1. Functions and Modeling
FSA- Algebra 2
Functions and Modeling

2. Sequences Explicit Formula: It is an ordered list of number.
MAFS.912.F-BF.1.2 
It is an ordered list of number.
Example:
𝟏, 𝟒, 𝟕, 𝟏𝟎
Explicit Formula:
Expresses the nth term of a sequence in terms of n.
Example:
In the sequence 𝟐, 𝟒, 𝟔, 𝟖, 𝟏𝟎,…, the nth term is twice the value of n.

3. MAFS.912.F-BF.1.2 
The function ƒ(𝑥) = 0.16𝑥 represents the number of U.S. dollars equivalent to x Chinese yuan. The function 𝑔(𝑥) = 13.60𝑥 represents the number of Mexican pesos equivalent to 𝑥 U.S. dollars. You can use 𝑔(ƒ(𝑥)) to find the number of Mexican pesos equivalent to 𝑥 Chinese yuan. What is the value, in Mexican pesos, of an item that costs 15 Chinese yuan?
Example
Since you are using the function 𝒈(ƒ(𝒙)) to find the correct answer, you must substitute 𝟎.𝟏𝟔𝒙 for ƒ(𝒙) in the function 𝒈(𝒙). The substitution would look like this:
𝒈(𝟎.𝟏𝟔𝒙) = 𝟏𝟑.𝟔𝟎𝒙
= 𝟏𝟑.𝟔𝟎 (𝟎.𝟏𝟔𝒙)
Using the given value of 15, the number of Mexican pesos, the expression would look like:
𝒈(𝒇(𝒙)) = 𝟏𝟑.𝟔𝟎(𝟎.𝟏𝟔(𝟏𝟓))
=𝟑𝟐.𝟔𝟒

4. MAFS.912.F-BF.1.2 
A bank offers a savings account that accrues simple interest annually based on an initial deposit of $500.  If S(t) represents the money in the account at the end of t years and 𝑆(5) = 575, write a function that could be used to determine the amount of money in the account over time.  Show your work or explain your reasoning.
Example
For the information given, I know the initial deposit $500 and accrues simple interest annually. I also know that in 5 years, the account will have $575 in it.
$𝟓𝟕𝟓−𝟓𝟎𝟎=$𝟕𝟓, which means over 5 years, the total interest the account accrues is $75.
$𝟕𝟓 𝟓 =$𝟏𝟓.𝟎𝟎, which means every years the account accrues is $15 in interest.
Thus the function s(t) would equal the initial deposit plus $15.00 multiplied by the numbers of years the account is open
𝑺 𝒕 =𝟓𝟎𝟎+𝟏𝟓𝒕

5. 𝑰 = 𝒑𝒓𝒕 𝐼 = 𝑃 (𝟏 + 𝑟) 𝑡 Example 𝟔𝟓𝟎 = 𝑃 (𝟏 +𝟎.𝟎𝟑) 𝟏𝟎
MAFS.912.F-BF.1.2 
Example
Another bank offers a savings account that accrues compound interest annually at a rate of 3%.  The bank in the previous problem offered $15 per year in interest.
What is the initial amount needed in this account so that it will have the same amount of money at the end of 10 years as the account in the previous problems at the end of 10 years?  Show your work or explain your reasoning.
Remember the simple interest formula and the compound interest formula.
𝑰 = 𝒑𝒓𝒕
𝐼 = 𝑃 (𝟏 + 𝑟) 𝑡
𝒑 = principal
𝒓 = interest rate
𝑆(𝑡) = 𝑡 (previous problem)
𝒕 = time in years
Using the simple interest rate to determine the amount after 10 years:
Using 3% as the compound interest and t = 10.
𝟔𝟓𝟎 = 𝑃 (𝟏 +𝟎.𝟎𝟑) 𝟏𝟎
𝑆(𝟏𝟎) = 𝟓𝟎𝟎 + 𝟏𝟓(𝟏𝟎)
𝟔𝟓𝟎 = 𝑃 (𝟏.𝟎𝟑) 𝟏𝟎
𝑆(𝟏𝟎) = 𝟔𝟓𝟎
𝑃≈𝟒𝟖𝟑.𝟔𝟔
Starting with about $ is the compound interest savings account will yield about the same amount of money depositing $500 into the simple interest account for 10 years.

