Arithmetic and Geometric Sequences - 1.1 - Cosenza ...

2 C H A P T E R 1 : A L G E B R A I C PAT T E R N S

SEQUENCE 2:

Term 5 Term 6

16 counters 32 counters

TEKSAR.2A Determine the patterns that identify the relationship between a function and its common ratio or related finite dif-ferences as appropriate, including linear, quadrat-ic, cubic, and exponential functions.

A.12C Identify terms of arithmetic and geometric sequences when the se-quences are given in func-tion form using recursive processes.

A.12D Write a for-mula for the nth term of arithmetic and geometric sequences, given the value of several of their terms.

MATHEMATICAL PROCESS SPOTLIGHTAR.1D Communi-cate mathematical ideas, reasoning, and their im-plications using multiple representations, includ-ing symbols, diagrams, graphs, and language as appropriate.

ELPS3G Express opin-ions, ideas, and feelings ranging from communicat-ing single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academ-ic topics.

VOCABULARYarithmetic sequence, geo-metric sequence, common difference, common ratio, recursive, additive relation-ship, multiplicative rela-tionship

SEQUENCE 1:

Term 4

Term 5

MATERIALS• 50 toothpicks for each

student group• 65 two-color counters

(or other round objects like pennies) for each student group


ENGAGEBrenda is at the farmer’s market. There are several baskets of to-matoes on a table. Each basket contains 6 tomatoes. What sequence would Brenda create if she listed the number of tomatoes in a set of baskets (1 basket, 2 baskets, 3 baskets, etc.)?

6, 12, 18, 24, 30, …

1.1FOCUSING QUESTION How are arithmetic and geometric sequences alike? How are they different?

LEARNING OUTCOMES■■ 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)■■ I can write a formula for the nth term of an arithmetic or geometric sequence.

(Algebra 1)■■ I can use symbols, tables, and language to communicate mathematical ideas.

Arithmetic and Geometric Sequences

EXPLORE

The first few terms of two different sequences are shown.

SEQUENCE 1

SEQUENCE 2

Use toothpicks or counters to build the next two terms of each sequence. Record your information in the table.See margin.

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2. Possible answer: The number of toothpicks was 1 more than 3 times the term number.

3. Possible answer based on response to previous question: For the next term, I started with what I had before and then added 3 more toothpicks to place another square onto the end of the term. The first term begins with 1 toothpick on the left edge of the square and 3 toothpicks added to complete the square. Note: additional patterns are possible, such as n sets of 2 horizontal toothpicks and n + 1 vertical toothpicks.

6. The number of counters is one power of 2 less than the power of two that corresponds to the term number.

7. For the next term, I count-ed the number of counters in the previous term and built a stack that was twice as high.

9. In Sequence 1, you add 3 to the previous term to get the next term. In Sequence 2, you multiply the pre-vious term by 2 to get the next term. Both sequences have a recognizable pat-tern, but Sequence 1 is an additive relationship while Sequence 2 is a multiplica-tive relationship. Sequence 2 grows much faster.

INSTRUCTIONAL HINTSComparing and Contrast-ing is a high-yield instruc-tional strategy identified by Robert Marzano and his colleagues (Classroom In-struction That Works, 2001). To support this strategy, consider asking students to make a Venn diagram comparing properties of Sequence 1 to properties of Sequence 2. This strate-gy helps students identify differences between linear and exponential relation-ships, which they will spend time during this chapter studying.

ELL STRATEGYExpressing ideas and opinions (ELPS: c3G) is an important part of stu-dent-student discourse about mathematics. Placing English language learners in small groups with their peers, as in the activity in this lesson, provides a safe environment for them to sharpen their skills using the English language while they are learning about mathematics.

1 . 1 • A R I T H E m E T I C A N d G E o m E T R I C S E q u E N C E S 3

SEQUENCE 1 SEQUENCE 2

TERM NUMBER NUMBER OF TOOTHPICKS TERM NUMBER NUMBER OF

COUNTERS

1 4 1 1

2 7 2 2

3 10 3 4

4 13 4 8

5 16 5 16

6 19 6 32

10 31 10 512

1. What patterns do you observe in Sequence 1? The number of toothpicks increases by 3 (add 3) for the next term.

2. What relationships do you observe between the term number and the number of toothpicks required to build each term in Sequence 1? (Hint: Record the multiples of 3 next to the number of toothpicks.) See margin

3. How does the pattern that you mentioned in question 2 appear in the figures that you constructed? See margin

4. How many toothpicks would you need to build the 10th term of Sequence 1? 31

5. What patterns do you observe in Sequence 2? The number of counters doubles (multiply by 2) for the next term.

6. What relationships do you observe between the term number and the number of counters required to build each term in Sequence 2? (Hint: Record powers of 2 next to the number of counters.) See margin

7. How does the pattern that you mentioned in question 6 appear in the figures that you constructed? See margin

8. How many counters would you need to build the 10th term of Sequence 2? 512

9. Compare the two sequences. How are they alike? How are they different? See margin

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REFLECT ANSWERS:Sequence 1 has a constant difference because to build or get to the next term, you add 3 toothpicks to the previous term every time.

Sequence 2 has a constant ratio because to build or get to the next term, you multiply the previous number of counters by 2 every time.

QUESTIONING STRATEGYIn mathematics, difference is defined as the result of subtraction, yet in the “ex-plain” problem, the com-mon difference is 4. Why is an “addend of 4” consid-ered a difference?


REFLECT

■■ Which sequence has a constant difference between the numbers of ob-jects for successive terms? How do you know?

See margin.■■ Which sequence has a constant ratio between the numbers of objects

for successive terms? How do you know?See margin.

EXPLAIN

A sequence is a set of numbers that are listed in order and that follow a particular pattern. An arithmetic sequence is a sequence that has a constant difference between consecutive terms.

For example, suppose Bernar-do earns tickets from the local arcade. He starts out with 11 tickets and earns 4 more tickets for each game he plays. Bernar-do can create an arithmetic se-quence to show the number of tickets he has after each game.

11, 15, 19, 23 27, 31, ...

+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-tation to generalize an arithmetic sequence like Bernardo’s.

a1 = 11an = an – 1 + 4

The addend of 4 is used to generate the next term in this arith-metic sequence. In general, the addend in an arithmetic se-quence is called a common difference, since it is also the dif-ference between consecutive terms.

Watch Explain andYou Try It Videos

or click here

A subscript is used to in-dicate a special case of a variable. a1 indicates the first term of a sequence, a2 indicates the second term of a sequence, and so on. a1 is read “a-sub one,” where “sub” indi-cates a subscript.

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INSTRUCTIONAL HINTSStudents process new skills better when they write about them in their own words. Take a moment to have students write about the difference between arithmetic and geometric sequences. Then have stu-dents share their explana-tions with one or two other students.

ADDITIONAL EXAMPLE Consider the scenario: Ericka and her mother bake 120 cookies for a bake sale. Every 15 minutes they sell half of the cookies on the table. They can create a geometric sequence to show their cookie sale.

120, 60, 30, 15, …

Ask students what the constant multiplier in this sequence would be. Listen for students to say “divide by 2.” Ask them how to write “divide by 2” as mul-tiplication.

The multiplier of 1–2 is used to generate the next term in the sequence.

Ask students to write a recursive rule for the se-quence.

a1 = 120

an = 1–2an – 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


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 se-quence 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.

a1 = 0.02an = 2an – 1

ARITHMETIC AND GEOMETRIC SEQUENCES

An arithmetic sequence has a constant addend or common difference. It is an additive relationship be-tween terms of the sequence.

A geometric sequence has a constant multiplier or common ratio. It is a multiplicative relationship be-tween terms of the sequence.

If you have a sequence, you can use the constant difference or constant multiplier to determine subsequent terms of the sequence.

EXAMPLE 1

Anthony worked all summer to save $1950 for spending money during the school year. He plans to withdraw the same amount from his savings account at the end of each week. Anthony can create an arithmetic sequence that shows the balance of his savings account at the beginning of each week of the school year.

$1950, $1885, $1820

How much money will be in Anthony’s savings account at the beginning of the fourth week of the school year? How much money will be in Anthony’s savings account at the beginning of the fifth week of the school year?

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YOU TRY IT! #1 ANSWER:The common difference is 41–

2 and the paper chain will be 81 inches long after Rachel has added six links to it.

QUESTIONING STRATEGY FOR YOU TRY IT #1Some students may expect the table of values to start with (1, 54). Ask students why the dependent variable starts with 581–

2 rather than 54. What column could be added to the table to account for the 54 inches of paper chain that Rachel started with?

ADDITIONAL EXAMPLES1. Daniel’s grandmother gave him a $50 gift card for music downloads for his birthday. If he down-loads one song per day, he can create an arithmetic sequence that shows the balance of his gift card each day.

$50, $48.55, $47.10, $45.65,...

What is the common difference, and what does it represent? How much money will be left on Dan-iel’s gift card at the end of one week?

The common difference is -$1.45.

At the end of one week (7 days), Daniel will have $39.85 on his gift card.

2. Keniesha went for a hike. She hiked at a steady pace, noting her distance, in miles, every 15 minutes. She can create an arithme-tic sequence to reflect her distance traveled every 15 minutes of her hike.

0.75, 1.5, 2.25, 3, …

What is the common difference, and what does it represent? How far will Keneisha have hiked if she hikes for one and a half hours?

The common difference is 0.75. It represents the distance in miles that Keneisha hiked every 15 minutes. In one and a half hours, Keneisha will have hiked 4.5 miles.


STEP 1 First, use the existing data to determine the common difference.

TIME (WEEKS)

SAVINGS BALANCE (DOLLARS)

1 $1950

2 $1885

3 $1820

The common difference is –$65.

STEP 2 Next, apply the common difference to the balance at the beginning of the third week to determine the balance at the beginning of the fourth week.

$1820 + (–$65) = $1755

STEP 3 Then, apply the common difference to determine the balance in Anthony’s savings account at the beginning of the fifth week of school.

$1755 + (–$65) = $1690

The balance in Anthony’s savings account is $1755 at the beginning of the fourth week of school and $1690 at the beginning of the fifth week of school.

–$65 = $1885 − $1950

–$65 = $1820 − $1855

YOU TRY IT! #1

Rachel helps the student council create a paper chain that contains students’ writ-ten pledges not to bully or tolerate bullying. When Rachel begins stapling, there are 54 inches of paper chain. She measures after adding each link and records the results in a table.

NUMBER OF LINKS RACHEL ADDED 1 2 3

LENGTH OF PAPER CHAIN (INCHES) 58

1—2 63 67

1—2

What is the common difference in this situation, and how long will the paper chain be after Rachel has added a total of six links to it?

See margin.

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QUESTIONING STRATEGIESHow is 20% written as a decimal?

The fund pays 20% annual-ly, yet the common ratio is 1.2, not 0.2. What does 1.2 represent?

ADDITIONAL EXAMPLES1. Jack purchased a new car for $25,120. Each year the value of the car will de-crease by 30%. Jack project-ed how much his car will be worth at the end of each year for the next four years.

$25,120, $17,584, $12,308.80, $8,616.16

What is the common ratio in this situation? Is the common ratio what you ex-pected? Why or why not?

The common ratio is 0.7. It 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 at the end of 8 years.

2. Montrell gets a job icing 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 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 at the bakery? How many total 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.


EXAMPLE 2

At the beginning of the year, an investor puts $1000 into a fund that pays 20% annu-ally. The investor projects how much will be in the fund at the end of each year for the next three years.

$1200, $1440, $1728

How much money will be in the investment fund at the end of four years? How much money will be in the investment fund at the end of five years?

STEP 1 First, determine the common ratio in this situation.

$1440 ÷ $1200 = 1.2$1728 ÷ $1440 = 1.2

The common ratio is 1.2.

STEP 2 Next, multiply the third value in the geometric sequence by the common ratio to determine the fourth value in the geometric sequence.

($1728)(1.2) = $2073.60

STEP 3 Then, multiply by the common ratio to determine the fifth value in the geometric sequence.

($2073.60)(1.2) = $2488.32

There will be $2073.60 in the fund after four years and $2488.32 in the fund after five years.

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YOU TRY IT! #2 ANSWER:The common ratio is 0.8 and there are approximately 164 milligrams of medicine in the patient’s system after five hours.

QUESTIONING STRATEGIESAssist students’ thinking as they process the differ-ence between recursive and explicit rules by asking the following questions.

• Explain how a recur-sive rule differs from an explicit rule?

• For which type of rule must you know the previous term in order to find the next?

• For which type of rule can you find any term in the sequence without needing to continue the sequence?

• Which rule do you pre-fer and why?

• How does the explicit rule relate to a function rule?


EXAMPLE 3

For the sequence shown, write a recursive rule and an explicit rule.

4, 6.5, 9, 11.5, 14, …

STEP 1 First, determine the common difference or ratio in this situation.

4, 6.5, 9, 11.5, 14, ...

+2.5 +2.5 +2.5 +2.5

The common difference is +2.5.

STEP 2 Next, write the first term in the sequence, a1. Use the common difference to write a recursive rule relating an to the previous term, an – 1.

a1 = 4an = an – 1 + 2.5

A recursive rule shows how to determine the nth term, an, using the value of the previous term. An ex-plicit rule, like a function, shows how to determine the nth term, an, using the term number, n.

YOU TRY IT! #2

A patient takes 500 milligrams of medicine. A nurse charts the amount of medication in the patient’s system.

TIME SINCE DOSAGE (HOURS)

MEDICINE IN PATIENT’S SYSTEM (MG)

1 400

2 320

3 256

4 204.8

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?

See margin.

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ADDITIONAL EXAMPLESWrite the recursive, explic-it, and function rules for the additional examples on page. 7.

1. Jack’s car

recursive rule: a1 = 25,120, an = 0.70an–1

explicit rule: an = 25,120(0.70)n

function rule: f(x) = 25,120(0.70)x

2. Montrell’s job

recursive rule: a1 = 5, an = 2an–1

explicit rule: an = 2.5(2)n

function rule: f(x)=2.5(2)x

YOU TRY IT! #3 ANSWER:recursive rule: a1 = 6, an = an – 1 + 11–

3

explicit rule: an = 42–

3 + 11–3n


EXAMPLE 3

For the sequence shown, write a recursive rule and an explicit rule.

4, 6.5, 9, 11.5, 14, …

STEP 1 First, determine the common difference or ratio in this situation.

4, 6.5, 9, 11.5, 14, ...

+2.5 +2.5 +2.5 +2.5

The common difference is +2.5.

STEP 2 Next, write the first term in the sequence, a1. Use the common difference to write a recursive rule relating an to the previous term, an – 1.

a1 = 4an = an – 1 + 2.5

A recursive rule shows how to determine the nth term, an, using the value of the previous term. An ex-plicit rule, like a function, shows how to determine the nth term, an, using the term number, n.

YOU TRY IT! #2

A patient takes 500 milligrams of medicine. A nurse charts the amount of medication in the patient’s system.

TIME SINCE DOSAGE (HOURS)

MEDICINE IN PATIENT’S SYSTEM (MG)

1 400

2 320

3 256

4 204.8

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?

See margin.


STEP 3 Use the common difference to work backwards to determine the value of term 0.

1.5, 4, 6.5, 11.5, 14, ...

+2.5

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 difference as the rate of change.

an = 1.5 + 2.5n

YOU TRY IT! #3

Write a recursive rule and an explicit rule for the sequence 6, 71—3 , 8

2—3, 10, 111—3, …

See margin.

PRACTICE/HOMEWORKFor questions 1 – 4 write an explicit rule that describes the number of items used to construct the pattern in terms of the term number, n.

1.

an = 2 + 2n

2.

an = n2

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3.

an = 1 + 2n

4.

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 addi-tional 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


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 se-quence 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 of bounces, n, the ball makes.f(n) = 1000(0.8)n

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11. arithmetic a1 = 1; an = an – 1 + 7 an = −6 + 7n

12. geometric a1 = 2; an = 3an – 1 an = 2(3)n – 1

13. arithmetic a1 = −10; an = an – 1 + 3.5 an = −13.5 + 3.5n

14. geometric a1 = 1.5; an = 5an – 1 an = 1.5(5)n – 1

15. geometric a1 = 64; an = an – 1 ÷ 4 = 1–4an – 1 an = 64(1–

4)n – 1

16. arithmetic a1 = 147; an = an – 1 – 20 an = 167 – 20n



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.

13. −10, −6.5, −3, 0.5, 4, 7.5, … 14. 1.5, 7.5, 37.5, 187.5 …See margin. See margin.

15. 64, 16, 4, 1, 0.25, … 16. 147, 127, 107, 87, 67, …See margin. See margin.

For questions 17 – 20 for each recursive rule and explicit rule given below, write the first 4 terms in the sequence.

17. a1 = 9.5; an = an – 1 + 6.5 18. a1 = 3; an = 4an – 1 an = 3 + 6.5n an = 3(4)n – 1

9.5, 16. 22.5, 29 3, 12, 48, 192

19. a1 = 625; an = an – 1 ÷ 5 = 1—5 an – 1 20. a1 = 140; an = an – 1 – 30

an = 625( 1—5)n – 1

an = 170 – 30n

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AR.2C Determine the function that models a giv-en table of related values using finite differences and its restricted domain and range.

MATHEMATICAL PROCESS SPOTLIGHTAR.1F Analyze math-ematical relationships to connect and communicate mathematical ideas.

ELPS1A Use prior knowl-edge and experiences to understand meanings in English.

VOCABULARYfinite differences, slope, y-intercept, linear function, restrictions

MATERIALS• N/A

ENGAGE ANSWER:Marcus could place the number of rows in one column and the number of trees in the other column.


1.2 Writing Linear Functions

FOCUSING QUESTION What are the characteristics of a linear function?

LEARNING OUTCOMES■■ I can determine patterns that identify a linear function from its related finite

differences.■■ I can determine the linear function from a table using finite differences, includ-

ing any restrictions on the domain and range.■■ I can analyze patterns to connect the table to a function rule and communicate

the linear pattern as a function rule.

ENGAGEMarcus works in an orchard. Each row in the orchard contains 20 trees. How can Marcus use this informa-tion to make a table of values to represent the number of trees in the orchard?

See margin.

EXPLORE

Miranda and her family will spend their summer vacation on the beach. They plan 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.

NUMBER OF DAYS RENTAL COST

1 $170

2 $285

3 $400

4 $515

5 $630

6 $745

7 $860

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4. The cleaning fee is $55, because that is the amount that you start with on day 0 in the table.

5. The daily rental fee is $115, because that is the amount that is added to the total rental cost for each day that Miranda’s family rents the beach house.

QUESTIONING STRATEGIESAs students move through the second scenario, call attention to how they did the first scenario. Ask ques-tions such as:

• How does your answer for this situation com-pare to the beach house rental?

• Which of the two situ-ations is an increasing function? Decreasing function?

1 . 2 • W R I T I N G L I N E A R F u N C T I o N S 13

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 how you know. See margin.

5. Based on the pattern in the table, what do you think the daily rental fee is? Explain how you know. See margin.

6. Use the pattern in the table to write an equation that shows the relationship between n, the number of days the beach house will be rented and r, the total rental cost. r = 55 + 115n

Miranda and her sister have pooled their money for meals. From the initial amount of money they placed in an envelope, they will spend a certain amount each day on food. The table below shows the balance of money remaining in the envelope after a certain number of days.

