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Exponential Growth and Exponential Decay
Section 8.1 and 8.2
WHAT YOU WILL LEARN:
1. How to graph exponential growth functions.
2. How to graph exponential decay functions.
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.
Exponential Growth
W 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 .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
1310.72
2 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 $11 $12 $13
$14
$15
$16
$17
$18
• 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:
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.
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.
Let’s Graph an Example
x-10 -5 5 10
y
-10
-5
5
10
Question: Will the graph ever pass below y of 0?
𝑦=2𝑥
Let’s Graph an Example
x-10 -5 5 10
y
-10
-5
5
10
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.
Let’s Graph an Example
x-10 -5 5 10
y
-10
-5
5
10
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.
Try the following on your graphing calculator
x
x
x
y
y
y
2
23
23
1
x
x
x
y
y
y
2
25
25
1
xaby
Group 1: Group 2:
How does “a” in the function affect the graph?
A Definition
y = abx is an exponential growth function. When a is greater than 0 and b is greater than 1.
Graphing Examplesxy 3
2
1• Graph
x-10 -5 5 10
y
-10
-5
5
10
Another Example
xy )2
3(Graph
x-10 -5 5 10
y
-10
-5
5
10
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.
An Example of Graphing by Translation
423 1 xyGraph
x-10 -5 5 10
y
-10
-5
5
10
You Try
132 2 xy• Graph
x-10 -5 5 10
y
-10
-5
5
10
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.
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?
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.
19
Example 1
x
x
x
xf
xf
xf
)3(10)(.3
)2
3(8)(.2
)3
2(5)(.1
20
• State whether the function is an exponential growth or exponential decay function.
You Try
x
x
xf
xf
)8
5(4)(.2
)2(3
1)(.1
• State whether the function is an exponential decay or growth function.
A Basic Graphx
y
2
1• A graph of
x-10 -5 5 10
y
-10
-5
5
10
Graphing Exponential Functions…again
x
y
4
13• Graph:
x-10 -5 5 10
y
-10
-5
5
10
Another Examplex
y
3
25• Graph:
x-10 -5 5 10
y
-10
-5
5
10
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.
Graphing by Translation
12
13
2
x
y• Graph:
x-10 -5 5 10
y
-10
-5
5
10
An Exponential Decay Word Problem• We will use the formula:
y = a(1 - r)t
(1-r) is called the decay factor.
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.
Homework
:
Page 469, 14-18 even, 19-24 all, 34, 36, 38, 43-45 allPage 477, 12, 16, 18, 19-24 all, 36, 40, 42, 47-49 all