6. Arithmetic Sequence
MAFS.912.F-BF.1.2 
It is a sequence where the difference between consecutive terms is constant. This difference is the common difference.

7. Arithmetic Sequence Example Identifying Arithmetic Sequences
MAFS.912.F-BF.1.2 
Identifying Arithmetic Sequences
Example
Is the sequence an arithmetic sequence?
Part A 3, 6, 9, 12, 15,…
Each difference is 3. the sequence has a common difference. The sequence is an arithmetic sequence.

8. Arithmetic Sequence Example Identifying Arithmetic Sequences
MAFS.912.F-BF.1.2 
Identifying Arithmetic Sequences
Example
Is the sequence an arithmetic sequence?
Part B 1, 4, 9, 16, 25,…
There is no common difference. The sequence is NOT an arithmetic sequence.

9. Arithmetic Sequence Example
MAFS.912.F-BF.1.2 
Using the Explicit Formula for an Arithmetic Sequences
Example
Sport Arena The number of seats in the first 13 rows in a section of an arena form an arithmetic sequence. Row 1 and 2 are shown in the diagram below. How many seats are in Row 13.

10. The Fibonacci Sequence
MAFS.912.F-BF.1.2 
One famous mathematical sequence is the Fibonacci sequence. You can find each term of the sequence using addition, but the sequence is not arithmetic.

11. Geometric Sequence
MAFS.912.F-BF.1.2 
It is a sequence with a constant ratio between consecutive terms.

12. Geometric Sequence Example Identifying Geometric Sequences
MAFS.912.F-BF.1.2 
Identifying Geometric Sequences
Example
Is the sequence geometric? If it is, what are a1 and r?
Part A 3, 6, 12, 24, 48,…
There is common ratio is 2. The sequence is geometric with a1 = 3 and r = 2.

13. Geometric Sequence Example Identifying Geometric Sequences
MAFS.912.F-BF.1.2 
Identifying Geometric Sequences
Example
Is the sequence geometric? If it is, what are a1 and r?
Part B 3, 6, 9, 12, 15,…
The ratios are different. With no common ratio, the sequence is not geometric.

14. Geometric Sequence Example
MAFS.912.F-BF.1.2 
Example
Using the Explicit Formula for a Geometric Sequences
What are the indicated terms of the geometric sequence?
Part A the 10th term of the geometric sequence 4, 12, 36,…

15. Geometric Sequence Example
MAFS.912.F-BF.1.2 
Example
Using the Explicit Formula for a Geometric Sequences
What are the indicated terms of the geometric sequence?
Part B the 2nd and 3rd term of the geometric sequence 2, , , -54,…

16. MAFS.912.F-BF.1.2
1. Which equation can be used to find the nth term for the sequence below? n
𝒏 𝟐 +𝟏
t=Term Value 1
𝟏 𝟐 +𝟏 2
𝟐 𝟐 +𝟏 5 3
𝟑 𝟐 +𝟏 10 4
𝟒 𝟐 +𝟏 17
A. 𝑡=𝑛+3
B. 𝑡= 𝑛 2 +1
C. 𝑡=2𝑛+1
D. 𝑡=3𝑛−1

