 y = ≈ 0.559 n-1 71.2 1 MARCH 26, 2011 39.8 yn 22.3 ∆x = 2 – 1 = 1 y = ≈ 0.560 n-1 39.8 2 MARCH 30, 2011 22.3 yn 12.5 ∆x = 3 – 2 = 1 y = ≈ 0.561 n-1 22.3 3 APRIL 3, 2011 12.5 yn 7.1 ∆x = 4 – 3 = 1 y = ≈ 0.568 n-1 12.5 4 APRIL 7, 2011 7.1 yn 3.9 ∆x = 5 – 4 = 1 y = ≈ 0.549 n-1 7.1 5 APRIL 11, 2011 3.9 yn 2.2 ∆x = 6 – 5 = 1 y = ≈ 0.564 n-1 3.9 6 APRIL 15, 2011 2.2

STEP 3 Determine whether the finite differences or the ratios between successive values of f(x) are approximately constant.

■■ The finite differences range in value from –31.4 to –1.7. This is a wide range, so the finite differences are not even approximately constant. ■■ The ratios between successive values of f(x) range from 0.549 to 0.568. These values are all close together, so the ratios between successive values of f(x) are approximately constant.

The set of data represents an exponential function, rather than a linear function, because the differences in x are constant and the ratios between successive values of f(x) are approximately constant. DRAFT FOR

1.5 • Modeling with e xponential Functions 59

ADDITIONAL EXAMPLE Students in a Biology class were studying how water evaporates over time. They measured the volume of the water in milliliters each day and recorded the data shown below. Determine if the relationship is linear or exponential.

DAY, x 0 1 2 PREVIEW3 4 WATER LEVEL (ML), f(x) 15 13.75 12.5 11.25 10 ONLY Exponential Remind students that the first step is to check the finite differences and then check ratios if the finite differences are not constant.

59 YOU TRY IT! #1 ANSWER: YOU TRY IT! #1 The set of data represents a A major league baseball player’s average in successive seasons is recorded in the table. linear function, rather than an exponential function, because 1-YEAR YEAR BATTING the finite differences in x are INTERVAL, x AVERAGE, f(x) constant and the finite differences in f(x) are approximately 0 2009 0.213 constant. 1 2010 0.242

2 2011 0.271

3 2012 0.301

4 2013 0.330

5 2014 0.359

Determine whether the relationship is linear or exponential. See margin.

EXAMPLE 2 The table below shows the average viewership for the Super Bowl, in numbers of households.

3-YEAR YEAR VIEWERSHIP, INTERVAL, x f(x)

0 1970 23,050

1 1973 27,670

2 1976 29,440

3 1979 35,090 DRAFT4 1982 40,020 FORData Source: http://www.nielsen.com/us/en.html Generate an exponential function model for Super Bowl viewership. How many households does your model predict will watch the Super Bowl in the year 2018.

60 chapteR 1: algeBRaic patteR ns

ADDITIONAL EXAMPLE Have students write the exponential function to match the data that was given in the exponential additional examples on pages 58 and 59. pg. 58: f(x) = 1.89(1.24)x pg. 59 f(x) = 15(0.9)PREVIEWx ONLY

60 CHAPTER 1: ALGEBRAIC PATTERNS STEP 1 Determine the finite differences in x-values and the ratios between successive values of f(x).

3-YEAR YEAR VIEWERSHIP, INTERVAL, x f(x)

0 1970 23,050 yn 27,670 ∆x = 1 – 0 = 1 y = ≈ 1.20 n-1 23,050 1 1973 27,670 y n = 29,440 ≈ 1.06 ∆x = 2 – 1 = 1 yn-1 27,670 2 1976 29,440 yn 35,090 ∆x = 3 – 2 = 1 y = ≈ 1.19 n-1 29,440 3 1979 35,090 y n = 40,020 ≈ 1.14 ∆x = 4 – 3 = 1 yn-1 35,090 4 1982 40,020

STEP 2 Calculate the average of the ratios and use this value for the base, b, in your exponential function model.

1.20 + 1.06 + 1.19 + 1.14 ≈ 1.15 4

STEP 3 Using a, the initial viewership value from the table, and the calculated value for b, write an exponential function model.

f(x) = 23,050(1.15)x

STEP 4 Use your exponential function model to predict the 2018 viewership, in number of households.

2018 – 1970 = 48, so the number of 3-year intervals since 1970, x = 16.

f(16) = 23,050(1.15)16 ≈ 215,693.2

According to the exponential function model f(x) = 23,050(1.15)x, approximately 215,693 householdsDRAFT will view the Super Bowl in the year 2018. FOR

1.5 • Modeling with e xponential Functions 61

ADDITIONAL EXAMPLE Alan purchased a new luxury car. He recorded the value of his car each year after his purchase in the table shown. Generate an exponential function model for the value of Alan’s car, in dollars.

YEARS SINCE PURCHASE, x 2 3 4 5 6 VALUE (DOLLARS), f(x) 41,115 36,181PREVIEW33,287 30,291 26,353

ONLYx Using the average ratio of 0.89 to find x = 0 to the nearest whole number, the function would be f(x) = 51907(0.89) . Based on your model, how much is Alan’s car depreciating in value each year? Alan’s car is decreasing about 11% in value each year. How much did Alan pay for his car originally? Alan paid $51,907 for his car. What will the value of Alan’s car be in 10 years? After 10 years, Alan’s car will be worth about $16,186.

61 YOU TRY IT! #2 ANSWER: YOU TRY IT! #2 f(x) = (2.113)x; According to A biologist places a single paramecium in a petri dish to observe the rate of population the exponential function model, growth of this single-celled organism. The biologist’s observations of the number of there will be approximately 840 paramecia in the petri dish over time are recorded in the table below. paramecia in the petri dish on the tenth day of the experiment. 1-DAY DAY OF POPULATION, INTERVAL, x EXPERIMENT f(x) 0 1ST 1

1 2ND 2

2 3RD 5

3 4TH 10

4 5TH 22

5 6TH 44

6 7TH 87

Generate an exponential function model for the paramecium population. How many paramecia does your model predict will be in the petri dish on the tenth day of the experiment? See margin.

EXAMPLE 3 The population of Throckmorton County, Texas, in each census since 1950 is shown in the table below.

10-YEAR POPULATION, YEAR INTERVAL, x f(x)

0 1950 3,618

1 1960 2,767

2 1970 2,205 DRAFT3 1980 2,053 4 1990 1,880 FOR5 2000 1,850 6 2010 1,641

Data Source: U.S. Census Bureau

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62 CHAPTER 1: ALGEBRAIC PATTERNS INSTRUCTIONAL HINTS Problems like these take YOU TRY IT! #2 Use the data set to generate an exponential model. Use your model to predict the mathematical stamina and population of Throckmorton County, Texas, in the year 2020. attention to detail. Have A biologist places a single paramecium in a petri dish to observe the rate of population growth of this single-celled organism. The biologist’s observations of the number of STEP 1 Determine the finite differences in x-values and the ratios students write the steps paramecia in the petri dish over time are recorded in the table below. between successive values of f(x). they would use to write an exponential function 1-DAY DAY OF POPULATION, 10-YEAR YEAR POPULATION, from real world data. Then INTERVAL, x f(x) INTERVAL, x EXPERIMENT f(x) have students compare this 0 1ST 1 0 1950 3,618 y n = 2,767 ≈ 0.765 writing to the steps they ∆x = 1 – 0 = 1 yn-1 3,618 1 2ND 2 wrote in the last section for 1 1960 2,767 y n = 2,205 ≈ 0.797 RD ∆x = 2 – 1 = 1 y exponential functions with 2 3 5 n-1 2,767 2 1970 2,205 constant ratios. Finally, yn 2,053 3 4TH 10 ∆x = 3 – 2 = 1 y = ≈ 0.931 n-1 2,205 give students time to brief- TH 3 1980 2,053 y 4 5 22 n 1,880 ly explain the steps to one ∆x = 4 – 3 = 1 y = ≈ 0.916 n-1 2,053 5 6TH 44 4 1990 1,880 to two other students. yn 1,850 ∆x = 5 – 4 = 1 y = ≈ 0.984 6 7TH 87 n-1 1,880 5 2000 1,850 yn 1,641 ∆x = 6 – 5 = 1 y = ≈ 0.887 n-1 1,850 Generate an exponential function model for the paramecium population. How many 6 2010 1,641 paramecia does your model predict will be in the petri dish on the tenth day of the experiment? See margin. STEP 2 Calculate the average of the ratios and use this value for the base, b, in your exponential function model.

0.765 + 0.797 + 0.931 + 0.916 + 0.984 + 0.887 ≈ 0.88 EXAMPLE 3 6 The population of Throckmorton County, Texas, in each census since 1950 is shown in the table below. STEP 3 Using a, the initial population value from the table, and the calculated value for b, write an exponential function model.

10-YEAR POPULATION, YEAR x INTERVAL, x f(x) f(x) = 3618(0.88)

0 1950 3,618 STEP 4 Use your exponential function model to predict the 2020 population. 1 1960 2,767

2 1970 2,205 2020 – 1950 = 70, so the number of 10-year intervals since 1950, x = 7. 3 1980 2,053 DRAFTf(7) = 3618(0.88)7 ≈ 1,478.59 4 1990 1,880 According to the exponential function model f(x) = 3618(0.88)x, the population of 5 2000 1,850 Throckmorton County, Texas, will be approximatelyFOR 1,479 people in the year 2020. 6 2010 1,641

Data Source: U.S. Census Bureau

62 chapteR 1: algeBRaic patteR ns 1.5 • Modeling with e xponential Functions 63 PREVIEWONLY

63 YOU TRY IT! #3 ANSWER: YOU TRY IT! #3 f(x) = 19.9(0.949)x ; According Percentages of adults in the United States who smoke cigarettes on a daily basis are to the exponential function recorded in the table. model, approximately 11.8% of American adults will smoke PERCENTAGE cigarettes on a daily basis in the 2-YEAR YEAR OF AMERICAN INTERVAL, x year 2015. ADULTS, f(x) 0 1995 19.9

1 1997 19.1

2 1999 18.0

3 2001 17.4

4 2003 16.9

5 2005 15.3

6 2007 14.5

Data Source: Center for Disease Control (CDC)

Use the data set to generate an exponential model. Use your model to predict the percentage of adults in the United States who smoke cigarettes on a daily basis in the year 2015. See margin.

PRACTICE/HOMEWORK

For questions 1 and 2, determine whether a linear model or an exponential model would be most appropriate for the data. Explain how you made your decision.

1. An exponential model 1. x y 2. x y would be best; first dif- 0 45 0 209.5 ference values range from 1 67.5 1 184.6 2 94.5 2 159.6 22.5 to 102.816, while suc- DRAFT3 151.2 3 134.4 cessive ratio values range 4 257.04 4 109.6 from 1.4 to 1.7 (these 5 359.856 5 84.5 values are approximately FOR constant). See margin. See margin. 2. A linear model would be best; successive ratio values range from 0.77 to 0.88, while first difference 64 chapteR 1: algeBRaic patteR ns values range from 24.8 to 25.2 (these values are approximately constant). PREVIEWONLY

64 CHAPTER 1: ALGEBRAIC PATTERNS For questions 3 - 5, calculate the average ratio between successive y-values.

3. x y 0 425.6 1 766.08 2 1225.73 3 2083.74 4 3750.73 1.725

4. x 0 1 2 3 4 5 6 y 2300.6 1173.31 586.65 287.46 137.98 70.37 36.59

0.5

5. x 0 1 2 3 4 y 1810.4 2172 2389.2 3105.96 3727.15

1.2

For questions 6 – 8, identify whether the data shows exponential growth or exponential decay. Then, determine an exponential function to model the situation. SCIENCE 6. The population of gray squirrels in a local park has been recorded every year since 2005.

1-YEAR SQUIRREL INTERVAL, YEAR POPULATION, x y 0 2005 62

1 2006 87

2 2007 113

3 2008 170

4 2009 204

5 2010 265

Exponential growth; as time passes, the squirrel population increases. y = 62(1.34)DRAFTx, where x represents the number of years since 2005 FOR

1.5 • Modeling with e xponential Functions 65 PREVIEWONLY

65 FINANCE 7. Exponential growth; as 7. Kristal noticed that her favorite painting in a museum has been increasing in time passes, the value of value over the years. The changing value of the painting is shown in the table. the painting increases. 10-YEAR YEAR VALUE OF THE f(x) = 2200(3.39)x, where INTERVAL, x PAINTING, f(x) x represents the number 0 1960 $2200 of 10-year intervals since 1 1970 $7500 1960. 2 1980 $25,000 8. Exponential decay; as trials numbers increase, 3 1990 $86,000 the number of pennies 4 2000 $292,000 decreases. 5 2010 $992,000 f(x) = 61(0.53)x, where x represents the trial See margin. number SCIENCE 8. Mrs. Montgomery’s class is doing an experiment with pennies. They empty a cup of pennies onto a table, and remove all the pennies that landed “heads-up.” Then, they put the other pennies back in the cup, and repeat the process four more times.

NUMBER TRIAL OF PENNIES NUMBER, x REMAINING, fix) 0 61

1 28

2 16

3 9

4 7

5 2 DRAFTSee margin. FOR

66 chapteR 1: algeBRaic patteR ns PREVIEWONLY

66 CHAPTER 1: ALGEBRAIC PATTERNS For questions 9 – 12 use the following situation. FINANCE Most cars decrease in value over time. The table below shows the value of Carla’s car from the time of its purchase.

VALUE OF 1-YEAR YEAR INTERVAL, x CAR, f(x) 0 2007 $29,870

1 2008 $24,180

2 2009 $20,480

3 2010 $17,420

4 2011 $14, 585

5 2012 $12,124

9. Use the data set to generate an exponential model. f(x) = 29,870 (0.84)x; where x represents the number of years since the car was purchased in 2007 10. The y-intercept, $29,870, 10. What do the y-intercept and base from your function rule mean in context of represents the value of the the situation? See margin. car when Carla purchased it in 2007. 11. In what year will the car be worth about $5900? The base, 0.835, or 84% See margin. means that the value of the car is 84% of its value the 12. Use your model to predict the value of Carla’s car in the year 2020. previous year. See margin. 11. f(x) = 29,870(0.835)x = 5900, x ≈ 9 years; 2007 + 9 = 2016. According to the exponential function model, the car will be worth about $5900 in the year 2016. DRAFT 12. 2020 – 2007 = 13, so the number of years since purchase, x = 13. FOR f(13) = 29,870(0.835)13 ≈ $2865.20 According to the exponential function model, the car will be worth about $2865 in the year 2020. 1.5 • Modeling with e xponential Functions 67 Using the base of (0.84), the car would be worth $3096 in the year 2020. PREVIEWONLY

67 For questions 13 – 16 use the following situation. FINANCE Ella sells hair ribbons and decided to start marketing them on the internet hoping to increase her sales. The table shows the total number of ribbons she has sold.

NUMBER OF TOTAL WEEKS SINCE NUMBER OF 15. f(x) = 310(1.08)x = 1000; MARKETING ON THE RIBBONS x ≈ 15 weeks INTERNET, x SOLD, f(x) According to the expo- 0 310 nential function model, 1 336 Ella will have sold 1,000 ribbons in about 15 weeks. 2 365 3 388 16. A year is 52 weeks; f(52) = 310(1.08)x; 4 425 x ≈ 16,959 5 445

According to the exponen- 6 496 tial function model, Ella will sell close about 16,95 13. Use the data set to determine an exponential function that models the situation. ribbons in a year. f(x) = 310(1.08)x; where x represents the number of weeks since she started marketing her ribbons on the internet. 14. What is the y-intercept of this function, and what does it mean in context of the problem? The y-intercept, 310, represents the number of ribbons Ella had sold before she started marketing on the internet. 15. Use your function model to determine approximately how many weeks it will take to sell 1,000 ribbons. See margin.

16. Use your function model to predict how many ribbons Ella will sell in a year. See margin.

For questions 17 – 20 use the following situation. SCIENCE A biologist is recording the population 2-HR NUMBER OF NUMBER OF of a certain bacteria in a petri dish. He INTERVAL, x HOURS BACTERIA, f(x) determines the number of bacteria in 0 0 8 DRAFTthe dish every 2 hours, as shown in the table below. 1 2 12 17. Does this data show exponential growth or exponential decay? Ex- 2 4 17 FORplain. 3 6 24 Exponential growth; as time increases, the number of bacteria 4 8 34 increases 5 10 50

68 chapteR 1: algeBRaic patteR ns PREVIEWONLY

68 CHAPTER 1: ALGEBRAIC PATTERNS 19. 1 day is 24 hours, which is 18. At some point, it becomes impossible to count all the bacteria, so an equation is 12 2-hr intervals; x = 12; necessary. Use the data set to generate an exponential model. 12 f(x) = 8(1.44)x; where x represents the number of 2-hour time intervals f(12) = 72(0.41) ≈ 636 bacteria 19. Use your exponential model to determine approximately how many bacteria According to the exponen- will be in the petri dish after 1 day. See margin. tial function model, there will be about 636 bacteria 20. Approximately how many days will elapse before there are 451,000 bacteria? after 1 day. See margin. x 20. f(x) = 8(1.44) = 451,000; For questions 21 – 24 use the following situation. x ≈ 30, and 30 2-hr inter- CRITICAL THINKING vals makes it 60 hours, or 2.5 days Beth enjoys running for exercise. She has started a training plan that will gradually increase her weekly mileage as she prepares for a half-marathon. The table shows her According to the exponen- training plan. tial function model, there will be 451,000 bacteria in NUMBER OF MILES PER about 2.5 days. WEEKS, x WEEK, f(x) 10 0 18 23. f(10) = 18(1.1) = 65; x ≈ 46.7 miles 1 20 According to the expo- 2 22 nential function model, 3 24 she should run about 46.7 4 26.5 miles in week 10. 5 29 24. f(x) = 18(1.1)x = 65; 6 32 x ≈ 14 weeks

7 36 According to the exponential function model, she 21. Using the data given above, generate an exponential function that models the should reach the goal of 65 situation. miles in week 14. f(x) = 18(1.1)x; where x represents the number of weeks on the training plan 22. What is the percent increase of her mileage from week to week on the plan? b = 1.1 is the average ratio; the 0.10 in this value indicates an increase of 10% each week. 23. According to your model, how much should Beth run in week 10 of her training DRAFTplan? See margin.

