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 = 85

2 85 – 0.15 * 85 = 72.25

3

4

5

Fill in the rest of the

table

What 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 r

emai

nin

g

Growth Factor = 0.85

Note: 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.