17. MAFS.912.F-BF.1.2
2. Paul started to train for a marathon. The table shows the number of miles Paul ran during each of the first three weeks after he began training.
If this pattern continues, which of the listed statements could model the number of miles Paul runs 𝑎𝑛, in terms of the number of weeks, 𝑛, after he began training? Select ALL that apply. n
𝒂 𝒏 =𝟐+ 𝒂 𝒏−𝟏
t=Term Value 1
𝒂 𝟏 =𝟏𝟎 10 2
𝟐+ 𝒂 𝟐−𝟏 =𝟐+𝟏𝟎 12 3
𝟐+ 𝒂 𝟑−𝟏 =𝟐+𝟏𝟐
14.4 n
𝒂 𝒏 =𝟏.𝟐 𝒂 𝒏−𝟏
t=Term Value 1
𝒂 𝟏 =𝟏𝟎 10 2
𝟏.𝟐 𝒂 𝟐−𝟏 =𝟏.𝟐(𝟏𝟎) 12 3
𝟏.𝟐 𝒂 𝟑−𝟏 =𝟏.𝟐(𝟏𝟐)
14.4 n
𝟏𝟎+𝟐(𝒏−𝟏)
t=Term Value 1
𝟏𝟎+𝟐 𝟏−𝟏 =𝟏𝟎 10 2
𝟏𝟎+𝟐 𝟐−𝟏 =𝟏𝟎 12 3
𝟏𝟎+𝟐 𝟑−𝟏 =𝟏𝟒
14.4 n
𝟏𝟎 𝒏 𝟐
t=Term Value 1
𝟏𝟎 (𝟏) 𝟐 =𝟏𝟎 10 2
𝟏𝟎 (𝟐) 𝟐 =𝟒𝟎 12 3
𝟏𝟎 (𝟑) 𝟐 =𝟗𝟎
14.4 n
𝟏𝟎 𝟏.𝟐 𝒏−𝟏
t=Term Value 1
𝟏𝟎 𝟏.𝟐 𝟏−𝟏 =𝟏𝟎 10 2
𝟏𝟎 𝟏.𝟐 𝟐−𝟏 =𝟏𝟐 12 3
𝟏𝟎 𝟏.𝟐 𝟑−𝟏 =𝟏𝟒.𝟒
14.4

18. MAFS.912.F-BF.1.2
3. Every day commuting to and from work, Jay drives his car a total of 45 miles. His car already has 2,700 miles on it. Which function shows the total number of miles Jay's car will have been driven after n more days?
3. Every day commuting to and from work, Jay drives his car a total of 45 miles. His car already has 2,700 miles on it. Which function shows the total number of miles Jay's car will have been driven after n more days?
3. Every day commuting to and from work, Jay drives his car a total of 45 miles. His car already has 2,700 miles on it. Which function shows the total number of miles Jay's car will have been driven after n more days?
A. 𝑑(𝑛) = 60
B. 𝑑(𝑛) = 60𝑛
C. 𝑑(𝑛) = ,700𝑛
D. 𝑑(𝑛) = 2, 𝑛
This is just saying that a distance is 60????
This is just saying that a distance of 60 times n days???
2,700 constant plus the miles he drives daily times the number of n days.

19. 𝒉(𝒙) = 𝒇(𝟐𝒙)𝒈(−𝟐𝒙) 𝟐 = (𝟐𝒙) 𝟐 ∙𝟐(−𝟐𝒙) 𝟐 = 𝟒𝒙 𝟐 ∙(−𝟒𝒙) 𝟐
MAFS.912.F-BF.1.2
4. The functions 𝑓 and 𝑔 are defined by 𝑓(𝑥)= 𝑥 2 and 𝑔(𝑥) = 2𝑥, respectively. Which equation is equivalent to ℎ(𝑥) = 𝑓(2𝑥)𝑔(−2𝑥) 2 ?
𝒉(𝒙) = 𝒇(𝟐𝒙)𝒈(−𝟐𝒙) 𝟐
A. ℎ(𝑥)=−2 𝑥 3
B. ℎ(𝑥)=−8 𝑥 3
C. ℎ(𝑥)= 𝑥 2 −2𝑥
D. ℎ(𝑥)=2 𝑥 2 +2𝑥
= (𝟐𝒙) 𝟐 ∙𝟐(−𝟐𝒙) 𝟐
= 𝟒𝒙 𝟐 ∙(−𝟒𝒙) 𝟐
= 𝟒𝒙 𝟐 𝟐 ∙ −𝟒𝒙 𝟐
= 𝟐𝒙 𝟐 −𝟐𝒙
= −𝟖𝒙 𝟑