NUMBER OF DAYS BALANCE

1 $225

2 $190

3 $155

4 $120

5 $85

6 $50

7 $15

7. What is the difference between the numbers of days in consecutive rows in the table? 1 day

8. What is the difference between the balances in consecutive rows in the table? −$35

9. Use the patterns in the table to predict the balance on day 0. $260

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10 Miranda and her sister initially pooled $260, because that is the amount that you start with on Day 0 in the table.

11. Miranda and her sister spent $35 each day on meals, because that is the amount that is subtracted from the balance each day.

REFLECT ANSWERS:The differences in values for successive table rows are con-stant.

The ratio of the differences is the same as the rate of change in the equation.

ELL STRATEGYConnecting to prior knowl-edge (ELPS: c1A) helps stu-dents create understand-ing of new mathematical topics. Connecting finite differences to independent and dependent variables helps students recall im-portant ideas from Algebra 1 and extend them to new content.

QUESTIONING STRATEGIESThink back to Miranda and her family.

• Which value(s) repre-sented slope?

• Which value(s) repre-sented the y-intercept?


10. Based on the patterns in the table, how much money do you think Miranda and her sister initially pooled? Explain how you know. See margin.

11. Based on the patterns in the table, how much money did Miranda and her sister spend on meals each day? Explain how you know. See margin.

12. Use the patterns in the table to write an equation that shows the relationship between n, the number of days of the vacation and b, the balance of pooled money remaining. b = 260 – 35n

REFLECT

■■ What do you notice about the differences in values for successive table rows for both the independent and dependent variable? See margin.

■■ What relationship exists between the ratio of the differences in the de-pendent variable to the differences in the independent variable and the equations that you have written? See margin.

EXPLAIN

The differences in values for successive table rows are called finite differences. When you have a table of data, you can use finite differences to determine the type of function the data represents and to write a function representing the relationship between the variables in the table.

Let’s look more closely at a linear func-tion. The table on the next page shows the relationship between x and f(x). In a linear function, f(x) = mx + b, m represents the slope or rate of change, and b represents the y-coordinate of the y-intercept, or starting point.

The slope, which is the rate of change, of a linear function connecting two points, (x1, y1) and (x2, y2) is found using the slope formula.

m = ∆y

= y2–y1

∆x x2–x1


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INTEGRATE TECHNOLOGYUse technology such as a graphing calculator or spreadsheet app on a display screen to show students how, no matter the numbers present in the linear function, the finite differences will always be constant.

INSTRUCTIONAL HINTSRemind students of pri-or learning in Algebra I. Students learned multiple ways to test if data repre-sented a linear function.


x y = f(x)

0 b

1 b + m

2 b + 2m

3 b + 3m

4 b + 4m

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 of y-values is m. The finite differences in y-values, ∆y, for a linear function are the same, so we can say that the finite differences are constant.

FINITE DIFFERENCES AND LINEAR FUNCTIONS

In a linear function, the finite differences between suc-cessive y-values, ∆y, are constant if the differences be-tween successive x-values, ∆x, are also constant.

If the finite differences in a table of values are constant, then the values represent a linear function.

You can also use the finite differences to write a linear function describing the relation-ship between the independent and dependent variables.

∆y = (b + m) – b = b + m – b = b – b + m = m

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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

EXAMPLE 1Does the set of data shown below represent a linear function? Justify your answer.

x y

0 11

1 17

2 23

3 29

4 35

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ADDITIONAL EXAMPLESProvide students with the lists of ordered pairs of numbers. Have them place the numbers in tables and answer the following question.

Does the data set represent a linear function? Justify your answer.

1. (2, 13), (3, 6), (4, -1), (5, -8), (6, -15)

Yes, the data set represents a linear function because the finite differences in y-values are constant when the finite differences in x-values are also constant.

2. (1, 3), (2, 6), (4, 9), (8, 12), (16, 15)

No, data set does not represent a linear function because while the finite differences in y-values are constant the finite differenc-es in x-values are not constant.

3. (-1, -12), (1, -6.5), (3, -1), (5, 4.5), (7, 10)

Yes, the data set represents a linear function because the finite differences in y-values are constant when the finite differences in x-values are also constant.

Note: Often students forget to check the finite differences between successive x-values. Draw special attention to the second additional example.

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


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

x y

0 11

1 17

2 23

3 29

4 35

STEP 2 Determine whether or not the ratios of the differences are constant.

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

The differences in y, ∆y, are all 6, so they are constant.

∆y—∆x =

6—1 = 6 for all pairs of ∆x and ∆y.

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

Yes, the set of data represents a linear function because the finite differences in y-values are constant when the finite differences in x-values are also constant.

∆y = 17 – 11 = 6

∆y = 23 – 17 = 6

∆y = 29 – 23 = 6

∆y = 35 – 29 = 6

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

YOU TRY IT! #1

Does the set of data shown below represent a linear function? Justify your answer.

x y

0 6.4

1 7.2

2 8.0

3 8.8

4 9.6

See margin.

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QUESTIONING STRATEGIESExample 2 does not rep-resent a linear function. How does this data look on a scatterplot? How can you tell from looking at the graph, that this is not a linear function?


EXAMPLE 2


x y

0 17

1 15.5

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

1 15.5

2 14

3 11.5

4 10

STEP 2 Determine whether or not the differences are constant.


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


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

∆x = 4 – 3 = 1∆x = 3 – 2 = 1

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆y = 15.5 – 17 = -1.5

∆y = 14 – 15.5 = -1.5

∆y = 11.5 – 14 = -2.5

∆y = 10 – 11.5 = -1.5

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YOU TRY IT! #2 ANSWER:No, the set of data does not represent a linear function, because the finite differences in x-values are constant but the finite differences in y-values are not constant.

QUESTIONING STRATEGIESAfter reading Example 3’s scenario, guide students to plan their steps.

How will the finite differ-ences help you determine the slope?

How can you identify the y-intercept from a table of values?

Do you need to draw a graph to write a function rule?


YOU TRY IT! #2


x y

0 7.1

1 7.5

2 8.1

3 8.9

4 9.9

See margin.

EXAMPLE 3

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

x y

1 7.5

2 10

3 12.5

4 15

5 17.5


∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x y

∆y = 10 – 7.5 = 2.5

∆y = 12.5 – 10 = 2.5

∆y = 15 – 12.5 = 2.5

∆y = 17.5 – 15 = 2.5

1 7.5

2 10

3 12.5

4 15

5 17.5

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ADDITIONAL EXAMPLESFor the following data sets, determine if the relation-ship is a linear function. If so, determine a function relating the variables.

1.

Linear, y = 1.5x + 4.5

2.

Not Linear

3. (1, 2), (2, 5), (3, 8), (4, 12), (5, 16)

Not Linear

4. (1, -5), (2, -13), (3, -21), (4, -29), (5, -37)

Linear, y = -8x - 13

x -3 -1 1 3 5

y 0 3 6 9 12

x 0 1 2 3 4

y -4 -8 -16 -32 -64


STEP 2 Determine whether or not the relationship is a linear function.

The differences in x, ∆x, are all 1, so they are constant.The differences in y, ∆y, are all 2.5, so they are constant.Since the finite differences are all constant, the relationship is a linear function.

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

Slope = ∆y—∆x =

2.5—1 = 2.5

STEP 4 Determine the y-intercept of the linear function. Work backwards from x = 1 and y = 7.5.

x y

0 b

1 7.5

2 10

7.5 – b = 2.5 7.5 – 7.5 – b = 2.5 – 7.5 −b = −5 b = 5

The y-intercept is (0, 5).

STEP 5 Use the slope and the y-coordinate of the y-intercept to write the function in slope-intercept form.

y = mx + b y = 2.5x + 5

1 – 0 = 1

2 – 1 = 1

7.5 – b = 2.5

10 – 7.5 = 2.5

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INSTRUCTIONAL HINTEncourage students to write the finite differences on the table of values for every problem. When x = 0, highlight or circle the y-intercept in the table.


YOU TRY IT! #3

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

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


x y

2 22

4 21

6 20

8 19

10 18

∆x = 4 – 2 = 2

∆x = 6 – 4 = 2

∆x = 8 – 6 = 2

∆x = 10 – 8 = 2

∆y = 21 – 22 = –1

∆y = 20 – 21 = –1

∆y = 19 – 20 = –1

∆y = 18 – 19 = –1

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ADDITIONAL EXAMPLESSome students may need additional practice with identifying slope and y-in-tercept from an equation and with writing functions given slope and y-intercept.

Identify the slope, m, and y-intercept, b, of the follow-ing linear functions.

1. y = 0.5x + 2

m = 0.5, b = 2

2. y = 3 – 4x

m = -4, b = 3

3. y = -2–3x – 5.5

m = -2–3, b = -5.5

4. y = 0.6x

m = 0.6, b = 0

5. y = 7

m = 0, b = 7

Write the function given the slope and y-intercept.

1. m = -0.25, b = -3

y = -0.25x – 3

2. slope = 4–5, contains the point (0, 2)

y = 4–5x + 2

3. slope = 1–2, contains the point (0, 0)

y = 1–2x

4. slope = -4, contains the point (2, -3)

y = -4x + 5

5. slope = -0.7, contains the point (10, -10)

y = -0.7x – 3


STEP 2 Determine whether or not the relationship is a linear function.

The differences in x, ∆x, are all 2, so they are constant.The differences in y, ∆y, are all −1, so they are constant.Since the finite differences are all constant, the relationship is a linear function.

STEP 3 Determine the slope of the linear function.

Slope = ∆y—∆x =

–1—2 = – 1–

2

STEP 4 Determine the y-intercept of the linear function. Work backwards from x = 2 and y = 22.

x y

0 b2 22

4 21

22 – b = −1 22 – 22 – b = −1 – 22 −b = −23 b = 23


STEP 5 Use the slope and the y-coordinate of the y-intercept to write the function in slope-intercept form.

y = mx + b y = −1–2x + 23

2 – 0 = 2

4 – 2 = 2

22 – b = –1

21 – 22 = –1

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YOU TRY IT! #4

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

x y

6 11

9 16

12 21

15 26

18 31

Answer: y = 5—3x + 1

PRACTICE/HOMEWORKFor 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. slope = 2—3 , y-intercept = (0, 3

1—3)

y = 2—3x + 3

1—3

3. slope = − 2—5 , contains the point (10, 3)

y = − 2—5x + 7

4. slope = 1—4 , contains the point (−8, 1)

y = 1—4x + 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.

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

5. x y1 1

2 4

3 9

4 16

5 25

6. x y1 5.5

2 7.5

3 9.5

4 11.5

5 13.5

7. x y1 8

2 11

3 14

4 17

5 20

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linear; y = –4x + 28 linear; y = –0.6x + 1.7 not linear

not linear linear; y = 1—2x + 7 linear; y = 4x – 10

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 = 1 and ∆y = −0.5, so the situation is linear.

y = −0.5x + 164

8. x y1 24

2 20

3 16

4 12

5 8

11. x y2 4

4 5

6 7

8 10

10 14

14. x y1 10

2 8

3 6

4 4

5 2

9. x y0 1.7

1 1.1

2 0.5

3 –0.1

4 –0.7

12. x y2 8

4 9

6 10

8 11

10 12

15. x y1 16

2 15

3 13

4 10

5 6

10. x y0 2

2 4

4 8

6 16

8 32

13. x y3 2

5 10

7 18

9 26

11 34

16. x y1 120

2 60

3 40

4 30

5 24

x y0 1641 163.52 1633 162.54 162

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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 = 5 and ∆y = 75, so the situation is linear.

y = 15x + 2,000

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

∆x = 2 but ∆y = is not constant, so the situation is not linear.

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 = 1 and ∆y = 3.25, so the situation is linear.

y = 3.25x + 85

x y0 2,0005 2,07510 2,15015 2,22520 2,300

x y0 7502 1,5004 3,0006 6,0008 12,000

x y0 85.001 88.252 91.503 94.754 98.00DRAFT

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TEKSAR.2D Determine a function that models real-world data and math-ematical contexts using finite differences such as the age of a tree and its circumference, figurative numbers, average velocity, and average acceleration.

MATHEMATICAL PROCESS SPOTLIGHTAR.1A Apply mathe-matics to problems arising in everyday life, society, and the workplace.

ELPS5B Write using newly acquired basic vo-cabulary and content-based grade-level vocabulary.

VOCABULARYlinear function, finite differ-ences

MATERIALS• graphing calculator,

spreadsheet, or a graphic application

ENGAGE ANSWER:Possible answer: The rings are formed because, as the tree grows, each year it adds a new layer of cells under the bark. The amount of rain might cause the rings to be closer together or farther apart.

1 . 3 • M o d e l i n g W i t h l i n e a r F u n c t i o n s 25

FOCUSING QUESTION How can you use finite differences to construct a linear model for a data set?

LEARNING OUTCOMES■■ I can use finite differences to write a linear function that describes a data set.■■ I can apply mathematics to problems that I see in everyday life, in society, and

in the workplace.

ENGAGEMariette, a dendrochronologist, observed that some tree stumps have rings that are close to-gether while other tree stumps have rings that are farther apart. Why does a tree stump have rings? What might cause the rings to be closer together or farther apart?

See margin.

EXPLORE

Each year during the growing season, trees grow larger by adding another layer of cells just beneath the bark. This layer is called a tree ring. Because a tree ring is added each year, scientists can determine the age of a tree by counting the number of tree rings that are present.

However, not all tree rings have the same width. Trees grow more when there is plenty of rain and the soil is fertile. Scientists can draw conclusions about tem-perature and rainfall for a particular year based on the width of the tree ring for that year.

Mariette measured the width of tree rings from a core sample she took from a post oak tree in the Brazos River valley of central Texas. From the tree ring width, she calculated the radius of the tree. The table below shows her results.

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

1.3 Modeling with Linear Functions

Image credit: Adrian Pingstone, Tree ring, Wikimedia Commons

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1.

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

+1 +1 +1 +1 +1 +1

+0.6 +0.4 +0.4 +0.5 +0.5 +0.6

5. The slope, 0.5 centimeters per year, represents the growth rate of the tree, or the number of centimeters that the radius of the tree grows each year.

The y-intercept, 2.5 cen-timeters, represents the radius of the tree in the year 2000, when the data were first collected.

9. The function model con-nects several of the data values and is very close to the remaining data values. The function model appears to closely predict the actual data values.

See page 27.

TECHNOLOGY INTEGRATION When modeling with real-world data, a scatterplot of the data set and the function model helps students visualize the relationship between the two variables. Scatterplots can be made using graphing calculators, spreadsheets, or graphing apps. ELL STRATEGY Writing with newly acquired vocabulary (ELPS: c5B) helps English language learners internalize the vocabulary terms that they have recently learned. Using vocabulary from current and past learning experiences (e.g., linear function, finite differences) to explain their thinking and mathematical reasoning reinforces how these terms are consistent through a variety of settings and contexts.

REFLECT ANSWERS:Use an average value of the first differences as the slope of a linear function model.

Use your linear function model to write an equation where the linear function is equal to a particular value. Then, solve for x.

26 c h a P t e r 1 : a l g e B r a i c Pat t e r n s

1. Calculate the finite differences between the year number and the radius. See margin

2. Are the first differences in the radius constant? Explain how you know. The first differences in radius are not exactly constant but are very close in value.

3. What is the average finite difference in radius? 0.5 cm

4. Use the information from the table to write a function rule that models the data. f(x) = 2.5 + 0.5x, where x represents the year number, or number of years since 2000.

5. What do the slope and y-intercept from your function rule mean in the context of this situation? See margin

6. Use your model to predict the radius of the tree in 2015. f(15) = 2.5 + 0.5(15) = 10 centimeters

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

8. What would the circumference of the tree be in 2015? C = 2πr = 2π(10) = 20π ≈ 62.8 centimeters

9. Make a scatterplot of your data set and graph the function model over the scatterplot. How well would you say the function model predicts the actual values in the data set? Explain your reasoning. See margin

REFLECT

■■ How can you determine a linear function model for a data set if the first differences are not exactly the same, but are almost constant?

See margin.

■■ Once you have your linear function model, how can you use the model to determine a value of the independent variable that generates a par-ticular value of the dependent variable?

See margin.

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EXPLAIN

A linear function model can be used to represent sets of math-ematical and real-world data. Dendrochronologists use core 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-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


RADIUS (CM) 2.5 3.1 3.5 3.9 4.4 4.9 5.5

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 cir-cle 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


RADIUS (CM) 2.5 3.1 3.5 3.9 4.4 4.9 5.5

CIRCUMFERENCE (CM) 15.7 19.5 22.0 24.5 27.6 30.8 34.5

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

+1 +1 +1 +1 +1 +1


CIRCUMFERENCE (CM) 15.7 19.5 22.0 24.5 27.6 30.8 34.5

+3.8 +2.5 +2.5 +3.1 +3.2 +3.7


or click here

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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.

∆y = 3.8 + 2.5 + 2.5 + 3.1 + 3.2 + 3.7 ≈ 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.

f(x) = 15.7 + 3.13x

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ADDITIONAL EXAMPLEGenerate 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 sec-onds, is recorded in the table. If he does not get tackled, how many seconds will it take him to reach the end zone? Why would the finite differences not be constant in this scenario?

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

TIME (S) 0 1 2 3 4

DISTANCE (YD) 0 8.2 16.9 25.4 34


EXAMPLE 1

A student takes small steps away from a motion detector at an approximately con-stant rate. The time, in seconds, for which the student walks and the distance, in me-ters, 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 Calculatethefinitedifferencesinthetable.

+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 Calculatetheaveragefinitedifferenceinthetableandusethis 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 y = 6.5

According to the linear function model, the student will be 6.5 meters away from the motion detector after 10 seconds.

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YOU TRY IT! #1 ANSWER:y = 8.325x + 0.175; according to the linear function model, Tracy will run the half-mar-athon in approximately 109 minutes.


YOU TRY IT! #1

Tracy, a long distance runner, times herself as she runs a half-marathon, which is 13.1 miles long. The distance is measured in miles and the time is measured in minutes.

DISTANCE (MI) 1 2 3 4 5

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 2Engineers 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 Calculatethefirstfinitedifferencesinthetable.

+1 +1 +1 +1

TIME (S) 1 2 3 4 5

SPEED (MPH) 65 62 58 56 53

-3 -4 -2 -3

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ADDITIONAL EXAMPLEMrs. 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 first 4 months of the school year are shown in the table below.

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.

TIME (MONTHS) 1 2 3 4

PENCILS 195 185 173 159


STEP 2 Calculatetheaveragefirstfinitedifferenceinthetableandusethis to determine the slope, or average acceleration, for a linear function model.

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

STEP 3 Usetheaverageaccelerationandfirstfinitedifferencesinthe 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

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YOU TRY IT! #2 ANSWER:y = –189.75x + 1550; the check-ing account Caleb’s parents set up will have a balance below $100 after 8 months.


YOU TRY IT! #2

Caleb’s parents set up a checking account for him before college so that he will be able to pay the utilities for his apartment. Caleb keeps track of his spending in the table below. Time represents the number of months he has been in his apartment and the checking account balance is measured in dollars.

