Transcript 2CH12L2 - VincentPienaar

12-2 Find a Pattern in Sequences
Warm Up
Problem of the Day
Lesson Presentation
Course 2
12-2
Find a Pattern in Sequences
Warm Up
Find the output for each input value.
Course 2
Input
Rule
Output
x
–3x + 2
y
–4
14
0
2
4
–10
12-2
Find a Pattern in Sequences
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.
Course 2
12-2
Find a Pattern in Sequences
Learn to find patterns to complete
sequences using function tables.
Course 2
12-2
Insert
Lesson Title
Here
Find
a Pattern
in Sequences
Vocabulary
sequence
Course 2
12-2
Find a Pattern in Sequences
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.
Course 2
12-2
Find a Pattern in Sequences
You can use a variable to represent a number’s
place in a sequence.
n
(place in list)
1st number 2nd number 3rd number 4th number
Number
in sequence:
You can use a function table to help identify the
pattern in a sequence.
Course 2
12-2
Find a Pattern in Sequences
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
Rule
y
1
1·4
4
2
2·4
8
3
3·4
4·4
12
4
Course 2
16
Multiply n by 4.
12-2
Find a Pattern in Sequences
Additional Example 1A Continued
Use the function to find the next three terms
in the sequence.
A. 4, 8, 12, 16, . . .
y = 4n
Write the function.
y = 4(5) = 20
Substitute for n to find
the next three terms.
y = 4(6) = 24
y = 4(7) = 28
Course 2
12-2
Find a Pattern in Sequences
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
Rule
y
1
1–5
–4
2
2–5
–3
3
3–5
4–5
–2
4
Course 2
–1
Subtract 5 from n.
12-2
Find a Pattern in Sequences
Additional Example 1B Continued
Use the function to find the next three
numbers in the sequence.
B. –4, –3, –2, –1, . . .
y=n–5
Write the function.
y = (5) – 5 = 0
Substitute for n to find
the next three terms.
y = (6) – 5 = 1
y = (7) – 5 = 2
Course 2
12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
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
Rule
y
1
1·3
3
2
2·3
6
3
3·3
4·3
9
4
Course 2
12
Multiply n by 3.
12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
Try This: Example 1A Continued
Use the function to find the next three terms
in the sequence.
A. 3, 6, 9, 12, . . .
y = 3n
Write the function.
y = 3 (5) = 15
Substitute for n to find
the next three numbers
y = 3 (6) = 18
y = 3 (7) = 21
Course 2
12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
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
Rule
y
1
1–9
–8
2
2–9
–7
3
3–9
4–9
–6
4
Course 2
–5
Subtract 9 from n.
12-2
Find a Pattern in Sequences
Try This: Example 1B Continued
Use the function to find the next three
numbers in the sequence.
B. –8, –7, –6, –5, . . .
y=n–9
Write the function.
y = (5) – 9 = –4
Substitute for n to find
the next three terms.
y = (6) – 9 = –3
y = (7) – 9 = –2
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12-2
Find a Pattern in Sequences
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.
Course 2
Find a Pattern in Sequences
12-2
Additional Example 2 Continued
1
Understand the Problem
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.
Course 2
Find a Pattern in Sequences
12-2
Additional Example 2 Continued
2
Make a Plan
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.
Course 2
12-2
Find a Pattern in Sequences
Additional Example 2 Continued
3
Solve
n
Rule
y
1
1 · 3.50 + 8
11.50
2
2 · 3.50 + 8
15.00
3
3 · 3.50 + 8
18.50
4
4 · 3.50 + 8
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.
Course 2
12-2
Find a Pattern in Sequences
Additional Example 2 Continued
4
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.
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12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
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.
Course 2
12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
Try This: Example 2 Continued
1
Understand the Problem
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.
Course 2
12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
Try This: Example 2 Continued
2
Make a Plan
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.
Course 2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
12-2
3
Try This: Example 2 Continued
Solve
n
Rule
y
1
1 · 6.50 + 12.5
19.00
2
2 · 6.50 + 12.5
25.50
3
3 · 6.50 + 12.5
32.00
4
4 · 6.50 + 12.5
38.50
5
5 · 6.50 + 12.5
45.00
After 1, 2, 3, 4 and 5 days, Bob will have
spent $19.00, $25.50, $32.00, $38.50,
and $45.00, respectively.
Course 2
12-2
Insert
Here
Find a Lesson
Pattern Title
in Sequences
Try This: Example 2 Continued
4
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.
Course 2
12-2
Find
Insert
a Pattern
Lesson Title
in Sequences
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
y = 6n; 30, 36, 42
2. –3, –2, –1, 0
y = n – 4; 1, 2, 3
3. 24, 21, 18, 15
y = 27 – 3n; 12, 9, 6
4. 7, 14, 21, 28
y = 7n; 35, 42, 49
Course 2
12-2
Find
Insert
a Pattern
Lesson Title
in Sequences
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.
Course 2