WARM UP

1. Given the function f(x) = 3ex: a. Fill in the following table of values:

b. Sketch the graph of the function.

c. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

x -2 0 1 2 3

F(x)

TODAY’S OBJECTIVES

To be able to use exponential functions to model real world data

BUT FIRST…HOW DO EXPONENTIAL FUNCTIONS MODEL REAL WORLD SCENARIOS?

For this, you will be in groups of 3 to conduct a brief experiment.

M&M experiment

WE JUST MODELED EXPONENTIAL GROWTH AND DECAY.

Look at the patterns of the numbers you wrote down.

What’s happening in the left column? Right column?

Use L1 and L2 and Stat Plot to look at the shape your points made.

THERE ARE 6 MAJOR TYPES OF EXPONENTIAL FUNCTIONS THAT WE NEED TO BE AWARE OF:

Half Life Doubling Time Compound Interest Continuous Compound Interest Exponential Growth/Decay Continuous Exponential Growth/Decay

COMPOUND INTEREST

P = principal r = rate (decimal) n = number of times per year t = time in years

Use it when you see: compounded yearly, quarterly, monthly, semi-annually, etc.

COMPOUND INTEREST - EXAMPLE

Kristy invests $300 in an account with a 6% interest rate, making no other deposits or withdrawals. What will Kristy’s account balance be after 20 years if the interest is compounded a. Semiannually? b. Monthly? c. Daily?

CONTINUOUS COMPOUND INTEREST

A = Pert

P = Principal r = rate (decimal) t = time (years)

Use it when: you see the words “compounded continuously”

CONTINUOUS COMPOUND INTEREST - EXAMPLE

Suppose Kristy finds an account that will allow her to invest her $300 at a 6% interest rate compounded continuously. If there are no other deposits or withdrawals, what will Kristy’s account balance be after 20 years?

EXPONENTIAL GROWTH/DECAY

N = N0(1 + r)t

N0 = initial amount r = rate (decimal)

growth: r is positive decay: r is negative

t = time (years) Use it when: growth/decay is “per year”

or “annual”

EXPONENTIAL GROWTH/DECAY EXAMPLE Mexico has a population of

approximately 110 million. If Mexico’s population continues to grow at 1.42% annually, predict the population of Mexico in 10 and 20 years.

CONTINUOUS EXPONENTIAL GROWTH/DECAY

N = N0ert

N0 = starting amount r = rate (decimal) t = time Use it when you see “continuous

growth or decay”

CONTINUOUS EXPONENTIAL GROWTH/DECAY EXAMPLE

The population of a town is declining at a rate of 6%. If the current population is 12,426 people, predict the population in 5 and 10 years using the continuous model.

HALF LIFE

A = P(1/2)t/HL

P = Principal (starting amount) t = # of years HL = half life Use it when: you see the words “half

life” in the problem

HALF LIFE EXAMPLE

The half life of a certain radioactive substance is 20 days and there are 5 grams present initially. How much will be left after 30 days?

DOUBLING TIME (“DOUBLE LIFE”)

A = P(2)t/DT

P = Principal (starting amount) t = time DT = doubling time Use it when you see the words

“doubling time”

DOUBLING TIME EXAMPLE

During springtime, the rabbit population of a certain forest has a doubling time of 40 days. Suppose the forest contains 100 rabbits to begin with. How many rabbits will be in the forest after 25 days, and 90 days?

PRACTICE PROBLEMS

Use your notes and study guide to help you complete the practice problems on the back of your sheet.

p. 166 # 21, 23, 25, 31, 36, 37 – 40 What you don’t finish you must finish

for homework.