1. Exponential Growth and Exponential Decay
Section 8.1 and 8.2

2. What You Will Learn: How to graph exponential growth functions.
How to graph exponential decay functions.

3. Exponential Growth
This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week.

4. Exponential Growth
This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Although the second option, growing at a constant rate of $1/week, pays more in the short run, the first option eventually grows much larger: W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
.01
.02
.04
.08
.16
.32
.64
1.28
2.56
5.12
10.24
20.48
40.96
81.92
163.84
327.68
655.36 $1 $2 $3 $4 $5 $6 $7 $8 $9
$10
$11
$12
$13
$14
$15
$16
$17
$18

5. Why! Exponential Growth!
The equation for option 1 is: y = 2n where n is the number of weeks.
The equation for option 2 is y = 1 + n where n is the number of weeks.

6. Oh Boy! Vocabulary
An exponential function involves the expression bx where the base “b” is a positive number other than 1. The variable is going to be in the “position” of the exponent.

7. Let’s Graph an Example
Question: Will the graph ever pass below y of 0?
𝑦= 2 𝑥

8. Let’s Graph an Example
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.

9. Let’s Graph an Example
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.
An asymptote is a line that a graph approaches as you move away from the origin.

10. Try the following on your graphing calculator
Group 1: Group 2:
How does “a” in the function affect the graph?

11. A Definition
y = abx is an exponential growth function. When a is greater than 0 and b is greater than 1.

12. Graphing Examples
Graph

13. Another Example
Graph

14. Graphing by Translation
The generic form of an exponential function is: y = abx-h + k Where h is movement along the x axis and k is movement along the y axis.

15. An Example of Graphing by Translation

16. You Try
Graph

17. Exponential Growth Model
We will use the formula:
y = a(1 + r)t
a is the initial amount, r is the percent increase expressed as a decimal and t is the number of years.
The term 1 + r is called the growth factor.

18. An Example Problem
In January 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year.
Write a model.
How many hosts were there in 1996?
Graph the model.
When will there be 30 million hosts?

19. Section 8.2 – Exponential Decay
These functions will have the form y = abx where a is greater than zero and b is between 0 and 1.

20. Example 1
State whether the function is an exponential growth or exponential decay function.

21. You Try
State whether the function is an exponential decay or growth function.

22. A Basic Graph
A graph of

23. Graphing Exponential Functions…again

24. Another Example
Graph:

25. Graphing by Translation
The generic form of an exponential function is: y = abx-h + k Where h is movement along the x axis and k is movement along the y axis.

26. Graphing by Translation

27. An Exponential Decay Word Problem
We will use the formula:
y = a(1 - r)t
(1-r) is called the decay factor.

28. The Word Problem
You buy a new car for $24,000. The value y of the car decreases by 16% each year.
1. Write an exponential decay model for the value of the car.
2. Use the model to estimate the value after 2 years.
3. Graph the model.
4. When will the car have a value of $12,000.

29. Homework Page 469, 14-18 even, 19-24 all, 34, 36, 38, 43-45 all:
Page 469, even, all, 34, 36, 38, all
Page 477, 12, 16, 18, all, 36, 40, 42, all