24. If the training plan limits the mileage to 65 miles per week, when should she reach this goal? FOR See margin.

1.5 • Modeling with e xponential Functions 69 PREVIEWONLY

69 Chapter 1 Mid-Chapter Review

1.1 Arithmetic and Geometric Sequences 1a. geometric 1. For each sequence of numbers below, determine whether it is arithmetic, geo- 3 metric or neither. Then, write a recursive rule and an explicit rule, if possible. – a1 = 128; an = 4an – 1 a) 128, 96, 72, 54, 40.5, … 3 See margin. – n-1 an = 128(4) b) 1c. 400, 300, 150, 75, 22.5, … arithmetic Neither arithmetic nor geometric

a1 = −5; an = an – 1 + 2.25 c) −5, −2.75, −0.5, 1.75, 4, …

an = -7.25 + 2.25n See margin

2. Given each recursive rule, write the first 4 terms of the sequence. Indicate whether each sequence is arithmetic or geometric.

a) a1 = 200; an = an-1 -15 200, 185, 170, 155 arithmetic 2 – b) a1 = 12; an = 3an-1 1 5 — — 12, 8, 53, 39 geometric 3. Given each explicit rule, write the first 4 terms of the sequence. Indicate whether each sequence is arithmetic or geometric. 1 n-1 – a) an = 243(3) 243, 81, 27, 9 geometric

b) an = 32 − 1.5n 30.5, 29, 27.5, 26 arithmetic 4. Write a rule to describe the number of shaded squares in each figure as a function of the figure number, n. DRAFT FORf(n) = 4n

70 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

70 CHAPTER 1: ALGEBRAIC PATTERNS 5. Evan’s parents tell him he can choose one of two ways to earn his allowance for the coming year. He can either: a) receive $1 in January, $2 in February, $4 in March and so on, doubling the amount he receives each month; or b) receive $20 in January, $25 in February, $30 in March and so on, increasing his monthly allowance by $5 each month.

How much will Evan’s December allowance be under each plan? Write the explicit rule and then show how the answer is derived using the explicit rule. n – 1 a) an = 1(2) b) an = 15 + 5n 11 a12 = 1(2) = $2,048 a12 = 15 + 12(5) = $75

1.2 Writing Linear Functions Write an equation in y=mx+b form for each linear function in problems 1-3 described below. 6. slope = 1.5, y-intercept = (0, 5) y = 1.5x + 5

-4 — 7. slope = 3 , contains the point (−3, 7) -4 — y = 3 x + 3

8. contains the points (-2, -10), (4, -7), and (20, 1) 1 — y = 2x - 9

9. For each table, determine whether the relationship shows a linear function. If so, write the function. a) x y b) x y c) x y 1 9 0 3.2 1 0 2 6 2 4.2 2 4 3 3 4 5.2 3 8 4 0 6 6.2 4 16 5 -3 8 7.2 5 24 linear; y = -3x+12 linear: y = 0.5x+3.2 not linear

10. Jerome weighed 210 pounds when he started on a diet. He plans to lose 2.5 pounds per week. Complete the table below to represent Jerome’s weight over time and write a function to represent this situation.

# OF WEEKS WEIGHT DRAFT0 210 1 207.5 2 205 3 202.5 FOR 4 200 f(x) = -2.5x + 210

C HAPTER 1 MI d-CHAPTER R E v IEw 71 PREVIEWONLY

71 1.3 Modeling with Linear Functions Use the following situation to answer the questions below. Amanda was given a 5-week-old kitten for her birthday. The table below shows how the kitten’s weight changed over the next 5 weeks.

TIME (WEEKS) 5 6 7 8 9 10 WEIGHT (GRAMS) 480 540 605 682 754 820

11. Write a function rule to model the situation. f(x) = 140 + 68x or f(x) = 68x + 140 where x represents the number of weeks. 12. What is the y-intercept from your function rule and what does it mean in the context of this situation? The y-intercept, 140 grams, represents the weight of the kitten at birth.

13. What does the slope from your function rule mean in the context of this situation? The slope, 68 grams per week, represents the growth rate of Amanda’s kitten (how much its weight increases each week). 14. Use your model to predict the weight of Amanda’s kitten when it is 22 weeks old. f(22) = 140 + 68(22) = 1,636 grams

15. In what week will the kitten weigh approximately 1300 grams? 1300 = 140 + 68x , x ≈ 17 weeks

1.4 Writing Exponential Functions For each table below, state whether the table represents an exponential function or not. If the table represents an exponential function, write the common ratio and the function relating the variables. 16. x y 17. x y 18. x y 0 240 0 12 1 750 1 180 1 24 2 900 2 135 2 36 3 1080 3 101.25 3 64 4 1296

exponential not exponential exponential DRAFT ratio = .75 ratio = 1.2 FOR y = 240(.75)x y = 625(1.2)x

72 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

72 CHAPTER 1: ALGEBRAIC PATTERNS For questions 4 – 5 use the situation below. In its first 4 years of operation, a certain company increased its revenue by 2.5% each month. NUMBER MONTHS, x MONTHLY REVENUE, y 0 $800.00

1 $820.00

2 $840.00

3 $861.51

4 $883.05

19. Write the function relating the variables. y = 800(1.025)x

20. If this pattern continues, what would be the company’s monthly revenue at the end of 1 year? For x = 12 months, the revenue would be about $1,075.91.

1.5 Modeling with Exponential Functions For questions 1 – 4 use the situation below. Evan made a fresh pot of coffee then measured its temperature in 3-minute intervals over a period of 15 minutes, as shown in the table below.

NUMBER NUMBER OF TEMPERATURE OF 3-MIN MINUTES (F°), INTERVALS, x f(x)

0 0 180.2

1 3 172.5

2 6 163.8

3 9 155.6

4 12 147.9

5 15 140.8

21. Does this data show exponential growth or exponential decay? Explain. ExponentialDRAFT decay; as time increases, the temperature decreases 22. Write an exponential model for this data. f(x) = 180.2(0.95)x; where x represents the number of 3-minute time intervals FOR 1 23. Use your exponential model to determine the approximate temperature after 23. – hour is 30 minutes, 1 2 – 2 hour. which is 10 3-min See margin. intervals; x = 10; f(10) = 180.2(0.95)10 ≈ C HAPTER 1 MI d-CHAPTER R E v IEw 73 107.89 According to the exponential function model, the temperature will be about 108° Fahrenheit after ½ PREVIEWONLYhour.

73 24. f(x) = 180.2(0.95)x = 80; 24. Approximately how many minutes will elapse before the temperature is 80° F? x ≈ 16, and 16 3-min See margin. intervals makes it about 48 For questions 5 – 7 use the situation below. minutes. The population of Harris County, TX over a 7-year period is shown in the table below, According to the exponen- where x represents the number of years since 2000 and y represents the population in tial function model, the millions. temperature will be 80° F POPULATION 25. Using the data given above, generate an expo- YEARS SINCE (IN MILLIONS), 2000, x in about 48 minutes. nential function that models the situation. f(x) x 15 f(x) = 3.4(1.0176) ; where x represents the 0 3.4 26. f(15) = 3.4(1.0176) ≈ number of years since 2000 4.42 million 26. According to your model, what is a reasonable 1 3.46 estimate of the population in the year 2015? According to the expo- 2 3.52 See margin. nential function model, 3 3.58 the population of Harris 27. According to your model, in what year is the County would be about population expected to reach 5 million? 4 3.64 4.42 million in 2015. See margin. 5 3.71

27. f(x) = 3.4(1.0176)x = 5; 6 3.78

x ≈ 22 years 7 3.84 According to the exponen- MULTIPLE CHOICE tial function model, the 28. An auditorium has seating in which the front row has 25 seats, the 2nd row 28 population would reach 5 seats, the 3rd row 31 seats, etc. If this pattern continues, how many seats will be million in the year 2022. in the 12th row? A. 46 B. 52 C. 55 D. 58

29. A t-shirt company charges a $20.00 set up fee plus $8.95 for each t-shirt that is screen-printed with a school’s logo. Does this situation represent a linear relationship? If so, which function can be used to find the total cost, C, for purchasing and screen-printing t number of t-shirts? A. yes, C = 20t + 8.95 B. yes, C = 20 + 8.95t C. yes, C = 20t + 8.95t DRAFTD. Not a linear function FOR

74 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

74 CHAPTER 1: ALGEBRAIC PATTERNS 30. Which of the following graphs shows the best linear function model for the given data?

x 0 1 2 3 4 5 6 y 6.4 5.8 5.2 4 3.2 2.9 2

A. B.

C. D.

31. Look at the table shown below.

x y 0 64

1 80

2 100

3 125

Which of the following statements is NOT true about the table? A. The function relating the variables is y = 64(1.25)x. B. The function is exponential. C. The common ratio is 1.25. D. The function represents exponential decay. 32. WhichDRAFT of the following functions best models the given data? x 0 1 2 3 4 5 6 y 12.2 18.5 26.7FOR 40 60.6 91.9 135.9 A. y = 120 – 16.5x B. y = 120 – 29x C. y = 12.2(1.5)x D. y = 120(0.25)x

C HAPTER 1 MI d-CHAPTER R E v IEw 75 PREVIEWONLY

75 TEKS AR.2A Determine the patterns that identify the Writing Quadratic Functions relationship between a 1.6 function and its common ratio or related finite differences as appropriate, FOCUSING QUESTION What are the characteristics of a quadratic function? including linear, quadratic, cubic, and exponential LEARNING OUTCOMES functions. ■■ I can determine patterns that identify a quadratic function from its related finite AR.2C Determine the differences. function that models a giv- ■■ I can determine the quadratic function from a table using finite differences, en table of related values including any restrictions on the domain and range. using finite differences and ■■ I can use finite differences to determine a quadratic function that models a its restricted domain and mathematical context. range. ■■ I can analyze patterns to connect the table to a function rule and communicate AR.2D Determine the quadratic pattern as a function rule. a function that models real-world data and mathematical contexts using ENGAGE finite differences such as the age of a tree and its Square numbers can be represented using circumference, figurative counters as shown. numbers, average velocity, and average acceleration. MATHEMATICAL PROCESS SPOTLIGHT AR.1F Analyze mathematical relationships to TERM 1: 1 TERM 2: 4 TERM 3: 9 TERM 4: 16 connect and communicate mathematical ideas. What patterns do you see in the geometric arrangements of square numbers? See margin. ELPS 5C Spell familiar English words with in- EXPLORE creasing accuracy, and employ English spelling A figurative number, sometimes called a figurate number, is a number that can patterns and rules with in- be represented by a regular geometric arrangement of dots or other objects. For ex- creasing accuracy as more ample, triangular numbers can be represented using arrangements of dots that are English is acquired. DRAFTshaped like regular triangles. VOCABULARY finite differences, domain, range, vertex, axis of sym- FOR metry, x-intercept, y-intercept, quadratic function ENAGE ANSWER: 1 3 6 10 Possible answer: The number of dots is equal to the term num- 76 CHAPTER 1: ALGEBRAIC PATTERNS ber squared.

2. In the table, ∆x = 1.

TERM TRIANGULAR TERM TRIANGULAR NUMBER NUMBER NUMBER NUMBER

1 1 1 1 y ∆x = 2 – 1 = 1 ∆y = 3 – 1 = 2 n 3 —y = — = 3 2 3 PREVIEWn -1 1 2 3 y ∆x = 3 – 2 = 1 ∆y = 6 – 3 = 3 n 6 — = — = 2 3 6 ONLY yn -1 3 3 6 ∆y = 10 – 6 = 4 y ∆x = 4 – 3 = 1 n 10 2 4 10 —y = — = 1 — 4 10 n -1 6 3 ∆x = 5 – 4 = 1 ∆y = 15 – 10 = 5 y n 15 1 The data set is 5 15 —y = — = 1— not exponential ∆y = 21 – 15 = 6 5 15 n -1 10 2 ∆x = 6 – 5 = 1 y 6 21 n 21 2 because the succes- —y = — = 1 — 6 21 n -1 15 5 sive ratios are not The data set is not linear because the finite differences in the constant. triangular numbers are not constant.

76 CHAPTER 1: ALGEBRAIC PATTERNS Use counters to build a sequence of the first six triangular numbers. Record the numbers in a table like the one shown.

TRIANGULAR TERM NUMBER NUMBER 1 1

2 3 3 6 4 10 5 15 6 21

1. As you build the sequence, what patterns do you see in each successive term? 1 For each successive term, See margin. you add a row of dots to the bottom of the trian- 2. Does the data set follow a linear or an exponential function? Explain your reasoning. gle that is one dot longer See margin. than the row of dots at the bottom of the previous A set of numbers, such as 3. What patterns do you see in the finite differences or x-values or y-values, can be triangle. the successive ratios? represented with brackets See margin. using set notation. The set of 2. See the bottom of page 76. whole numbers less than 10 3. Possible answers may 4. Calculate the second finite difference. What do you is represented as {0, 1, 2, 3, 4, notice? 5, 6, 7, 8, 9, 10}. If this set is a include: See margin. set of x-values, it can be written as {x|x = 0, 1, 2, 3, 4, 5, 6, The finite differences are 7, 8, 9, 10} which is read “the 2 not constant, but they do 5. The quadratic parent function is y = x . Generate a set of all x such that x equals sequence with y-values for {x|x = 1, 2, 3, 4, 5, 6}. zero, one, two, …” increase by 1 each time. 1, 4, 9, 16, 25, 36 The successive ratios can 6. all be represented as Calculate the second finite differences for the quadratic parent function. What yn 2 y = 1 + —n. do you notice? n - 1

x y 4. See below.

1 1 ∆x = 2 – 1 = 1 ∆y = 4 – 1 = 3 2 4 5 – 3 = 2 ∆x = 3 – 2 = 1 ∆y = 9 – 4 = 5 DRAFT3 9 7 – 5 = 2 ∆x = 4 – 3 = 1 ∆y = 16 – 9 = 7 4 16 9 – 7 = 2 ∆x = 5 – 4 = 1 ∆y = 25 – 16 = 9 5 25 FOR11 – 9 = 2 ∆x = 6 – 5 = 1 ∆y = 36 – 25 = 11 6 36

The second differences are all equal to 2 and are all constant.

1.6 • WRITING QuA d RATIC FuNCTIo NS 77

4. TERM TRIANGULAR NUMBER NUMBER

1 1 ∆x = 2 – 1 = 1 ∆y = 3 – 1 = 2 2 3 PREVIEW3 – 2 = 1 ∆x = 3 – 2 = 1 ∆y = 6 – 3 = 3 3 6 4 – 3 = 1 ONLY ∆x = 4 – 3 = 1 ∆y = 10 – 6 = 4 4 10 5 – 4 = 1 ∆x = 5 – 4 = 1 ∆y = 15 – 10 = 5 5 15 6 – 5 = 1 ∆x = 6 – 5 = 1 ∆y = 21 – 15 = 6 6 21

The second differences are all equal to 1 and are all constant.

