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Warm Up Activity: Put yourselves into groups of 2-4 Complete the Dice Activity together Materials needed: Worksheet 36 Die. Exponential Functions. Let’s compare Linear Functions and Exponential Functions. Suppose you have a choice of two different jobs when you graduate college: - PowerPoint PPT Presentation
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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 Functionsand 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?
At end of year Amount of interest earned New balance in
account
1 100 * 0.05 = $5.00 $105.00
2 105 * 0.05 = $5.25 $110.25
3 110.25 * 0.05 = $5.51 $115.76
4
5
At end of year
Amount of interest earned
New balance in account
0 0 $100.001 $5.00 $105.002 $5.25 $110.253 $5.51 $115.764 $5.79 $121.555 $6.08 $127.636 $6.38 $134.017 $6.70 $140.718 $7.04 $147.759 $7.39 $155.1310 $7.76 $162.89
Savings Accounts
• Simple Interest
• I = interest accrued• P = Principle• r = interest rate• t = time
• Compound Interest
• A = Current Balance• P = Principle• r = interest rate• n = number of times
compounded yearly• t = time in years
How do they differ?
Linear 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?
• 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 for
Example:
Exponential Decay for
Example:
Can you automatically conclude that an exponential function models decay if the
base of the power is a fraction or decimal?
or
No– some fractions and decimals have a value greater than one, such as 3.5 and , and these 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…
At end of hour Amount remaining
1 100 – 0.15 * 100 = 852 85 – 0.15 * 85 = 72.25345
Fill in the rest of the
tableWhat is the
growth factor?
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
At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06
Amount Remaining
0.00
20.00
40.00
60.00
80.00
100.00
0 1 2 3 4 5 6 7 8
At End of HourM
g re
mai
ning
Growth Factor = 0.85Note: when growth
factor < 1, exponential is a decreasing function
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.