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Find a Pattern in Sequences. 12-2. Course 2. Warm Up. Problem of the Day. Lesson Presentation. Find a Pattern in Sequences. 12-2. Course 2. Warm Up Find the output for each input value. Input. Rule. Output. x. –3 x + 2. y. 14. –4. 2. 0. 4. –10. Find a Pattern in Sequences.

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Presentation Transcript
slide1

Find a Pattern in Sequences

12-2

Course 2

Warm Up

Problem of the Day

Lesson Presentation

slide2

Find a Pattern in Sequences

12-2

Course 2

Warm Up

Find the output for each input value.

Input

Rule

Output

x

–3x + 2

y

14

–4

2

0

4

–10

slide3

Find a Pattern in Sequences

12-2

Course 2

Problem of the Day

Function Rule A: Square the input. Divide by 2. Subtract 3.

Function Rule B: Square the input. Subtract 6. Divide by 2.

If the input value for each rule is 222, what is the difference of the two output values? Why?

0; they are equivalent rules.

slide4

Find a Pattern in Sequences

12-2

Course 2

Learn to find patterns to complete sequences using function tables.

slide5

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Vocabulary

sequence

slide6

Find a Pattern in Sequences

12-2

Course 2

A sequence is an ordered list of numbers. One of the most well-known sequences is the Fibonacci sequence. In this sequence, each term after the second term is the sum of the two terms before it.

1, 1, 2, 3, 5, 8, 13,…

When the list follows a pattern, the numbers in the sequence are the output values of a function, and the value of each number depends on the number’s place in the list.

slide7

Find a Pattern in Sequences

12-2

Course 2

You can use a variable to represent a number’s

place in a sequence.

1 2 3 4

n

(place in list)

3rd number

4th number

1st number

2nd number

2 4 6 8

Number

in sequence:

You can use a function table to help identify the

pattern in a sequence.

slide8

Find a Pattern in Sequences

12-2

Course 2

Additional Example 1A: Identifying Functions in a Sequence

Find a function that describes each sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence.

A. 4, 8, 12, 16, . . .

n

y

Rule

Multiply n by 4.

1

1 ·4

4

2

2 ·4

8

3

3 ·4

12

4 ·4

4

16

slide9

Find a Pattern in Sequences

12-2

Course 2

Additional Example 1A Continued

Use the function to find the next three terms in the sequence.

A. 4, 8, 12, 16, . . .

Write the function.

y = 4n

y = 4(5) = 20

Substitute for n to find

the next three terms.

y = 4(6) = 24

y = 4(7) = 28

slide10

Find a Pattern in Sequences

12-2

Course 2

Additional Example 1B: Identifying Functions in Sequences

Find a function that describes the sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence.

B. –4, –3, –2, –1, . . .

n

y

Rule

Subtract 5 from n.

1

1 – 5

–4

2

2 – 5

–3

3

3 – 5

–2

4 – 5

4

–1

slide11

Find a Pattern in Sequences

12-2

Course 2

Additional Example 1B Continued

Use the function to find the next three numbers in the sequence.

B. –4, –3, –2, –1, . . .

Write the function.

y = n – 5

y =(5) – 5= 0

Substitute for n to find

the next three terms.

y =(6) – 5= 1

y =(7) – 5 = 2

slide12

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Try This: Example 1

Find a function that describes each sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence.

A. 3, 6, 9, 12, . . .

n

y

Rule

Multiply n by 3.

1

3

1 · 3

2

6

2 · 3

3

9

3 · 3

4

12

4 · 3

slide13

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Try This: Example 1A Continued

Use the function to find the next three terms in the sequence.

A. 3, 6, 9, 12, . . .

Write the function.

y = 3n

y = 3(5) = 15

Substitute for n to find

the next three numbers

y = 3(6) = 18

y = 3(7) = 21

slide14

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Try This: Example 1B

Find a function that describes the sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence.

B. –8, –7, –6, –5, . . .

n

y

Rule

Subtract 9 from n.

1

1 – 9

–8

2

2 – 9

–7

3

3 – 9

–6

4 – 9

4

–5

slide15

Find a Pattern in Sequences

12-2

Course 2

Try This: Example 1B Continued

Use the function to find the next three numbers in the sequence.