77 7. What type of function do you think represents the relationship between the triangular number and the term number, or its position in the sequence? Possible answer: quadratic function

REFLECT REFLECT ANSWERS: The first finite differences in a ■■ In a linear function, the first finite differences are constant. What is true quadratic function are not con- about the finite differences for a quadratic function? See margin. stant, but increase or decrease ■■ A linear function contains a polynomial with degree one (mx + b) and a with a particular pattern. The quadratic function contains a polynomial with degree two (ax2 + bx + c). second finite differences are What relationship is there between the degree of the polynomial and constant in a quadratic func- the level of finite differences that are constant? tion. See margin. The level of finite differences that are constant is the same as the degree of the polynomial EXPLAIN (linear: degree one and first In a linear function, the first finite differences, or the dif- There are many forms of a differences constant; quadratic: ference between consecutive values of the dependent vari- quadratic function. Polyno- degree two and second differ- able, are constant. But for a quadratic function, the first fi- mial form, also called stan- ences constant). nite differences are not constant. They do, however, have a dard form, expresses the pattern in that they increase or decrease by the same num- function as a polynomial ber. As a result, the second finite differences, or the differ- with exponents in decreas- ences between the first finite differences, are constant. ing order. ELL STRATEGY 2 Watch Explain and f(x) = ax + bx + c Writing with familiar You Try It Videos Let’s look more closely at a quadratic func- English language words tion. The table below shows the relation- In standard form, a, b, and c (ELPS: c5C) helps students ship between x and f(x) in a quadratic func- are rational numbers. tion written in polynomial or standard, both deepen their under- f(x) = ax2 + bx + c. standing of the mathematical content as well as become more comfortable or click here with the English language. Having students use the re- x PROCESS y = f(x) flect questions for a journal 0 a(0)2 + b(0) + c c entry in their interactive ∆x = 1 – 0 = 1 ∆y = (a + b + c) – c = a + b 1 a(1)2 + b(1) + c a + b + c math notebooks providesDRAFT ∆x = 2 – 1 = 1 ∆y = (4a + 2b + c) – (a + b + c) = 3a + b 2 a(2)2 + b(2) + c 4a + 2b + c an opportunity to reinforce ∆x = 3 – 2 = 1 ∆y = (9a + 3b + c) – (4a + 2b + c) = 5a + b this language proficiency 3 a(3)2 + b(3) + c 9a + 3b + c skill. ∆x = 4 – 3 = 1 ∆y = (16a + 4b + c) – (9a + 3b + c) = 7a + b FOR4 a(4)2 + b(4) + c 16 a + 4 b + c ∆x = 5 – 4 = 1 ∆y = (25a + 5b + c) – (16a + 4b + c) = 9a + b 5 a(5)2 + b(5) + c 25a + 5b + c

78 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

78 CHAPTER 1: ALGEBRAIC PATTERNS The first differences are not constant, but there is a pattern as the differences increase from a + b to 3a + b, from 3a + b to 5a + b, and so on. So let’s look at the second differences. When you look at the second differences, three patterns emerge.

x y = f(x)

0 c ∆y = (a + b + c) – c = a + b 1 a + b + c ∆2y = (3a + b) – (a + b) = 2a ∆y = (4a + 2b + c) – (a + b + c) = 3a + b 2 4a + 2b + c ∆2y = (5a + b) – (3a + b) = 2a ∆y = (9a + 3b + c) – (4a + 2b + c) = 5a + b 3 9a + 3b + c ∆2y = (7a + b) – (5a + b) = 2a ∆y = (16a + 4b + c) – (9a + 3b + c) = 7a + b 4 16a + 4b + c ∆2y = (9a + b) – (7a + b) = 2a ∆y = (25a + 5b + c) – (16a + 4b + c) = 9a + b 5 25a + 5b + c

You can use these three patterns to determine the quadratic function from the table of data. ■■ The value of c is the y-coordinate of the y-intercept, (0, c). ■■ The second difference is equal to a2 . ■■ The first difference between the y-values for x = 0 and x = 1 is equal to a + b.

FINITE DIFFERENCES AND QUADRATIC FUNCTIONS INTEGRATE TECHNOLOGY In a quadratic function, the second differences between successive y-values are constant if the dif- Use technology such as ferences between successive x-values, ∆x, are also a graphing calculator constant. or spreadsheet app on a If the second differences between consecutive display screen to show y-values in a table of values are constant, then the val- students how, no matter ues represent a quadratic function. the numbers present in the quadratic function, the sec- The formulas for finding the values of a, b, and c to write the quadratic function only ond differences will always work when ∆x = 1. When ∆x ≠ 1, there are other formulas that can be used to determine be constant. the valuesDRAFT of a, b, and c for the quadratic function. FOR

1.6 • WRITING QuA d RATIC FuNCTIo NS 79 PREVIEWONLY

79 ADDITIONAL EXAMPLES What type of function (lin- EXAMPLE 1 ear, exponential, or qua- What type of function would best model the data set below? Justify your answer. dratic) would best model the data sets below? Justify x y your answer. 0 0 1. 1 1 x 1 3 5 7 9 2 6 y 15 135 1215 10935 98415 3 15

Exponential 4 28 2. STEP 1 Determine whether or not the set of data represents a linear x 1 2 3 4 5 function. y 6 -4 -18 -36 -58 x y

Quadratic 0 0 ∆x = 1 – 0 = 1 ∆y = 1 – 0 = 1 1 1 ∆x = 2 – 1 = 1 ∆y = 6 – 1 = 5 2 6 ∆x = 3 – 2 = 1 ∆y = 15 – 6 = 9 3 15 ∆x = 4 – 3 = 1 ∆y = 28 – 15 = 13 4 28

The differences inx , ∆x, are all 1, so they are constant.

The differences iny , ∆y, are not all the same, so they are not constant.

Therefore, the set of data does not represent a linear function because the first finite differences are not constant. DRAFT FOR

80 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

80 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES STEP 2 Determine whether or not the set of data represents an exponential function. What type of function (linear, exponential, or qua-

x y dratic) would best model the data sets below? Justify 0 0 y1 1 your answer. ∆x = 1 – 0 = 1 — = – (undefined) y0 0 1 1 y2 6 ∆x = 2 – 1 = 1 — = – = 6 1. y1 1 2 6 y3 15 ∆x = 3 – 2 = 1 — = — = 2.5 y2 6 -7 -4 -1 2 5 3 15 x y4 28 ∆x = 4 – 3 = 1 — = — = 1.8666... y3 15 2 -16 -34 -52 -70 4 28 y Linear The data set is not exponential because the successive ratios are not constant. 2. STEP 3 Determine whether or not the set of data represents a quadratic function. x 0 1 2 3 4 y 4 8 16 20 24 x y None of the functions 0 0 ∆x = 1 – 0 = 1 ∆y = 1 – 0 = 1 1 1 ∆2y = 5 – 1 = 4 ∆x = 2 – 1 = 1 ∆y = 6 – 1 = 5 2 6 ∆2y = 9 – 5 = 4 INSTRUCTIONAL HINTS ∆x = 3 – 2 = 1 ∆y = 15 – 6 = 9 3 15 ∆2y = 13 – 9 = 4 Help students organize ∆x = 4 – 3 = 1 ∆y = 28 – 15 = 13 their thoughts by creat- 4 28 ing their own flowchart of steps for determining The second finite differences are all 4, so the set of data represents a quadratic function. whether or not a set of data represents a linear, exponential, or quadratic function. YOU TRY IT! #1 Within the flowchart ask students to include the Determine if the function rule for the set of data is linear, exponential, or quadratic. questions they ask them- x y selves as they look at data such as, “are the differ- DRAFT0 0 ences in x constant?” or 1 1 “are the successive rations 2 FOR7 constant?” 3 19

4 37

Answer: The set of data does not represent any of these functions. 1.6 • WRITING QuA d RATIC FuNCTIo NS 81 PREVIEWONLY

81 INSTRUCTIONAL HINTS If students are struggling with “triangular numbers,” EXAMPLE 2 have them turn back to Determine the function rule for the set of triangular numbers shown below. the Explore on pg. 76 for a visual. x y

Give students isomet- 0 0 ric graph paper to draw 1 1 square or hexagonal numbers. Have them create a ta- 2 3 ble of values and determine 3 6 the function rule for the set 4 10 of numbers they drew. STEP 1 Determine the finite differences between successive x-values and successive y-values. INSTRUCTIONAL HINTS Encourage students to x y add abbreviated direc- 0 0 ∆x = 1 – 0 = 1 ∆y = 1 – 0 = 1 tions for writing function 1 1 rules on their flowcharts ∆x = 2 – 1 = 1 ∆y = 3 – 1 = 2 2 3 from the Instructional ∆x = 3 – 2 = 1 ∆y = 6 – 3 = 3 Hints on pg. 82. 3 6 ∆x = 4 – 3 = 1 ∆y = 10 – 6 = 4 Summarizing learning in 4 10 one flowchart will help students study and process STEP 2 Determine whether or not the differences are constant. the material. The differences inx , ∆x, are all 1, so they are constant. The differences iny , ∆y, are not constant. STEP 3 Determine whether or not the second finite differences in ADDITIONAL successive y-values are constant. EXAMPLES Determine the function x y rule for the Additional 0 0 Examples from pages 80 ∆x = 1 – 0 = 1 ∆y = 1 – 0 = 1 1 1 ∆2y = 2 – 1 = 1 and 81. ∆x = 2 – 1 = 1 ∆y = 3 – 1 = 2 DRAFT ∆2y = 3 – 2 = 1 2 3 pg. 80 ∆x = 3 – 2 = 1 ∆y = 6 – 3 = 3 3 6 ∆2y = 4 – 3 = 1 x 1. y = 5(3) ∆x = 4 – 3 = 1 ∆y = 10 – 6 = 4 FOR4 10 2. y = -2x2 - 4x + 12 pg. 81 The second differences are all equal to 1 and are constant. 1. y = -6x - 40 2. Not linear, exponential, 82 CHAPTER 1: ALGEBRAIC PATTERNS or quadratic PREVIEWONLY

82 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES STEP 4 Calculate a, b, and c for the quadratic function f(x) = ax2 + bx + c. Create a table of values for For x = 0, y = 0. So c = 0. square numbers from the 1 – Engage diagram on pg. 76. The second finite difference isa 2 , so 2a = 1 and a = 2. Then determine the func- The first difference between the y-values for x = 0 and x = 1 is equal to tion rule. 1 1 – – a + b, so a + b = 1. Since a = 2, b must also equal 2. f(x) = x2 STEP 5 Write the function with the values for a, b, and c: Determine the function 1 1 x2 + x rule for the set of hexag- f(x) = – x2 + – x + 0 or f(x) = 2 2 2 onal numbers using the

values in the table.

x 0 1 2 3 4 y 0 1 6 15 28 YOU TRY IT! #2 f(x) = 2x2 - x Determine the function rule for the set of pentagonal numbers using the values in the table.

x y 0 0

1 1

2 5

3 12

4 22 1 5 12 22 35

3 1 3x2 – x Answer: f(x) = —x2 – —x + 0 = 2 2 2

EXAMPLE 3 Write a quadratic function where the second finite difference is 4, the y-intercept is (0,1), andDRAFT a + b is 5. STEP 1 Determine the values of aFOR, b, and c. The second finite difference is 4. Soa 2 = 4, and a = 2. The y-value of the y-intercept, c, is 1. Since a + b = 5 and a = 2, then b = 3.

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ADDITIONAL EXAMPLES 3. 1. Write a quadratic function where the second finite dif- x 1 2 3 4 5 ference is 3, the y-intercept is (0, -2), and a + b is 13. y 16 -4 -28 -56 -88 f(x) =1.5x2 - 11.5x - 2 PREVIEWf(x) = -2x2 - 14x + 32 For the data sets shown, write a function rule relating the variables. ONLY 2. 4. x 1 2 3 4 5 x 0 1 2 3 4 y -12 -10 -4 6 20 y 3 9 17 27 39 2 f(x) = x2 + 5x + 3 f(x) = 2x - 4x - 10

83 STEP 2 Write a quadratic function in standard form: ax2 + bx + c.

f(x) = 2x2 + 3x + 1

INSTRUCTIONAL HINT YOU TRY IT! #3 If students struggle with finding c in YOU TRY IT #3 For the data set below, write a function relating the variables. help then work through a few of the Additional Ex- x y amples on page 83 before 1 1 revisiting the YOU TRY IT. 2 9

3 23

4 43

5 69

Answer: y = 3x2 – x – 1

PRACTICE/HOMEWORK

For questions 1 – 8, use finite differences to determine if the data sets shown in the tables below represent a linear, exponential, quadratic, or other type of function.

1. x y = f(x) 2. x y = f(x) 3. x y = f(x) 1 5 1 5 1 5 2 11 2 11 2 9 3 21 3 17 3 16 4 35 4 23 4 29 5 53 5 29 5 52 Quadratic Linear Exponential

4. x y = f(x) 5. x y = f(x) 6. x y = f(x) 1 5 1 5 1 5 DRAFT2 14 2 12 2 8 3 29 3 31 3 13 4 50 4 68 4 20 FOR5 77 5 129 5 29 Quadratic Other Quadratic

84 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

84 CHAPTER 1: ALGEBRAIC PATTERNS 7. x y = f(x) 8. x y = f(x) 1 5 1 5 2 11 2 9 3 24 3 13 4 53 4 17 5 117 5 21 Exponential Linear

For questions 9 – 12, the data sets shown in the tables represent quadratic functions. Use finite differences to determine the values of a, b, and c and then write the function in standard form. 9. x y = f(x) 10. x y = f(x) 0 7 0 3 1 10 1 6 2 19 2 13 3 34 3 24 a = 3, b = 0, c = 7 a = 2, b = 1, c = 3 f(x) = 3x2 + 7 f(x) = 2x2 + x + 3

11. x y = f(x) 12. x y = f(x) 0 –1 0 –6 1 5 1 –1 2 19 2 14 3 41 3 39 a = 4, b = 2, c = -1 a = 5, b = 0, c = -6 f(x) = 4x2 + 2x – 1 f(x) = 5x2 – 6

For questions 13 – 16, the data sets shown in the tables represent quadratic functions. Use finite differences to determine f(0), the values of a, b, and c and then write the function in standard form.

13. x y = f(x) 14. x y = f(x) 0 ? 0 ? 1 –1 1 3 2 5 2 16 3 13 3 41 4 23 4 78 f(0) = -5; a = 1, b = 3, c = -5 f(0) = 2; a = 6, b= -5, c = 2 f(x) =DRAFT x2 + 3x – 5 f(x) = 6x2 – 5x + 2 FOR

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85 15. x y = f(x) 16. x y = f(x) 0 ? 0 ? 1 –9 1 7 2 –8 2 22 3 –1 3 45 4 12 4 76 f(0) = -4; a = 3, b = -8, c = -4 f(0) = 0; a = 4, b = 3, c = 0 f(x) = 3x2 – 8x – 4 f(x) = 4x2 + 3x

For questions 17 – 20 use the situation below. CRITICAL THINKING Toothpicks were used to create the pattern below.

1 2 3

17. Relate the length of one side of the figure, x, to the area of the figure, y, by completing the table below. The first row has been completed for you.

LENGTH AREA x y 1 1 2 4 3 9

18. Write the function relating the variables in problem 17. y = x2

19. If the pattern continues, what would be the area of a figure with a side length of 7? DRAFT49 FOR

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86 CHAPTER 1: ALGEBRAIC PATTERNS 20. Relate the figure number, x, to the total number of toothpicks needed to create the figure, y, by completing the table below. The first row has been completed for you.

FIGURE TOTAL NUMBER TOOTHPICKS x y 1 4 2 12 3 24

21. Write the function relating the variables in problem 20. y = 2x2 + 2x

22. If the pattern continues, how many toothpicks would be needed to create Figure 5? 60

For questions 23 – 24 use the situation below. SCIENCE GRAVITY EXPERIMENT An experiment is conducted by dropping an object from a height of 150 feet and measuring the distance it has fallen at 1-second intervals. Identical objects were used to perform the experiment on Venus, Earth, and Mars. The tables below show the results of each experiment.

VENUS EARTH MARS TIME DISTANCE TIME DISTANCE TIME DISTANCE (SEC) (FEET) (SEC) (FEET) (SEC) (FEET) x y x y x y 0 0 0 0 0 0 1 14.8 1 16 1 6.2 2 59.2 2 64 2 24.8 3 133.2 3 144 3 55.8

23. Determine if each table represents a linear, exponential, or quadratic function. Venus: Quadratic Earth: Quadratic Mars: Quadratic

24. WriteDRAFT a function relating the variables in each of the tables above. Venus: y = 14.8x2 Earth: y = 16x2 Mars: y = 6.2x2 FOR

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87 TEKS AR.2B Classify a func- Modeling with Quadratic tion as linear, quadratic, 1.7 Functions cubic, and exponential when a function is represented tabularly using fi- FOCUSING QUESTION How can you use finite differences to construct a nite differences or common quadratic model for a data set? ratios as appropriate. LEARNING OUTCOMES AR.2D Determine ■■ I can use finite differences or common ratios to classify a function as linear, qua- a function that models dratic, or exponential when I am given a table of values. real-world data and mathematical contexts using ■■ I can use finite differences to write a quadratic function that describes a data set. finite differences such as ■■ I can apply mathematics to problems that I see in everyday life, in society, and the age of a tree and its in the workplace. circumference, figurative numbers, average velocity, and average acceleration. ENGAGE MATHEMATICAL Mrs. Hernandez wants to construct a rectangular sandbox for PROCESS SPOTLIGHT her niece and nephew. She has AR.1A Apply mathe- 36 feet of lumber to use as the matics to problems arising border. What are some possible in everyday life, society, dimensions that Mrs. Hernan- and the workplace. dez could use to construct the sandbox? ELPS See margin. 5B Write using newly acquired basic vocabulary and content-based EXPLORE grade-level vocabulary. Use color tiles to build rectangles that represent a sandbox with a perimeter of 36. VOCABULARY Recall that the area of a rectangle can be found using the formula A = lw and the quadratic function, finite perimeter of a rectangle can be found using the formula P = 2(l + w). Record the dimensions and the area of each rectangle in a table like the one shown. differences, maximum, minimum MATERIALS DRAFT • color tiles • square-inch grid paper or chart paper FOR STRATEGIES FOR SUCCESS If students have difficul- ty creating rectangles or visualizing the area or 88 CHAPTER 1: ALGEBRAIC PATTERNS perimeter of the sandbox, use square-inch grid paper as a template or outline on which students can 2. 5. For a quadratic function, organize the color tiles to • The value of c is the create rectangles. Perim- y-coordinate of the eter can be traced out on y-intercept, (0, c). the paper and counted. Area can be counted from PREVIEW• The second difference the tiles themselves using ONLYis equal to 2a. skip-counting or multipli- • The first difference be- cation of rows and col- tween the y-values for umns. x = 0 and x = 1 is equal to a + b. ENGAGE ANSWER: Work backwards in the ta- Possible answers: 3 ft. by 15 ft., ble to reach the row when 4 ft. by 14 ft.,....9 ft. by 9 ft. x = 0. See table page 89.

