Chapter 2: Functions and Exponential Models

Lesson 5: Exponential Models

Mrs. Parziale

Do Now

• Pick up the M&M Activity Sheet and read through it carefully to understand what you will be doing.

• You will partner with another person in doing this activity.

• If necessary, move your seat to sit with your partner.

Finding Exponential Models

• In the past, we have been given the exponential model (function notation) and been asked to find the value of the function. Today, we are given values of the function and attempting to find the exponential model.

Vocabulary

• Initial starting value: in the exponential function, “a” is the starting value.

• Growth factor: in the exponential function, “b” is the growth factor (by how much the function is growing).

• Exponential growth: when a > 0 and b > 1• Exponential decay: when a > 0 and 0 <b < 1

xy ab

More Terms

• Doubling time: the amount of time it takes for a quantity to double

• Half-life: the amount of time it takes to half the original amount.

Finding an Exponential Model

• When 2 points are available – Use the By Hand Method to determine the model.

• Example 1: Suppose algae grow in a pond exponentially. There are 100 algae, and 3 hours later, there are 200. Fit an exponential model to this data. Show work. (Use 2 points)

Example 2:

• A certain substance has a half-life of 24 years. If a sample of 80 grams is being observed,

(a) How much will remain in 50 years? _________________________

(b) When will only 5 gram remain?

____________________________

Example 3:

A substance has a doubling time of 8 hours. Suppose you start with 3 grams.

(a) How much will you have after 2 full days? ___________________

(b) How long will it take to reach 384 grams? _____________________

(2)time

DoublingTimeInitialValue

Example 4: • The half-life of a certain radioactive substance

is 40 days. If 10 grams of the substance are present initially, how much of the substance will be present in 90 days?

Number of Half-Life Periods 0 1 2 3

t = Number of Days After Decay Starts

0 40 80 120

f(t) = Amount Present (grams) 10 5 2.5 1.25

1

2

time

HalfLifeInitialValue

b. Calculate the Exponential Model for this data.c. What is the initial value? What is the growth

factor?d. Answer the question being presented: How much

of the substance will be present after 90 days?

Number of Half-Life Periods 0 1 2 3

t = Number of Days After Decay Starts

0 40 80 120

f(t) = Amount Present (grams) 10 5 2.5 1.25

Example 5:

• After injecting 14 mg of anesthetic into the bloodstream of a lab rat, observers monitored how much was left in the bloodstream each hour. The data is below:

Time (hr.) 0 1 2 3 4 5 6 7 8 9

Anesthetic (mg)

14 9.38 6.28 4.21 2.82 1.89 1.27 .85 .57 .38

(a) Use your calculator to make a scatterplot

(b) Fit a linear model to this data: ____________________ r = _______

(c) Fit an exponential model: _________________________ r = _______

(d) Which fits better? ___________________________

(e) How much anesthetic will be left after 12 hours? _________________Is this interpolation or extrapolation?

(f) How much anesthetic will be left after 5.5 hours? _________________ Is this interpolation or extrapolation?

Time (hr.) 0 1 2 3 4 5 6 7 8 9

Anesthetic (mg)

14 9.38 6.28 4.21 2.82 1.89 1.27 .85 .57 .38

Closure

• In an exponential function, – What is the “a”?– What is the “b”?– What is half life?– What is doubling?– Given the following function,

a) Starting point? b) growth or decay?

1( ) 24

2

x

f x

( ) xf x ab