Reporting Strand 7: Exponential Functions

CCSS Instructional Focus

Write Sequences (F.IF.3) Write arithmetic and geometric sequences recursively and explicitly to model real

world situations.

Interpret Functions (F.IF.8b) Differentiate between exponential growth and exponential decay and identify the

percent rate of change in exponential functions.

Determine whether each table or rule represents an exponential function. Explain why or why not.

1. 𝑦 = 7𝑥 + 3 2. 3. 𝑦 = 12𝑥2 4.

Graph each exponential function.

5. 𝑦 = −4𝑥 6. 𝑦 = − (1

3)

𝑥

Graph the following functions.

EXPONENTIAL FUNCTIONS

7. 𝑓(𝑥) = −2

3𝑥 + 4 8. 𝑓(𝑥) = −1 ∙ 2𝑥

Determine whether each rule represents a linear or an exponential function.

9. 𝑦 = 6(2𝑥) 10. 𝑦 = 3𝑥 + 42 11. 𝑦 = 20 12. 𝑦 = (1

3)

𝑥

Create a linear or exponential function for each situation.

13. (−1, 7), (1, −1), (2, −5) 14. (−2,0.25), (−1,0.5), (0,1), (1,2)

15. Jacquelyn’s earnings m are a function of the number of lawns n she mows at a rate of $12 per lawn.

16. An investment of $5000 doubles in value every decade.

17. 18.

19. Graph A represents the exponential equation 𝑦 = 3(5)𝑥 and Graph B represents 𝑦 = 5(3)𝑥 . Which

statements are true about Graph A and Graph B? Select all that apply.

A. Both graphs rise at the same rate.

B. Graph A rises faster than Graph B.

C. Graph B rises faster than Graph A.

D. The y-intercepts of both graphs are the same.

E. The y-intercept of Graph A is greater than the y-intercept of Graph B.

F. The y-intercept of Graph B is greater than the y-intercept of Graph A.

20. The graphs of 𝑓(𝑥) and 𝑔(𝑥) are shown. The graph of 𝑔(𝑥) is the reflection of the graph of 𝑓(𝑥) across

the x-axis. If 𝑓(𝑥) = 3 (1

2)

𝑥

, what is the equation of 𝑔(𝑥)?

EXPONENTIAL GROWTH VS. EXPONENTIAL DECAY

21. Which of the equations below represent exponential decay? Select all that apply.

A. 𝑦 = (3

4) 2𝑥 B. 𝑦 = 700(1 − 0.35)𝑥 C. 𝑦 = (

4

3)

𝑥 D. 𝑦 = (0.01)𝑥 E. 𝑦 = (6.35)𝑥

22. From the list below, indicate whether each exponential function is an increasing function or a

decreasing function.

A. 𝑓(𝑥) = 5 ∙ 2𝑥 B. 𝑔(𝑥) = 0.25 ∙ 6𝑥 C. ℎ(𝑥) = 7(0.9)𝑥 D. 𝑗(𝑥) = 2(1 + 0.03)𝑥

E. 𝑘(𝑥) = 10 (3

5)

𝑥

F. 𝑚(𝑥) = 3(4)𝑥 − 5 G. 𝑛(𝑥) = 4 (7

6)

𝑥

H. 𝑝(𝑥) = −10(8)𝑥

23. A population of fish in a lake decreases 6% annually. What is the decay factor?

24. The number of students enrolled at a college is 15,000 and grows 4% each year. How many students

will be enrolled after 25 years?

25. A population of 75 foxes in a wildlife preserve quadruples in size every 15 years. Create the function

to represent the population growth making x the number of 15 year periods. How many foxes will there

be after 45 years?

26. Velma takes 100 mg of a medicine, which dissipates at a constant rate of 10% per hour. Write the

function to represent this situation.

27. Margaret sells 1,500 tickets for her school’s first baseball game, but then she notices that attendance

changes by a certain rate at each game. If the function giving the population as a function of the game

number is 𝑦 = 1500(0.95)𝑥, what is the change in attendance?

28. Derek places 1,000 bacteria in a vial. Their change in population per hour is modeled by the function

𝑓(𝑥) = 1000(1.03)𝑥. Explain the base & the growth or decay rate in context of the situation.

29. Assume that half of the representatives in Congress at any given time will not be serving in 8 years;

that is, the half-life of the number of original representatives is 8 years. Of the 435 reps serving this year,

how many will still be serving in 32 years?

30. Brianna learns in science class that a typical cup of coffee has 130 mg of caffeine. She also discovers

that her body eliminates 15% of the caffeine each hour. Brianna quickly drinks a full cup of coffee on the

way to school.

a. Write an exponential function to represent the situation.

b. To the nearest milligram, 8 hours after drinking the coffee, Brianna still has __________________ mg of

caffeine from the coffee in her system.

31. The population of a city is 45,000 and decreases 2% each year. If the trend continues, what will the

population be after 15 years?

32. In 1971, there were 294,105 females that number has tripled each year. Write a function to represent

the number of females participating in high school sports if the trend continued.

33. The value of Jayme’s car (in dollars) as a function of time (in years) is given by 𝑓(𝑥) = 24,000(0.83)𝑥.

What is the rate of depreciation?

