warm up activity: put yourselves into groups of 2-4 complete the dice activity together

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.