1. Exponential Growth Exponential Decay
Algebra-2
7-1, and 7-2
Exponential Growth
Exponential Decay

2. Lesson Objectives
1. Be able to explain how a power is similar to and exponential function and how it is different.
2. Be able to explain why the graph of an exponential function has the shape it does.
3. Know what input value results in the initial value of the function.
4. Explain how the “initial value” got its name.

3. Lesson Objectives
4. Describe how a given function is a transformation of the parent exponential function.
5. Describe the difference between exponential “growth” and exponential “decay.”
6. Know how the base determines whether the function exhibits growth or decay.
7. Solve simple problems involving exponential growth or decay.

4. Arrange the following equations into 3 different groups8. 1. 5. 2. 6. 9. 7. 3.
10. 4.

5. Your turn Exponent coefficient Base 1. Define what a power is.
Power: An expression formed by repeated
Multiplication of the same factor.
2. Give an example of a power. Label the parts.
Exponent
coefficient
Base

6. Exponential Function Power Function
3. Describe how you can tell the power function from the exponential function.
Power: “x” (input variable) is the base and a number is the exponent.
Exponential: “x” is the exponent and a number is the base.

7. Your turn: 4. What happens to population as time goes by?
5. The population of the USA is now about 300 (million). Make a graph that shows how the population will change as time goes by.

8. Your turn:
6. “Plug in” the following input values in order to “fill in” the table below. Round to the 0.1 decimal position. x -5 -3
-0.5 1 2 y
0.1
0.4
2.1 3 6 12
7. Graph the points on an x-y plot.

9. Vocabulary
Asymptote: a line that the graph of a function approaches but never reaches.

10. Negative Input values x -5 -3 -1 1 0.5 1 2 Will f(x) ever equal zero?1
Will f(x) ever equal zero?
0.5
0.031
0.125 1 2
Horizontal asymptote: y = 0

11. Your turn:
8. Graph the exponential function on your calculator then copy the graph to your answer sheet.
9. Adjust your window to “ZOOM in”. Copy this to your answer sheet
10. What is the horizontal asymptote?

12. Your turn: 11. What is the “base” of the exponential function?

13. The “Initial” Value
If the input variable was time, the previous function
would look like:
Since “negative time” doesn’t make sense,
what is the “domain” of this function?
( what input values are allowed?)
The initial value occurs when t = 0.
What is the initial value of f(t) ??
 f(0) = ?

14. Vocabulary: The “Initial” Value
The initial value of the function is the coefficient
of the power.
What is he initial value of the following functions ?

15. Your turn: 0.5 14. What is the initial value of: a
16. The ‘y’ intercept is a point on the y-axis. What input value (for x) causes a y-intercept ?
17. Find f(0) for the following function:
f(0) = 2

16. Transformations of the “exponential function”
The parent function:
Replacing x with (x – 2) (in the parent function):
Translates right 2
Adding 2 (to the parent function):
Translates up 2
Multiplying the parent function by 3:
Vertically stretches by a factor of 3

17. Identifying the Parts of the function:
‘a’ is the initial value  f(0) = ‘a’
‘b’ is called the growth factor
‘d’ shifts graph up/down and is the horizontal asymptote
Initial value: f(0) = = 12
Growth factor: 4
Horizontal asymptote: y = 2

18. Transformations of the “exponential function”
The parent function:
Multiplying the parent function by -1:
Reflects across x-axis
Combinations of transformations:
Reflects across x-axis
Translated left 3
Vertically stretched by a factor of 4
Translated down 5

19. Your turn: What is the transformation of the parent function:
Reflected across x-axis and vertically stretched by a factor of 2
18.
Translated right 7 and up 5
19.
vertically stretched by a factor of ½ and translated down 4
20.

20. Your turn: What is the horizontal asymptote?
21.
y = 5
22.
y = -4
23.

21. Exponential Growth Table of values x f(x) 3 2 6 1 2 12 2 2 24 3 2 4 483 2 6 1 2 12 2 2 24 3 2 4 48 -1
‘a’ is the initial value  f(0) = ‘a’
1.5
‘b’ is called the growth factor -2
0.75
‘d’ is the horizontal asymptote

22. Exponential Growth Horizontal asymptote “ Growth” occurs
when the growth
factor ‘b’ > 1
Horizontal asymptote
Does the output value ever reach ‘0’ ?
What do we call the line: y = 0 ?