88 CHAPTER 1: ALGEBRAIC PATTERNS 2. See bottom of page 88. Sample data: 5. See bottom of pages 88-89. AREA WIDTH (IN.) LENGTH (IN.) (SQ. IN.) 6. See bottom of page 89. 3 15 45 +1 +11 7. Data set: 4 14 56 -2 +1 +9 domain: whole numbers, 5 13 65 -2 +1 +7 1 ≤ x ≤ 17; range: {17, 32, 6 12 72 -2 +1 +5 45, 56, 65, 72, 77, 80, 81} 7 11 77 -2 +1 +3 8 10 80 -2 Function: +1 +1 9 9 81 -2 +1 -1 domain: all real numbers; 10 8 80 -2 +1 -3 range: y ≤ 81 11 7 77 -2 +1 -5 The domain and range of 12 6 72 -2 +1 -7 13 5 65 -2 the data set are subsets of +1 -9 14 4 56 -2 the domain and range of +1 -11 15 3 45 -2 the function rule. +1 -13 16 2 32 The domain and range of the data set are limited 1. What do you think a scatterplot of area versus width for your set of rectangles to whole numbers, but would look like? the domain and range of Answers may vary. Possible response: The graph will increase until the the function rule include width reaches 9 inches and then the graph will decrease. 2. Sketch a scatterplot of area versus width for your set of rectangles. additional real numbers. See margin for graph. 8. A sandbox with dimensions of 9 feet by 9 feet 3. Based on the shape of your graph, what type of function best models the data? Quadratic gives an area of 81 square feet, the maximum area of 4. Calculate the finite differences between the width and the area. Do the data a sandbox with a perimeter appear to be linear or quadratic? How do you know? of 36 feet. The data appear to be quadratic since the first differences are not constant but the second differences are constant. See table above. An area of 81 square feet 5. Use the patterns in the finite differences to write a function rule that describes is the largest value of the the data set. area in the table. f(x) = −x2 + 18x See margin for solution. The vertex of the graph, (9, 6. Use a graphing calculator to graph the function rule over the scatterplot. What 81), represents the width do you notice about the function rule and the scatterplot? that generates the largest The graph of the function rule connects each of the data points. area. The ordered pair (9, SeeDRAFT margin for graph. 7. Compare the domain and range of the data set and the domain and range of the 81) represents a width of function rule. How are they alike? How are they different? 9 feet with an area of 81 See margin. FOR square feet. 8. What dimensions will give Mrs. Hernandez a sandbox with the greatest area? Use the table and graph to justify your answer. See margin

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a + b = 17 6. -1 + b = 17 WIDTH LENGTH AREA b = 18 (IN.) (IN.) (SQ. IN.) c = 0 0 18 0 PREVIEW +1 +17 -2 +1 1 17 17 +15 ONLY -2 2a = -2 +1 2 16 32 +13 a = -1 -2 +1 3 15 45 +11 4 14 56

89 9. No, because the x-values 9. Are the x-intercepts of the function included in your data set? Why or why not? represent the width of a See margin. rectangle and the y-values represent the area of a 10. The perimeter of a rectangle is found using the formula P = 2l + 2w. If you know rectangle. You cannot have the perimeter is 36 feet, how are the length and width related? a rectangle with a width of See margin. 0 feet or an area of 0 11. square feet. Use the relationship between the length and width of a rectangle with a fixed perimeter to write an equation for the area of the rectangle, A = lw. How does 10. 36 = 2l + 2w this equation compare with the function rule that you generated from finite differences? 36 ÷ 2 = 2l ÷ 2 + 2w ÷ 2 See margin. 18 = l + w l = 18 – w The length and width REFLECT must add up to 18, so the ■■ How can you determine a quadratic function model for a data set? length is the difference See margin. between 18 and the width. 11. A = lw = (18 – w)w ■■ How do the parameters obtained from finite differences relate to the data set being modeled? If you use x to represent See margin. the width, w, and f(x) to represent the area, A: f(x) = (18 – x)x f(x) = (18 – x)x = EXPLAIN 18x – x2 = −x2 + 18x, which is the same as the Quadratic function mod- function rule generated els can be used to repre- from finite differences. sent sets of mathematical The two equations are and real-world data. A equivalent. quadratic function has several key attributes REFLECT ANSWERS: that are important to consider when using and Confirm that the second finite interpreting models that differences are constant. If they are based on quadrat- are, then work backward in the ic functions. The graph table, if necessary, to DRAFTidentify of the quadratic func- function values for x = 0 and tion, which is a parabo- x = 1. Use the patterns in the la, helps to explain how these attributes relate to table to identify the values of theFOR quadratic function a, b, and c to write a quadratic model. function in standard (polynomial) form, f(x) = ax2 + bx + c. b = 18, which is half of the perimeter of 36. The perime- 90 CHAPTER 1: ALGEBRAIC PATTERNS ter is divided in half because the perimeter of a rectangle is twice the sum of the length and ELL SUPPORT width. Asking students to write using newly acquired basic a = −1 because the parabola vocabulary and content-based grade-level vocabulary needs to be reflected vertically (ELPS c(5)(b)) helps English language learners make con- in order to generate a maxi- nections amongPREVIEW vocabulary terms and important mathe- mum height instead of a mini- matical ideas. Students encountered quadratic functions mum height. in Algebra 1 and are using finite differencesONLY to deepen their understanding of how to use quadratic functions to model real-world phenomena. Writing with previous and new vocabulary terms helps students merge these ideas while they are learning the English language.

90 CHAPTER 1: ALGEBRAIC PATTERNS The vertex of the parabola represents the data values that gen- Watch Explain and You Try It Videos erate a minimum or maximum value. In the case of the sandbox problem, the vertex reveals the width, or x-coordinate, that generates the maximum area, or y-coordinate. No other x-value in this model will generate a function value greater than the y-value of the vertex.

The axis of symmetry is a vertical line that represents the divid- ing line between the part of the parabola with increasing y-values or click here and the part of the parabola with decreasing y-values. The axis of symmetry passes through the vertex.

The x-intercepts represent the points with function values that are equal to 0.

In the sandbox problem, you can identify the vertex, axis of symmetry, and x-intercepts from the table of values. ■■ The vertex is the row containing the greatest function value, or the greatest area. ■■ The axis of symmetry is represented by the x-value in the row where the function values change from increasing from row to row to decreasing from row to row. ■■ The x-intercept is a row in which the function value is 0.

WIDTH LENGTH AREA (IN.) (IN.) (SQ. IN.) x-INTERCEPT 0 18 0 +1 +17 1 17 17 -2 +1 +15 2 16 32 -2 +1 +13 3 15 45 -2 +1 +11 4 14 56 -2 +1 +9 5 13 65 -2 +1 +7 6 12 72 -2 +1 +5 7 11 77 -2 +1 +3 8 10 80 -2 +1 +1 AXIS OF VERTEX 9 9 81 -2 SYMMETRY +1 ₋1 10 8 80 -2 +1 ₋3 11 7 77 -2 +1 ₋5 12 6 72 -2 DRAFT+1 ₋7 13 5 65 -2 +1 ₋9 14 4 56 -2 +1 ₋11 15 3FOR45 -2 +1 ₋13 16 2 32 -2 +1 ₋15 17 1 17 -2 +1 ₋17 x-INTERCEPT 18 0 0

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91 You can use the values of the key attributes of a quadratic function in order to inter- pret the model. In the sandbox problem, the function f(x) = −x2 + 18x models the data set. The vertex of this function, (9, 81), reveals that a width of 9 feet generates a maximum area of 81 square feet. All x-values to the left of the vertex represent the part of the function where the area increases as the width increases. All x-values to the right of the vertex represent the part of the function where the area decreases as the width increases.

In the sandbox problem, the x-intercepts do not make sense. An x-intercept of (18, 0) means that when the width of the sandbox is 18 feet, the area of the sandbox is 0 square feet. A rectangle cannot have an area of 0 square feet. The x-intercepts represent domain restrictions on the function model when it is applied to the situation.

DOMAIN RANGE

FUNCTION x y ≤ 81 (ALL REAL NUMBERS) 0 < ∈x < 18 SITUATION (INCLUDES WHOLE AND 0 < y ≤ 81 REAL NUMBERS)

In the real world, you cannot have lengths that are negative or 0; they must be positive numbers. When using color tiles to represent the situation, you are further limiting the domain to only whole numbers. But in reality, Mrs. Hernandez could create a sandbox with fractional side lengths, such as 3.5 feet by 14.5 feet. Such a sandbox has a perimeter of 36 feet and meets the criteria of the problem.

MODELING WITH QUADRATIC FUNCTIONS

Real-world data rarely follows exact patterns, but you can use patterns in data to look for trends. If the data set has a constant or approximately constant second finite difference, then a quadratic function model may be appropriate for the data set.

The domain and range for the situation may be a subset of the domain and range of the quadratic function model. DRAFT FOR

92 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

92 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES EXAMPLE 1 Determine whether the sets of data represent a linear, A ball is thrown from the top of a building. The table below shows the height of a ball above the ground at one-second intervals. Determine whether the set of data rep- quadratic, or exponential resents a linear, quadratic, or exponential function. function.

TIME IN HEIGHT IN 1. SECONDS, x METERS, f(x) x 2 3 4 5 6 0 100 y 4.2 -0.3 -4.8 -9.3 -13.8 1 105.1 Linear 2 100.4 2. 3 85.9 x 3 4 5 6 7 4 61.6 y -4.12 12.54 32.98 47.15 125.77

5 27.5 None

STEP 1 Determine the finite differences in values ofx and values of f(x).

TIME IN HEIGHT IN SECONDS, x METERS, f(x)

0 100 ∆x = 1 – 0 = 1 ∆f(x) = 105.1 − 100 = 5.1 1 105.1 ∆x = 2 – 1 = 1 ∆f(x) = 100.4 – 105.1 = −4.7 2 100.4 ∆x = 3 – 2 = 1 ∆f(x) = 85.9 – 100.4 = −14.5 3 85.9 ∆x = 4 – 3 = 1 ∆f(x) = 61.6 – 85.9 = –24.3 4 61.6 ∆x = 5 – 4 = 1 ∆f(x) = 27.5 – 61.6 = –34.1 DRAFT5 27.5 FOR

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93 ADDITIONAL EXAMPLES Determine whether the sets STEP 2 Determine the ratios between successive values of f(x). of data represent a linear, TIME IN HEIGHT IN quadratic, or exponential SECONDS, x METERS, f(x) function. 0 100 y n = 105.1 ≈ 1.051 ∆x = 1 – 0 = 1 yn-1 1. 100 1 105.1 y n = 100.4 ≈ 0.955 0 1 2 3 4 ∆x = 2 – 1 = 1 yn-1 x 105.1 2 100.4 y 85.9 y 45.9 49.1 58.7 74.7 97.1 n = ≈ 0.856 ∆x = 3 – 2 = 1 yn-1 100.4 3 85.9 y Quadratic n = 61.6 ≈ 0.717 ∆x = 4 – 3 = 1 yn-1 85.9 4 61.6 y 2. n = 27.5 ≈ 0.446 ∆x = 5 – 4 = 1 yn-1 61.6 x 1 2 3 4 5 5 27.5 y 4.9 5.9 7.1 8.5 10.2 STEP 3 Determine the second finite differences in thef(x) values. Exponential TIME IN HEIGHT IN SECONDS, x METERS, f(x)

0 100 ∆x = 1 – 0 = 1 +5.1 1 105.1 ₋9.8 ∆x = 2 – 1 = 1 ₋4.7 2 100.4 ₋9.8 ∆x = 3 – 2 = 1 ₋14.5 3 85.9 ₋9.8 ∆x = 4 – 3 = 1 ₋24.3 4 61.6 ₋9.8 ∆x = 5 – 4 = 1 ₋34.1 5 27.5

STEP 4 Determine whether the finite differences or the ratios between successive values of f(x) are approximately constant.

■■ The first finite differences range in value from 5.1 to –34.1. This is a wide range, so the first finite differences are not approximately constant. DRAFT■■ The ratios between successive values of f(x) range from 0.446 to 1.051. This is also a wide range, so the ratios between successive values of f(x) are not approximately constant. FOR■■ The second finite differences are all –9.8 and are constant. The set of data represents a quadratic function, rather than a linear or exponential function, because the differences in x are constant and the second finite differences in f(x) are constant.

94 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

94 CHAPTER 1: ALGEBRAIC PATTERNS YOU TRY IT! #1 YOU TRY IT! #1 ANSWER:

A softball pitcher throws a ball to her catcher. The ball’s path is tracked in the table. The set of data represents a quadratic function because the finite differences in x and the HORIZONTAL VERTICAL DISTANCE second finite differences in f(x) FROM PITCHER HEIGHT IN FEET, IN FEET, x f(x) are both constant. 0 2 5 5.6 10 8.4 15 10.4 20 11.6 25 12

Determine whether the relationship is linear, exponential, or quadratic. See margin.

INSTRUCTIONAL HINT EXAMPLE 2 As students are learning to model real world scenarios The total amount in a savings account is shown in the table. Determine whether the interest that is being earned in the savings account follows a linear, quadratic, or ex- with linear, exponential, ponential function. and quadratic functions, ask them to predict what 1-YEAR INTEREST IN kind of function the sce- INTERVAL, x DOLLARS, f(x) nario might use before 0 500 they test the data. Key words like percent and 1 530 interest might steer stu-

2 561.80 dents toward exponential functions. Scenarios that 3 595.51 mention a constant rate of change might suggest a DRAFT4 631.24 linear function. FOR

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95 STEP 1 Determine the finite differences in values ofx and values of f(x).

1-YEAR INTEREST IN INTERVAL, x DOLLARS, f(x)

0 500 ∆f(x) = 530 − 500 = 30 1 530 ∆f(x) = 561.80 − 530 = 31.80 2 561.80 ∆f(x) = 595.51 − 561.80 = 33.71 3 595.51 ∆f(x) = 631.24 − 595.51 = 35.73 4 631.24

The differences in the x-values are constant and the first finite differences in the values for f(x) range from 30 to 35.73.

STEP 2 Determine the ratios between successive values of f(x).

1-YEAR INTEREST IN INTERVAL, x DOLLARS, f(x)

0 500 yn 530 y = ≈ 1.06 n-1 500 1 530 yn 561.80 y = ≈ 1.06 n-1 530 2 561.80 yn 595.51 y = ≈ 1.061 n-1 561.80 3 595.51 yn 631.24 y = ≈ 1.059 n-1 595.51 4 631.24

The ratios between successive values of f(x) are approximately 1.06. These values are all close together, so the ratios between successive values of f(x) are approxi- DRAFTmately constant, and the data set represents an exponential function. FOR

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96 CHAPTER 1: ALGEBRAIC PATTERNS INSTRUCTIONAL HINT Justification of why a data YOU TRY IT! #2 represents a specific type of function is important for The total amount in a savings account is shown in the table. Determine whether the interest that is being earned in the savings account follows a linear, quadratic, or ex- students to demonstrate ponential function. mastery of the concept. Have students write com- 1-YEAR TOTAL AMOUNT plete sentences with expla- INTERVAL, x IN DOLLARS, f(x) nations of their answers. 0 750 1 780 2 810 ADDITIONAL 3 840 EXAMPLES 4 870 Turn back to pages 93 and 94. Write the functions Answer: The data set represents a linear function because the finite dif- for each of the Additional ference in x and the first finite difference in f(x) is constant. Examples. pg. 93, AE 1: y = -4.5x + 13.2 EXAMPLE 3 pg. 93, AE 2: none Possum Kingdom Lake in Palo Pinto County, Texas, was the setting for a world-class cliff diving in 2014. The champion diver’s approximate position during the dive is pg. 94, AE 1: recorded in the table. y = 3.2x2 - 4.1x + 50

DISTANCE HEIGHT pg. 94, AE 2: AWAY FROM ABOVE THE y = 4.1(1.2)x THE CLIFF IN WATER IN METERS, x METERS, f(x)

0 27

1 28.1

2 27.4

3 24.8

DRAFT4 20.0 5 FOR12.9 Data Source: Redbullcliffdiving.com

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97 Use the data set to generate a quadratic function that best models the data.

Use the table to estimate the height of the cliff, the height of the diver at his highest point, and his distance from the cliff when he entered the water.

STEP 1 Determine the finite differences inx -values and the second finite differences in the values off (x).

