1. Warm Up Activity: Put yourselves into groups of 2-4
Complete the Dice Activity together
Materials needed:
Worksheet
36 Die

2. Exponential Functions

3. Let’s compare Linear Functions and Exponential Functions
Change at a constant rate
Rate of change (slope) is a constant
Change at a changing rate
Change at a constant percent rate

4. Suppose you have a choice of two different jobs when you graduate college:
Start at $30,000 with a 6% per year increase
Start at $40,000 with $1200 per year raise
Which should you choose?

5. Which Job? When is Option A better? When is Option B better?
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
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

6. Let’s look at another example

7. Amount of interest earned
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
* 0.05 = $5.51
$115.76 4 5

8. Savings Accounts Linear Exponential How do they differ?
Simple Interest
𝑰=𝑷𝒓𝒕
I = interest accrued
P = Principle
r = interest rate
t = time
Compound Interest
𝑨=𝑷 𝟏+ 𝒓 𝒏 𝒏∙𝒕
A = Current Balance
P = Principle
r = interest rate
n = number of times compounded yearly
t = time in years
Linear
Exponential

9. Where else in our world do we see exponential models?

10. Examples of Exponential Models
Money/Investments
Appreciation/Depreciation
Radioactive Decay/Half Life
Bacteria Growth
Population Growth

11. How can you determine whether an exponential function models growth or decay just by looking at its graph?
Graph 1
Graph 2

12. Exponential growth functions increase from left to right
Exponential decay functions decrease from left to right

13. How Can We Define Exponential Functions Symbolically?
𝑓 𝑥 =𝑎 𝑏 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)

14. Comparing Exponential Growth/Decay in Terms of Their Equations
Exponential Decay
𝑓 𝑥 = 𝑎∙𝑏 𝑥 for 𝑏>1 Example: 𝑓 𝑥 = 2 𝑥
𝑓 𝑥 = 𝑎∙𝑏 𝑥 for 0<𝑏<1 Example: 𝑓 𝑥 = 1 2 𝑥

15. Can you automatically conclude that an exponential function models decay if the base of the power is a fraction or decimal? 𝑓 𝑥 = 𝑥 or 𝑓 𝑥 = 𝑥
No– some fractions and decimals have a value greater than one, such as 3.5 and 7 2 , and these bases produce exponential growth functions

16. 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

17. One More Example…

18. 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 remaining 1
100 – 0.15 * 100 = 85 2
85 – 0.15 * 85 = 72.25 3 4 5
Fill in the rest of the table
What is the growth factor?

19. Note: when growth factor < 1, exponential is a decreasing function

20. Here are Some Videos to Further Explain Exponential Models

21. 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

22. The Science of Overpopulation
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.