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 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: Start at $30,000 with a 6% per year increase 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? 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

Let’s look at another example

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 3 110.25 * 0.05 = $5.51 $115.76 4   5

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

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 =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)

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

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 2 𝑥 or 𝑓 𝑥 =3 2.5 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

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 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?

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 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.