DISTANCE HEIGHT AWAY FROM ABOVE THE THE CLIFF IN WATER IN METERS, x METERS, f(x)

0 27 ∆x = 1 – 0 = 1 +1.1 1 28.1 ₋1.8 ∆x = 2 – 1 = 1 ₋0.7 2 27.4 ₋1.9 ∆x = 3 – 2 = 1 ₋2.6 3 24.8 ₋2.2 ∆x = 4 – 3 = 1 ₋4.8 4 20.0 ₋2.3 ∆x = 5 – 4 = 1 ₋7.1 5 12.9 TECHNOLOGY INTEGRATION STEP 2 Calculate the average of the second finite differences and use Have students use a graph- this value to determine a in your quadratic function model, f(x) = ax2 + bx + c. ing calculator or app to graph the scatterplot of the -1.8 -1.9 -2.2 - 2.3 given data and the func- 2a = = -2.05 4 tion. Then, explore various points on the function us- So a = −1.025 ing tools to find intercepts, maximum and minimum STEP 3 Calculate the value of b. points, and other data points along the path of the The difference between the values of f(x) for x = 0 and 1 is (a + b). diver from Example 3 and DRAFT a + b = 1.1 the baseball from the Addi- (−1.025) + b = 1.1 tional Example above. FOR b = 2.125

98 CHAPTER 1: ALGEBRAIC PATTERNS

ADDITIONAL EXAMPLES HEIGHT TIME IN ABOVE The data set below represents the height of a baseball, f(x), over time in seconds, x. SECONDS, x GROUND 1. Use the data set to generate a quadratic function that best models the baseball’s path. IN FEET, f(x) f(x) = -16.45x2 + 154.25x + 2.5 PREVIEW0 2.5 2. Use the table to estimate the height of the ball when it was hit, the highestONLY point in the 1 140.3 ball’s path, and the time when the ball will hit the highest point. 2 245.3 According to the quadratic function model, the ball was 2.5 feet off the ground when it was hit. The highest point in the ball’s path was about 364 feet around 4.7 seconds. 3 311.7

4 353.9

5 360.1

98 CHAPTER 1: ALGEBRAIC PATTERNS STEP 4 Determine the value of c.

The value of f(0) = c. f(0) = 27, so c = 27.

STEP 5 Substitute the values of a, b, and c into the general form to determine the function model.

The quadratic function model is f(x) = – 1.025x2 + 2.125x + 27.

Using the table, the top of the cliff must have been about 27 meters above the water. The highest point of the dive was a little more than 28 meters, and the diver entered the water at a distance of a little more than 6 meters from the cliff.

YOU TRY IT! #3 YOU TRY IT! #3 ANSWER: A study compared the speed x (in miles per hour) and the average fuel economy f(x) f(x) = −0.8125x2 + 4.3125x +24.5 (in miles per gallon) for cars. The results in 10 mile per hour increments over 20 mph are shown in the table. According to the quadratic function model, a car would get approximately 21.125 miles per 10-MILE MILES PER GASOLINE USAGE PER HOUR HOUR IN MILES PER gallon driven at 80 miles per INTERVAL, x GALLON, f(x) hour. 0 20 24.5

1 30 28.

2 40 30.0

3 50 30.2

4 60 28.8

5 70 25.8

Use the data set to generate a quadratic model. Use your model to predict the fuel economy at 80 miles per hour. SeeDRAFT margin. FOR

1.7 • ModELING w ITH QuA d RATIC FuNCTIo NS 99 PREVIEWONLY

99 PRACTICE/HOMEWORK

For questions 1-6 determine whether the set of data represents a linear, quadratic, or exponential function.

1. x y = f(x) 2. x y = f(x) 3. x y = f(x) 1 7 1 -4 1 -13 2 16 2 -1 2 -28 3 27 3 2 3 -45 4 40 4 5 4 -64 5 55 5 8 5 -85 Quadratic Linear Quadratic 4. x y = f(x) 5. x y = f(x) 6. x y = f(x) 1 2 1 -4 1 0.2 2 4 2 -6 2 0.04 3 8 3 -6 3 0.008 4 16 4 -4 4 0.0016 5 32 5 0 5 0.00032 Exponential Quadratic Exponential

For questions 7-12 use the data set to generate a quadratic function that best models the data.

7. x y = f(x) 8. x y = f(x) 9. x y = f(x) 1 3 1 2 1 -12 2 12 2 2 2 -20 3 27 3 0 3 -24 4 48 4 -4 4 -24 5 75 5 -10 5 -20 f(x) = 3x2 f(x) = -x2 + 3x f(x) = 2x2 - 14x

10. x y = f(x) 11. x y = f(x) 12. x y = f(x) 1 8.5 1 1 1 6 2 18 2 -8 2 28 3 28.5 3 -23 3 58 4 40 4 -44 4 96 5 52.5 5 -71 5 142 1 — 2 2 2 DRAFTf(x) = 2x + 8x f(x) = -3x + 4 f(x) = 4x + 10x - 8 FOR

100 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

100 CHAPTER 1: ALGEBRAIC PATTERNS For questions 13 and 14, use the following information. SCIENCE The Texas Department of Public Safety can use the length of skid marks to help determine the speed of a vehicle before the brakes were applied. The quadratic function x2 — that best models the data is f(x) = 24 where x represents the speed of the vehicle and f(x) is the length of the skid mark. The speeds of a vehicle and the length of the corre- sponding skid marks are shown in the table below.

SPEED OF A DISTANCE OF VEHICLE IN MILES THE SKID IN PER HOURS, x FEET, f(x)

30 37.5

36 54

42 73.5

48 96

54 121.5

60 150

13. Use the table of data to determine the length of a skid mark of a vehicle that was traveling at a speed of 72 miles when it applied the brakes. 216 feet

14. Use the table of data to determine how fast a vehicle was traveling if the length of the skid mark was 24 feet. 24 miles per hour

For questions 15 - 17, use the following information. SCIENCE A ball is thrown upward with an initial velocity of 35 meters per second. The position of the ball over time is recorded in the table below.

15. Use the data in the table to generate a DISTANCE FROM TIME IN THE GROUND IN quadratic function that models the data. SECONDS, x f(x) = -5x2 + 35x METERS, f(x) 0 0 16. Use the data in the table to find the DRAFT1 30 height of the ball after 7 seconds. 0 meters 2 50 17. Use the data in the table to determineFOR 3 60 after how many seconds the ball will be 4 60 30 meters high. 1 second and 6 seconds 5 50

1.7 • ModELING w ITH QuA d RATIC FuNCTIo NS 101 PREVIEWONLY

101 For questions 18 - 20, use the following information. GEOMETRY Judy wants to construct a rectangular pen for her puppy, but only has 56 feet of fenc- ing to use for the pen. The table below shows the width, length, and area of different size pens.

AREA WIDTH (FT) LENGTH (FT) (SQ. FT.) 10 18 180

11 17 187

12 16 192

13 15 195

14 14 196

15 13 195

16 12 192

18. Use the data in the table to generate a quadratic function that models the data. f(x) = -x2 + 28x

19. Use the data in the table to determine the dimensions that would create a pen with an area of 160 ft2. 8 feet and 20 feet

20. Use the data in the table to determine the area of a pen where one of the dimensions measures 20 feet. 160 square feet DRAFT FOR

102 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

102 CHAPTER 1: ALGEBRAIC PATTERNS TEKS AR.2A Determine the 1.8 Writing Cubic Functions patterns that identify the relationship between a function and its common FOCUSING QUESTION What are the characteristics of a cubic function? ratio or related finite differences as appropriate, LEARNING OUTCOMES including linear, quadrat- ■■ I can determine patterns that identify a cubic function from its related finite ic, cubic, and exponential differences. functions. ■■ I can determine the cubic function from a table using finite differences, including any restrictions on the domain and range. AR.2C Determine the ■■ I can use finite differences to determine a cubic function that models a mathe- function that models a giv- matical context. en table of related values ■■ I can analyze patterns to connect the table to a function rule and communicate using finite differences and the cubic pattern as a function rule. its restricted domain and range. AR.2D Determine ENGAGE a function that models A tomato sauce can is in the shape of a cylinder and the diam- real-world data and math- eter of the base is equal to the height of the can. Generate a ematical contexts using sequence showing the volumes of a series of cans with a radi- finite differences such as us of 1 inch, 2 inches, 3 inches, and 4 inches. What patterns do the age of a tree and its you notice in the sequence? circumference, figurative See margin. numbers, average velocity, and average acceleration. MATHEMATICAL PROCESS SPOTLIGHT EXPLORE AR.1F Analyze mathematical relationships to The volume of a prism is found using the area of the base and the height of the connect and communicate prism, V = Bh. If the prism is a rectangular prism, then the base is a rectangle and its mathematical ideas. area is the product of the length and width of the rectangle, A = lw. Combining these formulas generates a formula you can use to determine the volume of a rectangular ELPS prims, V = lwh. DRAFT 3D Speak using Use cubes to build the first two terms in a sequence of rectangular prisms. For this grade-level content area sequence, the term number is the length of the prism, the width of the prism is dou- vocabulary in context to ble the term number, and the height of theFOR prism is triple the term number. internalize new English words and build academic 1. Sketch the first two terms that you built with the cubes. See margin. language proficiency. VOCABULARY finite differences, cubic 1.8 • Writing CubiC FunC tions 103 function MATERIALS • 60 building blocks per 1. Term 1 Term 2 student group

ENGAGE ANSWER: PREVIEWPossible answer: 2 , 16 , 54 , 128 ONLYπ π π Theπ number that is multiplied Length by is twice the value of the radius cubed. π

Length

103 ELL STRATEGY 2. Complete a table like the one shown using the relationships among the dimen- Encourage all students, sions of the prism (length = x, width = 2x, and height = 3x). especially English language learners, to speak using TERM NUMBER PROCESS VOLUME appropriate mathematics 1 1(2)(3) 6 vocabulary (ELPS: c(3) (d)). Doing so helps them 2 2(4)(6) 48 internalize new English 3 3(6)(9) 162 words and make connec- tions between old and new 4 4(8)(12) 384 content. 5 5(10)(15) 750

6 6(12)(18) 1296

3. Does the data set follow a linear or an exponential function? Explain your reasoning. See margin.

4. Calculate the second finite differences. What do you notice? The second differences are not constant. See margin for details.

5. Calculate the third finite differences. What do you notice? The third differences are constant. See margin for details.

6. Use the relationships among the dimensions that you were originally given to calculate the volume of a rectangular prism with a length of x units. What type of function does this appear to be? V = x(2x)(3x) = 6x3 The volume equation is a cubic function.

REFLECT ANSWERS: REFLECT The level of finite differences ■■ A cubic function is a function of the form f(x) = ax3 + bx2 + cx + d. What is that are constant is the same the degree of this function (i.e., the power of the greatest exponent)? as the degree of the polynomi- A cubic function has a degree of 3. al (linear: degree one and the first differences are constant; ■■ A linear function contains a polynomial with degree one (mx + b) and a quadratic: degree two and the quadratic function contains a polynomial with degree two (ax2 + bx + c). DRAFTWhat relationship is there between the degree of the polynomial and second differences are constant; cubic: degree three and the the level of finite differences that are constant? See margin. third differences are constant) FOR

104 CHAPt E r 1: ALg E brA i C PAttE rns

3. TERM TERM PROCESS VOLUME PROCESS VOLUME NUMBER NUMBER

1 1(2)(3) 6 1 1(2)(3) 6 y n 48 ∆x = 2 – 1 = 1 ∆y = 42 — = — = 8 yn -1 6 2 2(4)(6) 48 2 2(4)(6) 48 y PREVIEWn 162 ∆x = 3 – 2 = 1 ∆y = 114 — = — = 3.375 yn -1 48 3 3(6)(9) 162 3 3(6)(9) 162 y ONLYn 384 ∆x = 4 – 3 = 1 ∆y = 222 — = — ≈ 2.37 yn -1 162 4 4(8)(12) 384 4 4(8)(12) 384 y n 750 ∆x = 5 – 4 = 1 ∆y = 366 — = — ≈ 1.95 yn -1 384 5 5(10)(15) 750 5 5(10)(15) 750 y n 1296 ∆x = 6 – 5 = 1 ∆y = 546 — = = 1.728 yn -1 750 6 6(12)(18) 1296 6 6(12)(18) 1296

In the table, ∆x = 1. The data set is not linear because the The data set is not exponential because the successive ratios first differences in the volume are not constant. are not constant 104 CHAPTER 1: ALGEBRAIC PATTERNS EXPLAIN

In a linear function, the first finite differences, or the There are many forms of a cubic difference between consecutive values of the depen- function. Polynomial form, also dent variable, are constant. For a quadratic function, called standard form, expresses the second finite differences are constant. In a cubic a function as a polynomial with function, the third finite differences are constant. exponents in decreasing order.

3 2 Let’s look more closely at a cubic function. The table f(x) = ax + bx + cx + d below shows the relationship between x and f(x). In In standard form, a, b, c, and d a cubic function written in polynomial or standard, are rational numbers. f(x) = ax3 + bx2 + cx + d.

x PROCESS y = f(x)

0 a(0)3 + b(0)2 + c(0) + d d ∆x = 1 – 0 = 1 ∆y = a + b + c 1 a(1)3 + b(1)2 + c(1) + d a + b + c + d ∆x = 2 – 1 = 1 ∆y = 7a + 3b + c 2 a(2)3 + b(2)2 + c(2) + d 8a + 4b + 2c + d ∆x = 3 – 2 = 1 ∆y = 19a + 5b + c 3 a(3)3 + b(3)2 + c(3) + d 27a + 9b + 3c + d ∆x = 4 – 3 = 1 ∆y = 37a + 7b + c 4 a(4)3 + b(4)2 + c(4) + d 64a + 16b + 4c + d ∆x = 5 – 4 = 1 ∆y = 61a + 9b + c 5 a(5)3 + b(5)2 + c(5) + d 125a + 25b + 5c + d

The first differences are not constant, so let’s look at the second differences.

x PROCESS y = f(x)

0 a(0)3 + b(0)2 + c(0) + d d ∆x = 1 – 0 = 1 ∆y = a + b + c 1 a(1)3 + b(1)2 + c(1) + d a + b + c + d 6a + 2b ∆x = 2 – 1 = 1 ∆y = 7a + 3b + c 2 a(2)3 + b(2)2 + c(2) + d 8a + 4b + 2c + d 12a + 2b ∆x = 3 – 2 = 1 ∆y = 19a + 5b + c 3 a(3)3 + b(3)2 + c(3) + d 27a + 9b + 3c + d 18a + 2b ∆x = 4 – 3 = 1 ∆y = 37a + 7b + c 4 a(4)3 + b(4)2 + c(4) + d 64a + 16b + 4c + d 24a + 2b ∆x = 5 – 4 = 1 DRAFT∆y = 61a + 9b + c 5 a(5)3 + b(5)2 + c(5) + d 125FORa + 25b + 5c + d

1.8 • Writing CubiC FunC tions 105

4. TERM 5. TERM PROCESS VOLUME PROCESS VOLUME NUMBER NUMBER

1 1(2)(3) 6 1 1(2)(3) 6 ∆x = 2 – 1 = 1 ∆y = 42 ∆x = 2 – 1 = 1 ∆y = 42 2 2(4)(6) 48 PREVIEW72 2 2(4)(6) 48 72 ∆x = 3 – 2 = 1 ∆y = 114 ∆x = 3 – 2 = 1 ∆y = 114 36 3 3(6)(9) 162 108 ONLY3 3(6)(9) 162 108 ∆x = 4 – 3 = 1 ∆y = 222 ∆x = 4 – 3 = 1 ∆y = 222 36 4 4(8)(12) 384 144 4 4(8)(12) 384 144 ∆x = 5 – 4 = 1 ∆y = 366 ∆x = 5 – 4 = 1 ∆y = 366 36 5 5(10)(15) 750 180 5 5(10)(15) 750 180 ∆x = 6 – 5 = 1 ∆y = 546 ∆x = 6 – 5 = 1 ∆y = 546 6 6(12)(18) 1296 6 6(12)(18) 1296

The second differences are not constant. The third differences are constant.

105 INTEGRATE The second differences are not constant, either. But, you can see some Watch Explain and TECHNOLOGY patterns emerging. Notice that every time you take another round of fi- You Try It Videos Use technology such as nite differences, the last constant term drops off because it is subtracted a graphing calculator out. For the second differences, the coefficients of the a term are mul- or spreadsheet app on a tiples of 6. Also, each second difference has the same b term, 2b. Let’s calculate the third differences. display screen to show students how, no matter the numbers present in the or click here x PROCESS y = f(x) cubic function, the third differences will always be 0 a(0)3 + b(0)2 + c(0) + d d constant. ∆x = 1 – 0 = 1 ∆y = a + b + c 1 a(1)3 + b(1)2 + c(1) + d a + b + c + d 6a + 2b ∆x = 2 – 1 = 1 ∆y = 7a + 3b + c 6a 2 a(2)3 + b(2)2 + c(2) + d 8a + 4b + 2c + d 12a + 2b ∆x = 3 – 2 = 1 ∆y = 19a + 5b + c 6a 3 a(3)3 + b(3)2 + c(3) + d 27a + 9b + 3c + d 18a + 2b ∆x = 4 – 3 = 1 ∆y = 37a + 7b + c 6a 4 a(4)3 + b(4)2 + c(4) + d 64a + 16b + 4c + d 24a + 2b ∆x = 5 – 4 = 1 ∆y = 61a + 9b + c 5 a(5)3 + b(5)2 + c(5) + d 125a + 25b + 5c + d

The third differences are, indeed, constant. Each third difference is 6a. You can use patterns from the table to determine the quadratic function from the table of data. ■■ The value of d is the y-coordinate of the y-intercept, (0, d). ■■ The third difference is equal to6 a. ■■ The second difference between the first two pairs of y-values is equal to 6a + 2b. ■■ The first difference between the y-values for x = 0 and x = 1 is equal to a + b + c.

FINITE DIFFERENCES AND CUBIC FUNCTIONS

In a cubic function, the third differences between successive y-values are constant if the differences between successive x-values, ∆x, are also constant.