34. When Mitch invested $1,000, he believed that it would increase at a rate of 2%. The actual function

that describes the growth of this investment over time is 𝑓(𝑥) = 1,000(1.04)𝑥. What is the actual rate of

growth?

35.

36.

SEQUENCES

Identify each sequence as arithmetic or not. If it is arithmetic, identify the common difference.

37. −9, −17, −26, −33, . .. 38. 6, 9, 12, 15, . .. 39. 10, 8, 6, 4, . ..

40. 4, 12, 26, 108, . .. 41. 10, 24, 36, 52, . ..

ARITHMETIC SEQUENCES

EXPLICIT VS. RECURSIVE FORMULA

Write a recursive formula and an explicit formula to represent the sequence.

42. −3, −7, −11, −15, . .. 43. 7, 9, 11, 13, . .. 44. 6, 13, 20, 27, . ..

45. You have a cafeteria card worth $50. After you buy lunch on Monday, its value is $46.75. After you buy

lunch on Tuesday, its value is $43.50.

a. Write a rule to represent the amount of money left on the card as an arithmetic sequence.

b. What is the value of the card after you buy 12 lunches?

Write an explicit formula for each recursive formula.

46. 𝐴(𝑛) = 𝐴(𝑛 − 1) + 3.4; 𝐴(1) = 7.3

47. 𝐴(𝑛) = 𝐴(𝑛 − 1) − 0.3; 𝐴(1) = 0.3

Write a recursive formula for each explicit formula.

48. 𝐴(𝑛) = −3 + (𝑛 − 1)(−5) 49. 𝐴(𝑛) = 4 + (𝑛 − 1)(1)

50. Consider the following sequence: 9, 1, −7, −15, . ..

a) Write an explicit formula for the sequence.

b) Write a recursive formula for

the sequence.

c) Find the 38th term of the sequence.

51. An accountant starts off with a base salary of $53,000. Their salary increases by $2,000 each year.

a. Write an explicit formula that represents this situation.

b. Find the accountant's salary if he/she has been working there for 6 years.

c. What year will you he/she be making $93,000?

GEOMETRIC SEQUENCES

Identify each sequence as geometric or not. If it is geometric, identify the common ratio.

52. 0.1, 0.5, 2.5, 12.5, . .. 53. 4, 12, 36, 108, . .. 54. 25, 35, 45, 55, . ..

Write the recursive formula and the explicit formula for each situation.

55. 500, 100, 20, 4, . .. 56. 5, 15, 45, 135, . .. 57. 40, 10,5

2,

5

8, . ..

58. Suppose you want to make a reduced copy of a photograph that has an actual length of 8 inches. Each

time you press the reduce button on the copier the copy is reduced by 12%. What is the length of the

photograph's copy if you press the reduce button 5 times? Round to the nearest hundredth.

59. The starting salary at a company is $42,000 per year. The company automatically gives a raise of 3%

per year. Write an explicit and recursive formula for the geometric sequence formed by the salary

increase. What will you make after you work for this company for 4 years?

EXPLICIT VS. RECURSIVE FORMULA

60. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling.

If we start with only one bacteria which can double every hour, how many bacteria will we have by the

end of one day?

61. Each year the local country club sponsors a tennis tournament. Play starts with 128 participants.

During each round, half of the players are eliminated. How many players remain after 5 rounds?

Tell whether each sequence is arithmetic or geometric. Justify your answer. Write a recursive

formula and an explicit formula to represent the situation.

62. 6, −12, 24, −48, . .. 63. 3, 3.5, 4, 4.5, . ..

64. Find the 17th term for the sequence listed in question #62.

65. Find the 50th term for the sequence listed in question #63.

66. A child puts $1.00 into a piggy bank. One week later he puts $1.25 in the bank. The next week he puts

$1.50 in the bank, and so on. How much money does he put in the bank on the 25th week?

67. Heavy rain in Brianne's town caused the river to rise. The river rose three inches the first day, and

each day twice as much as the previous day. How much did the river rise on the fifth day?

68. You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is

16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic

sequence. What is the distance the object will fall the sixth second?

69. After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests

jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase

the time by 6 minutes. How many weeks will it be before you are up to jogging 60 minutes per day?

70. You complain that the hot tub in your hotel suite is not hot enough. The hotel tells you that they will

increase the temperature by 10% each hour. If the current temperature of the hot tub is 75 degrees

Fahrenheit, what will the temperature of the hot tub be after 3 hours to the nearest tenth of a degree?

71. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of

subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville

in 1994?

RECAP ON EQUATIONS

If asked to write an exponential function (or growth/decay function):

𝒇(𝒙) = 𝒂 ∙ 𝒃𝒙

If asked to write the recursive formula:

Arithmetic Geometric

RECAP ON b (growth or decay rate)

If given a growth rate as a percent:

𝒃 = 𝟏 + (𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆 𝒂𝒔 𝒂 𝒅𝒆𝒄𝒊𝒎𝒂𝒍)

If given a decay rate as a percent:

𝒃 = 𝟏 − (𝒅𝒆𝒄𝒂𝒚 𝒓𝒂𝒕𝒆 𝒂𝒔 𝒂 𝒅𝒆𝒄𝒊𝒎𝒂𝒍)

If given a number that’s used for repeated multiplication:

𝒃 = 𝒕𝒉𝒂𝒕 𝒏𝒖𝒎𝒃𝒆𝒓