20. 𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏 𝒂 𝟏 =𝟏 𝒓=𝟐 𝒏=𝟗 𝒂 𝟏 =𝟏 𝒂 𝟐 =𝟐 𝒂 𝟑 =𝟒 𝒂 𝟒 =𝟏∙ 𝟐 𝟒−𝟏 =𝟖
MAFS.912.F-BF.1.2
5. A board is made up of 9 squares. A certain number of pennies is placed in each square, following a geometric sequence. The first square has 1 penny, the second has 2 pennies, the third has 4 pennies, etc. When every square is filled, how many pennies will be used in total?
5. A board is made up of 9 squares. A certain number of pennies is placed in each square, following a geometric sequence. The first square has 1 penny, the second has 2 pennies, the third has 4 pennies, etc. When every square is filled, how many pennies will be used in total?
𝒂 𝟏 =𝟏
A. 512
B. 511
C. 256
D. 81
𝒂 𝟐 =𝟐
𝒂 𝟑 =𝟒
𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏
𝒂 𝟒 =𝟏∙ 𝟐 𝟒−𝟏 =𝟖
𝒂 𝟏 =𝟏
𝒂 𝟓 =𝟏∙ 𝟐 𝟓−𝟏 =𝟏𝟔
4÷2=𝟐
𝒓=𝟐
𝒂 𝟔 =𝟏∙ 𝟐 𝟔−𝟏 =𝟑𝟐
2÷1=𝟐
𝒏=𝟗
𝒂 𝟕 =𝟏∙ 𝟐 𝟕−𝟏 =𝟔𝟒
𝒂 𝟖 =𝟏∙ 𝟐 𝟖−𝟏 =𝟏𝟐𝟖
𝒂 𝟖 =𝟏∙ 𝟐 𝟗−𝟏 =𝟐𝟓𝟔
𝑻𝒐𝒕𝒂𝒍=𝟓𝟏𝟏

21. MAFS.912.F-BF.1.2
6. A rabbit population doubles every 4 weeks. There are currently five rabbits in a restricted area. If 𝑡 represents the time, in weeks, and 𝑃(𝑡) is the population of rabbits with respect to time, about how many rabbits will there be in 98 days?
A. 56
B. 152
C. 3688
D. 81,920

22. MAFS.912.F-BF.1.2
7. DeShawn is in his fifth year of employment as a patrol officer for the Metro Police. His salary for his first year of employment was $47,000. Each year after the first, his salary increased by 4% of his salary the previous year.
Part A
What is the sum of DeShawn's salaries for his first five years of service?
A. $101,983
B. $188,000
C. $219,932
D. $254,567

23. MAFS.912.F-BF.1.2
7. DeShawn is in his fifth year of employment as a patrol officer for the Metro Police. His salary for his first year of employment was $47,000. Each year after the first, his salary increased by 4% of his salary the previous year.
Part B
If DeShawn continues his employment at the same rate of increase in yearly salary, for which year will the sum of his salaries first exceed $1,000,000? Give your answer to the nearest integer.
Enter your answer in the box. 𝟏𝟔

24. MAFS.912.F-BF.1.2
8. Monthly mortgage payments can be found using the formula below:
The Banks family would like to borrow $120,000 to purchase a home. They qualified for an annual interest rate of 4.8%. Algebraically determine the fewest number of whole years the Banks family would need to include in the mortgage agreement in order to have a monthly payment of no more than $720.
𝟐𝟑 𝒚𝒆𝒂𝒓𝒔