TIME (MONTHS) 0 1 2 3 4

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/HOMEWORKFor 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 y1 15.3

2 25.3

3 35.2

4 45.4

5 55.2

6 65.3

10

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

-2.917

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

0.575

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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 (DOLLARS) 40 35.68 31.22 26.97 22.65 18.40

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 (MONTHS) 1 2 3 4 5 6

MILEAGE(MILES) 11,540 12,482 13,570 14,670 15,682 16,757

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 APPS PURCHASED 1 2 3 4 5

COST (DOLLARS) 1.25 2.50 3.75 5.00 6.25

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8. The slope, 0.25 inches per month, represents the growth rate of Char-lie’s brother (how much his height increases each month). The y-intercept, 54 inches, represents the height of Charlie’s brother when he started tracking his height.

12. The slope represents a decrease of 1.12 ounces per serving (each serving is 1.12 ounces). The y-inter-cept, 14 ounces, represents the starting weight of the box of cereal.

13. No.

f(x) = 14 – 1.12x = 0; x = 12.5. He will finish the cereal at about 12.5 servings, instead of 14 servings. However, he may have used more than the suggested number of ounces per serving, so his results are not conclusive.


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 (MONTHS) 0 1 2 3 4 5

HEIGHT (INCHES) 54 54.20 54.45 54.85 55.10 55.25

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.


9. Use your model to predict the height of Charlie’s brother after a year. f(12) = 54 + 0.25(12) = 57 inches

10. In what month will his height be approximately 56 inches?f(x) = 54 + 0.25x = 56, x = 8 months


CRITICAL THINKING

Jeff noticed that the nutrition information on his box of cereal states that there are 14 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.

NUMBER OF SERVINGS 0 1 2 3 4 5

WEIGHT OF REMAINING

CEREAL (OUNCES)

14 12.7 11.6 10.7 9.5 8.4

11. Write a function rule that models the situation. f(x) = 14 – 1.12x or f(x) = -1.12x + 14, where x represents the number of ce-real servings


13. Was Jeff able to confirm the claim on the cereal box by eating 14 servings? Ex-plain your answer.See margin

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14. f(x) = 704 – 19.9x or f(x) = -19.9x + 704, where x represents the time (in hours) since Bob started tracking the hurricane.

15. The slope, -19.9, means that the distance of the hurricane to land is decreasing by 19.9 miles each hour. The y-inter-cept, 704, gives us the original distance from land when Bob started tracking the hurricane.

19. No.

f(10) = 3.8(10) = 38. She will finish the race in about 38 minutes, which is a half minute slower than her previous best time.



SCIENCE

Bob is tracking a hurricane moving toward the coast of Florida. The table below shows its distance from land over time.

TIME (HOURS) 0 1 2 3 4

DISTANCE (MILES) 704 684 663.7 644.2 624.4

14. Write a function rule that models the situation. See margin

15. What do the slope and y-intercept from your function rule mean in the context of this situation?See margin

16. About how far will the hurricane be from land after 24 hours?

f(24) = 704 – 19.9(24) = 226.4 miles from land

17. Approximately when will the hurricane make landfall?f(x) = 704 – 19.9x = 0; x = 35.38; in a little over 35 hours


CRITICAL THINKING

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, as shown in her chip data below.

DISTANCE (KILOMETERS) 0 1 2 3 4 5

TIME (MINUTES) 0 4.1 7.9 11.8 15.5 19

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? Explain your answer.See margin

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20. Yes. Her equation models the data closely. Not every point contained in the function will fit the actual data; it is only an approx-imation of the relationship between the data

21. Her new equation will be approximately f(x) = 11.3x. This equa-tion also models the data closely, but now the function goes through the point (0, 0) specifically.



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 (HOURS) 1 2 3 4 5

TOTAL EARNED (DOLLARS) 9.3 19.3 30.73 43.23 56.33

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 func-tion equation, since she didn’t earn -$2.50 for working 0 hours.

20. Is her equation correct? Explain why or why not.See margin

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 the situation?See margin

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AR.2B Classify a func-tion as linear, quadratic, cubic, and exponential when a function is repre-sented tabularly using fi-nite differences or common ratios as appropriate



ELPS5F Write using a va-riety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increas-ingly accurate ways as more English is acquired.

VOCABULARYcommon ratio, base, expo-nent, multiplier, exponen-tial relationship

MATERIALS• 2 sheets of paper for

each student

1 . 4 • W r i t i n g E x p o n E n t i a l F u n c t i o n s 37

FOCUSING QUESTION What are the characteristics of an exponential function?

LEARNING OUTCOMES■■ I can determine patterns that identify an exponential function from its related

common ratios.■■ I can classify a function as linear or exponential when I am given a table.■■ I can determine the exponential function from a table using common ratios,

including any restrictions on the domain and range.■■ I can analyze patterns to connect the table to a function rule and communicate

the exponential pattern as a function rule.

ENGAGEMiranda shared a cookie recipe on social media with three friends. Each of Mi-randa’s friends shared the cookie recipe with three of their friends. If this trend continues, how many people will receive a cookie recipe in the fifth round?

243 people.

EXPLORE

Begin with a sheet of paper. Fold it in half and record the number of layers of paper after the fold in a table like the one shown.

NUMBER OF FOLDS NUMBER OF LAYERS

0 1

1 2

2 4

3 8

4 16

5 32

6 64

1.4 Writing Exponential Functions

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4. Possible response: The finite differences in the number of layers is the same as the number of lay-ers shifted down one row. For each fold, the number of layers doubles.

8. Possible response: An exponential function best describes this relationship because each fold doubles the number of layers, so the number of layers is multiplied by the same number, 2, for every fold you make.

QUESTIONING STRATEGIESAs 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 of layers?

• Which of the two situations is an increasing function? Decreasing function?

38 c H a p t E r 1 : a l g E B r a i c pat t E r n s

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 FOLDS NUMBER OF LAYERS

∆y = 2 – 1 = 1

∆y = 4 – 2 = 2

∆y = 8 – 4 = 4

∆y = 16 – 8 = 8

∆y = 32 – 16 = 16

∆y = 64 – 32 = 32

0 1

1 2

2 4

3 8

4 16

5 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. What patterns do you observe in the differences in the table?See margin.

5. Is this a linear relationship? How do you know?No, because the finite differences are not constant for the same ∆x.

6. What is the ratio between successive numbers of layers?2–1 = 2

4–2 = 2

8–4 = 2

16—8 = 2

32—16 = 2

64—32 = 2

7. How many layers would there be after the 7th fold? 10th fold?7th fold: 128, 10th fold: 1024

8. What type of function best describes this relationship? Explain your reasoning.See margin.

9. Write an equation that could be used to determine y, the number of layers, if you know x, the number of folds.y = 2x

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INSTRUCTIONAL HINTSSuggest that students write measurements along the edges of the piece of paper each time they fold it.

Create a need for an equa-tion rather than continuing the table. Ask students why an equation in helpful in both scenarios.


Begin with a new sheet of paper.

10. What is area of the sheet of paper without any folds?Possible response: If the paper is 8.5 × 11 inches, then the area is 93.5 square inches.

11. Fold the paper in half and record the area of the region showing after the fold in a table like the one shown. If necessary, round to the nearest tenth of a square inch.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.

NUMBER OF FOLDS

NUMBER OF 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 FOLDS

NUMBER OF LAYERS

∆y = 46.8 – 93.5 = −46.7

∆y = 23.4 – 46.8 = −23.4

∆y = 11.7 – 23.4 = −11.7

∆y = 5.8 – 11.7 = −5.9

∆y = 2.9 – 5.8 = −2.9

∆y = 1.5 – 2.9 = −1.4

0 93.5

1 46.8

2 23.4

3 11.7

4 5.8

5 2.9

6 1.5

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15. Possible response: The finite differences in the number of layers is the same as the number of layers shifted down one row. For each fold, the area of the new region is about half of the area of the previous region.

19. Possible response: An exponential function best describes this relationship because each fold gener-ates a region that has half the area of the previous region.

REFLECT ANSWERS:When the difference in x-values is 1, the successive ratios are constant.

The successive ratio is the same as the base in the equation.

HIGHER-ORDER THINKING STRATEGYUse a graphic organizer such as a Venn diagram to help students compare and contrast linear relationships and exponential relationships. The Venn diagram could be construct-ed in an interactive math notebook or math journal and students could add more infor-mation as they learn more about linear and exponential relationships in this chapter.


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. What patterns do you observe in the differences in the table?See margin.

16. Is this a linear relationship? How do you know?No, because the finite differences are not constant for the same ∆x.

17. What is the ratio between successive areas of regions?46.8

93.5 ≈ 0.5 23.4

46.8 = 0.5 11.7

23.4 = 0.5 5.8

11.7 ≈ 0.5 2.9

5.8 = 0.5 1.5

2.9 ≈ 0.5

18. What would be the area of the region present after the 7th fold? 7th fold: 0.8

19. What type of function best describes this relationship? Explain your reasoning.See margin.

20. Write an equation that could be used to determine y, the area of the region, if you know x, the number of folds.

y = 93.5( 1–2) x

REFLECT

■■ What do you notice about the successive ratios in each relationship?See margin.

■■ What relationship exists between the successive ratios in the depen-dent variable and the equations that you have written?

See margin.

EXPLAIN

When you found finite differences that were constant for the de-pendent variable, y, and the differences between values of the independent variable, x, were the same, the relationship was lin-ear. But as you have seen, this is not true for every functional relationship.

If the finite differences are not constant, look at the ratios be-tween successive rows in the table. If these ratios are constant, then the constant ratio is called a common ratio and the relation-ship is an exponential function. An exponential relationship is


or click here

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one in which there is repeated multiplication. For example, a geometric sequence has a constant multiplier, so the sequence can be represented as an exponential function with a domain of whole numbers.

Let’s look more closely at an exponential function. The table below shows the relationship between x and f(x). In an exponential function, f(x) = abx, b represents the base of the expo-nential function, which is also a common ratio or constant mul-tiplier. The parameter a represents an initial value or y-intercept.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x y = f(x)

∆y = ab – a = a(b – 1)

∆y = ab(b) – ab = ab(b – 1)

∆y = ab(b)(b) – ab(b) = ab2(b – 1)

∆y = ab(b)(b)(b) – ab(b)(b) = ab3(b – 1)

∆y = ab(b)(b)(b)(b) – ab(b)(b)(b) = ab4(b – 1)

0 a

1 ab

2 ab(b)

3 ab(b)(b)

4 ab(b)(b)(b)

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 = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x y = f(x)

yn = ab = byn-1 a

yn = a(b)(b) = byn-1 a(b) yn = a(b)(b)(b) = byn-1 a(b)(b) yn = a(b)(b)(b)(b) = byn-1 a(b)(b)(b) yn = a(b)(b)(b)(b)(b) = byn-1 a(b)(b)(b)(b)

0 a

1 ab

2 ab(b)

3 ab(b)(b)

4 ab(b)(b)(b)

5 ab(b)(b)(b)(b)

In an exponential function, if the base is greater than 1, then the function represents exponential growth. If the base is between 0 and 1, then the function represents exponential decay.

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INTEGRATE TECHNOLOGYUse technology such as a graphing calculator or spreadsheet app on a display screen to show students how geometric sequences are related to exponential functions. If the exponential function’s domain is restricted to whole numbers, then the resulting points represent a geometric sequence.


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-sive y-values,

yn yn-1, are constant if the differences between

successive x-values, ∆x, are also constant.

If the ratios of successive values of the dependent vari-able in a table of values are constant, then the values rep-resent an exponential function.

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

EXAMPLE 1Does the set of data shown below represent an exponential function? Justify your answer.

x y

0 1

1 5

2 25

3 125

4 625DRAFT

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ADDITIONAL EXAMPLESDetermine if each of the data sets below represent an exponential function. Justify your answer.

1.

Exponential

2.

Exponential

3.

Not Exponential

Note: Pay special attention to students’ justification for the third additional example. Be mindful that some students may say “no, because the differences between successive values of x are not 1.” The differences are constant, but the ratio between successive values of y are not constant.

x 0 1 2 3 4

y 2 6 18 54 162

x -1 0 1 2 3

y 8 4 2 1 0.5

x 3 6 9 12 15

y 12 17 22 27 32


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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

yn = 5 = 5yn-1 1 yn = 25 = 5yn-1 5 yn = 125 = 5yn-1 25 yn = 625 = 5yn-1 125

0 1

1 5

2 25

3 125

4 625

STEP 2 Determine whether or not the differences between successive x-values and ratios between successive y-values are constant.

The differences between successive values of x, ∆x, are all 1, so they are constant.

The ratios between successive values of y, yn yn-1

, are all 5, so they are constant.

yn yn-1 =

5

1

= 5 for all pairs of ∆x and

ynyn-1

. ∆x

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

Yes, the set of data represents an exponential function because the differences between successive x-values and the ratios between successive y-values are constant.DRAFT

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YOU TRY IT! #1 ANSWER:Yes, the set of data represents an exponential function, because the finite differences in x and the successive ratios in y are both constant.


EXAMPLE 2Does 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

YOU TRY IT! #1

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

x y

0 1.2

1 1.44

2 1.728

3 2.0736

See margin.

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YOU TRY IT! #2 ANSWER:No, the set of data does not rep-resent an exponential function, because although the first dif-ferences in x are constant, the ratios between successive values of y are not constant.


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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

yn = 4 = 1yn-1 8 2 yn = 2 = 1yn-1 4 2 yn = 0.8 = 2yn-1 2 5 yn = 0.4 = 1yn-1 0.8 2

0 8

1 4

2 2

3 0.8

4 0.4



The ratios of successive values of y, yn yn-1

, are not all the same, so they are not constant.


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 be-tween successive values of y are not constant.

YOU TRY IT! #2

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

x y

0 9.2

1 18.4

2 46

3 92

4 230

See margin.

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ADDITIONAL EXAMPLESDetermine if each of the data sets below represent an exponential function. If so, write a function relating the variables. Justify your answer.

1.

Exponential, y = -2x

2.

Not Exponential

3.

Not Exponential

4.

Exponential, y = 3(1.5)x

x 0 1 2 3 4

y -1 -2 -4 -8 -16

x 2 3 5 8 12

y -2—3 1 -1.5 2.25 -3.375

x 1 2 3 4

y 4.5 6.75 10.125 15.1875

x 18 15 12 9

y 24 22 20 18


EXAMPLE 3For the data set below, determine if the relationship is an exponential function. If so, write a function relating the variables.

x y

1 4

2 10

3 25

4 62.5

5 156.25

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

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x y

yn = 10 = 2.5yn-1 4 yn = 25 = 2.5yn-1 10 yn = 62.5 = 2.5yn-1 25 yn = 156.25 = 2.5yn-1 62.5

1 4

2 10

3 25

4 62.5

5 156.25

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



, are all 2.5, so they are constant.

Since the first differences in x and the successive ratios in y are all con-stant, the relationship is an exponential function.

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ADDITIONAL EXAMPLESAfter completing the addi-tional examples on pg. 46, draw students’ attention back to examples 2 and 3. Based on prior lessons, ask if they can write a function rule for either of those sets of data?

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


STEP 3 Determine the y-intercept of the exponential function.

Work backwards from x = 1 and y = 4.

1 – 0 = 1

2 – 1 = 1

x y

4 = 2.5 a 10 = 2.5 4

0 a

1 4

2 10

4 = 2.5

a

a( 4 ) = 2.5(a) a

4 = 2.5a

4—2.5 =

2.5a—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

1 14

2 98

3 686

4 4802

Answer: y = 2(7)x

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QUESTIONING STRATEGIESAssist students in process-ing their learning by hav-ing students write the steps in their own words.

• How would you ex-plain to a friend how you know if a set of data represents an ex-ponential function?

• Explain how to write an exponential func-tion based on a table of values to someone who has never seen this math lesson.


EXAMPLE 4For the data set below, determine if the relationship is an exponential function. If so, write a function relating the variables.

x y

1 1000

2 400

3 160

4 64

5 25.6

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

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x y

yn = 400 = 0.4yn-1 1000 yn = 160 = 0.4yn-1 400 yn = 64 = 0.4yn-1 160 yn = 25.6 = 0.4yn-1 64

1 1000

2 400

3 160

4 64

5 25.6

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



, are all 0.4, so they are constant.

Since the first differences in x and the successive ratios in y are all con-stant, the relationship is an exponential function.

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STEP 3 Determine the y-intercept of the exponential function.

Work backwards from x = 1 and y = 1000.

1 – 0 = 1

2 – 1 = 1

x y

1000 = 0.4

a 400 = 0.4 1000

0 a

1 1000

2 400

1000

a = 0.4

a( 1000a ) = 0.4(a)

1000 = 0.4a

1000

0.4 = 0.4a

0.4

2500 = a


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

1 81

2 27

3 9

4 3

5 1

Answer: y = 243( 1—3) x

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

Yes No

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.

Exponential Function? No Exponential Function? Yes Common Ratio: None Common Ratio: 2

Exponential Function? Yes Exponential Function? Yes Common Ratio: 1.5 Common Ratio: 0.25

1. x y0 2

1 6

2 18

3 54

5. x y1 2

2 4

3 6

4 8

3. x y0 0

1 1

2 8

3 27

7. x y1 3

2 4.5

3 6.75

4 10.125

2. x y0 3

1 4

2 7

3 12

6. x y1 2

2 4

3 8

4 16

4. x y0 3

1 6

2 12

3 24

8. x y1 4

2 1

3 0.25

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10. Exponential function

While ∆x is constant, ∆y is not constant. The situ-ation does have a common ratio of 2.


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.

Exponential Function? No Exponential Function? Yes Function: N/A Function: y = 40(0.2)x

• ∆x = 1• The situation is an example

of exponential decay.• The function is increasing.• The common ratio is 2.• The y-intercept is (0, 0.1).

• The function is linear.• The function is decreasing. • ∆y = 0.1• The common ratio is 0.2. • The situation is an example of ex-

ponential growth.

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 thick-ness of the layers of paper is 0.4 mm.

9. Complete the table below to represent the situation.

NUMBER OF FOLDS

x

TOTAL THICKNESS OF LAYERS

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-fy your answer.See margin.

11. Which of the following represents the function that models this situation?A. y = x + 0.1 C. y = 0.1 • 2x

B. y = 2 • 0.1x D. y = 2x + 0.1

12. Which of the following statements are true about the situation?

13. x y0 0

1 4

2 32

3 108

14. x y0 40

1 8

2 1.6

3 0.32

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Exponential Function? Yes Exponential Function? No Function: y = 50(0.5)x Function: N/A

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 CUTS

x

AREA OF ONE PIECE

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.y = 100(0.5)x

15. x y0 50

1 25

2 12.5

3 6.25

17. x y1 4500

2 6750

3 10,125

4 15,187.5

16. x y1 300

2 150

3 100

4 75

18. x y1 14

2 56

3 224

4 896

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TEKSAR.2B Classify a func-tion as linear, quadratic, cubic, and exponential when a function is repre-sented tabularly using fi-nite differences or common ratios as appropriate.

