Transcript Exponential Functions PowerPoint

Warm Up Activity:
• Put yourselves into groups of 2-4
• Complete the Dice Activity together
o Materials needed:
 Worksheet
 36 Die
Exponential Functions
Let’s compare Linear Functions
and Exponential Functions
Linear Function
Exponential Function
Change at a constant rate
 Rate of change (slope) is a constant


Change at a changing rate
 Change at a constant percent rate
Suppose you have a choice of two different
jobs when you graduate college:
o Start at $30,000 with a 6% per
year increase
o Start at $40,000 with $1200 per
year raise
• Which should you choose?
Which Job?
• When is Option A better?
• When is Option B better?
• Rate of increase changing
• Percent of increase is a constant
• Ratio of successive years is 1.06
• Rate of increase a
constant $1200
Year
Option A
Option B
1
$30,000
$40,000
2
$31,800
$41,200
3
$33,708
$42,400
4
$35,730
$43,600
5
$37,874
$44,800
6
$40,147
$46,000
7
$42,556
$47,200
8
$45,109
$48,400
9
$47,815
$49,600
10
$50,684
$50,800
11
$53,725
$52,000
12
$56,949
$53,200
13
$60,366
$54,400
14
$63,988
$55,600
Let’s look at another
example
Consider a savings account with
compounded yearly income
• What does compounded
yearly mean?
• You have $100 in the account
• You receive 5% annual
interest
• Complete the table
• Find an equation to model the
situation.
• How much will you have in
your account after 20 years?
end of of
New balance
in
AtAt end
Amount
of New
balance
in
Amount
of interest earned
year
account
year
interest earned
account
1
100 * 0.05 = $5.00
$105.00
2
105 * 0.05 = $5.25
$110.25
$100.00
$115.76
$105.00
$110.25
$115.76
$121.55
$127.63
$134.01
$140.71
$147.75
$155.13
$162.89
0
3
1
4
52
3
4
5
6
7
8
9
10
0
110.25 * 0.05 = $5.51
$5.00
$5.25
$5.51
$5.79
$6.08
$6.38
$6.70
$7.04
$7.39
$7.76
Savings Accounts
How do they differ?
• Simple Interest
• Compound Interest
𝑰 = 𝑷𝒓𝒕
•
•
•
•
I = interest accrued
P = Principle
r = interest rate
t = time
Linear
•
•
•
•
•
𝒓 𝒏∙𝒕
𝑨=𝑷 𝟏+
𝒏
A = Current Balance
P = Principle
r = interest rate
n = number of times compounded
yearly
t = time in years
Exponential
Where else in our world do
we see exponential models?
Examples of Exponential Models
• Money/Investments
• Appreciation/Depreciation
• Radioactive Decay/Half Life
• Bacteria Growth
• Population Growth
How can you determine whether an
exponential function models growth or decay
just by looking at its graph?
Graph 1
Graph 2
• Exponential growth functions increase
from left to right
• Exponential decay functions decrease
from left to right
How Can We Define Exponential Functions
Symbolically?
• 𝑓 𝑥 = 𝑎𝑏 𝑥
• Notice the variable is in the exponent?
• The base is b and a is the coefficient.
• This coefficient is also the initial value/y-intercept (when
x=0)
Comparing Exponential Growth/Decay in
Terms of Their Equations
Exponential Growth
Exponential Decay
𝑓 𝑥 = 𝑎 ∙ 𝑏 𝑥 for 𝑏 > 1
𝑓 𝑥 = 𝑎 ∙ 𝑏 𝑥 for 0 < 𝑏 < 1
Example:
𝑓 𝑥 = 2𝑥
Example:
1
𝑓 𝑥 =
2
𝑥
Can you automatically conclude that an exponential
function models decay if the base of the power is a
fraction or decimal?
𝑓 𝑥 =3
1 𝑥
2
or 𝑓 𝑥 = 3 2.5
𝑥
7
No– some fractions and decimals have a value greater than one, such as 3.5 and , and these
2
bases produce exponential growth functions
Fry's Bank Account (clip 1)
Fry’s Bank Account (clip 2)
• On the TV show “Futurama” Fry checks his bank statement
• Since he is from the past his bank account has not been touched
for 1000 years
• Watch the clip above to see how Fry’s saving’s account balance has
changed over time
• Answer the questions on your worksheet following each clip
One More Example…
Consider a medication:
• The patient takes 100 mg
• Once it is taken, body filters medication out
over period of time
• Suppose it removes 15% of what is present in
the blood stream every hour
Fill in the
rest of the
table
At end of hour
Amount remaining
1
100 – 0.15 * 100 = 85
2
85 – 0.15 * 85 = 72.25
3
4
5
What is the
growth factor?
At end of hour Amount Remaining
1
85.00
2
72.25
3
61.41
4
52.20
5
44.37
6
37.71
7
32.06
Growth Factor = 0.85
Note: when growth factor
< 1, exponential is a
decreasing function
Amount Remaining
Mg remaining
100.00
80.00
60.00
40.00
20.00
0.00
0
1
2
3
4
5
At End of Hour
6
7
8
Here are Some Videos to
Further Explain Exponential
Models
The Magnitude of an Earthquake
• Exponential Functions: Earthquakes Explained (2:23)
• In this clip, students explore earthquakes using exponential models. In particular, students analyze
the earthquake that struck the Sichuan Province in China in 2008
The Science of Overpopulation
• The Science of Overpopulation (10:18)
• This clip shows how human population grows exponentially.
There is more of an emphasis on science in this clip then there
is about mathematics as a whole.