If the third differences between consecutive y-values in a table of values are constant, then the values rep- DRAFTresent a cubic function.

The formulas for finding the values of a, b, c, and d to write the cubic function only workFOR when ∆x = 1. When ∆x ≠ 1, there are other formulas that can be used to determine the values of a, b, c, and d for the cubic function.

106 CHAPt E r 1: ALg E brA i C PAttE rns PREVIEWONLY

106 CHAPTER 1: ALGEBRAIC PATTERNS QUESTIONING STRATEGIES EXAMPLE 1 Have students read the scenario for Example 1 again Use the table from the Explore activity to determine the cubic function rule for the volume of a set of rectangular prisms with width = x, length = 2x, and height = 3x. after completing the steps to write the cubic function TERM rule. Draw their attention PROCESS VOLUME, y NUMBER, x to the process column. 1 1(2)(3) 6 • How did the scenario 2 2(4)(6) 48 translate into the cubic function rule? 3 3(6)(9) 162 • How is the cubic function rule reflected in the 4 4(8)(12) 384 process column on the 5 5(10)(15) 750 table? • Could you have written 6 6(12)(18) 1296 this cubic function rule from the scenario and STEP 1 Analyze the first, second, and third finite differences to calculate table without the steps the values of a, b, c, and d for the function rule f(x) = ax3 + bx2 + cx + d. below? Use the finite differences to calculate the value ofy for x = 0.

TERM PROCESS VOLUME, NUMBER, x y

0 0(0)(0) 0 ∆x = 1 – 0 = 1 ∆y = 6 1 1(2)(3) 6 36 ∆x = 2 – 1 = 1 ∆y = 42 36 2 2(4)(6) 48 72 ∆x = 3 – 2 = 1 ∆y = 114 36 3 3(6)(9) 162 108 ∆x = 4 – 3 = 1 ∆y = 222 36 4 4(8)(12) 384 144 ∆x = 5 – 4 = 1 ∆y = 366 36 5 5(10)(15) 750 180 ∆x = 6 – 5 = 1 ∆y = 546 DRAFT6 6(12)(18) 1296 STEP 2 The value of d is the y-coordinate of the y-intercept, (0, d), and for this data set, d = 0.

STEP 3 The third difference is equalFOR to 6a. Since 6a = 36, a = 6.

1.8 • Writing CubiC FunC tions 107

ADDITIONAL EXAMPLE

Veronica makes cylindrical TERM NUMBER, x PROCESS VOLUME (EXACT), y VOLUME (ESTIMATED), y candles to sell on a popular 2 crafting website. To order 0 (0 )(3·0) 0 0 the proper amount of wax, PREVIEW1 Π (12)(3·1) 3 9.42 she calculates the volume 2 Π (22)(3·2) 24Π 75.4 of each candle. Use the ta- ONLY ble below to determine the 3 Π (32)(3·3) 81 Π 254.47 cubic function rule for the 4 Π (42)(3·4) 192Π 603.19 volume of her candles with 2 radius = x and height = 3x. 5 Π (5 )(3·5) 375 Π 1178.1 Π Π y = 3(x3) π

107 STEP 4 The second difference between the first two pairs ofy -values (x = 0 and x = 1; x = 1 and x = 2) is 6a + 2b.

6a + 2b = 36 36 + 2b = 36 2b = 0 b = 0

STEP 5 The first difference between they -values for x = 0 and x = 1 is equal to a + b + c.

a + b + c = 6 6 + 0 + c = 6 c = 0

STEP 6 Write the cubic function rule.

y = 6x3 + 0x2 + 0x + 0, or simply y = 6x3.

INSTRUCTIONAL HINT YOU TRY IT! #1 When completing YOU Popcorn is sold in boxes in the shape of square prisms. The dimensions of the boxes TRY IT #1 some students and their volumes are shown in the table. Write an equation to describe the volume, may struggle with writing y, related to the width of the box, x, and verify it with a function rule using the finite this function rule. If so, differences in the table. draw their attention back to #6 in the Explore exercise, WIDTH, VOLUME BOX BOX WIDTH LENGTH, IN CUBIC and encourage them to fo- NUMBER IN INCHES, x AND HEIGHT IN INCHES INCHES, y cus on the “width, length, and height in inches” col- 1 (SAMPLER) 1 1(1)(2) 2 umn of the table. 2 (KID’S) 2 2(2)(4) 16 3 (SMALL) 3 3(3)(6) 54 4 (MEDIUM) 4 4(4)(8) 128 DRAFT5 (LARGE) 5 5(5)(10) 250 6 (SUPER) 6 6(6)(12) 432

The formula V = lwh becomes y = x(x)(2x) = 2x3. The function rule, using FOR 3 2 3 finite differences in the table, is f(x) = 2x + 0x + 0x + 0, or simply y = 2x .

108 CHAPt E r 1: ALg E brA i C PAttE rns PREVIEWONLY

108 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES EXAMPLE 2 Determine whether the data sets shown are linear, Determine whether the data set shown is linear, quadratic, exponential, or cubic. quadratic, exponential, or cubic. x y 1. -1 34 x 0 1 2 3 4 0 50 y -8 -2 18 70 172 1 56 Cubic 2 58 2.

3 62 x -2 -1 0 1 2 3 y 0.125 0.5 2 8 32 128 4 74 Exponential 5 100

STEP 1 Determine the first, second, and third differences between successive x-values and successive y-values.

x y

-1 34 ∆x = 0 – -1 = 1 ∆y = 50 − 34 = 16 0 50 -10 ∆x = 1 – 0 = 1 ∆y = 56 − 50 = 6 +6 1 56 -4 ∆x = 2 – 1 = 1 ∆y = 58 − 56 = 2 +6 2 58 +2 ∆x = 3 – 2 = 1 ∆y = 62 − 58 = 4 +6 3 62 +8 ∆x = 4 – 3 = 1 ∆y = 74 − 62 = 12 +6 4 74 +14 ∆x = 5 – 4 = 1 ∆y = 100 − 74 = 26 DRAFT5 100 FOR

1.8 • Writing CubiC FunC tions 109 PREVIEWONLY

109 ADDITIONAL EXAMPLES Determine whether the STEP 2 Determine the ratios between successive y-values. data sets shown are linear, quadratic, exponential, or x y cubic. -1 34 yn 50 y = ≈ 1.47 1. n-1 34 0 50 yn 56 y = ≈ 1.12 x 2 3 4 5 6 n-1 50 1 56 yn 58 y 15 23.5 36 52.5 73 y = ≈ 1.04 n-1 56 2 58 yn 62 Quadratic y = ≈ 1.07 n-1 58 3 62 yn 74 2. y = ≈ 1.19 n-1 62 4 74 yn 100 x -3 0 3 6 9 y = ≈ 1.35 n-1 74 y 8 6 4 2 0 5 100 Linear STEP 3 Determine whether the set of data represents a linear, quadratic, exponential, or cubic function.

INSTRUCTIONAL HINT The differences in x, ∆x, are all 1, so they are constant.

When learning quadratic The first differences are not constant, so the set of data does not represent a linear models, the Instructional function. Strategies suggested that students create a flowchart The second differences are not constant, so the set of data does not represent a qua- for testing data to deter- dratic function. mine the type of function The ratios between successive y-values are not constant, so the set of data does not represented by a set of represent an exponential function. data. Have students return to the flowchart they creat- The third differences are constant, so the set of data represents a cubic function. ed to add testing for cubic functions. DRAFT FOR

110 CHAPt E r 1: ALg E brA i C PAttE rns PREVIEWONLY

110 CHAPTER 1: ALGEBRAIC PATTERNS YOU TRY IT! #2 YOU TRY IT! #2 ANSWER:

Does the set of data shown below represent a cubic function? Justify your answer. No, the set of data does not represent a cubic function, because x y the differences in x are constant but the third finite differences 1 2 in y are not constant. 2 5

3 11

4 23

5 47

6 99

7 191

See margin.

EXAMPLE 3 ADDITIONAL EXAMPLES Write a cubic function for the values in the table. Write the function rules for the tables in the Addi- x y tional Examples on pages 0 0 109 and 110. pg. 109, AE #1: 1 1 y = 3x3 - 2x2 + 5x - 8 2 2 pg. 109, AE #2: x 3 4 y = 2(4) pg. 110, AE #1: 4 8 y = 2x2 - 1.5x + 10 5 15 pg. 110, AE #2: 2 DRAFT6 26 y = - –x + 6 FOR 3

1.8 • Writing CubiC FunC tions 111 PREVIEWONLY

111 INSTRUCTIONAL HINT Have students add the steps for writing a cubic STEP 1 Determine the first differences between successivex -values and the third finite differences in successivey -values. function to their flowchart (page 110). Encourage students to use their flowchart x y to study for the test and 0 0 guide their practice. ∆x = 1 – 0 = 1 ∆y = 1 − 0 = 1 1 1 0 ∆x = 2 – 1 = 1 ∆y = 2 − 1 = 1 1 2 2 1 ADDITIONAL ∆x = 3 – 2 = 1 ∆y = 4 − 2 = 2 1 EXAMPLES 3 4 2 Write a cubic function for ∆x = 4 – 3 = 1 ∆y = 8 − 4 = 4 1 4 8 3 the values in the tables. ∆x = 5 – 4 = 1 ∆y = 15 − 8 = 7 1 1. 5 15 4 ∆x = 6 – 5 = 1 ∆y = 26 − 15 = 11 x 0 1 2 3 4 5 6 26 y 12 11.33 20.67 42 77.33 128.67 3 2 1 STEP 2 Calculate the values for a, b, c, and d in f(x) = ax + bx + cx + d. y = –x3 + 4x 2 – 5x + 12 3 ■■ The value of d is the y-coordinate of the y-intercept, (0, d). For this data set, d = 0. 2. 1 – ■■ The third difference is equal to6 a. Since 6a = 1, a = 6. 0 1 2 3 4 5 x ■■ The second difference between the first two pairs of y-values (x = 0 y 0 -6.5 -45 -139.5 -314 -592.5 and x = 1; x = 1 and x = 2) is 6a + 2b. 3 3 2 – y = -4x – 4x + 2 x 6a + 2b = 0 1 + 2b = 0 2b = -1 1 – b = -2

■■ The first difference between the y-values for x = 0 and x = 1 is equal to a + b + c.

a + b + c = 1 1 1 – – 6 – 2 + c = 1 8 4 – – DRAFT c = 6 = 3 STEP 3 Write the cubic function rule with the values of a, b, c, and d:

1 1 4 – 3 – 2 – FOR f(x) = 6x − 2x + 3x.

112 CHAPt E r 1: ALg E brA i C PAttE rns PREVIEWONLY

112 CHAPTER 1: ALGEBRAIC PATTERNS ADDITIONAL EXAMPLES YOU TRY IT! #3 For the tables below, determine if the relationship is a For the data set below, determine if the relationship is a cubic function. If so, write a function relating the variables. cubic function. If so, write a function relating the x y variables.

0 −6 1. x y 1 — 1 72 1 14.6

2 18 0 10 1 — 1 5.4 3 282 2 3.2 4 42 1 3 5.8 — 5 612 4 15.6 6 90 2 Cubic, y = -–x3 + 5x – 10 1 5 y = —x3 − 3x2 + 16x – 6 2 2. x y -2 -143.25 PRACTICE/HOMEWORK -1 -25.75 0 -1.25 For each table below, determine whether the set of data represents a linear, exponential, quadratic, or cubic function. 1 2.25

2 56.75 1. x f(x) 2. x y 3. x f(x) -1 0.2 0 -1.25 -1 -5 3 234.25 0 1 1 -1 0 0 1 5 2 -0.75 1 5 Cubic, 2 25 3 -0.5 2 40 y = 12x 3 - 10.5x 2 + 2x - 1.25 3 125 4 -0.25 3 135 4 625 5 0 4 320 Exponential Linear Cubic 4. x f(x) 5. x y 6. x y -1DRAFT-2 -1 8 1 40 0 -8 0 5 2 38 1 -2 1 10 3 36 2 16 2 29 4 34 3 46 3 FOR68 5 32 4 88 4 133 6 30 Quadratic Cubic Linear

1.8 • Writing CubiC FunC tions 113 PREVIEWONLY

113 7. Does the set of data shown below represent a cubic function? Justify your response. x y 0 0 1 -4 2 -28 3 -76 4 -148 5 -244

No, the data do not represent a cubic function; the differences in x are constant, but the third differences in y are not constant.

For questions 8 – 10, determine if the given relationship is a cubic function. If it is, write a function relating the variables.

8. x y 9. x f(x) 10. x y -1 8 0 0 0 -7 0 5 1 0.5 1 -5 1 6 2 4 2 9 2 11 3 13.5 3 47 3 20 4 32 4 121 4 33 5 62.5 5 243 Not a cubic function cubic; f(x) = 0.5x3 cubic; y = 2x3 – 7

For questions 11 – 16, the data sets shown in the tables represent cubic functions. Write a cubic function for the values in the table.

11. x y 12. x f(x) 13. x y 0 0 0 -5 0 1 1 0.25 1 -4.8 1 9 2 2 2 -3.4 2 57 3 6.75 3 0.4 3 181 4 16 4 7.8 4 417 5 31.25 5 20 5 801 y = 0.25x3 f(x) = 0.2x3 – 5 y = 6x3 + 2x2 + 1

14. x y 15. x f(x) 16. x y 0 0 0 -1 0 0 DRAFT1 -4 1 0.3 1 -27 2 0 2 7.4 2 -60 3 18 3 25.1 3 -63 FOR4 56 4 58.2 4 0 5 120 5 111.5 5 165 y = x3 + x2 – 6x f(x) = 0.8x3 + 0.5x2 – 1 y = 6x3 – 21x2 – 12x

114 CHAPt E r 1: ALg E brA i C PAttE rns PREVIEWONLY

114 CHAPTER 1: ALGEBRAIC PATTERNS Use the situation below to answer questions 17 – 18. GEOMETRY The volume of a set of rectangular prisms with a base length of x inches, is shown below.

LENGTH OF VOLUME, BASE, x v(x) (INCHES) (CUBIC INCHES) 0 0

1 1.5

2 12

3 40.5

4 96

5 187.5

17. Write the cubic function relating the length of the base to the volume. v(x) = 1.5x3

18. Use your function to predict the length of the base of the prism when the volume is 1500 cubic inches. The length of the base of the prism will be 10 inches.

Use the situation below to answer questions 19 – 20. FINANCE A local mail service charges different rates, based on the weight of the package being mailed. A sample of their prices is shown in the table below.

WEIGHT OF PRICE TO MAIL PACKAGE, w PACKAGE, p (POUNDS) ($) 0 0

1 3.45

2 6.60

3 10.65 DRAFT4 16.80 5 FOR26.25 19. Write a cubic function to represent the given data. p = 0.2w3 – 0.75w2 + 4w

1.8 • Writing CubiC FunC tions 115 PREVIEWONLY

115 20. Use your cubic function to determine the cost to mail a 6-pound package. The price to mail a 6-pound package would be $40.20.

Use the situation below to answer questions 21 – 23. CRITICAL THINKING Blocks were stacked to create the pattern below.

21. Relate the number of layers in a stack, x, to the total number of blocks, y, by completing the table below. The first few rows have been completed for you.