AR.2D Determine a function that models real-world data and math-ematical contexts using finite differences such as the age of a tree and its circumference, figurative numbers, average velocity, and average acceleration.



VOCABULARYexponential function, com-mon ratios, exponential decay, exponential growth

MATERIALS• 1 ball for each student

group (variety of balls, such as soccer balls, basketballs, tennis balls, and racquetballs, keeping in mind that not all balls will bounce on carpet)

• 2 meter sticks per stu-dent group

• masking tapeENAGE ANSWER:Possible sketch

1 . 5 • M o d e l i n g w i t h e x p o n e n t i a l F u n c t i o n s 53 1 . 5 • M o d e l i n g w i t h e x p o n e n t i a l F u n c t i o n s 53

FOCUSING QUESTION How can you use common ratios to construct an ex-ponential model for a data set?

LEARNING OUTCOMES■■ I can use finite differences or common ratios to classify a function as either lin-

ear or exponential when I am given a table of values.■■ I can use common ratios to write an exponential function that describes a data

set.■■ I can apply mathematics to problems that I see in everyday life, in society, and

in the workplace.

ENGAGEDeAnna dropped a basketball and let it bounce several times. What would a graph of the height of the basketball versus time look like?

See margin.

EXPLORE

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 rebound to a height of at least 1.2 meters but no more than 1.4 meters.

Bounce height is an important aspect to making sure that a ball used in any sport is properly inflated or is not worn out.

DIRECTIONS■■ Tape two meter sticks to the wall and use them to mea-

sure the bounce height, or the height to which the ball bounces, when dropped from a given height.

■■ Begin with 180 centimeters. Select a spot on the ball, such as the highest point on the top of the ball, to use as a consistent reference point. Drop the ball from this height and record the height of the first bounce.

■■ Repeat for two more trials and calculate the average bounce height for a drop from 180 centimeters.

1.5 Modeling with Exponential Functions

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STRATEGIES FOR SUCCESSIt is always best to have students collect data on their own. However, if it is not possible to collect data for this activity, provide students with the sample data so that they can use it to answer the questions with this activity.

Provide different student groups with different balls; e.g., basketballs, racquet-balls, tennis balls (only if used on tile floors), golf balls, etc. Each type of ball will generate a different model. After the activity, discuss with students why the models are different and how the parameters a and b change with different types of balls.

TECHNOLOGY INTEGRATIONMotion detectors and calculators can be used to capture and graph the height of a ball as it bounc-es in real time. You can use a motion detector to record the height versus time of a series of bounces, and then use the graph and trace features of the calculator to determine the height of each of the first 5 or 6 bounces and use that data to complete the activity.

2.

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

BOUNCENUMBER

0 1 2 3 4 5 6

AVERAGEBOUNCEHEIGHT

(CM)

180 135 100 75 55 40 30

+1 +1 +1 +1 +1 +1

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

54 c h a p t e R 1 : a l g e B R a i c pat t e R n s

■■ Use the average bounce height as the starting point for the second bounce. Record the bounce height when the ball is released from this drop height for three trials. Calculate the average bounce height.

■■ Repeat for a total of 6 bounces.■■ Record your information in a table like the one shown.

Sample data:

DROP HEIGHT

(CM)

BOUNCE HEIGHT 1

(CM)

BOUNCE HEIGHT 2

(CM)

BOUNCE HEIGHT 3

(CM)

AVERAGE BOUNCE HEIGHT

(CM)

180 135 130 140 135

135 95 105 100 100

100 70 70 85 75

75 50 55 60 55

55 45 35 40 40

40 30 25 35 30

1. If you were to drop the ball once and let it continue bouncing until it stopped, a height versus time graph of the ball might look like the figure shown. Use an ordered pair (bounce number, height of bounce) to label each point shown on the graph.

2. Calculate the finite differences between the bounce number and the average bounce height. Do the data appear to be linear? How do you know?See margin.

3. Calculate the ratios between the average bounce heights for successive bounces. Do the data appear to be exponential? How do you know?See margin.

4. What is the starting point, or y-intercept, of the data? The starting point is 180 centimeters.

HT(CM)(0, 180)

(1, 135)

(2, 100)

(3, 75)

(4, 55)(5, 40)

(6, 30)

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3.

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

BOUNCENUMBER

0 1 2 3 4 5 6

AVERAGEBOUNCEHEIGHT

(CM)

180 135 100 75 55 40 30

+1 +1 +1 +1 +1 +1

6. The y-intercept, 180 centimeters, represents the height of the ball when it was first dropped.

The base, 0.75 or 75%, means that the ball will bounce back 75% of the height of the previous bounce.

9. Answers may vary de-pending on the ball used and the data collected. For a basketball with the sample data:

f(7) = 180(0.75)7 ≈ 24 cen-timeters, so the 7th bounce will be the same height as the diameter of the ball, and should be the last “bounce” that is observed.

REFLECT ANSWERS:Use an average value of the common ratios as the base of an exponential function model.

Make a graph of the exponential function and then locate the ordered pair along the curve that contains the value of the dependent variable, usually y. The x-coordinate of this ordered pair is the value of the inde-pendent variable that generates that y-value.

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

1 . 5 • M o d e l i n g w i t h e x p o n e n t i a l F u n c t i o n s 55

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 rule mean in the context of this situation?See margin.

7. Use your model to predict the height of the 8th bounce.Answer using sample data:f(8) = 180(0.75)8 ≈ 18 centimeters

8. What is the diameter of the ball that you used? (You may need to use the for-mula C = πd to calculate the diameter of the ball.)Answers may vary. A basketball has an approximate diameter of about 24 centimeters.

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?See margin.

REFLECT

■■ How can you determine an exponential function model for a data set if the common ratios are not exactly the same, but are very close to each other?

See margin.■■ Once you have determined your exponential function model, how can

you use the model to determine a value of the independent variable that generates a particular value of the dependent variable?

See margin.

EXPLAIN

Exponential function models can be used to represent sets of math-ematical and real-world data. If an exponential function is decreas-ing, then the function is called an exponential decay function, since the values of the dependent variable, f(x), decay, or become smaller, as the values of the independent variable, x, increase. The relationship between bounce height and drop height is an expo-nential decay relationship.

Other exponential functions are increasing and are called expo-nential growth functions. The values of the dependent variable, f(x), grow, or become larger, as the values of the independent variable, x, increase. Population growth is sometimes exponential when the population of a city or county grows at the same percent each year.


or click here

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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.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

5-YEAR INTERVAL, x YEAR POPULATION,

f(x)

yn = 65,767 ≈ 1.17yn-1 56,225 yn = 78,956 ≈ 1.20yn-1 65,767 yn = 99,070 ≈ 1.25yn-1 78,956 yn = 126,470 ≈ 1.28yn-1 99,070 yn = 158,289 ≈ 1.25yn-1 126,470 yn = 197,298 ≈ 1.25yn-1 158,289

0 1985 56,225

1 1990 65,767

2 1995 78,956

3 2000 99,070

4 2005 126,470

5 2010 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.

f(x) = 56,225(1.23)x

STARTING POINT

COMMON RATIO

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QUESTIONING STRATEGIESThe scatterplot of data and graphed function rule are shown to the left.

• When looking at the two together, how do you know that the graph is a good repre-sentation of the data?

• Which point(s) from the data are included in the graph? (Listen for students to mention the y-intercept.)

• Why is the y-intercept an exact point both from the data and func-tion?


Make a scatterplot of the data and graph the function rule over the scatterplot.

Notice that not all of the data points lie on the curve representing the function model. That is because the successive ratios are not exactly equal, but are close to 1.23.

Community leaders need to know how many people will live in the community in order to decide how many fire stations, schools, or restaurants to build. Demographers, or people who study population trends, use population models like this one to make predictions about how many people will live in a com-munity in a particular year. Com-munity and business leaders rely on demographers in order to help them make better decisions for people liv-ing in a particular community.

A graph helps you to visualize the 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 ap-propriate 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 of its value each year, then the depreciation is an exponential decay.

POPULATION OF HAYS COUNTY, TEXAS

5-YEAR INTERVALS SINCE 1985

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PU

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ADDITIONAL EXAMPLEThe average salary, in millions of dollars, for professional players on a national sports team is shown in the table below. Use the data set to de-termine if the relationship is linear or exponential.

Exponential

Encourage students to follow the steps from Ex-ample 1 to complete the additional example.

5-YEAR INTERVAL, x YEAR

SALARY (MILLIONS OF DOLLARS), f(x)

0 1995 1.89

1 2000 2.4

2 2005 2.88

3 2010 3.8

4 2015 4.38


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

4-DAY INTERVAL, x DATE RADIATION

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 Determinethefinitedifferencesinvaluesofxandvaluesoff(x).

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1


LEVEL, f(x)

∆f(x) = 39.8 – 71.2 = –31.4

∆f(x) = 22.3 – 39.8 = –17.5

∆f(x) = 12.5 – 22.3 = –9.8

∆f(x) = 7.1 – 12.5 = –5.4

∆f(x) = 3.9 – 7.1 = –3.2

∆f(x) = 2.2 – 3.9 = –1.7

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

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

Exponential

Remind students that the first step is to check the finite differences and then check ratios if the finite differences are not constant.

DAY, x 0 1 2 3 4

WATER LEVEL (ML), f(x) 15 13.75 12.5 11.25 10


STEP 2 Determinetheratiosbetweensuccessivevaluesoff(x).

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1


LEVEL, f(x)

yn = 39.8 ≈ 0.559yn-1 71.2 yn = 22.3 ≈ 0.560yn-1 39.8 yn = 12.5 ≈ 0.561yn-1 22.3 yn = 7.1 ≈ 0.568yn-1 12.5 yn = 3.9 ≈ 0.549yn-1 7.1 yn = 2.2 ≈ 0.564yn-1 3.9

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

STEP 3 Determinewhetherthefinitedifferencesortheratiosbetweensuccessivevaluesoff(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.

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YOU TRY IT! #1 ANSWER:The set of data represents a linear function, rather than an exponential function, because the finite differences in x are constant and the finite differ-ences in f(x) are approximately constant.

ADDITIONAL EXAMPLEHave 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)x


YOU TRY IT! #1

A major league baseball player’s average in successive seasons is recorded in the table.

1-YEAR INTERVAL, x YEAR BATTING

AVERAGE, f(x)

0 2009 0.213

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

3-YEAR INTERVAL, x YEAR VIEWERSHIP,

f(x)

0 1970 23,050

1 1973 27,670

2 1976 29,440

3 1979 35,090

4 1982 40,020

Data 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.

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ADDITIONAL EXAMPLEAlan 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.

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)x.

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.

YEARS SINCE PURCHASE, x 2 3 4 5 6

VALUE (DOLLARS), f(x) 41,115 36,181 33,287 30,291 26,353


STEP 1 Determinethefinitedifferencesinx-valuesandtheratiosbetweensuccessivevaluesoff(x).

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

3-YEAR INTERVAL, x YEAR VIEWERSHIP,

f(x)

yn = 27,670 ≈ 1.20yn-1 23,050 yn = 29,440 ≈ 1.06yn-1 27,670 yn = 35,090 ≈ 1.19yn-1 29,440 yn = 40,020 ≈ 1.14yn-1 35,090

0 1970 23,050

1 1973 27,670

2 1976 29,440

3 1979 35,090

4 1982 40,020

STEP 2 Calculatetheaverageoftheratiosandusethisvalueforthebase, b, in your exponential function model.

1.20 + 1.06 + 1.19 + 1.14 ≈ 1.15 4

STEP 3 Usinga,theinitialviewershipvaluefromthetable,andthecalculatedvalueforb, write an exponential function model.

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

STEP 4 Use your exponential function model to predict the 2018 viewership,innumberofhouseholds.

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

f(16) = 23,050(1.15)16 ≈ 215,693.2

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YOU TRY IT! #2 ANSWER:f(x) = (2.113)x; According to the exponential function model, there will be approximately 840 paramecia in the petri dish on the tenth day of the experiment.


YOU TRY IT! #2A 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 paramecia in the petri dish over time are recorded in the table below.

1-DAY INTERVAL, x

DAY OF EXPERIMENT

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


f(x)

0 1950 3,618

1 1960 2,767

2 1970 2,205

3 1980 2,053

4 1990 1,880

5 2000 1,850

6 2010 1,641

Data Source: U.S. Census Bureau

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INSTRUCTIONAL HINTSProblems like these take mathematical stamina and attention to detail. Have students write the steps they would use to write an exponential function from real world data. Then have students compare this writing to the steps they wrote in the last section for exponential functions with constant ratios. Finally, give students time to brief-ly explain the steps to one to two other students.


Use the data set to generate an exponential model. Use your model to predict the population of Throckmorton County, Texas, in the year 2020.

STEP 1 Determinethefinitedifferencesinx-valuesandtheratiosbetweensuccessivevaluesoff(x).

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1


f(x)

yn = 2,767 ≈ 0.765yn-1 3,618 yn = 2,205 ≈ 0.797yn-1 2,767 yn = 2,053 ≈ 0.931yn-1 2,205 yn = 1,880 ≈ 0.916yn-1 2,053 yn = 1,850 ≈ 0.984yn-1 1,880 yn = 1,641 ≈ 0.887yn-1 1,850

0 1950 3,618

1 1960 2,767

2 1970 2,205

3 1980 2,053

4 1990 1,880

5 2000 1,850

6 2010 1,641

STEP 2 Calculatetheaverageoftheratiosandusethisvalueforthebase, b, in your exponential function model.

0.765 + 0.797 + 0.931 + 0.916 + 0.984 + 0.887 ≈ 0.88 6

STEP 3 Usinga,theinitialpopulationvaluefromthetable,andthecalculatedvalueforb, write an exponential function model.

f(x) = 3618(0.88)x

STEP 4 Use your exponential function model to predict the 2020 population.

2020 – 1950 = 70, so the number of 10-year intervals since 1950, x = 7.

f(7) = 3618(0.88)7 ≈ 1,478.59

According to the exponential function model f(x) = 3618(0.88)x, the population of Throckmorton County, Texas, will be approximately 1,479 people in the year 2020.


YOU TRY IT! #2A 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 paramecia in the petri dish over time are recorded in the table below.

1-DAY INTERVAL, x

DAY OF EXPERIMENT

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


f(x)

0 1950 3,618

1 1960 2,767

2 1970 2,205

3 1980 2,053

4 1990 1,880

5 2000 1,850

6 2010 1,641

Data Source: U.S. Census Bureau

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YOU TRY IT! #3 ANSWER:f(x) = 19.9(0.949)x ; According to the exponential function model, approximately 11.8% of American adults will smoke cigarettes on a daily basis in the year 2015.

1. An exponential model would be best; first dif-ference values range from 22.5 to 102.816, while suc-cessive ratio values range from 1.4 to 1.7 (these values are approximately constant).

2. A linear model would be best; successive ratio values range from 0.77 to 0.88, while first difference values range from 24.8 to 25.2 (these values are approximately constant).


YOU TRY IT! #3

Percentages of adults in the United States who smoke cigarettes on a daily basis are recorded in the table.


PERCENTAGE OF AMERICAN

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/HOMEWORKFor 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.

See margin. See margin.

1. x y0 45

1 67.5

2 94.5

3 151.2

4 257.04

5 359.856

2. x y0 209.5

1 184.6

2 159.6

3 134.4

4 109.6

5 84.5

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For questions 3 - 5, calculate the average ratio between successive y-values.

1.725

0.5

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.


SQUIRREL POPULATION,

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)x, where x represents the number of years since 2005

3. x y0 425.6

1 766.08

2 1225.73

3 2083.74

4 3750.73

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

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

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7. Exponential growth; as time passes, the value of the painting increases.

f(x) = 2200(3.39)x, where x represents the number of 10-year intervals since 1960.

8. Exponential decay; as trials numbers increase, the number of pennies decreases.

f(x) = 61(0.53)x, where x represents the trial number


FINANCE

7. Kristal noticed that her favorite painting in a museum has been increasing in value over the years. The changing value of the painting is shown in the table.

10-YEAR INTERVAL, x YEAR VALUE OF THE

PAINTING, f(x)

0 1960 $2200

1 1970 $7500

2 1980 $25,000

3 1990 $86,000

4 2000 $292,000

5 2010 $992,000

See margin.

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.

TRIAL NUMBER, x

NUMBER OF PENNIES

REMAINING, fix)

0 61

1 28

2 16

3 9

4 7

5 2

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10. The y-intercept, $29,870, represents the value of the car when Carla purchased it in 2007.

The base, 0.835, or 84% means that the value of the car is 84% of its value the previous year.

11. f(x) = 29,870(0.835)x = 5900, x ≈ 9 years; 2007 + 9 = 2016.

According to the exponen-tial function model, the car will be worth about $5900 in the year 2016.

12. 2020 – 2007 = 13, so the number of years since purchase, x = 13.

f(13) = 29,870(0.835)13 ≈ $2865.20

According to the exponen-tial function model, the car will be worth about $2865 in the year 2020.

Using the base of (0.84), the car would be worth $3096 in the year 2020.


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.

1-YEAR INTERVAL, x YEAR VALUE OF

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. What do the y-intercept and base from your function rule mean in context of the situation? See margin.

11. In what year will the car be worth about $5900?See margin.

12. Use your model to predict the value of Carla’s car in the year 2020. See margin.

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15. f(x) = 310(1.08)x = 1000; x ≈ 15 weeks

According to the expo-nential function model, Ella will have sold 1,000 ribbons in about 15 weeks.

16. A year is 52 weeks; f(52) = 310(1.08)x; x ≈ 16,959

According to the exponen-tial function model, Ella will sell close about 16,95 ribbons in a year.


For questions 13 – 16 use the following situation. FINANCEElla 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 WEEKS SINCE MARKETING

ON THE INTERNET, x

TOTAL NUMBER OF

RIBBONS SOLD, f(x)

0 310

1 336

2 365

3 388

4 425

5 445

6 496

13. Use the data set to determine an exponential function that models the situation. 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 be-fore 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.


SCIENCE

A biologist is recording the population of a certain bacteria in a petri dish. He determines the number of bacteria in the dish every 2 hours, as shown in the table below.17. Does this data show exponential

growth or exponential decay? Ex-plain.Exponential growth; as time in-creases, the number of bacteria increases

2-HR INTERVAL, x

NUMBER OF HOURS

NUMBER OF BACTERIA, f(x)

0 0 8

1 2 12

2 4 17

3 6 24

4 8 34

5 10 50

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19. 1 day is 24 hours, which is 12 2-hr intervals; x = 12; f(12) = 72(0.41)12 ≈ 636 bacteria

According to the exponen-tial function model, there will be about 636 bacteria after 1 day.

20. f(x) = 8(1.44)x = 451,000; x ≈ 30, and 30 2-hr inter-vals makes it 60 hours, or 2.5 days

According to the exponen-tial function model, there will be 451,000 bacteria in about 2.5 days.

23. f(10) = 18(1.1)10 = 65; x ≈ 46.7 miles

According to the expo-nential function model, she should run about 46.7 miles in week 10.

24. f(x) = 18(1.1)x = 65; x ≈ 14 weeks

According to the exponen-tial function model, she should reach the goal of 65 miles in week 14.