B. –8, –7, –6, –5, . . .

Write the function.

y = n – 9

y =(5) – 9= –4

Substitute for n to find

the next three terms.

y =(6) – 9= –3

y =(7) – 9 = –2

slide16

Find a Pattern in Sequences

12-2

Course 2

Additional Example 2: Problem Solving Application

Holli earned $3.50 a day pet sitting for her neighbor’s fish. Holli pet sat for 5 days. Her neighbor also paid her $8 for picking up the mail. Write a sequence showing how much money Holli earned after 1, 2, 3, 4, and 5 days.

slide17

Find a Pattern in Sequences

12-2

1

Understand the Problem

Course 2

Additional Example 2 Continued

List the important information:

• Holly earns $3.50 each day.

• She pet sits for 5 days.

• She gets paid $8 for picking up the mail.

The answer will be the amount of money

she has earned after 1, 2, 3, 4, and 5 days.

slide18

Find a Pattern in Sequences

12-2

Make a Plan

2

Course 2

Additional Example 2 Continued

You can find a pattern, then write a rule that

can be used to find a sequence.

After the first day, Holli will have earned $3.50 + $8.

After the second day, Holli will have earned 2 · $3.50 + $8.

The rule n ·$3.50 + $8 can be used to find

the terms in the sequence.

slide19

Find a Pattern in Sequences

12-2

3

Solve

Course 2

Additional Example 2 Continued

1 · 3.50 + 8

11.50

1

2

2 · 3.50 + 8

15.00

3

3 · 3.50 + 8

18.50

4 · 3.50 + 8

4

22.00

5

5 · 3.50 + 8

25.50

After 1, 2, 3, 4, and 5 days, Holli will have earned $11.50, $15.00, $18.50, $22.00, and $25.50, respectively.

slide20

Find a Pattern in Sequences

12-2

4

Course 2

Additional Example 2 Continued

Look Back

If Holli earns $4 per day, she will have

earned about $12, $16, $20, $24, and $28 after each of the 5 days respectively.

The answer is reasonable.

slide21

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Try This: Example 2

Bob spends $12.50 each week for a lunch ticket. In addition, he usually spends $6.50 a day for incidentals. Write a sequence showing how much money Bob had spent after 1, 2, 3, 4, and 5 days.

slide22

Find a Pattern in Sequences

12-2

1

Understand the Problem

Course 2

Insert Lesson Title Here

Try This: Example 2 Continued

List the important information:

• Bob spends $12.50 each week for meals.

• He spends $6.50 a day on incidentals.

The answer will be the amount of money

he has spent after 5 days.

slide23

Find a Pattern in Sequences

12-2

Make a Plan

2

Course 2

Insert Lesson Title Here

Try This: Example 2 Continued

You can find a pattern, then write a rule that

can be used to find a sequence.

After the first day, Bob will have spent $6.50 + $12.50.

After the second day, Bob will have spent

2 · $6.50 + $12.50.

The rule n · $6.50 + $12.50 can be used to find

the terms in the sequence.

slide24

Find a Pattern in Sequences

12-2

3

Solve

Course 2

Insert Lesson Title Here

Try This: Example 2 Continued

n

Rule

y

1 · 6.50 + 12.5

19.00

1

2

2 · 6.50 + 12.5

25.50

32.00

3

3 · 6.50 + 12.5

38.50

4 · 6.50 + 12.5

4

5

45.00

5 · 6.50 + 12.5

After 1, 2, 3, 4 and 5 days, Bob will have spent $19.00, $25.50, $32.00, $38.50, and $45.00, respectively.

slide25

Find a Pattern in Sequences

12-2

4

Course 2

Insert Lesson Title Here

Try This: Example 2 Continued

Look Back

If Bob spends 6.50 per day, he will have spent $19.00, $25.50, $32.00, $38.50, and $45.00 after each of the 5 days respectively.

The answer is reasonable.

slide26

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Lesson Quiz: Part 1

Find a function that describes each sequence. Use the function to find the next three terms in the sequence.

1. 6, 12, 18, 24

2. –3, –2, –1, 0

3. 24, 21, 18, 15

4. 7, 14, 21, 28

y = 6n; 30, 36, 42

y = n – 4; 1, 2, 3

y = 27 – 3n; 12, 9, 6

y = 7n; 35, 42, 49

slide27

Find a Pattern in Sequences

12-2

Course 2

Insert Lesson Title Here

Lesson Quiz: Part 2

5. Arisha bought a 5-pound bag of chocolate chips. She adds 0.5 lb of chips to each batch of cookies she makes. Find a function that describes this relationship. How many pounds of chips remain after she makes 5 batches of cookies?

y = 5 – 0.5x; 2.5 lb remain.