NUMBER OF TOTAL NUMBER LAYERS, x OF BLOCKS, p 0 0

1 1

2 5

3 14

4 30 5 55

22. Write the function relating the variables in problem 21. 1 1 1 — 3 — 2 — y = 3x + 2x + 6x

23. If the pattern continues, how many blocks would it take to create a 7-layer stack? DRAFTIt would take 140 blocks to create a 7-layer stack. FOR

116 CHAPt E r 1: ALg E brA i C PAttE rns PREVIEWONLY

116 CHAPTER 1: ALGEBRAIC PATTERNS TEKS AR.2B Classify a func- 1.9 Modeling with Cubic Functions tion as linear, quadratic, cubic, and exponential when a function is repre- FOCUSING QUESTION How can you use finite differences to construct a sented tabularly using fi- cubic model for a data set? nite differences or common ratios as appropriate. LEARNING OUTCOMES ■■ I can use finite differences or common ratios to classify a function as linear, AR.2C Determine the quadratic, cubic, or exponential when I am given a table of values. function that models a giv- ■■ I can determine the cubic function from a table using finite differences, includ- en table of related values ing any restrictions on the domain and range. using finite differences and ■■ I can use finite differences to write a cubic function that describes a data set. its restricted domain and ■■ I can apply mathematics to problems that I see in everyday life, in society, and range. in the workplace. AR.2D Determine a function that models real-world data and math- ENGAGE ematical contexts using A diving board is a platform that someone finite differences such as can use to dive into a deep water pool. Div- ing boards typically have enough flexibili- the age of a tree and its ty to allow a diver to bounce before leaving circumference, figurative the board and entering the pool. Diving numbers, average velocity, boards have a heel end on the pool deck, and average acceleration. which is where the person steps onto the diving board, and a toe end that hangs over the pool. Diving boards also have a fulcrum that balances the board. MATHEMATICAL When a person stands on a diving board, it bends or deflects slightly due to the per- PROCESS SPOTLIGHT son’s weight. What questions could you ask about the deflection of a diving board AR.1A Apply mathemat- that could be answered by collecting data? ics to problems arising Work with a partner to give information you know about diving boards and this in everyday life, society, situation. and the workplace. ELPS EXPLORE 3F Ask and give When a person stands on a information ranging from diving DRAFTboard, the diving board using a very limited bank bends beneath the weight of of high-frequency, high- the person. The farther the per- need, concrete vocabulary, son stands from the fulcrum of including key words and the diving board, the more the FOR diving board bends, or deflects, expressions needed for from the horizontal. basic communication in academic and social contexts, to using abstract and content-based vocabulary 1.9 • Modeling with CubiC FunC tions 117 during extended speaking assignments. VOCABULARY SUPPORTING ENGLISH LANGUAGE LEARNERS cubic function, finite dif- For the Engage, pair students up so they may alternate giving information about what ferences, maximum, mini- questions they could ask for the diving board situation. Encourage them to use key mum words and expressions related to functions and diving boards/swimming pools. En- courage students to think about bothPREVIEW academic and social contexts (ELPS 3F). MATERIALS ONLY• graphing calculator

117 1. No, because as x increases The table below shows the amount of deflection, in thousandths of an inch, when the by 1 foot, the amount of same person stands x feet from the fulcrum of the diving board. deflection increases by a greater amount each time. DISTANCE FROM DEFLECTION The increase from a dis- FULCRUM (FT), (0.001 IN.), tance of 0 feet to 1 foot is x f(x) 0.116 inches of deflection, 0 0 but the increase from a 1 116 distance of 1 foot to 2 feet is 0.332 inches of deflec- 2 448 tion. 3 972

2. No. The deflection between 4 1664 a distance of 1 foot and 2 5 2500 feet increases by a factor of about 4, but the deflection 1. Does the data set appear to have a constant rate of change in deflection? Ex- between a distance of 2 feet plain how you know. and 3 feet increases by a See margin. factor of about 2. 2. Does the amount of deflection appear to increase by the same factor each time 3. See margin below. the distance increases by 1 foot? Explain how you know. See margin. 4. See margin bottom of pages 118-119. 3. Calculate the finite differences between the deflection and the distance from the fulcrum. Do the data appear to be linear, quadratic, or cubic? How do you 5. See margin and bottom of know? page 119. The data appear to be cubic because the third differences are constant. See margin for details. 6. Data set - domain: whole 4. Use the patterns in the finite differences to write a function rule that describes numbers, 0 ≤ x ≤ 5; range: the data set. {0, 116, 448, 972, 1664, f(x) = −4x3 + 120x2 2500} See margin for details. 5. Use a graphing calculator to graph the function rule over a scatterplot of the Function - domain: all real data. What do you notice about the function rule and the scatterplot? numbers; range: all real The graph of the function rule connects each of the data points. See numbers margin for graph. 6. Compare the domain and range of the data set and the domain and range of the The domain and range of function rule. How are they alike? How are they different? the data set are subsets of See margin. the domain and range of 7. What will be the deflection if the person stands 10 feet from the fulcrum? the function rule.DRAFTSee margin. The domain and range of 8. Are the x-intercepts of the function included in your data set? Why or why not? the data set are limited to See margin. whole numbers, but the FOR domain and range of the 9. What dimension of the diving board represents the greatest possible x-value function rule include all that could be contained in the data set? What limit does that place on the do- real numbers. main of the data set? See margin. 7-9. See margin page 119. 118 C h APteR 1: AlgebRAi C PAtteR ns

3. DEFLECTION 4. 3 2 DISTANCE FROM For a cubic function of the form f(x) = ax + bx + cx + d, (0.001 IN.), FULCRUM (FT), x f(x) • The value of d is the y-coordinate of the y-intercept, (0, d). • The third difference is equal to 6a. 0 0 • The second difference between the first two pairs of +1 +116 y-values, x = 0 and x = 1, and x = 1 and x = 2, is equal to 1 116 PREVIEW+216 6a + 2b. +1 +332 -24 2 448 +192 • The firstONLY difference between the y-values for x = 0 and +1 +524 -24 x = 1 is equal to a + b + c. 3 972 +168 +1 +692 -24 4 1664 +144 +1 +836 5 2500

118 CHAPTER 1: ALGEBRAIC PATTERNS 5. The graph of the function rule connects each of the REFLECT data points.

■■ How can you determine a cubic function model for a data set? See graph below. See margin. 7. f(x) = −4x3 + 120x2 f(10) = −4(10)3 + 120(10)2 ■■ What other factors could influence the deflection of the diving board? Explain how they would do so. f(10) = −4,000 + 12,000 See margin. f(10) = 8,000 At a distance of 10 feet from the fulcrum, the EXPLAIN diving board will deflect 8.000 inches. Watch Explain and Cubic function models can be used to represent sets of mathematical You Try It Videos 8. One x-intercept, (0, 0), is and real-world data. Cubic functions have several attributes that should included in the data set be- be considered when they are used for mathematical models. cause it makes senses that The domain and range of most cubic functions are all real numbers. Un- when the person stands 0 like quadratic functions, when you raise a negative number to the third feet from the fulcrum that power, you can have a negative number as a result. the diving board deflects

or click here 0.000 inches. The other Cubic functions have as many as x-intercept, (30, 0), is not 3 x-intercepts and 1 y-intercept. As included in the data set with other function types, the x-intercepts represent x-values that because it does not make generate a function value equal to sense that if a diving board 0. The y-intercept is also a starting were over 30 feet long that point, or y-value when x = 0. a person standing 30 feet from the fulcrum would Quadratic functions have a maxi- not deflect the diving mum or minimum point at the ver- board. tex. These are absolute maximum or minimum values. A cubic function, 9. The distance from the however, may have what are called fulcrum to the end of local maximum or local minimum the diving board is the values. These points are not the ab- greatest possible x-value solute maximum or minimum values for the function. Instead, they that could be contained in are maximum or minimum values the data set. This x-value for a nearby,DRAFT or local, part of the represents the maximum function. for the domain of the data set. Let’s look more closely at the data set and function model for the div- FOR REFLECT ANSWERS: ing board problem. See margin page 120.

1.9 • Modeling with CubiC FunC tions 119

4. DEFLECTION 5. DISTANCE FROM a + b + c = 116 (0.001 IN.), (-4) + (120) + c = 116 FULCRUM (FT), x d = 0 f(x) 116 + c = 116 c = 0 0 0 +1 +116 6a + 2b = 216 1 116 PREVIEW+216 6(-4) + 2b = 216 -24 + 2b = 216 +1 +332 -24 2b = 240 2 448 +192 ONLY b = 120 +1 +524 -24 3 972 +168 6a = -24 +1 +692 -24 a = -4 4 1664 +144 +1 +836 5 2500

f(x) = −4x3 + 120x2

119 REFLECT ANSWERS: Confirm that the third finite differences are constant. If they are, then work backward in the table if necessary to identify function values for x = 0 and x = 1. Use the patterns in the table to identify the values of a, b, c, and d to write a cubic function in standard (polynomial) form, f(x) = ax3 + bx2 + cx + d.

Answers may vary. Possible responses may include: In the diving board problem, the data points represent The weight of the person a small portion of the function. The shaded region on With polynomial function models, frequently the function mod- standing on the diving board the graph shows the part of the function that would influences the deflection. If the el only represents the data set for model the deflection of a diving board that has a dis- a certain interval of the function. person weighs less than the per- tance of 15 feet between the end of the diving board In the diving board problem, son for whom the data was col- and the fulcrum. there is a local maximum at lected, then the deflection will (20, 16,000). This means that a The function has two x-intercepts, one of which makes person standing 20 feet from be less. If the person weights the fulcrum causes a deflection sense in the context of the diving board problem. When more, then the deflection will be of 16.000 inches. The function the person stands 0 feet from the fulcrum, you would greater. model for x-values greater than expect the diving board to deflect 0 inches. However, 20 then begins to decrease. It is The flexibility or rigidity of it is not likely that a person standing 30 feet from the not likely that a person standing fulcrum would generate a deflection of 0 inches. farther than 20 feet from the ful- the diving board influences the crum would generate less deflec- deflection. If the diving board is tion, so the model likely does not more rigid and doesn’t bend as represent the situation beyond a domain of 20 feet. easily, then the deflection will be less. MODELING WITH CUBIC FUNCTIONS

If the data set has a constant or approximately constant third finite difference, then a cubic function model may be appropriate for the data set.

DRAFTThe domain and range for the situation may be a subset of the domain and range of the cubic function model. Cubic functions can have intervals within their domain when they are increasing and intervals within FORtheir domain when they are decreasing.

120 C h APteR 1: AlgebRAi C PAtteR ns

TECHNOLOGY TIP Some students may not be familiar with the shape of a graph of a cubic function. Use graphing technology such as a graphing calculator or a graphing app to allow students to explore the parent function of a cubic func- 3 PREVIEW tion, f(x) = x , as well as several other cubic functions that generate different shapes, including local maxima and ONLY minima. For the function in the diving board situation, use graphing technology to zoom out and show students the shape of the complete graph.

120 CHAPTER 1: ALGEBRAIC PATTERNS EXAMPLE 1 A display of large juice cans takes the shape of a square-based pyramid with 1 can in the top level, 4 cans in the second level, 9 in the third level, 16 in the fourth level, etc. To predict the number of cans y for a display of x number of levels, determine if this situation represents a linear, quadratic, or cubic function.

LEVELS, PROCESS NUMBER OF x CANS, y

1 1 1

2 1 + 4 5

3 1 + 4 + 9 14

4 1 + 4 + 9 + 16 30

5 1 + 4 + 9 + 16 + 25 55

6 1 + 4 + 9 + 16 + 25 + 36 91

Use the data set to determine if the relationship is linear, quadratic, or cubic.

STEP 1 Consider the value of y, the number of cans, in the “zero” level of the display. The “zero” level would logically have no cans.

STEP 2 Determine the finite differences in values ofx and the first, second, and third finite differences in values off (x).

LEVELS, PROCESS NUMBER OF x CANS, y

0 0 0 ∆x = 1 – 0 = 1 +1 1 1 1 +3 ∆x = 2 – 1 = 1 +4 +2 2 1 + 4 5 +5 ∆x = 3 – 2 = 1 +9 +2 3 1 + 4 + 9 14 +7 ∆x = 4 – 3 = 1 +16 +2 4 1 + 4 + 9 + 16 30 +9 ∆x = 5 – 4 = 1 +25 +2 DRAFT5 1 + 4 + 9 + 16 + 25 55 +11 ∆x = 6 – 5 = 1 +36 6 1 + 4 + 9 + 16FOR + 25 + 36 91 The set of data represents a cubic function because the differences in x are constant and the third finite differences between successive values of f(x) are constant.

1.9 • Modeling with CubiC FunC tions 121

ADDITIONAL EXAMPLES Determine if the situations below represent a linear, quadratic, or cubic function. 1. A rocket is fired into the air. Its height above the ground, h, in meters at a given time, t, in seconds is shown in the table. PREVIEW TIME (SECONDS), t 0 1 2 3 4 HEIGHT (METERS), h 6.5 79.6 142.9 196.4 240.1 ONLY

Quadratic

2. As a balloon was inflated, its radius, r, and volume, V, were recorded in the table shown.

RADIUS (IN), r 0 1 2 3 4 5 VOLUME (IN3), V 0 4.19 33.51 113.1 268.08 523.6

Cubic

121 YOU TRY IT! #1

The number of squares on a checkerboard, including the number of 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on, are shown in the table. If the checkerboard is 1 × 1, there is only one square possible. In a 2 × 2 board, there are four 1 × 1 squares and one 4 × 4 square, for a total of five squares. The chart below shows the total number of squares contained in two checkerboards.

SIDE TOTAL LENGTH OF NUMBER OF BOARD IN SQUARES, SQUARES, x f(x) 1 2

2 10

3 28

4 60

5 110

6 182

7 280

8 408

Determine whether the relationship is linear, quadratic, or cubic. The set of data represents a cubic function because the differences in x values and the third finite differences in f(x) values are both constant.

ADDITIONAL EXAMPLE EXAMPLE 2 Write the cubic function for Determine the cubic function rule to model the display of cans in Example 1. Compare Additional Example #2 on the domain and range of the data set and the function. page 121. 4 Step 1 Calculate the values of a, b, c, and d in the cubic function rule – 3 3 V = 3 r or V = 4.19r f(x) = ax3 + bx2 + cx + d using the data in the table and finite differences. π DRAFT ■■ There would be no cans in the “zero” level. The value for y when x = 0 is 0. So d = 0. FOR■■ The third finite difference in the set of data is 2. This equals a6 . 1 – If 6a = 2, then a = 3.

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122 CHAPTER 1: ALGEBRAIC PATTERNS ■■ The second difference between the first two pairs of y-values (x = 0 and x = 1; x = 1 and x = 2) is equal to 6a + 2b.

6a + 2b = 3 2 + 2b = 3

1 – 2b = 1 and b = 2

■■ The first difference between the y-values for x = 0 and x = 1 is equal to a + b + c. a + b + c = 1 1 1 – – 3 + 2 + c = 1 1 – c = 6

STEP 3 Write the cubic function rule with the values of a, b, c, and d:

1 1 1 – 3 – 2 – f(x) = 2x + 3x + 6x

STEP 4 Enter the x- and y-values from the table and then graph the function with technology. The graph connects the data points. The data points are limited to values for levels 1 to 8.

■■ Considering the weight of the cans in several levels, it doesn’t make sense to add more levels. Also, the function rule produces fractional parts of cans for many values of x. ■■ Data set - domain: whole numbers, 1 ≤ x ≤ 8; range: {1, 5, 14, 30, 55, 91, 140, 204} ■■ Function - domain: all real numbers; range: all real numbers

The domain and range of the data set are subsets of the domain and range of the function rule.

The domain and range of the data set are limited to whole numbers, but the domain and rangeDRAFT of the function rule include all real numbers. FOR

1.9 • Modeling with CubiC FunC tions 123

ADDITIONAL EXAMPLES Determine the cubic function rule to model the given data. 1. x 0 1 2 3 4 PREVIEW5 y 13.75 14.75 29.75 76.75 173.75 338.75 ONLY 1 3 2 – f(x) = 3x – 2x + 2x + 13.25 2. x 0 1 2 3 4 5 y -12 -19.5 -8 31.5 108 230.5

3 – 3 2 f(x) = 2x + 5x - 14x - 12

123 YOU TRY IT! #2 ANSWER: YOU TRY IT! #2 CUBE WEIGHT IN f(x) = 4x3 – 48x2 + 144x. NUMBER, x OUNCES, f(x) A tray is made of a 12 in. x 12 in. square piece of cardboard 2 12.5 A 2-inch cutout yields a tray with squares cut out of the four corners. The resulting sides with the greatest volume. In are folded up and taped to form a square prism (open box). 4 34.3 this situation, the domain The volume of the tray related to the size of the squares cut 6 72.9 includes whole numbers from the corners. between 0 and 6, 8 133.1 SIZE OF {x|x W, 0 < x < 6} VOLUME 10 219.7 SQUARE PROCESS IN CUBIC (There cannot be a cutout as CUTS IN (12 – 2x)(12 – 2x)(x) INCHES, INCHES, x f(x) 12 337.5 great∈ as half the width of the cardboard), and the range 0 (12 – 0)(12 – 0)(0) 0 1 (12 – 2)(12 – 2)(1) 100 STEP 1 Determine the finite differences in values ofx and the first, includes whole numbers, second, and third finite differences in values off (x). 0 < y <128. 2 (12 – 4)(12 – 4)(2) 128 3 (12 – 6)(12 – 6)(3) 108 CUBE WEIGHT IN NUMBER, x OUNCES, f(x) 4 (12 – 8)(12 – 8)(4) 64 2 12.5 INSTRUCTIONAL HINTS 5 (12 – 10)(12 – 10)(5) 20 ∆x = 4 – 2 = 2 +21.8 With every set of data, have 6 (12 – 12)(12 – 12)(6) 0 4 34.3 +16.8 students enter the function ∆x = 6 – 4 = 2 +38.6 +4.8 6 72.9 +21.6 rule in their graphing cal- Generate a cubic function model for the volume of the tray. What size cutout produc- +60.2 +4.8 culators in order to see the es the tray with the greatest volume? Identify the domain and range that most appro- ∆x = 8 – 6 = 2 8 133.1 +26.4 priately models the data. graph. Once graphed, have +86.6 +4.8 See margin. ∆x = 10 – 8 = 2 students zoom out until 10 219.7 +31.2 they see the “bigger pic- ∆x = 12 – 10 = 2 +117.8 ture.” Students might look 12 337.5 at a small portion of a cubic function and mistake it for EXAMPLE 3 The set of data represents a cubic function because the differences in x are constant, although the ∆x is 2, not 1, and the third finite differences a quadratic function. A set of graduated cubes comes with a large industrial scale. The cubes are numbered between successive values of f(x) are constant. according to increasing size. The weights of some of the cubes are given in the table. Help students process the Determine if the data set represents a linear, quadratic, or cubic function. Identify the STEP 2 From the information, it is unclear if there are larger or odd- differences and similari- domain and range of the model that most appropriately models the data. numbered cubes. ties in quadratic and cubic functions by writing about The set of data represents a cubic function. them. The domain is a subset of whole numbers, 0 < x ≤ 12. The range is a subset of real numbers, 0 < y ≤ 337.5. Have students drawDRAFT a Venn Diagram to compare and contrast the two types of functions. FOR

124 C h APteR 1: AlgebRAi C PAtteR ns 1.9 • Modeling with CubiC FunC tions 125 PREVIEWONLY

124 CHAPTER 1: ALGEBRAIC PATTERNS CUBE WEIGHT IN NUMBER, x OUNCES, f(x)

2 12.5

4 34.3

6 72.9

8 133.1

10 219.7

12 337.5

STEP 1 Determine the finite differences in values ofx and the first, second, and third finite differences in values off (x).