18. At some point, it becomes impossible to count all the bacteria, so an equation is necessary. Use the data set to generate an exponential model. f(x) = 8(1.44)x; where x represents the number of 2-hour time intervals

19. Use your exponential model to determine approximately how many bacteria will be in the petri dish after 1 day.See margin.

20. Approximately how many days will elapse before there are 451,000 bacteria?See margin.


CRITICAL THINKING

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 training plan.

NUMBER OF WEEKS, x

MILES PER WEEK, f(x)

0 18

1 20

2 22

3 24

4 26.5

5 29

6 32

7 36

21. Using the data given above, generate an exponential function that models the situation. 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 train-ing plan?See margin.

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

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1a. geometric

a1 = 128; an = 3–4an – 1

an = 128(3–4)n-1

1c. arithmetic

a1 = −5; an = an – 1 + 2.25

an = -7.25 + 2.25n

70

Chapter 1 Mid-Chapter Review

Arithmetic and Geometric Sequences

1. For each sequence of numbers below, determine whether it is arithmetic, geo-metric or neither. Then, write a recursive rule and an explicit rule, if possible.a) 128, 96, 72, 54, 40.5, … See margin.

b) 400, 300, 150, 75, 22.5, … Neither arithmetic nor geometric

c) −5, −2.75, −0.5, 1.75, 4, … 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 arithmeticb) a1 = 12; an =

2–3an-1

12, 8, 51—3, 3

5—9

geometric3. Given each explicit rule, write the first 4 terms of the sequence. Indicate wheth-

er each sequence is arithmetic or geometric.

a) an = 243(1–3)n-1

243, 81, 27, 9 geometricb) 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 func-tion of the figure number, n.

f(n) = 4n

1.1


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C H A P T E R 1 M I d - C H A P T E R R E v I E w 71

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 ex-plicit rule and then show how the answer is derived using the explicit rule. a) an = 1(2)n – 1 b) an = 15 + 5n a12 = 1(2)11 = $2,048 a12 = 15 + 12(5) = $75

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

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

y =-4—3 x + 3

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

1—2x - 9

9. For each table, determine whether the relationship shows a linear function. If so, write the function.

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.

f(x) = -2.5x + 210

1.2

a) x y1 9

2 6

3 3

4 0

5 -3

b) x y0 3.2

2 4.2

4 5.2

6 6.2

8 7.2

c) x y1 0

2 4

3 8

4 16

5 24

# OFWEEKS WEIGHT

0 210

1 207.5

2 205

3 202.5

4 200

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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

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.

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

1.3

1.4

16. x y0 240

1 180

2 135

3 101.25

17. x y0 12

1 24

2 36

3 64

18. x y1 750

2 900

3 1080

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23. 1–2 hour is 30 minutes, which is 10 3-min intervals; x = 10; f(10) = 180.2(0.95)10 ≈ 107.89

According to the exponen-tial function model, the temperature will be about 108° Fahrenheit after ½ hour.


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.

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.

Modeling with Exponential Functions


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 OF 3-MIN

INTERVALS, xNUMBER OF

MINUTESTEMPERATURE

(F°), 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.Exponential 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

23. Use your exponential model to determine the approximate temperature after 1–2 hour.See margin.

1.5

NUMBER MONTHS, x MONTHLY REVENUE, y

0 $800.00

1 $820.00

2 $840.00

3 $861.51

4 $883.05

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24. f(x) = 180.2(0.95)x = 80; x ≈ 16, and 16 3-min intervals makes it about 48 minutes.

According to the exponen-tial function model, the temperature will be 80° F in about 48 minutes.

26. f(15) = 3.4(1.0176)15 ≈ 4.42 million

According to the expo-nential function model, the population of Harris County would be about 4.42 million in 2015.

27. f(x) = 3.4(1.0176)x = 5; x ≈ 22 years

According to the exponen-tial function model, the population would reach 5 million in the year 2022.


24. Approximately how many minutes will elapse before the temperature is 80° F?See margin.

For questions 5 – 7 use the situation below.The population of Harris County, TX over a 7-year period is shown in the table below, where x represents the number of years since 2000 and y represents the population in millions.

25. Using the data given above, generate an expo-nential function that models the situation. f(x) = 3.4(1.0176)x; where x represents the number of years since 2000

26. According to your model, what is a reasonable estimate of the population in the year 2015?See margin.

27. According to your model, in what year is the population expected to reach 5 million?See margin.

MULTIPLE CHOICE28. An auditorium has seating in which the front row has 25 seats, the 2nd row 28

seats, the 3rd row 31 seats, etc. If this pattern continues, how many seats will be in the 12th row?A. 46B. 52C. 55D. 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 rela-tionship? If so, which function can be used to find the total cost, C, for purchas-ing and screen-printing t number of t-shirts?A. yes, C = 20t + 8.95B. yes, C = 20 + 8.95tC. yes, C = 20t + 8.95tD. Not a linear function

YEARS SINCE 2000, x

POPULATION (IN MILLIONS),

f(x)

0 3.4

1 3.46

2 3.52

3 3.58

4 3.64

5 3.71

6 3.78

7 3.84

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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. Which of the following functions best models the given data?

x 0 1 2 3 4 5 6

y 12.2 18.5 26.7 40 60.6 91.9 135.9

A. y = 120 – 16.5x B. y = 120 – 29xC. y = 12.2(1.5)x

D. y = 120(0.25)x

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TEKSAR.2A Determine the patterns that identify the relationship between a function and its common ratio or related finite dif-ferences as appropriate, including linear, quadrat-ic, cubic, and exponential functions.AR.2C Determine the function that models a giv-en table of related values using finite differences and its restricted domain and range.AR.2D Determine a function that models real-world data and math-ematical contexts using finite differences such as the age of a tree and its circumference, figurative numbers, average velocity, and average acceleration.


ELPS5C Spell familiar English words with in-creasing accuracy, and employ English spelling patterns and rules with in-creasing accuracy as more English is acquired.

VOCABULARYfinite differences, domain, range, vertex, axis of sym-metry, x-intercept, y-inter-cept, quadratic function

ENAGE ANSWER:Possible answer: The number of dots is equal to the term num-ber squared.

2. In the table, ∆x = 1.

The data set is not linear because the finite differences in the triangular numbers are not constant.

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

TERMNUMBER

TRIANGULARNUMBER

∆y = 3 – 1 = 2

∆y = 6 – 3 = 3

∆y = 10 – 6 = 4

∆y = 15 – 10 = 5

∆y = 21 – 15 = 6

1 1

2 3

3 6

4 10

5 15

6 21

TERMNUMBER

TRIANGULARNUMBER

yn— yn -1

= 3—1 = 3

yn— yn -1

= 6—3

= 2

yn— yn -1

= 10—6

= 1 2—3

yn— yn -1

= 15—10

= 1 1—2

yn— yn -1

= 21—15

= 1 2—5

1 1

2 3

3 6

4 10

5 15

6 21

The data set is not exponential because the succes-sive ratios are not constant.

76 C H A P T E R 1 : A L G E B R A I C PAT T E R N S76 C H A P T E R 1 : A L G E B R A I C PAT T E R N S

1.6 Writing Quadratic Functions

FOCUSING QUESTION What are the characteristics of a quadratic function?

LEARNING OUTCOMES■■ I can determine patterns that identify a quadratic function from its related finite

differences.■■ I can determine the quadratic function from a table using finite differences,

including any restrictions on the domain and range.■■ I can use finite differences to determine a quadratic function that models a

mathematical context.■■ I can analyze patterns to connect the table to a function rule and communicate

the quadratic pattern as a function rule.

ENGAGESquare numbers can be represented using counters as shown.

TERM 1: 1 TERM 2: 4 TERM 3: 9 TERM 4: 16

What patterns do you see in the geometric arrangements of square numbers?See margin.

EXPLORE

A figurative number, sometimes called a figurate number, is a number that can be represented by a regular geometric arrangement of dots or other objects. For ex-ample, triangular numbers can be represented using arrangements of dots that are shaped like regular triangles.

1 3 6 10

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

4.

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

TERMNUMBER

TRIANGULARNUMBER

∆y = 3 – 1 = 23 – 2 = 1

4 – 3 = 1

5 – 4 = 1

6 – 5 = 1

∆y = 6 – 3 = 3

∆y = 10 – 6 = 4

∆y = 15 – 10 = 5

∆y = 21 – 15 = 6

1 1

2 3

3 6

4 10

5 15

6 21

1 For each successive term, you add a row of dots to the bottom of the trian-gle that is one dot longer than the row of dots at the bottom of the previous triangle.

2. See the bottom of page 76.

3. Possible answers may include:

The finite differences are not constant, but they do increase by 1 each time.

The successive ratios can all be represented as ynyn - 1

= 1 + 2—n.

4. See below.

1 . 6 • W R I T I N G Q u A d R AT I C F u N C T I o N S 77

Use counters to build a sequence of the first six triangular numbers. Record the num-bers in a table like the one shown.

TERM NUMBER TRIANGULAR 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?See margin.

2. Does the data set follow a linear or an exponential function? Explain your rea-soning. See margin.

3. What patterns do you see in the finite differences or the successive ratios?See margin.

4. Calculate the second finite difference. What do you notice? See margin.

5. The quadratic parent function is y = x2. Generate a sequence with y-values for {x|x = 1, 2, 3, 4, 5, 6}.1, 4, 9, 16, 25, 36

6. Calculate the second finite differences for the quadratic parent function. What do you notice?

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

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

x y

∆y = 4 – 1 = 35 – 3 = 2

7 – 5 = 2

9 – 7 = 2

11 – 9 = 2

∆y = 9 – 4 = 5

∆y = 16 – 9 = 7

∆y = 25 – 16 = 9

∆y = 36 – 25 = 11

1 1

2 4

3 9

4 16

5 25

6 36

A set of numbers, such as x-values or y-values, can be represented with brackets using set notation. The set of whole numbers less than 10 is represented as {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. If this set is a set of x-values, it can be writ-ten as {x|x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} which is read “the set of all x such that x equals zero, one, two, …”

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REFLECT ANSWERS:The first finite differences in a quadratic function are not con-stant, but increase or decrease with a particular pattern. The second finite differences are constant in a quadratic func-tion.

The level of finite differences that are constant is the same as the degree of the polynomial (linear: degree one and first differences constant; quadratic: degree two and second differ-ences constant).

ELL STRATEGYWriting with familiar English language words (ELPS: c5C) helps students both deepen their under-standing of the mathe-matical content as well as become more comfortable with the English language. Having students use the re-flect questions for a journal entry in their interactive math notebooks provides an opportunity to reinforce this language proficiency skill.


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

■■ In a linear function, the first finite differences are constant. What is true about the finite differences for a quadratic function?

See margin.■■ A linear function contains a polynomial with degree one (mx + b) and a

quadratic function contains a polynomial with degree two (ax2 + bx + c). What relationship is there between the degree of the polynomial and the level of finite differences that are constant?

See margin.

EXPLAIN

In a linear function, the first finite differences, or the dif-ference between consecutive values of the dependent vari-able, are constant. But for a quadratic function, the first fi-nite differences are not constant. They do, however, have a pattern in that they increase or decrease by the same num-ber. As a result, the second finite differences, or the differ-ences between the first finite differences, are constant.

Let’s look more closely at a quadratic func-tion. The table below shows the relation-ship between x and f(x) in a quadratic func-tion written in polynomial or standard, f(x) = ax2 + bx + c.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x PROCESS y = f(x)

∆y = (a + b + c) – c = a + b

∆y = (4a + 2b + c) – (a + b + c) = 3a + b

∆y = (9a + 3b + c) – (4a + 2b + c) = 5a + b

∆y = (16a + 4b + c) – (9a + 3b + c) = 7a + b

∆y = (25a + 5b + c) – (16a + 4b + c) = 9a + b

0 a(0)2 + b(0) + c c

1 a(1)2 + b(1) + c a + b + c

2 a(2)2 + b(2) + c 4a + 2b + c

3 a(3)2 + b(3) + c 9a + 3b + c

4 a(4)2 + b(4) + c 16a + 4b + c

5 a(5)2 + b(5) + c 25a + 5b + c

There are many forms of a quadratic function. Polyno-mial form, also called stan-dard form, expresses the function as a polynomial with exponents in decreas-ing order.

f(x) = ax2 + bx + c

In standard form, a, b, and c are rational numbers.


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INTEGRATE TECHNOLOGYUse technology such as a graphing calculator or spreadsheet app on a display screen to show students how, no matter the numbers present in the quadratic function, the sec-ond differences will always be constant.


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 differenc-es. When you look at the second differences, three patterns emerge.

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

FINITE DIFFERENCES AND QUADRATIC FUNCTIONS

In a quadratic function, the second differences be-tween successive y-values are constant if the dif-ferences between successive x-values, ∆x, are also constant.

If the second differences between consecutive y-values in a table of values are constant, then the val-ues represent a quadratic function.

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

x y = f(x)

∆y = (a + b + c) – c = a + b∆2y = (3a + b) – (a + b) = 2a

∆2y = (5a + b) – (3a + b) = 2a

∆2y = (7a + b) – (5a + b) = 2a

∆2y = (9a + b) – (7a + b) = 2a

∆y = (4a + 2b + c) – (a + b + c) = 3a + b

∆y = (9a + 3b + c) – (4a + 2b + c) = 5a + b

∆y = (16a + 4b + c) – (9a + 3b + c) = 7a + b

∆y = (25a + 5b + c) – (16a + 4b + c) = 9a + b

0 c

1 a + b + c

2 4a + 2b + c

3 9a + 3b + c

4 16a + 4b + c

5 25a + 5b + c

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ADDITIONAL EXAMPLESWhat type of function (lin-ear, exponential, or qua-dratic) would best model the data sets below? Justify your answer.

1.

Exponential

2.

Quadratic

x 1 3 5 7 9

y 15 135 1215 10935 98415

x 1 2 3 4 5

y 6 -4 -18 -36 -58


EXAMPLE 1What type of function would best model the data set below? Justify your answer.

x y

0 0

1 1

2 6

3 15

4 28


∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

∆y = 1 – 0 = 1

∆y = 6 – 1 = 5

∆y = 15 – 6 = 9

∆y = 28 – 15 = 13

0 0

1 1

2 6

3 15

4 28


The differences in y, ∆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

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ADDITIONAL EXAMPLESWhat type of function (lin-ear, exponential, or qua-dratic) would best model the data sets below? Justify your answer.

1.

Linear

2.

None of the functions

INSTRUCTIONAL HINTSHelp students organize their thoughts by creat-ing their own flowchart of steps for determining whether or not a set of data represents a linear, exponential, or quadratic function.

Within the flowchart ask students to include the questions they ask them-selves as they look at data such as, “are the differ-ences in x constant?” or “are the successive rations constant?”

x -7 -4 -1 2 5

y 2 -16 -34 -52 -70

x 0 1 2 3 4

y 4 8 16 20 24



∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

y1—y0 =

1–0 (undefined)y2—y1

= 6–1 = 6

y3—y2 =

15—6 = 2.5y4—y3

= 28—15 = 1.8666...

0 0

1 1

2 6

3 15

4 28

The data set is not exponential because the successive ratios are not constant.

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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

∆y = 1 – 0 = 1 ∆2y = 5 – 1 = 4

∆2y = 9 – 5 = 4

∆2y = 13 – 9 = 4

∆y = 6 – 1 = 5

∆y = 15 – 6 = 9

∆y = 28 – 15 = 13

0 0

1 1

2 6

3 15

4 28

The second finite differences are all 4, so the set of data represents a quadratic function.

YOU TRY IT! #1

Determine if the function rule for the set of data is linear, exponential, or quadratic.

x y

0 0

1 1

2 7

3 19

4 37

Answer: The set of data does not represent any of these functions.

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INSTRUCTIONAL HINTSIf students are struggling with “triangular numbers,” have them turn back to the Explore on pg. 76 for a visual.

Give students isomet-ric graph paper to draw square or hexagonal num-bers. Have them create a ta-ble of values and determine the function rule for the set of numbers they drew.

INSTRUCTIONAL HINTSEncourage students to add abbreviated direc-tions for writing function rules on their flowcharts from the Instructional Hints on pg. 82.

Summarizing learning in one flowchart will help students study and process the material.

ADDITIONAL EXAMPLESDetermine the function rule for the Additional Examples from pages 80 and 81.

pg. 80

1. y = 5(3)x

2. y = -2x2 - 4x + 12

pg. 81

1. y = -6x - 40

2. Not linear, exponential, or quadratic 82 C H A P T E R 1 : A L G E B R A I C PAT T E R N S

EXAMPLE 2 Determine the function rule for the set of triangular numbers shown below.

x y

0 0

1 1

2 3

3 6

4 10


∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

∆y = 1 – 0 = 1

∆y = 3 – 1 = 2

∆y = 6 – 3 = 3

∆y = 10 – 6 = 4

0 0

1 1

2 3

3 6

4 10



The differences in y, ∆y, are not constant.

STEP 3 Determine whether or not the second finite differences in successive y-values are constant.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

x y

∆y = 1 – 0 = 1 ∆2y = 2 – 1 = 1

∆2y = 3 – 2 = 1

∆2y = 4 – 3 = 1

∆y = 3 – 1 = 2

∆y = 6 – 3 = 3

∆y = 10 – 6 = 4

0 0

1 1

2 3

3 6

4 10

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

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3.

f(x) = -2x2 - 14x + 32

4.

f(x) = 2x2 - 4x - 10

ADDITIONAL EXAMPLES1. Write a quadratic function where the second finite dif-ference is 3, the y-intercept is (0, -2), and a + b is 13.

f(x) =1.5x2 - 11.5x - 2

For the data sets shown, write a function rule relating the variables.

2.

f(x) = x2 + 5x + 3

ADDITIONAL EXAMPLESCreate a table of values for square numbers from the Engage diagram on pg. 76. Then determine the func-tion rule.

f(x) = x2

Determine the function rule for the set of hexag-onal numbers using the values in the table.

f(x) = 2x2 - x

x 0 1 2 3 4

y 0 1 6 15 28

x 0 1 2 3 4

y 3 9 17 27 39

x 1 2 3 4 5

y 16 -4 -28 -56 -88

x 1 2 3 4 5

y -12 -10 -4 6 20


STEP 4 Calculate a, b, and c for the quadratic function f(x) = ax2 + bx + c. For x = 0, y = 0. So c = 0.

The second finite difference is 2a, so 2a = 1 and a = 1–2.

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

1–2, b must also equal 1–2.

STEP 5 Write the function with the values for a, b, and c:

f(x) = 1–2 x2 +

1–2 x + 0 or f(x) = x2 + x

2

YOU TRY IT! #2

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

Answer: f(x) = 3—2x2 –

1—2x + 0 = 3x2 – x

2

EXAMPLE 3Write a quadratic function where the second finite difference is 4, the y-intercept is (0,1), and a + b is 5.

STEP 1 Determine the values of a, b, and c.

The second finite difference is 4. So 2a = 4, and a = 2.The y-value of the y-intercept, c, is 1.Since a + b = 5 and a = 2, then b = 3.

1 5 12 22 35

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INSTRUCTIONAL HINTIf students struggle with finding c in YOU TRY IT #3 help then work through a few of the Additional Ex-amples on page 83 before revisiting the YOU TRY IT.


