Name: _________________________Type equation here.
Unit 4 D8 TEST STUDY GUIDE
Exp functions, Rate of Change, and Intervals
Date: ______________ Per: ________
Write the correct formula for each exponential model
1. Exponential Growth
2. Exponential Decay
𝑦 = 𝑎(1 + 𝑟)𝑡
𝑦 = 𝑎(1 − 𝑟)𝑡
3. Compound Interest
4. Half-Life
𝑟
𝑛
𝐴 = 𝑃(1 + )𝑛𝑡
𝐴 = 𝑃(0.5)𝑡
If the following functions are exponential, fill in the missing y-values.
5.
x
-1
1
3
5
y
4
8
16
32
6. {(-1, ___1.5____), (2, ___6____), (3, 24), (4, 96)}
The function 𝒇(𝒙) = 𝟏𝟐𝟐𝟕(𝟏. 𝟎𝟗)𝒙 models a school population after x years.
8. What was the starting population?
____1227___________
9. Is the population increasing or decreasing?
_____increasing__________
10. By what percent?
______9_________
11. What will the population be at year 5?
______1887_________
12. When will the population exceed 2000?
______6 years_________
Find the domain and range for the following exponential function and give the equation of the asymptote.
13. Domain: _____(−∞, ∞)_______________
14. Range: _____(−4, ∞)________________
15. Asymptote: __y = -4___________________
16. You have just inherited a large amount of money and have decided to invest. The investment plan
you have chosen can be modeled by the function 𝑓(𝑥) = 10,000(1.075)𝑥 . What will be the value of the
investment in 20 years?
$42,478.51
In the year 2029, a certain species of turtle has begun to increase at an alarming rate, overflowing
the planet’s rivers and oceans with cuteness. The population is increasing by 15.7% per year. At
this rate, the function 𝒇(𝒙) = 𝟑(𝟏. 𝟏𝟓𝟕)𝒙 gives the population, in millions, x years after 2029.
17. What will the population be in 2034?
6.2 million
18. In what year will the population exceed 12 million?
19. Which graph represents the function 𝑓(𝑥) = 2 ∙ (2)𝑥 ?
2039
Write the exponential function to represent each situation.
20. The value of a house was $127,000 when it was purchased. The value depreciates at a rate of 11%
per year.
𝑦 = 127000(1 − .11)𝑡
21. Math club started the year with a surplus of petty cash at $65. Their balance will increase at a rate of
2.5% each month.
𝑦 = 65(1 + .025)𝑡
22. Jenni starts a savings account with $5600. The account is compounded monthly 1.6%.
𝑦 = 5600(1 +
. 016 𝑡
)
12
Write a model and solve the exponential function in each situation.
23. The half-life of Zn-71 is 2.4 minutes. If one had 100.0 g at the beginning, how many grams would be
left after 7.2 minutes has elapsed?
12.5 grams
24. The population of a city after x years can be represented by the exponential model
𝑓(𝑥) = 230,000(0.87)𝑥 . What is the population after 7 years?
86,768
25. Debbie invested $67,200 into a secured CD at a rate of 1.8% compounded quarterly. What is the total
amount of the CD in 6 years?
$74,845.87
26. You are performing an experiment. You start with 8,000 mg. of two different chemicals. When each
chemical has reached the time of its half-life, you record the results. Which of the two chemicals would
you have the most of after 5 minutes? (Explain or justify)
Chemical A has a half-life of 50 seconds. Chemical B has a half-life of 150 seconds.
300
5 minutes = 300 seconds
Chemical A: 8000(0.5) 50 = 125 𝑚𝑔
300
Chemical B: 8000(0.5)150 = 2000 𝑚𝑔
27. What is the average rate of change over
the interval from -1 to 1?
x
-1
0
1
2
y
3
5
8
11
𝑦2 − 𝑦1
8−3
5
=
= = 2.5
𝑥2 − 𝑥1 1 − −1 2
28. Find the average rate of change between 0 and 4 hours.
(0, 0) and (4, 120)
𝑦2 −𝑦1
𝑥2 −𝑥1
=
120−0
4−0
=
120
4
= 30
Give ALL the intervals of increase and decrease.
29. Increase: __(0, 10), (30, 45)_______________
30. Decrease: ___(25, 30), (45, 50)________________
For each function tell whether exponential growth or decay is modeled and give the percent rate of
growth/decay.
31. 𝑓(𝑥) = 200,000(0.95)𝑥
Decay, 5%
32. g(𝑥) = 17(1.15)𝑥
Growth, 15%
33. ℎ(𝑥) = 1.3(1.012)𝑥
Growth, 1.2%