23. Your turn: 24. Where does it cross the y-axis? y = 5
Graph the function:
24. Where does it cross the y-axis?
y = 5
25. What is the “intial value of f(x) ? 5
26. What is the horizontal asymptote?
y = 0
27. What is the growth factor? 2

24. Exponential Decay Table of values x f(x) 4 ½ 2 1 ½ 1 2 ½ 0.5 3 ½ 4 -14 2 1 1 2
0.5 3 4
0.25 -1
‘a’ is the initial value  f(0) = ‘a’ 8
‘b’ is called the decay factor -2 16
‘d’ shifts everything up or down

25. Your turn: 28. Is the following function growth or decay?
30. Is he following function growth or decay?

26. Population Growth It’s just a formula!!!
time
Population (as a
function of time)
Growth
rate
Initial
population
It’s just a formula!!!
The initial population of a colony of bacteria
is The population doubles every hour. What is the population after 5 hours?

27. Your turn:
Huntsville has a population of 600 people. The population increases by 3% every year. What will the population be in 50 years?
31.

28. Your turn:
The population of Detroit, Michigan decreases by 2% every year. If the population is 750,000 right now, what will the population be in 12 years
32.

29. Your turn
time
Amount (as a
function of time)
Initial amount
(“principle”)
Growth rate
33.
You spend 20% of your savings every month (80% remains at the end of each month). How much money will you have left in 10 months if you started with $500?

30. Your turn: 34. A bank account pays 3.5% interest per year.
Initial amount
(“principle”)
34.
A bank account pays 3.5% interest per year.
If you initially invest $200, how much money
will you have after 5 years?

31. Your turn: 35. A bank account pays 14% interest per year.
If you initially invest $2500, how much money
will you have after 7 years?

32. Your turn: 36. f(0) = ? 37. f(1) = ? 38. Horizontal asymptote = ?
38. Domain = ?
39. range = ?

33. Your turn:
The population of a small town was 1500 in the population increases by 3% every year.
37. What is the population in 2009?

34. Transforming Exponential Functions
The graph of can be obtained
from by reflecting it across
the y-axis.

35. Putting it all together:
vertical shift
If negative:
Reflect across x-axis
Horizontal shift
Initial value:
Crosses y-axis here
If negative:
Reflect across y-axis
Growth factor:

36. Your turn: 38. 39. 40. For each of the following what is the:
a. “initial value”?
b. “decay factor”?
c. “horizontal asymptote”
d. Any reflections (across x-axis or y-axis)
38.
39.
40.

37. Identifying the Parts of the function:
‘a’ is the initial value  f(0) = ‘a’ (plus ‘d’)
‘b’ is called the decay factor
‘d’ shifts graph up/down and is the horizontal asymptote
Initial value: f(0) = = 12
Decay factor: 0.4
Horizontal asymptote: 2

38. Graphing Exponential Decay Use the “power of the calculator” or:
2. Some other point
f(1) = ?
f(1) = 3
3. Horizontal
asymptote
y = 0
Domain = ?
Range = ?
All real #’s
y > 0

39. Exponential Growth and Decay
exponential growth: growth factor > 1
exponential decay: growth factor 0 < b < 1

40. What 3 things do you need to graph exponential growth?
1. f(0) = ? f(0) = 3 + 5
2. Some other point
f(1) = ? f(1) = = 11
3. Horizontal asymptote
y = 5
Domain = ?
Range = ?
All real #’s
y > 5

41. What 3 things do you need to graph exponential decay?
1. f(0) = ? f(0) =
2. Some other point
f(1) = ? f(-1) = = 17.5
3. Horizontal asymptote
y = 10
Domain = ?
Range = ?
All real #’s
y > 10

42. Your turn: 42. 43. For each of the following what is the:
a. “initial value”?
b. “growth factor”?
c. “horizontal asymptote”
41.
42.
43.