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

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

YOU TRY IT! #3

For the data set below, write a function relating the variables.

x y

1 1

2 9

3 23

4 43

5 69

Answer: y = 3x2 – x – 1

PRACTICE/HOMEWORKFor 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.

Quadratic Linear Exponential

Quadratic Other Quadratic

1. x y = f(x)1 5

2 11

3 21

4 35

5 53

4. x y = f(x)1 5

2 14

3 29

4 50

5 77

3. x y = f(x)1 5

2 9

3 16

4 29

5 52

6. x y = f(x)1 5

2 8

3 13

4 20

5 29

2. x y = f(x)1 5

2 11

3 17

4 23

5 29

5. x y = f(x)1 5

2 12

3 31

4 68

5 129

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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.

a = 3, b = 0, c = 7 a = 2, b = 1, c = 3 f(x) = 3x2 + 7 f(x) = 2x2 + x + 3

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.

f(0) = -5; a = 1, b = 3, c = -5 f(0) = 2; a = 6, b= -5, c = 2 f(x) = x2 + 3x – 5 f(x) = 6x2 – 5x + 2

7. x y = f(x)1 5

2 11

3 24

4 53

5 117

9. x y = f(x)0 7

1 10

2 19

3 34

11. x y = f(x)0 –1

1 5

2 19

3 41

13. x y = f(x)0 ?

1 –1

2 5

3 13

4 23

8. x y = f(x)1 5

2 9

3 13

4 17

5 21

10. x y = f(x)0 3

1 6

2 13

3 24

12. x y = f(x)0 –6

1 –1

2 14

3 39

14. x y = f(x)0 ?

1 3

2 16

3 41

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f(0) = -4; a = 3, b = -8, c = -4 f(0) = 0; a = 4, b = 3, c = 0f(x) = 3x2 – 8x – 4 f(x) = 4x2 + 3x


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 com-pleting the table below. The first row has been completed for you.

LENGTHx

AREAy

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?49

15. x y = f(x)0 ?

1 –9

2 –8

3 –1

4 12

16. x y = f(x)0 ?

1 7

2 22

3 45

4 76

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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 NUMBER

x

TOTAL TOOTHPICKS

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


SCIENCE

GRAVITY EXPERIMENTAn experiment is conducted by dropping an object from a height of 150 feet and mea-suring 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(SEC)

x

DISTANCE(FEET)

y

TIME(SEC)

x

DISTANCE(FEET)

y

TIME(SEC)

x

DISTANCE(FEET)

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. Write a function relating the variables in each of the tables above. Venus: y = 14.8x2

Earth: y = 16x2

Mars: y = 6.2x2

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VOCABULARYquadratic function, finite differences, maximum, minimum

MATERIALS• color tiles• square-inch grid paper

or chart paper

STRATEGIES FOR SUCCESSIf students have difficul-ty creating rectangles or visualizing the area or perimeter of the sandbox, use square-inch grid paper as a template or outline on which students can organize the color tiles to create rectangles. Perim-eter can be traced out on the paper and counted. Area can be counted from the tiles themselves using skip-counting or multipli-cation of rows and col-umns.

ENGAGE ANSWER:Possible answers: 3 ft. by 15 ft., 4 ft. by 14 ft.,....9 ft. by 9 ft.

2. 5. For a quadratic function,• The value of c is the

y-coordinate of the y-intercept, (0, c).

• The second difference is equal to 2a.

• The first difference be-tween the y-values for x = 0 and x = 1 is equal to a + b.

Work backwards in the ta-ble to reach the row when x = 0. See table page 89.


FOCUSING QUESTION How can you use finite differences to construct a quadratic model for a data set?

LEARNING OUTCOMES■■ I can use finite differences or common ratios to classify a function as linear, qua-

dratic, or exponential when I am given a table of values.■■ I can use finite differences to write a quadratic function that describes a data set.■■ I can apply mathematics to problems that I see in everyday life, in society, and

in the workplace.

1.7 Modeling with Quadratic Functions

ENGAGEMrs. Hernandez wants to con-struct a rectangular sandbox for her niece and nephew. She has 36 feet of lumber to use as the border. What are some possible dimensions that Mrs. Hernan-dez could use to construct the sandbox?

See margin.

EXPLORE

Use color tiles to build rectangles that represent a sandbox with a perimeter of 36. Recall that the area of a rectangle can be found using the formula A = lw and the perimeter of a rectangle can be found using the formula P = 2(l + w). Record the di-mensions and the area of each rectangle in a table like the one shown.DRAFT

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2. See bottom of page 88.

5. See bottom of pages 88-89.

6. See bottom of page 89.

7. Data set: domain: whole numbers,

1 ≤ x ≤ 17; range: {17, 32, 45, 56, 65, 72, 77, 80, 81}

Function: domain: all real numbers;

range: y ≤ 81

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 range of the function rule include additional real numbers.

8. A sandbox with dimen-sions of 9 feet by 9 feet gives an area of 81 square feet, the maximum area of a sandbox with a perimeter of 36 feet.

An area of 81 square feet is the largest value of the area in the table.

The vertex of the graph, (9, 81), represents the width that generates the largest area. The ordered pair (9, 81) represents a width of 9 feet with an area of 81 square feet.

6.

WIDTH (IN.)

LENGTH (IN.)

AREA (SQ. IN.)

0 18 0

1 17 17

2 16 32

3 15 45

4 14 56

+1+1+1+1

+17+15+13+11

-2-2-2

a + b = 17-1 + b = 17

b = 18c = 0

2a = -2a = -1

1 . 7 • M o d E L I N G w I T H Q u A d R AT I C F u N C T I o N S 89

Sample data:

WIDTH (IN.) LENGTH (IN.) AREA (SQ. IN.)

3 15 45

4 14 565 13 656 12 727 11 778 10 809 9 8110 8 8011 7 7712 6 7213 5 6514 4 5615 3 4516 2 32

1. What do you think a scatterplot of area versus width for your set of rectangles would look like?Answers may vary. Possible response: The graph will increase until the width reaches 9 inches and then the graph will decrease.

2. Sketch a scatterplot of area versus width for your set of rectangles.See margin for graph.

3. Based on the shape of your graph, what type of function best models the data?Quadratic

4. Calculate the finite differences between the width and the area. Do the data appear to be linear or quadratic? How do you know?The data appear to be quadratic since the first differences are not con-stant but the second differences are constant. See table above.

5. Use the patterns in the finite differences to write a function rule that describes the data set.f(x) = −x2 + 18x See margin for solution.

6. Use a graphing calculator to graph the function rule over the scatterplot. What do you notice about the function rule and the scatterplot?The graph of the function rule connects each of the data points.See margin for graph.

7. Compare the domain and range of the data set and the domain and range of the function rule. How are they alike? How are they different?See margin.

8. What dimensions will give Mrs. Hernandez a sandbox with the greatest area? Use the table and graph to justify your answer.See margin

+1+1+1+1+1+1+1+1+1+1+1+1+1

+11+9+7+5+3+1-1-3-5-7-9-11-13

-2-2-2-2-2-2-2-2-2-2-2-2

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9. No, because the x-values represent the width of a rectangle and the y-values represent the area of a rectangle. You cannot have a rectangle with a width of 0 feet or an area of 0 square feet.

10. 36 = 2l + 2w 36 ÷ 2 = 2l ÷ 2 + 2w ÷ 2 18 = l + w l = 18 – w

The length and width must add up to 18, so the length is the difference between 18 and the width.

11. A = lw = (18 – w)w If you use x to represent

the width, w, and f(x) to represent the area, A: f(x) = (18 – x)x

f(x) = (18 – x)x = 18x – x2 = −x2 + 18x, which is the same as the function rule generated from finite differences. The two equations are equivalent.

REFLECT ANSWERS:Confirm that the second 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, and c to write a quadratic function in standard (polyno-mial) form, f(x) = ax2 + bx + c.

b = 18, which is half of the perimeter of 36. The perime-ter is divided in half because the perimeter of a rectangle is twice the sum of the length and width.a = −1 because the parabola needs to be reflected vertically in order to generate a maxi-mum height instead of a mini-mum height.

ELL SUPPORTAsking students to write using newly acquired basic vocabulary and content-based grade-level vocabulary (ELPS c(5)(b)) helps English language learners make con-nections among vocabulary terms and important mathe-matical ideas. Students encountered quadratic functions in Algebra 1 and are using finite differences 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.


9. Are the x-intercepts of the function included in your data set? Why or why not?See margin.

10. The perimeter of a rectangle is found using the formula P = 2l + 2w. If you know the perimeter is 36 feet, how are the length and width related?See margin.

11. 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 this equation compare with the function rule that you generated from finite differences?See margin.

REFLECT

■■ How can you determine a quadratic function model for a data set?See margin.

■■ How do the parameters obtained from finite differences relate to the data set being modeled?

See margin.

EXPLAIN

Quadratic function mod-els can be used to repre-sent sets of mathematical and real-world data. A quadratic function has several key attributes that are important to consider when using and interpreting models that are based on quadrat-ic functions. The graph of the quadratic func-tion, which is a parabo-la, helps to explain how these attributes relate to the quadratic function model.

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The vertex of the parabola represents the data values that gen-erate a minimum or maximum value. In the case of the sandbox problem, the vertex reveals the width, or x-coordinate, that gener-ates 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 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-inter-cepts 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.

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

+1

WIDTH (IN.)

LENGTH (IN.)

AREA (SQ. IN.)

+17-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

+15

+13

+11

+9

+7

+5

+3

+1

₋1

₋3

₋5

₋7

₋9

₋11

₋13

₋15

₋17

0 18 0

1 17 17

2 16 32

3 15 45

4 14 56

5 13 65

6 12 72

7 11 77

8 10 80

9 9 81

10 8 80

11 7 77

12 6 72

13 5 65

14 4 56

15 3 45

16 2 32

17 1 17

18 0 0


or click here

x-INTERCEPT

VERTEX

x-INTERCEPT

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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 max-imum 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 do-main restrictions on the function model when it is applied to the situation.

DOMAIN RANGE

FUNCTION x ∈ (ALL REAL NUMBERS) y ≤ 81

SITUATION0 < x < 18

(INCLUDES WHOLE AND REAL NUMBERS)

0 < y ≤ 81

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 sub-set of the domain and range of the quadratic function model.DRAFT

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ADDITIONAL EXAMPLESDetermine whether the sets of data represent a linear, quadratic, or exponential function.

1.

Linear

2.

None


EXAMPLE 1A 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-resents a linear, quadratic, or exponential function.

TIME IN SECONDS, x

HEIGHT IN METERS, f(x)

0 100

1 105.1

2 100.4

3 85.9

4 61.6

5 27.5

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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

TIME IN SECONDS, x


∆f(x) = 105.1 − 100 = 5.1

∆f(x) = 100.4 – 105.1 = −4.7

∆f(x) = 85.9 – 100.4 = −14.5

∆f(x) = 61.6 – 85.9 = –24.3

∆f(x) = 27.5 – 61.6 = –34.1

0 100

1 105.1

2 100.4

3 85.9

4 61.6

5 27.5

x 2 3 4 5 6

y 4.2 -0.3 -4.8 -9.3 -13.8

x 3 4 5 6 7

y -4.12 12.54 32.98 47.15 125.77

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ADDITIONAL EXAMPLESDetermine whether the sets of data represent a linear, quadratic, or exponential function.

1.

Quadratic

2.

Exponential


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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

TIME IN SECONDS, x


yn = 105.1 ≈ 1.051yn-1 100 yn = 100.4 ≈ 0.955yn-1 105.1 yn = 85.9 ≈ 0.856yn-1 100.4 yn = 61.6 ≈ 0.717yn-1 85.9 yn = 27.5 ≈ 0.446yn-1 61.6

0 100

1 105.1

2 100.4

3 85.9

4 61.6

5 27.5

STEP 3 Determine the second finite differences in the f(x) values.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

TIME IN SECONDS, x


0 100

1 105.1

2 100.4

3 85.9

4 61.6

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 con-stant.

■■ 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.

■■ 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.

₋9.8

₋9.8

₋9.8

₋9.8

+5.1

₋4.7

₋14.5

₋24.3

₋34.1

x 0 1 2 3 4

y 45.9 49.1 58.7 74.7 97.1

x 1 2 3 4 5

y 4.9 5.9 7.1 8.5 10.2

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YOU TRY IT! #1 ANSWER:The set of data represents a quadratic function because the finite differences in x and the second finite differences in f(x) are both constant.

INSTRUCTIONAL HINTAs students are learning to model real world scenarios with linear, exponential, and quadratic functions, ask them to predict what kind of function the sce-nario might use before they test the data. Key words like percent and interest might steer stu-dents toward exponential functions. Scenarios that mention a constant rate of change might suggest a linear function.


YOU TRY IT! #1

A softball pitcher throws a ball to her catcher. The ball’s path is tracked in the table.

HORIZONTAL DISTANCE

FROM PITCHER IN FEET, x

VERTICAL HEIGHT IN FEET,

f(x)

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.

EXAMPLE 2The 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-ponential function.

1-YEAR INTERVAL, x

INTEREST IN DOLLARS, f(x)

0 500

1 530

2 561.80

3 595.51

4 631.24

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

1-YEAR INTERVAL, x


0 500

1 530

2 561.80

3 595.51

4 631.24

The differences in the x-values are constant and the first finite differ-ences 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 INTERVAL, x


yn = 530 ≈ 1.06yn-1 500 yn = 561.80 ≈ 1.06yn-1 530 yn = 595.51 ≈ 1.061yn-1 561.80 yn = 631.24 ≈ 1.059yn-1 595.51

0 500

1 530

2 561.80

3 595.51

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-mately constant, and the data set represents an exponential function.

∆f(x) = 530 − 500 = 30

∆f(x) = 561.80 − 530 = 31.80

∆f(x) = 595.51 − 561.80 = 33.71

∆f(x) = 631.24 − 595.51 = 35.73

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INSTRUCTIONAL HINTJustification of why a data represents a specific type of function is important for students to demonstrate mastery of the concept. Have students write com-plete sentences with expla-nations of their answers.

ADDITIONAL EXAMPLESTurn back to pages 93 and 94. Write the functions for each of the Additional Examples.

pg. 93, AE 1: y = -4.5x + 13.2

pg. 93, AE 2: none

pg. 94, AE 1: y = 3.2x2 - 4.1x + 50

pg. 94, AE 2: y = 4.1(1.2)x


YOU TRY IT! #2

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-ponential function.

1-YEAR INTERVAL, x

TOTAL AMOUNT IN DOLLARS, f(x)

0 750

1 780

2 810

3 840

4 870

Answer: The data set represents a linear function because the finite dif-ference in x and the first finite difference in f(x) is constant.

EXAMPLE 3Possum 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 recorded in the table.

DISTANCE AWAY FROM THE CLIFF IN

METERS, x

HEIGHT ABOVE THE WATER IN

METERS, f(x)

0 27

1 28.1

2 27.4

3 24.8

4 20.0

5 12.9

Data Source: Redbullcliffdiving.com

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TECHNOLOGY INTEGRATIONHave students use a graph-ing calculator or app to graph the scatterplot of the given data and the func-tion. Then, explore various points on the function us-ing tools to find intercepts, maximum and minimum points, and other data points along the path of the diver from Example 3 and the baseball from the Addi-tional Example above.

ADDITIONAL EXAMPLESThe data set below represents the height of a baseball, f(x), over time in seconds, x.

1. Use the data set to generate a quadratic function that best models the baseball’s path.

f(x) = -16.45x2 + 154.25x + 2.5

2. Use the table to estimate the height of the ball when it was hit, the highest point in the ball’s path, and the time when the ball will hit the highest point.

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.

TIME IN SECONDS, x

HEIGHT ABOVE

GROUND IN FEET, f(x)

0 2.5

1 140.3

2 245.3

3 311.7

4 353.9

5 360.1


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 in x-values and the second finite differences in the values of f(x).

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

DISTANCE AWAY FROM THE CLIFF IN

METERS, x

HEIGHT ABOVE THE WATER IN

METERS, f(x)

0 27

1 28.1

2 27.4

3 24.8

4 20.0

5 12.9

STEP 2 Calculate the average of the second finite differences and use this value to determine a in your quadratic function model, f(x) = ax2 + bx + c.

2a = -1.8 -1.9 -2.2 - 2.3 = -2.05 4

So a = −1.025

STEP 3 Calculate the value of b.

The difference between the values of f(x) for x = 0 and 1 is (a + b).

a + b = 1.1 (−1.025) + b = 1.1 b = 2.125

₋1.8

₋1.9

₋2.2

₋2.3

+1.1

₋0.7

₋2.6

₋4.8

₋7.1

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YOU TRY IT! #3 ANSWER:f(x) = −0.8125x2 + 4.3125x +24.5

According to the quadratic function model, a car would get approximately 21.125 miles per gallon driven at 80 miles per hour.


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

A study compared the speed x (in miles per hour) and the average fuel economy f(x) (in miles per gallon) for cars. The results in 10 mile per hour increments over 20 mph are shown in the table.

10-MILE PER HOUR

INTERVAL, xMILES PER

HOUR

GASOLINE USAGE IN MILES PER GALLON, f(x)

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.

See margin.

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

Quadratic Linear Quadratic

Exponential Quadratic Exponential

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

f(x) = 3x2 f(x) = -x2 + 3x f(x) = 2x2 - 14x

f(x) = 1—2x2 + 8x f(x) = -3x2+ 4 f(x) = 4x2+ 10x - 8

1. x y = f(x)1 7

2 16

3 27

4 40

5 55

7. x y = f(x)1 3

2 12

3 27

4 48

5 75

10. x y = f(x)1 8.5

2 18

3 28.5

4 40

5 52.5

4. x y = f(x)1 2

2 4

3 8

4 16

5 32

3. x y = f(x)1 -13

2 -28

3 -45

4 -64

5 -85

9. x y = f(x)1 -12

2 -20

3 -24

4 -24

5 -20

12. x y = f(x)1 6

2 28

3 58

4 96

5 142

6. x y = f(x)1 0.2

2 0.04

3 0.008

4 0.0016

5 0.00032

2. x y = f(x)1 -4

2 -1

3 2

4 5

5 8

8. x y = f(x)1 2

2 2

3 0

4 -4

5 -10

11. x y = f(x)1 1

2 -8

3 -23

4 -44

5 -71

5. x y = f(x)1 -4

2 -6

3 -6

4 -4

5 0

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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 de-termine the speed of a vehicle before the brakes were applied. The quadratic function that best models the data is f(x) =

x2—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 VEHICLE IN MILES

PER HOURS, x

DISTANCE OF THE SKID IN

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.

SCIENCEA 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 quadratic function that models the data.f(x) = -5x2 + 35x

16. Use the data in the table to find the height of the ball after 7 seconds.0 meters

17. Use the data in the table to determine after how many seconds the ball will be 30 meters high.1 second and 6 seconds

TIME IN SECONDS, x

DISTANCE FROM THE GROUND IN

METERS, f(x)

0 0

1 30

2 50

3 60

4 60

5 50

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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.

