12-2

1. Find a Pattern in Sequences 12-2 Course 2 Warm Up Problem of the Day Lesson Presentation

2. 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

3. 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.

4. Find a Pattern in Sequences 12-2 Course 2 Learn to find patterns to complete sequences using function tables.

5. Find a Pattern in Sequences 12-2 Course 2 Insert Lesson Title Here Vocabulary sequence

6. 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.

7. 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.

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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

27. 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.