CUBE WEIGHT IN NUMBER, x OUNCES, f(x)

2 12.5 ∆x = 4 – 2 = 2 +21.8 4 34.3 +16.8 ∆x = 6 – 4 = 2 +38.6 +4.8 6 72.9 +21.6 ∆x = 8 – 6 = 2 +60.2 +4.8 8 133.1 +26.4 ∆x = 10 – 8 = 2 +86.6 +4.8 10 219.7 +31.2 ∆x = 12 – 10 = 2 +117.8 12 337.5

The set of data represents a cubic function because the differences in x are constant, although the ∆x is 2, not 1, and the third finite differences between successive values of f(x) are constant.

STEP 2 From the information, it is unclear if there are larger or odd- numbered cubes.

The set of data represents a cubic function. The domain is a subset of whole numbers, 0 < x ≤ 12. The rangeDRAFT is a subset of real numbers, 0 < y ≤ 337.5. FOR

1.9 • Modeling with CubiC FunC tions 125

ADDITIONAL EXAMPLES Determine the cubic function rule to model the data sets below. 1. x 0 1 2 3 4 PREVIEW5 y 0 -0.7 -52.4 -227.1 -596.8 -1233.5 ONLY f(x) = -12x3 + 10.5x2 + 4.5x 2.

x 0 1 2 3 4 5 y 0 42.33 86.67 135 189.33 251.67 1 – 3 f(x) = 3x + 42x

125 YOU TRY IT! #3 ANSWER: YOU TRY IT! #3 The differences in the x-values The power, y, (in kilowatts) generated by a wind turbine is related to the wind speed. are constant. The third finite Determine if the data set represents a linear, quadratic, or cubic function. Identify the differences, which average 0.33, domain and range of the model that most appropriately models the data. are approximately constant, so the data set represents a cubic AVERAGE ANNUAL WIND ANNUAL ENERGY OUTPUT function model. The domain SPEED IN THE USA IN IN KWH/YEAR, METERS PER SECOND, x f(x) represents the annual average wind speeds in the USA, so the 4 3.40 domain of the data set is the 5 6.64 subset of whole numbers, 4 ≤ x ≤ 10. The annual energy 6 11.47 output reflects the data col- 7 18.22 lected for those average wind speeds, so the range of the set is 8 27.25 the subset of real numbers, 3.40 ≤ y ≤ 53.12. 9 38.75 10 53.12

Data Source: National Renewal Energy Laboratory and Energy.gov See margin.

PRACTICE/HOMEWORK

For questions 1 – 6, use finite differences to determine if the data sets shown in the tables below represent a linear, exponential, quadratic, or cubic function. 1. x y 2. x y 3. x y 0 3 0 -6 0 2.25 1 6 1 1 1 8.75 2 12 2 16 2 15.25 3 24 3 39 3 21.75 4 48 4 70 4 28.25 5 96 5 109 5 34.75 Exponential Quadratic Linear DRAFT4. x y 5. x y 6. x y 0 5 0 20 0 -8 1 10 1 50 1 3 2 35 2 125 2 44 FOR3 92 3 312.5 3 145 4 193 4 781.25 4 336 5 350 5 1953.125 5 647 Cubic Exponential Cubic

126 C h APteR 1: AlgebRAi C PAtteR ns PREVIEWONLY

126 CHAPTER 1: ALGEBRAIC PATTERNS For questions 7 – 12, the data sets shown in the tables represent cubic functions. Use finite differences to determine the function that relates the variables.

7. x y 8. x y 9. x y 0 -1 0 3 0 0 1 0 1 7 1 14 2 11 2 1 2 72 3 50 3 -27 3 198 4 135 4 -89 4 416 5 284 5 -197 5 750 f(x) = 3x3 – 4x2 + 2x – 1 f(x) = -2x3 + x2 + 5x + 3 f(x) = 4x3 + 10x2

10. x y 11. x y 12. x y 0 -4 0 1 0 -9 1 -9 1 6 1 -3 2 -48 2 15 2 39 3 -157 3 31 3 153 4 -372 4 57 4 375 5 -729 5 96 5 741 1 1 3 2 — 3 — 2 3 f(x) = -6x + x – 4 f(x) = 2x + 2x + 4x + 1 f(x) = 6x – 9

For questions 13 – 17 use the scenario below. GEOMETRY A box is created from a 20-inch by 24-inch rectangular piece of cardboard by cutting congruent squares from each corner. The squares are cut in 1-inch increments. The resulting sides are folded up and taped to form a rectangular prism (open box). The volume of the box is a function of the side length of the square removed from each corner. The table below relates the volume of the box to the side length of the square.

SIDE LENGTH VOLUME x y 0 0 1 396 2 640 3 756 4 768 5 700 DRAFT6 576 7 420 8 256 9 FOR108

1.9 • Modeling with CubiC FunC tions 127 PREVIEWONLY

127 13. Generate a cubic function model for the volume of tray when given the side length of the square cut from each of the corners. y = 4x3 – 88x2 + 480x

14. What side length of the square produces a tray with the greatest volume? 4 inches

15. The graph to the right represents the data in the table and the function that models the table.

What is the domain and range of this situation? Domain = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Range = {0, 108, 256, 396, 420, 576, 640, 700, 756, 768}

16. The graph to the right represents the data in the table and the function that models the table, but has been graphed with a different window setting.

Why are there no individual points plotted on the graph for x < 0 and x > 10? The side length of the cut-out square cannot be a negative number and it also cannot be greater than half the length or DRAFTwidth of the rectangle. FOR 17. Why does the domain and range only contain whole numbers? The squares cut from each corner of the rectangular piece of cardboard are cut in 1-inch increments.

128 C h APteR 1: AlgebRAi C PAtteR ns PREVIEWONLY

128 CHAPTER 1: ALGEBRAIC PATTERNS For questions 18 – 22 use the scenario below. CRITICAL THINKING An employee at a toy store is creating a display of soccer balls in the shape of a tetrahedron, or an equilateral triangular pyramid.

The table below shows the total number of soccer balls at each level of the display, with Level 1 being at the top of the display.

TOTAL NUMBER LEVEL, OF SOCCER BALLS, x y 1 1 2 4 3 10 4 20 5 35 6 56

18. Write a function using finite differences that models the data in the table. 1 1 1 — 3 — 2 — f(x) = 6x + 2x + 3x

19. What does the domain of the function represent in the situation? The level number

20. What does the range of the function represent in the situation? The total number of soccer balls used to build the display through that level. 21. Is 2.5 an element in the domain of this situation? Why or why not? No; there cannot be 0.5 of a level.

22. How many soccer balls would be needed to build a display 10 levels high? 220 soccer balls DRAFT FOR

1.9 • Modeling with CubiC FunC tions 129 PREVIEWONLY

129 Chapter 1 Review

1. ȱȱǰȱ ȱȱęȱŚȱȱȱȱǯȱ ȱ ȱȱ ȱȱȱȱǯ 3 ȱƺȱŗ – a) an = 256(Ś) c) aŗȱƽȱȬśŚǲȱan = aȱȮȱŗ + 7 üüüü geometric arithmetic

b) aŗȱƽȱŘǲȱan = 3aȱȮȱŗ d) anȱƽȱřŘśŖȱƺȱŝśn geometric arithmetic 2. ȱȱȱȱy = mx + bȱȱȱȱȱȱǯ 5 – a) ȱƽȱ2, yȬȱƽȱǻŖǰȱŗŖǼȱȱd) ȱ x y 5 — 2 21 y = 2x + 10 4 18 b) ȱƽȱƺŘǰȱȱȱȱǻśǰȱƺŝǼ 6 15 ^"ü] 8 12 10 9 c) ȱȱȱǻśǰȱȬřǼȱȱǻȬŚǰȱŖǼ ^"ü] -1 4 — — y = 3 ] ü 3

3. ȱȱȱ ȱȱ¡ȱǰȱ ȱȱȱȱ ȱȱȱȱȱǯ

a) x y b) x y 0 243 1 500 1 81 2 1000 2 27 3 2000 3 9 4 4000 1 – ratio = 3 ratio = 2 1 x y = 250(2)x — y = 243(3) 4a. Exponential, because the 4. ȱȱȱȱȱȱȱȱȱȱȱȱ ȱȱȱȱ successive ratios of the ǰȱ ȱxȱȱȱȱȱ¢ȱȱȱȂȱȱ ȱęȱȱȱyȱȱȱǯȱȱ y-values are approximately equal when Δx is NUMBER OF DAYS, x 0 1 2 3 4 5 6 7 constant. DRAFTPOPULATION, y 123 290 715 1,695 4,048 9,769 23,500 56,400 a. ȱȱǰȱǰȱǰȱǰȱȱ¡ǵȱSee margin. b. ȱȱȱȱȱȱǯȱf(x) = 123(2.4)x FORc. ȱȱ¢ȱǰȱ ȱ ȱȱȱȱȱȱ ȱȱ ȱȱȱȱȱȱşȱ¢ǵ 324,942 d. ȱȱ¢ȱǰȱ ȱ ȱȱȱȱȱȱ ȱȱȱȱȱȱ¡ȱŗŖǰŖŖŖǰŖŖŖǵȱBy the 13th day, the population is expected to be over 10,000,000.

130 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

130 CHAPTER 1: ALGEBRAIC PATTERNS For problems 5 – 10, determine if the function represented is linear, quadratic, cubic, or exponential, then write a function equation relating the variables. 5. x y 6. x y x y 1 11 0 12 0 8 2 22 2 9 1 12 3 37 4 6 2 18 4 56 6 3 3 27 5 79 8 0 4 40.5 Quadratic Linear Exponential y = 2x2] ^"ü] ^" x

8. x y 9. x y 10. x y 0 -8 -2 19 0 6 1 -6 -1 13 1 2 8 — 0 5 1 93 3 46 1 -5 2 — 4 120 2 263 2 -17 Cubic Quadratic y = 2x3ü 3 60 ^"ü]2ü] 1 — 4 1113

Cubic 1 — 3 2 y = 3x + 6x ü]

Use the following situation to answer questions 11-14. ȱ ȱȱȱȱęȱȱ ǯȱȱȱȱȱȱȱȱ ȱȱȱęȱȱȱȱȱȱȱǯ

FIGURE 1 2 3 4 5 n NUMBER, n NUMBER OF 1 6 15 28 DRAFTDOTS, D(n) FOR

C HAPTER 1 RE v IEw 131 PREVIEWONLY

131 11. ȱȱĴȱȱȱěǰȱȱȱȱȱȱȱ ȱȱȱȱśȱęǯȱ45

FIGURE 1 2 3 4 5 n NUMBER, n NUMBER OF 1 6 15 28 DOTS, D(n) 45

+5 +9 +13

+4 +4 +4 12. ȱȱȱȱȱȱȱȱ ȱnȱȱD(n)ǯ ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ D(n) = 2n2üS

13. ȱ¢ȱȱ ȱȱȱȱşthȱęǵȱȱȱ D(9) = 2(9)2 – 9 = 153 dots

14. ȱ ȱęȱ ȱȱȱŚřśȱǵȱȱȱ 2 2 435 = 2n – n, Enter Y1 = 2x å]NSYTLWFUMNSLHFQHZQFYTWFSIąSINS>1 HTQZRS S"\NQQ^NJQIXTYMJYMąLZWJ\TZQIMF[JITYX

Use the following situation to answer questions 15-17.

ȱȱȱ¡ȱȱȱȱȱȱȱȱ ȱȱȱ ȱřȱȱȱ ȱȱ ǯȱȱȱȱȱ¡ȱ ȱȱ¢ȱĴȱȱȱȱȱ ŗ – ȱȱȱȱȱȱ6ȱȱȱ ȱȱȱȱȱȱȱȱ ȱ ȱȱęǯ ¡ ¡

¡ Ŝ¡ Ś¡ Ś¡ Ś¡ȱƸȱř Ś¡ȱƸȱř

Ŝ¡ȱƸȱř

ȱȱȱ¡ȱȱ ȱȱȱȱȱȱęȱ¡ȱȱȱ ȱȱȱȱȱȱȱȱǯ

DRAFTHEIGHT OF BOX 1 2 3 4 5 x (INCHES), x

VOLUME OF BOX 28 176 540 1,216 2,300 FOR(CUBIC INCHES), V

132 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

132 CHAPTER 1: ALGEBRAIC PATTERNS 15. ȱȱĴȱȱȱěǰȱȱȱȱȱȱ¡ȱ ȱȱ ȱȱŜȱǯȱ3,888

HEIGHT OF BOX 1 2 3 4 5 x (INCHES), x

VOLUME OF BOX 28 176 540 1,216 2,300 3,888 (CUBIC INCHES), V

+148 +364 +1,084 +1,588 +216 +312 +408 +504

+96 +96 +96

16. ȱȱȱȱȱȱȱȱ ȱ¡ȱȱǯȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱ V = 16x3 + 12x2

ȱ ȱȱȱȱȱȱ¡ȱ ȱȱȱȱŘǯśȱǵȱȱȱ V = 16(2.5)3 + 12(2.5)2 = 325 cubic inches

Use the following situation to answer questions 18-20. ȱȱ¢ȱȱȱȱȱ¢ȱȱȱ ȱȱȱ ȱȱȱȱǯȱȱ£ȱȱȱęȱȱȱ ȱȱ ȱȱ ȱȱ ǯ

WEIGHT OF RADIUS OF CARTON, W CARTON, r (IN) (OUNCES)

1 6

2 50

3 170

4 402

5 785

6 1357

18. DRAFT ȱȱȱȱęȱȱǰȱǰȱǰȱȱ¡ȱǵȱ ¢ȱ ¢ȱ ǯ 9MJIFYFąYFHZGNHRTIJQXNSHJYMJYMNWIINKKJWJSHJXFWJFUUWT]NRFYJQ^ constant. FOR

C HAPTER 1 RE v IEw 133 PREVIEWONLY

133 19. Assume that for a radius 19. and height of 0, the ȱȱȱȱȱȱȱ ȱrȱȱWǯ weight = 0 so d = 0. RADIUS OF WEIGHT OF CARTON, W a + b + c = 6 and CARTON, r (IN) (OUNCES) 6a + 2b = 38. 0 0 The average 3rd difference 6 is 37.75 so 6a = 37.75. 1 6 řŞ Therefore, a ≈ 6.3, b = 0.1, ŚŚ řŞ c = −0.4 2 50 76 ŗŘŖ 36 The function is 3 170 ŗŗŘ W = 6.3r3 + 0.1r2 − 0.4r 232 39 4 402 ŗśŗ řŞř řŞ 5 785 ŗŞş 20. From a graphing calcu- 572 lator, a value close to 138 6 1357 in the y column yields an See margin. x-value of 2.8. Therefore, a carton weighing about 20. ȱȱ¢ȱǰȱ ȱ ȱȱȱ¡ȱȱȱȱȱ 138 oz, would have a ȱ ȱȱŗřŞȱǵ height of about 2.8 inches See margin.

MULTIPLE CHOICE 1. ȱȱǞŗśŖȱȱȱȱȱȱȱȱȱǞřŖȱȱ ȱȱ ¢ȱȱȱȱ¢Ĵǯȱȱȱȱȱȱȱȱ ȱǰȱSǰȱȱ ȱ ǵ A. SȱƽȱřŖȱƸȱŗśŖw B. SȱƽȱŗśŖǻřŖǼw C. SȱƽȱǻŗśŖȱƸȱřŖǼw D. SȱƽȱŗśŖȱƸȱřŖw

2. ȱȱȱ ȱȱȱxȱȱf(x)ȱȱȱȱǰȱ f(x) = ax2 + bx + cǰȱȱȱěȱȱȱǻ¡Ǽȱȱȱ¢ȱ ȱ ȱȱȱȱǵ A. a B. 2a C. 6a DRAFTD. ȱƸȱȱƸȱ FOR

134 CHAPTER 1: ALGEBRAIC PATTERNS PREVIEWONLY

134 CHAPTER 1: ALGEBRAIC PATTERNS 3. ȱȱȱ ȱȱȱȱȱȱȱȱ ǵ

x 0 1 2 3 4 5 6 y 15 19 35 63 103 155 219

A. 6x2ȱƺȱŗǯşxȱƸȱŗŚǯŞ B. 6x2ȱƸȱŗǯşxȱƸȱŗŚǯŞ C. x2ȱƺȱŗǯşxȱƸȱŚǯŘ D. 3x2ȱƺȱŗşxȱƸȱŗŞǯŚȱȱȱ

4. ȱȱȱ ȱȱȱȱ ǰȱ ȱȱȱĜȱȱȱx3ȱǵ

x 0 1 2 3 4 5 6 7 y = ax3 + bx2 + cx + d -200 -186 -144 -50 120 390 784 1326

A. ŘŚ B. ŗŘ C. 6 D. Ś

3n2 – n th 5. ȱȱȱȱȱǰȱȱn ȱęȱȱȱ 2ȱ ǯ

ȱ ȱ¢ȱȱȱȱȱşthȱȱęǵ A. 92 B. ŗŗŝ C. ŘřŚ D.DRAFT 360 FOR

C HAPTER 1 RE v IEw 135 PREVIEWONLY

135