WIDTH (FT) LENGTH (FT) AREA (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 dimen-sions measures 20 feet.160 square feet

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ELPS3D Speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency.

VOCABULARYfinite differences, cubic function

MATERIALS• 60 building blocks per

student group

ENGAGE ANSWER:Possible answer: 2π, 16π, 54π, 128π

The number that is multiplied by π is twice the value of the radius cubed.

1. Term 1 Term 2

Length

Length

1 . 8 • W r i t i n g C u b i C F u n C t i o n s 103

1.8 Writing Cubic Functions

FOCUSING QUESTION What are the characteristics of a cubic function?

LEARNING OUTCOMES■■ I can determine patterns that identify a cubic function from its related finite

differences.■■ I can determine the cubic function from a table using finite differences, includ-

ing any restrictions on the domain and range.■■ I can use finite differences to determine a cubic function that models a mathe-

matical context.■■ I can analyze patterns to connect the table to a function rule and communicate

the cubic pattern as a function rule.

ENGAGE A tomato sauce can is in the shape of a cylinder and the diam-eter of the base is equal to the height of the can. Generate a sequence showing the volumes of a series of cans with a radi-us of 1 inch, 2 inches, 3 inches, and 4 inches. What patterns do you notice in the sequence?

See margin.

EXPLORE

The volume of a prism is found using the area of the base and the height of the prism, V = Bh. If the prism is a rectangular prism, then the base is a rectangle and its 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 prims, V = lwh.

Use cubes to build the first two terms in a sequence of rectangular prisms. For this sequence, the term number is the length of the prism, the width of the prism is dou-ble the term number, and the height of the prism is triple the term number.

1. Sketch the first two terms that you built with the cubes.See margin.

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3.

In the table, ∆x = 1. The data set is not linear because the first differences in the volume are not constant.

The data set is not exponential because the successive ratios are not constant

TERM NUMBER

PROCESS VOLUME

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 6(12)(18) 1296

TERM NUMBER

PROCESS VOLUME

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 6(12)(18) 1296

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

∆y = 42

∆y = 114

∆y = 222

∆y = 366

∆y = 546

yn— yn -1

= 48—6

= 8

yn— yn -1

= 162—48

= 3.375

yn— yn -1

= 384—162

≈ 2.37

yn— yn -1

= 750—384

≈ 1.95

yn— yn -1

= 1296750

= 1.728

REFLECT ANSWERS:The level of finite differences that are constant is the same as the degree of the polynomi-al (linear: degree one and the first differences are constant; quadratic: degree two and the second differences are constant; cubic: degree three and the third differences are constant)

ELL STRATEGYEncourage all students, especially English language learners, to speak using appropriate mathematics vocabulary (ELPS: c(3)(d)). Doing so helps them internalize new English words and make connec-tions between old and new content.

104 C H A P t E r 1 : A L g E b r A i C PAt t E r n s

2. Complete a table like the one shown using the relationships among the dimen-sions of the prism (length = x, width = 2x, and height = 3x).

TERM NUMBER PROCESS VOLUME

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 6(12)(18) 1296

3. Does the data set follow a linear or an exponential function? Explain your rea-soning. 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

■■ A cubic function is a function of the form f(x) = ax3 + bx2 + cx + d. What is the degree of this function (i.e., the power of the greatest exponent)?

A cubic function has a degree of 3.

■■ A linear function contains a polynomial with degree one (mx + b) and a quadratic function contains a polynomial with degree two (ax2 + bx + c). What relationship is there between the degree of the polynomial and the level of finite differences that are constant?

See margin.

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4.

The second differences are not constant.

TERM NUMBER

PROCESS VOLUME

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 6(12)(18) 1296

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

∆y = 42

∆y = 114

∆y = 222

∆y = 366

∆y = 546

72

108

144

180

36

36

36

TERM NUMBER

PROCESS VOLUME

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 6(12)(18) 1296

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

∆y = 42

∆y = 114

∆y = 222

∆y = 366

∆y = 546

72

108

144

180

5.

The third differences are constant.


EXPLAIN

In a linear function, the first finite differences, or the difference between consecutive values of the depen-dent variable, are constant. For a quadratic function, the second finite differences are constant. In a cubic function, the third finite differences are constant.

Let’s look more closely at a cubic function. The table below shows the relationship between x and f(x). In a cubic function written in polynomial or standard, f(x) = ax3 + bx2 + cx + d.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x PROCESS y = f(x)

∆y = a + b + c

∆y = 7a + 3b + c

∆y = 19a + 5b + c

∆y = 37a + 7b + c

∆y = 61a + 9b + c

0 a(0)3 + b(0)2 + c(0) + d d

1 a(1)3 + b(1)2 + c(1) + d a + b + c + d

2 a(2)3 + b(2)2 + c(2) + d 8a + 4b + 2c + d

3 a(3)3 + b(3)2 + c(3) + d 27a + 9b + 3c + d

4 a(4)3 + b(4)2 + c(4) + d 64a + 16b + 4c + d

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 = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x PROCESS y = f(x)

∆y = a + b + c

∆y = 7a + 3b + c

∆y = 19a + 5b + c

∆y = 37a + 7b + c

∆y = 61a + 9b + c

0 a(0)3 + b(0)2 + c(0) + d d

1 a(1)3 + b(1)2 + c(1) + d a + b + c + d

2 a(2)3 + b(2)2 + c(2) + d 8a + 4b + 2c + d

3 a(3)3 + b(3)2 + c(3) + d 27a + 9b + 3c + d

4 a(4)3 + b(4)2 + c(4) + d 64a + 16b + 4c + d

5 a(5)3 + b(5)2 + c(5) + d 125a + 25b + 5c + d

6a + 2b

12a + 2b

18a + 2b

24a + 2b

There are many forms of a cubic function. Polynomial form, also called standard form, expresses a function as a polynomial with exponents in decreasing order.

f(x) = ax3 + bx2 + cx + d

In standard form, a, b, c, and d are rational numbers.

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INTEGRATE TECHNOLOGYUse technology such as a graphing calculator or spreadsheet app on a display screen to show students how, no matter the numbers present in the cubic function, the third differences will always be constant.


The second differences are not constant, either. But, you can see some patterns emerging. Notice that every time you take another round of fi-nite differences, the last constant term drops off because it is subtracted out. For the second differences, the coefficients of the a term are mul-tiples of 6. Also, each second difference has the same b term, 2b. Let’s calculate the third differences.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x PROCESS y = f(x)

∆y = a + b + c

∆y = 7a + 3b + c

∆y = 19a + 5b + c

∆y = 37a + 7b + c

∆y = 61a + 9b + c

6a + 2b

12a + 2b

18a + 2b

24a + 2b

6a

6a

6a

0 a(0)3 + b(0)2 + c(0) + d d

1 a(1)3 + b(1)2 + c(1) + d a + b + c + d

2 a(2)3 + b(2)2 + c(2) + d 8a + 4b + 2c + d

3 a(3)3 + b(3)2 + c(3) + d 27a + 9b + 3c + d

4 a(4)3 + b(4)2 + c(4) + d 64a + 16b + 4c + d

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 to 6a.■■ 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 be-tween 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-resent a cubic function.

The formulas for finding the values of a, b, c, and d to write the cubic function only work 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.


or click here

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QUESTIONING STRATEGIESHave students read the sce-nario for Example 1 again after completing the steps to write the cubic function rule. Draw their attention to the process column.

• How did the scenario translate into the cubic function rule?

• How is the cubic func-tion rule reflected in the process column on the table?

• Could you have written this cubic function rule from the scenario and table without the steps below?

ADDITIONAL EXAMPLE Veronica makes cylindrical candles to sell on a popular crafting website. To order the proper amount of wax, she calculates the volume of each candle. Use the ta-ble below to determine the cubic function rule for the volume of her candles with radius = x and height = 3x.

y = 3(x3) π

TERM NUMBER, x PROCESS VOLUME (EXACT), y VOLUME (ESTIMATED), y

0 Π (02)(3·0) 0 0

1 Π (12)(3·1) 3 Π 9.42

2 Π (22)(3·2) 24 Π 75.4

3 Π (32)(3·3) 81 Π 254.47

4 Π (42)(3·4) 192 Π 603.19

5 Π (52)(3·5) 375 Π 1178.1


EXAMPLE 1Use 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.

TERM NUMBER, x PROCESS VOLUME, y

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 6(12)(18) 1296

STEP 1 Analyze the first, second, and third finite differences to calculate the values of a, b, c, and d for the function rule f(x) = ax3 + bx2 + cx + d. Use the finite differences to calculate the value of y for x = 0.

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

TERM NUMBER, x PROCESS VOLUME, y

∆y = 6

∆y = 42

∆y = 114

∆y = 222

∆y = 366

∆y = 546

36

72

108

144

180

36

36

36

36

0 0(0)(0) 0

1 1(2)(3) 6

2 2(4)(6) 48

3 3(6)(9) 162

4 4(8)(12) 384

5 5(10)(15) 750

6 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 equal to 6a. Since 6a = 36, a = 6.

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INSTRUCTIONAL HINT When completing YOU TRY IT #1 some students may struggle with writing this function rule. If so, draw their attention back to #6 in the Explore exercise, and encourage them to fo-cus on the “width, length, and height in inches” col-umn of the table.


STEP 4 The second difference between the first two pairs of y-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 the y-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.

YOU TRY IT! #1

Popcorn is sold in boxes in the shape of square prisms. The dimensions of the boxes and their volumes are shown in the table. Write an equation to describe the volume, y, related to the width of the box, x, and verify it with a function rule using the finite differences in the table.

BOX NUMBER

BOX WIDTH IN INCHES, x

WIDTH, LENGTH,

AND HEIGHT IN INCHES

VOLUME IN CUBIC INCHES, y

1 (SAMPLER) 1 1(1)(2) 2

2 (KID’S) 2 2(2)(4) 16

3 (SMALL) 3 3(3)(6) 54

4 (MEDIUM) 4 4(4)(8) 128

5 (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 finite differences in the table, is f(x) = 2x3 + 0x2 + 0x + 0, or simply y = 2x3.

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ADDITIONAL EXAMPLESDetermine whether the data sets shown are linear, quadratic, exponential, or cubic.

1.

Cubic

2.

Exponential

x 0 1 2 3 4

y -8 -2 18 70 172

x -2 -1 0 1 2 3

y 0.125 0.5 2 8 32 128


EXAMPLE 2Determine whether the data set shown is linear, quadratic, exponential, or cubic.

x y

-1 34

0 50

1 56

2 58

3 62

4 74

5 100

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

∆x = 0 – -1 = 1

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

x y

∆y = 50 − 34 = 16

∆y = 56 − 50 = 6

∆y = 58 − 56 = 2

∆y = 62 − 58 = 4

∆y = 74 − 62 = 12

∆y = 100 − 74 = 26

-10

-4

+2

+8

+14

+6

+6

+6

+6

-1 34

0 50

1 56

2 58

3 62

4 74

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ADDITIONAL EXAMPLESDetermine whether the data sets shown are linear, quadratic, exponential, or cubic.

1.

Quadratic

2.

Linear

INSTRUCTIONAL HINTWhen learning quadratic models, the Instructional Strategies suggested that students create a flowchart for testing data to deter-mine the type of function represented by a set of data. Have students return to the flowchart they creat-ed to add testing for cubic functions.

x 2 3 4 5 6

y 15 23.5 36 52.5 73

x -3 0 3 6 9

y 8 6 4 2 0


STEP 2 Determine the ratios between successive y-values.

x y

yn = 50 ≈ 1.47yn-1 34 yn = 56 ≈ 1.12yn-1 50 yn = 58 ≈ 1.04yn-1 56 yn = 62 ≈ 1.07yn-1 58 yn = 74 ≈ 1.19yn-1 62 yn = 100 ≈ 1.35yn-1 74

-1 34

0 50

1 56

2 58

3 62

4 74

5 100

STEP 3 Determine whether the set of data represents a linear, quadratic, exponential, or cubic function.


The first differences are not constant, so the set of data does not represent a linear function.

The second differences are not constant, so the set of data does not represent a qua-dratic function.

The ratios between successive y-values are not constant, so the set of data does not represent an exponential function.

The third differences are constant, so the set of data represents a cubic function.DRAFT

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YOU TRY IT! #2 ANSWER:No, the set of data does not rep-resent a cubic function, because the differences in x are constant but the third finite differences in y are not constant.

ADDITIONAL EXAMPLESWrite the function rules for the tables in the Addi-tional Examples on pages 109 and 110.

pg. 109, AE #1:y = 3x3 - 2x2 + 5x - 8

pg. 109, AE #2: y = 2(4)x

pg. 110, AE #1: y = 2x2 - 1.5x + 10

pg. 110, AE #2: y = -2–

3x + 6


YOU TRY IT! #2

Does the set of data shown below represent a cubic function? Justify your answer.

x y

1 2

2 5

3 11

4 23

5 47

6 99

7 191

See margin.

EXAMPLE 3Write a cubic function for the values in the table.

x y

0 0

1 1

2 2

3 4

4 8

5 15

6 26

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INSTRUCTIONAL HINTHave students add the steps for writing a cubic function to their flowchart (page 110). Encourage stu-dents to use their flowchart to study for the test and guide their practice.

ADDITIONAL EXAMPLESWrite a cubic function for the values in the tables.

1.

y = 1–3x3 + 4x2 – 5x + 12

2.

y = -4x3 – 4x2 + 3–2x


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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

x y

∆y = 1 − 0 = 1

∆y = 2 − 1 = 1

∆y = 4 − 2 = 2

∆y = 8 − 4 = 4

∆y = 15 − 8 = 7

∆y = 26 − 15 = 11

0

1

2

3

4

1

1

1

1

0 0

1 1

2 2

3 4

4 8

5 15

6 26

STEP 2 Calculate the values for a, b, c, and d in f(x) = ax3 + bx2 + cx + d.

■■ The value of d is the y-coordinate of the y-intercept, (0, d). For this data set, d = 0.

■■ The third difference is equal to 6a. Since 6a = 1, a = 1–6.

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

6a + 2b = 0 1 + 2b = 0 2b = -1 b = -

1–2

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

a + b + c = 1

1–6 – 1–2 + c = 1

c = 8–6 =

4–3

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

f(x) = 1–6x3 −

1–2x2 + 4–3x.

x 0 1 2 3 4 5

y 12 11.33 20.67 42 77.33 128.67

x 0 1 2 3 4 5

y 0 -6.5 -45 -139.5 -314 -592.5

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ADDITIONAL EXAMPLESFor the tables below, deter-mine if the relationship is a cubic function. If so, write a function relating the variables.

1.

Cubic, y = -2–5x3 + 5x – 10

2.

Cubic, y = 12x3 - 10.5x2 + 2x - 1.25


YOU TRY IT! #3

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

x y

0 −6

1 71—2

2 18

3 281—2

4 42

5 611—2

6 90

y = 1—2x3 − 3x2 + 16x – 6

PRACTICE/HOMEWORKFor each table below, determine whether the set of data represents a linear, exponential, quadratic, or cubic function.

Exponential Linear Cubic

Quadratic Cubic Linear

1. x f(x)-1 0.2

0 1

1 5

2 25

3 125

4 625

4. x f(x)-1 -2

0 -8

1 -2

2 16

3 46

4 88

3. x f(x)-1 -5

0 0

1 5

2 40

3 135

4 320

6. x y1 40

2 38

3 36

4 34

5 32

6 30

2. x y0 -1.25

1 -1

2 -0.75

3 -0.5

4 -0.25

5 0

5. x y-1 8

0 5

1 10

2 29

3 68

4 133

x y1 14.6

0 10

1 5.4

2 3.2

3 5.8

4 15.6

x y-2 -143.25

-1 -25.75

0 -1.25

1 2.25

2 56.75

3 234.25

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7. Does the set of data shown below represent a cubic function? Justify your response.

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.

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.

y = 0.25x3 f(x) = 0.2x3 – 5 y = 6x3 + 2x2 + 1

y = x3 + x2 – 6x f(x) = 0.8x3 + 0.5x2 – 1 y = 6x3 – 21x2 – 12x

x y

0 0

1 -4

2 -28

3 -76

4 -148

5 -244

8. x y-1 8

0 5

1 6

2 11

3 20

4 33

11. x y0 0

1 0.25

2 2

3 6.75

4 16

5 31.25

14. x y0 0

1 -4

2 0

3 18

4 56

5 120

10. x y0 -7

1 -5

2 9

3 47

4 121

5 243

13. x y0 1

1 9

2 57

3 181

4 417

5 801

16. x y0 0

1 -27

2 -60

3 -63

4 0

5 165

9. x f(x)0 0

1 0.5

2 4

3 13.5

4 32

5 62.5

12. x f(x)0 -5

1 -4.8

2 -3.4

3 0.4

4 7.8

5 20

15. x f(x)0 -1

1 0.3

2 7.4

3 25.1

4 58.2

5 111.5

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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 BASE, x

(INCHES)

VOLUME, v(x)(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.


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 OFPACKAGE, w

(POUNDS)

PRICE TO MAILPACKAGE, p

($)

0 0

1 3.45

2 6.60

3 10.65

4 16.80

5 26.25

19. Write a cubic function to represent the given data.p = 0.2w3 – 0.75w2 + 4w

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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.


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 OFLAYERS, x

TOTAL NUMBEROF BLOCKS, p

0 0

1 1

2 5

3 14

4 30

5 55

22. Write the function relating the variables in problem 21.y =

1—3x3 +

1—2x2 +

1—6x

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

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MATHEMATICAL PROCESS SPOTLIGHTAR.1A Apply mathemat-ics to problems arising in everyday life, society, and the workplace.

ELPS3F Ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social con-texts, to using abstract and content-based vocabulary during extended speaking assignments.

VOCABULARYcubic function, finite dif-ferences, maximum, mini-mum

MATERIALS• graphing calculator

SUPPORTING ENGLISH LANGUAGE LEARNERSFor the Engage, pair students up so they may alternate giving information about what questions they could ask for the diving board situation. Encourage them to use key words and expressions related to functions and diving boards/swimming pools. En-courage students to think about both academic and social contexts (ELPS 3F).

1 . 9 • M o d e l i n g w i t h C u b i C F u n C t i o n s 117

1.9 Modeling with Cubic Functions

FOCUSING QUESTION How can you use finite differences to construct a cubic model for a data set?

LEARNING OUTCOMES■■ I can use finite differences or common ratios to classify a function as linear,

quadratic, cubic, or exponential when I am given a table of values.■■ I can determine the cubic function from a table using finite differences, includ-

ing any restrictions on the domain and range.■■ I can use finite differences to write a cubic function that describes a data set.■■ I can apply mathematics to problems that I see in everyday life, in society, and

in the workplace.

ENGAGEA diving board is a platform that someone can use to dive into a deep water pool. Div-ing boards typically have enough flexibili-ty to allow a diver to bounce before leaving the board and entering the pool. Diving boards have a heel end on the pool deck, 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.

When a person stands on a diving board, it bends or deflects slightly due to the per-son’s weight. What questions could you ask about the deflection of a diving board that could be answered by collecting data?

Work with a partner to give information you know about diving boards and this situation.

EXPLORE

When a person stands on a diving board, the diving board bends beneath the weight of the person. The farther the per-son stands from the fulcrum of the diving board, the more the diving board bends, or deflects, from the horizontal.

