Transcript Exponential Growth

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
• 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 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
10.2
4
20.4
8
40.
96
81.
92
163
.84
327
.68
655
.36
131
0.7
2
2
$1
$2
$3
$4
$5
$6
$7
$8
$9
$10
$11
$12
$13
$14
$15
$16
$17
$18
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
𝑦=
Question: Will the
graph ever pass
below y of 0?
y
2𝑥
10
5
-10
-5
5
-5
-10
10
x
Let’s Graph an Example
Question: Will the
graph ever pass
below y of 0?
y
10
We say that there
is an asymptote at
y = 0.
5
-10
-5
5
-5
-10
10
x
Let’s Graph an Example
Question: Will the
graph ever pass
below y of 0?
y
10
We say that there
is an asymptote at
y = 0.
5
-10
-5
An asymptote is a line that a
graph approaches as you move
away from the origin.
5
-5
-10
10
x
Try the following on your graphing calculator
Group 1:
y 
Group 2:
1
2
x
3
2
x
5
y  32
y 2
y
1
x
x
How does “a” in the function
y  5  2
y2
y  ab
x
x
x
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 Examples
• Graph
y
1
3
x
2
y
10
5
-10
-5
5
-5
-10
10
x
Another Example
Graph
3
y  ( )
2
x
y
10
5
-10
-5
5
-5
-10
10
x
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
Graph
y  3 2
x 1
4
y
10
5
-10
-5
5
-5
-10
10
x
You Try
• Graph
y  23
x2
1
y
10
5
-10
-5
5
-5
-10
10
x
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
• State whether the function is an
exponential growth or exponential decay
function.
2
x
3
x
1. f ( x )  5( )
3
2. f ( x )  8( )
2
3 . f ( x )  10 ( 3 )
x
20
You Try
• State whether the function is an
exponential decay or growth function.
1. f ( x) 
1
(2)
x
3
5
2. f ( x)  4( )
8
x
A Basic Graph
• A graph of
1
y 
2
x
y
10
5
-10
-5
5
-5
-10
10
x
Graphing Exponential Functions…again
• Graph:
1
y  3 
4
x
y
10
5
-10
-5
5
-5
-10
10
x
Another Example
• Graph:
2
y   5 
3
x
y
10
5
-10
-5
5
-5
-10
10
x
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
• Graph:
1
y   3 
2
x2
1
y
10
5
-10
-5
5
-5
-10
10
x
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 all
Page 477, 12, 16, 18, 19-24 all, 36, 40, 42,
47-49 all