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3.

1. No, because as x increases by 1 foot, the amount of deflection increases by a greater amount each time. The increase from a dis-tance of 0 feet to 1 foot is 0.116 inches of deflection, but the increase from a distance of 1 foot to 2 feet is 0.332 inches of deflec-tion.

2. No. The deflection between a distance of 1 foot and 2 feet increases by a factor of about 4, but the deflection between a distance of 2 feet and 3 feet increases by a factor of about 2.

3. See margin below.

4. See margin bottom of pages 118-119.

5. See margin and bottom of page 119.

6. Data set - domain: whole numbers, 0 ≤ x ≤ 5; range: {0, 116, 448, 972, 1664, 2500}

Function - domain: all real numbers; range: all real numbers


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

7-9. See margin page 119.

DISTANCE FROM FULCRUM (FT), x

DEFLECTION(0.001 IN.),

f(x)

0 0

1 116

2 448

3 972

4 1664

5 2500

+1

+1

+1

+1

+1

+116

+332

+524

+692

+836

+216

+192

+168

+144

-24

-24

-24

4. For a cubic function of the form f(x) = ax3 + bx2 + cx + d,

• The value of d is the y-coordinate of the y-intercept, (0, d).• The third difference is equal to 6a.• The second difference between the first two pairs of

y-values, x = 0 and x = 1, and x = 1 and x = 2, is equal to 6a + 2b.

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

118 C h A P t e R 1 : A l g e b R A i C PAt t e R n s

The table below shows the amount of deflection, in thousandths of an inch, when the same person stands x feet from the fulcrum of the diving board.

DISTANCE FROM

FULCRUM (FT), x


f(x)

0 0

1 116

2 448

3 972

4 1664

5 2500

1. Does the data set appear to have a constant rate of change in deflection? Ex-plain how you know.See margin.

2. Does the amount of deflection appear to increase by the same factor each time

the distance increases by 1 foot? Explain how you know.See margin.

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 know?The data appear to be cubic because the third differences are constant. See margin for details.

4. Use the patterns in the finite differences to write a function rule that describes the data set.f(x) = −4x3 + 120x2 See margin for details.

5. Use a graphing calculator to graph the function rule over a scatterplot of the data. What do you notice about the function rule and the scatterplot?The graph of the function rule connects each of the data points. See margin for graph.

6. Compare the domain and range of the data set and the domain and range of the function rule. How are they alike? How are they different?See margin.

7. What will be the deflection if the person stands 10 feet from the fulcrum?See margin.

8. Are the x-intercepts of the function included in your data set? Why or why not?See margin.

9. What dimension of the diving board represents the greatest possible x-value that could be contained in the data set? What limit does that place on the do-main of the data set?See margin.

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4.

f(x) = −4x3 + 120x2

5.

5. The graph of the function rule connects each of the data points.

See graph below.

7. f(x) = −4x3 + 120x2

f(10) = −4(10)3 + 120(10)2

f(10) = −4,000 + 12,000 f(10) = 8,000 At a distance of 10 feet

from the fulcrum, the diving board will deflect 8.000 inches.

8. One x-intercept, (0, 0), is included in the data set be-cause it makes senses that when the person stands 0 feet from the fulcrum that the diving board deflects 0.000 inches. The other x-intercept, (30, 0), is not included in the data set because it does not make sense that if a diving board were over 30 feet long that a person standing 30 feet from the fulcrum would not deflect the diving board.

9. The distance from the fulcrum to the end of the diving board is the greatest possible x-value that could be contained in the data set. This x-value represents the maximum for the domain of the data set.

REFLECT ANSWERS:See margin page 120.

DISTANCE FROM FULCRUM (FT), x


f(x)

0 0

1 116

2 448

3 972

4 1664

5 2500

+1

+1

+1

+1

+1

+116

+332

+524

+692

+836

+216

+192

+168

+144

-24

-24

-24

a + b + c = 116(-4) + (120) + c = 116

116 + c = 116c = 0

6a + 2b = 2166(-4) + 2b = 216-24 + 2b = 216

2b = 240b = 120

d = 0

6a = -24a = -4


REFLECT

■■ How can you determine a cubic function model for a data set?See margin.

■■ What other factors could influence the deflection of the diving board? Explain how they would do so.

See margin.


or click here

EXPLAIN

Cubic function models can be used to represent sets of mathematical and real-world data. Cubic functions have several attributes that should be considered when they are used for mathematical models.

The domain and range of most cubic functions are all real numbers. Un-like quadratic functions, when you raise a negative number to the third power, you can have a negative number as a result.

Cubic functions have as many as 3 x-intercepts and 1 y-intercept. As with other function types, the x-intercepts represent x-values that generate a function value equal to 0. The y-intercept is also a starting point, or y-value when x = 0.

Quadratic functions have a maxi-mum or minimum point at the ver-tex. These are absolute maximum or minimum values. A cubic function, however, may have what are called local maximum or local minimum values. These points are not the ab-solute maximum or minimum val-ues for the function. Instead, they are maximum or minimum values for a nearby, or local, part of the function.

Let’s look more closely at the data set and function model for the div-ing board problem.

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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:

The weight of the person standing on the diving board influences the deflection. If the person weighs less than the per-son for whom the data was col-lected, then the deflection will be less. If the person weights more, then the deflection will be greater.

The flexibility or rigidity of the diving board influences the deflection. If the diving board is more rigid and doesn’t bend as easily, then the deflection will be less.

TECHNOLOGY TIPSome 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-tion, f(x) = x3, as well as several other cubic functions that generate different shapes, including local maxima and minima.

For the function in the diving board situation, use graph-ing technology to zoom out and show students the shape of the complete graph.


In the diving board problem, the data points represent a small portion of the function. The shaded region on the graph shows the part of the function that would model the deflection of a diving board that has a dis-tance of 15 feet between the end of the diving board and the fulcrum.

The function has two x-intercepts, one of which makes sense in the context of the diving board problem. When the person stands 0 feet from the fulcrum, you would expect the diving board to deflect 0 inches. However, it is not likely that a person standing 30 feet from the fulcrum would generate a deflection of 0 inches.

MODELING WITH CUBIC FUNCTIONS

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

The domain and range for the situation may be a sub-set 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 their domain when they are decreasing.

With polynomial function mod-els, frequently the function mod-el only represents the data set for a certain interval of the function. In the diving board problem, there is a local maximum at (20, 16,000). This means that a person standing 20 feet from the fulcrum causes a deflection of 16.000 inches. The function model for x-values greater than 20 then begins to decrease. It is not likely that a person standing farther than 20 feet from the ful-crum would generate less deflec-tion, so the model likely does not represent the situation beyond a domain of 20 feet.DRAFT

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ADDITIONAL EXAMPLESDetermine 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.

Quadratic

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

Cubic

TIME (SECONDS), t 0 1 2 3 4

HEIGHT (METERS), h 6.5 79.6 142.9 196.4 240.1

RADIUS (IN), r 0 1 2 3 4 5

VOLUME (IN3), V 0 4.19 33.51 113.1 268.08 523.6


EXAMPLE 1A 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.

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 of x and the first, second, and third finite differences in values of f(x).

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.

LEVELS, x PROCESS NUMBER OF

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

∆x = 1 – 0 = 1

∆x = 2 – 1 = 1

∆x = 3 – 2 = 1

∆x = 4 – 3 = 1

∆x = 5 – 4 = 1

∆x = 6 – 5 = 1

LEVELS, x PROCESS NUMBER OF

CANS, y

+1

+4

+9

+16

+25

+36

+3

+5

+7

+9

+11

0 0 0

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

+2

+2

+2

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ADDITIONAL EXAMPLEWrite the cubic function for Additional Example #2 on page 121.

V = 4–3πr3 or V = 4.19r3


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 LENGTH OF BOARD IN

SQUARES, x

TOTAL NUMBER OF SQUARES,

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.

EXAMPLE 2Determine the cubic function rule to model the display of cans in Example 1. Compare the domain and range of the data set and the function.

Step 1 Calculate the values of a, b, c, and d in the cubic function rule f(x) = ax3 + bx2 + cx + d using the data in the table and finite differences.

■■ There would be no cans in the “zero” level. The value for y when x = 0

is 0. So d = 0.

■■ The third finite difference in the set of data is 2. This equals 6a. If 6a = 2, then a =

1–3.

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ADDITIONAL EXAMPLESDetermine the cubic function rule to model the given data.

1.

f(x) = 3x3 – 2x2 + 1–2x + 13.25

2.

f(x) = 3–2x3 + 5x2 - 14x - 12

x 0 1 2 3 4 5

y 13.75 14.75 29.75 76.75 173.75 338.75

x 0 1 2 3 4 5

y -12 -19.5 -8 31.5 108 230.5


■■ 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 2b = 1 and b =

1–2

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

a + b + c = 1 1–3 +

1–2 + c = 1 c =

1–6

STEP 3 Write the cubic function rule with the values of a, b, c, and d: f(x) =

1–2x3 + 1–3x2 +

1–6x

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


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YOU TRY IT! #2 ANSWER:f(x) = 4x3 – 48x2 + 144x. A 2-inch cutout yields a tray with the greatest volume. In this situation, the domain includes whole numbers between 0 and 6, {x|x∈ W, 0 < x < 6} (There cannot be a cutout as great as half the width of the cardboard), and the range includes whole numbers, 0 < y <128.

INSTRUCTIONAL HINTSWith every set of data, have students enter the function rule in their graphing cal-culators in order to see the graph. Once graphed, have students zoom out until they see the “bigger pic-ture.” Students might look at a small portion of a cubic function and mistake it for a quadratic function.

Help students process the differences and similari-ties in quadratic and cubic functions by writing about them.

Have students draw a Venn Diagram to compare and contrast the two types of functions.


YOU TRY IT! #2

A tray is made of a 12 in. x 12 in. square piece of cardboard with squares cut out of the four corners. The resulting sides are folded up and taped to form a square prism (open box). The volume of the tray related to the size of the squares cut from the corners.

SIZE OF SQUARE CUTS IN

INCHES, x

PROCESS (12 – 2x)(12 – 2x)(x)

VOLUME IN CUBIC

INCHES, f(x)

0 (12 – 0)(12 – 0)(0) 0

1 (12 – 2)(12 – 2)(1) 100

2 (12 – 4)(12 – 4)(2) 128

3 (12 – 6)(12 – 6)(3) 108

4 (12 – 8)(12 – 8)(4) 64

5 (12 – 10)(12 – 10)(5) 20

6 (12 – 12)(12 – 12)(6) 0

Generate a cubic function model for the volume of the tray. What size cutout produc-es the tray with the greatest volume? Identify the domain and range that most appro-priately models the data.

See margin.

EXAMPLE 3A set of graduated cubes comes with a large industrial scale. The cubes are numbered according to increasing size. The weights of some of the cubes are given in the table. Determine if the data set represents a linear, quadratic, or cubic function. Identify the domain and range of the model that most appropriately models the data.


CUBE NUMBER, x

WEIGHT IN OUNCES, f(x)

2 12.5

4 34.3

6 72.9

8 133.1

10 219.7

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 range is a subset of real numbers, 0 < y ≤ 337.5.

∆x = 4 – 2 = 2

∆x = 6 – 4 = 2

∆x = 8 – 6 = 2

∆x = 10 – 8 = 2

∆x = 12 – 10 = 2

CUBE NUMBER, x


+21.8

+38.6

+60.2

+86.6

+117.8

+16.8

+21.6

+26.4

+31.2

2 12.5

4 34.3

6 72.9

8 133.1

10 219.7

12 337.5

+4.8

+4.8

+4.8

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ADDITIONAL EXAMPLESDetermine the cubic function rule to model the data sets below.

1.

f(x) = -12x3 + 10.5x2 + 4.5x

2.

f(x) = 1–3x3 + 42x

x 0 1 2 3 4 5

y 0 -0.7 -52.4 -227.1 -596.8 -1233.5

x 0 1 2 3 4 5

y 0 42.33 86.67 135 189.33 251.67


CUBE NUMBER, x


2 12.5

4 34.3

6 72.9

8 133.1

10 219.7

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 range is a subset of real numbers, 0 < y ≤ 337.5.

∆x = 4 – 2 = 2

∆x = 6 – 4 = 2

∆x = 8 – 6 = 2

∆x = 10 – 8 = 2

∆x = 12 – 10 = 2

CUBE NUMBER, x


+21.8

+38.6

+60.2

+86.6

+117.8

+16.8

+21.6

+26.4

+31.2

2 12.5

4 34.3

6 72.9

8 133.1

10 219.7

12 337.5

+4.8

+4.8

+4.8

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YOU TRY IT! #3 ANSWER:The differences in the x-values are constant. The third finite differences, which average 0.33, are approximately constant, so the data set represents a cubic function model. The domain represents the annual average wind speeds in the USA, so the domain of the data set is the subset of whole numbers, 4 ≤ x ≤ 10. The annual energy output reflects the data col-lected for those average wind speeds, so the range of the set is the subset of real numbers, 3.40 ≤ y ≤ 53.12.


YOU TRY IT! #3The power, y, (in kilowatts) generated by a wind turbine is related to the wind speed. Determine if the data set represents a linear, quadratic, or cubic function. Identify the domain and range of the model that most appropriately models the data.

AVERAGE ANNUAL WIND SPEED IN THE USA IN

METERS PER SECOND, xANNUAL ENERGY OUTPUT

IN KWH/YEAR, f(x)

4 3.40

5 6.64

6 11.47

7 18.22

8 27.25

9 38.75

10 53.12

Data Source: National Renewal Energy Laboratory and Energy.govSee margin.

PRACTICE/HOMEWORKFor 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.

Exponential Quadratic Linear

Cubic Exponential Cubic

1. x y0 3

1 6

2 12

3 24

4 48

5 96

4. x y0 5

1 10

2 35

3 92

4 193

5 350

3. x y0 2.25

1 8.75

2 15.25

3 21.75

4 28.25

5 34.75

6. x y0 -8

1 3

2 44

3 145

4 336

5 647

2. x y0 -6

1 1

2 16

3 39

4 70

5 109

5. x y0 20

1 50

2 125

3 312.5

4 781.25

5 1953.125

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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.

f(x) = 3x3 – 4x2 + 2x – 1 f(x) = -2x3 + x2 + 5x + 3 f(x) = 4x3 + 10x2

f(x) = -6x3 + x2 – 4 f(x) = 1—2x3 +

1—2x2 + 4x + 1 f(x) = 6x3 – 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

xVOLUME

y

0 0

1 396

2 640

3 756

4 768

5 700

6 576

7 420

8 256

9 108

7. x y0 -1

1 0

2 11

3 50

4 135

5 284

10. x y0 -4

1 -9

2 -48

3 -157

4 -372

5 -729

9. x y0 0

1 14

2 72

3 198

4 416

5 750

12. x y0 -9

1 -3

2 39

3 153

4 375

5 741

8. x y0 3

1 7

2 1

3 -27

4 -89

5 -197

11. x y0 1

1 6

2 15

3 31

4 57

5 96

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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 func-tion 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 width of the rectangle.

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.

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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.

LEVEL,x

TOTAL NUMBER OF SOCCER BALLS,

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.f(x) =

1—6x3 +

1—2x2 +

1—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 ballsDRAFT

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4a. Exponential, because the successive ratios of the y-values are approxi-mately equal when Δx is constant.


Chapter 1 Review

1.

a) an = 256(3–) c) a an = a + 7

geometric arithmeticb) a an = 3a d) an n

geometric arithmetic2. y = mx + b

a) 5–2, y

y = 5—2x + 10

b)

c)

y = -1—3

4—3

3.

ratio = 1–3 ratio = 2

y = 243( 1—3) x

y = 250(2)x

4. x

y

NUMBER OF DAYS, x 0 1 2 3 4 5 6 7

POPULATION, y 123 290 715 1,695 4,048 9,769 23,500 56,400

a. See margin.b. f(x) = 123(2.4)x

c. 324,942

d. By the 13th day, the population is expected to be

over 10,000,000.

d) x y2 21

4 18

6 15

8 12

10 9

a) x y0 243

1 81

2 27

3 9

b) x y1 500

2 1000

3 2000

4 4000

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C H A P T E R 1 R E v I E w 131

For problems 5 – 10, determine if the function represented is linear, quadratic, cubic, or exponential, then write a function equation relating the variables.

Quadratic Linear Exponentialy = 2x2 x

Cubic

Cubic

Quadraticy = 2x3

y = 1—3x3 + 6x2

2

Use the following situation to answer questions 11-14.

FIGURE NUMBER, n 1 2 3 4 5 n

NUMBER OF DOTS, D(n) 1 6 15 28

5. x y1 11

2 22

3 37

4 56

5 79

8. x y0 -8

1 -6

2 8

3 46

4 120

x y0 8

1 12

2 18

3 27

4 40.5

10. x y-2 19

-1 13

0 5

1 -5

2 -17

6. x y0 12

2 9

4 6

6 3

8 0

9. x y

0 6

1 91—3

2 262—3

3 60

4 1111—3

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11. 45

FIGURE NUMBER, n 1 2 3 4 5 n

NUMBER OF DOTS, D(n) 1 6 15 28 45

12. n D(n)D(n) = 2n2

13. th

D(9) = 2(9)2 – 9 = 153 dots

14. 435 = 2n2 – n, Enter Y1 = 2x2

1


–6

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

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

+5 +9 +13

+4 +4 +4

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15. 3,888

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

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

16. V = 16x3 + 12x2

V = 16(2.5)3 + 12(2.5)2 = 325 cubic inches


RADIUS OF CARTON, r (IN)

WEIGHT OF CARTON, W (OUNCES)

1 6

2 50

3 170

4 402

5 785

6 1357

18.

constant.

+216 +312 +408 +504

+148 +364 +1,084 +1,588

+96 +96 +96

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19. r W

RADIUS OF CARTON, r (IN)

WEIGHT OF CARTON, W (OUNCES)

0 0

1 6

2 50

3 170

4 402

5 785

6 1357

See margin.

20.

See margin.

MULTIPLE CHOICE1.

SA. S wB. S wC. S wD. S w

2. x f(x) f(x) = ax2 + bx + c

A. aB. 2aC. 6aD.

6

232

572

36

39

76

19. Assume that for a radius and height of 0, the weight = 0 so d = 0.

a + b + c = 6 and 6a + 2b = 38.

The average 3rd difference is 37.75 so 6a = 37.75. Therefore, a ≈ 6.3, b = 0.1, c = −0.4

The function is W = 6.3r3 + 0.1r2 − 0.4r

20. From a graphing calcu-lator, a value close to 138 in the y column yields an x-value of 2.8. Therefore, a carton weighing about 138 oz, would have a height of about 2.8 inches

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3.

x 0 1 2 3 4 5 6

y 15 19 35 63 103 155 219

A. 6x2 xB. 6x2 xC. x2 xD. 3x2 x

4. x3

x 0 1 2 3 4 5 6 7

y = ax3 + bx2 + cx + d -200 -186 -144 -50 120 390 784 1326

A. B. C. 6D.

5. nth 3n2 – n 2

th

A. 92B. C. D. 360DRAFT

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