Notes #3-1: Exponential and Logistic Functions Go to page 252 and begin reading at the chapter overview.

In this chapter we explore three interrelated families of functions: ___________________________

_____________________, and __________________________ functions.

Exponential functions model _____________ and _____________ over time, such as

__________________________ population growth and ______________ of radioactive substances.

Logistic functions model ____________________ population growth, certain chemical reactions,

and the ________________ of ___________________ and diseases.

Logarithmic functions are the basis of the __________________ ________________ of earthquake

intensity, the pH acidity scale, and the ___________________________ measurement of sound.

Pg. 252 “exponential functions and their graphs”

Exponential Functions and Their Graphs

The functions _____________ and _____________ each involve a base raised to a power, but the roles

are reversed:

For _________________, the base is the _______________ , and the exponent is the ___________ ;

is the familiar monomial and _____________ function

For _________________, the base is the ______________ , and the exponent is the

_______________, ; is an _________________________ function. See Figure 3.1

Sketch figure 3.1 below

DEFINITION Exponential Functions

Let ____ and ____ be ________ number constants. An exponential function in ____ is a function that

can be written in the form:

Where _____ is a ___________, ____ is _________, and _____ .

The ____________ is the initial value of (the value at _____________); and is the base.

Remember

this for ex.3

Example 1: Identifying Exponential Functions

Determine if the following are exponential functions. If it is not, provide a reason why. If it is, identify

the initial value and base.

a.

Yes no

Reason for no:

Initial value: _______ base: _______

b.

Yes no

Reason for no:

Initial value: _______ base: _______

c.

Yes no

Reason for no:

Initial value: _______ base: _______

d.

Yes no

Reason for no:

Initial value: _______ base: _______

Example 2: Computing Exponential Function Values for Rational Number Inputs

For NO CALCULATOR

a. b.

c. d.

e.

f.

We can write an exponential function by looking at the table of values.

We will solve for and !

Watch a

video of

this

example

http://whs1

314pc.wee

bly.com

notes >>

chapter 3

part a >>

notes #3a-

1 video

Basic form of

exponential function:

Plug in Remember, is the value

when

Use the other point for and ….

is a fancy way of

writing , so the point , and

Solve for

Example 3: Finding an Exponential Function from Its Table of Values

Determine formulas ( ) for the exponential functions and whose values are given in the

table below.

(a) (b)

Go to page 254 and look at the very bottom of the page for the definition:

DEFINITION Exponential Growth and Decay

For _______ exponential function _____________ and any real number ,

Use this definition:

If ____ 0 and ____ 1, the function is _____________________ and is an exponential growth function.

The base is its _________________ __________________.

If ____ 0 and ____ 1, the function is _____________________ and is an exponential decay function.

The base is its _________________ __________________.

Remember from our definition of an exponential function, must be positive. For all exponential

decay functions; .

Look at the graphs at the bottom of page 255. Sketch both graphs, and the information below:

Exponential Growth

is greater than 1

Exponential Decay

is positive, but less than1

Sketch

Label both sets of points

Coordinates of points _____) and _____) _____) and _____)

Each exponential

function, passes

through the

point and

!

Example 4: Growth or decay

(a)

(b)

(c) (d)

How do you know? How do you know? How do you know? How do you know?

Properties of Exponents (pg. 7)

Let ___ and ___ be ________ numbers, variables, or algebraic expressions and and be -

________________. All bases are assumed to be nonzero.

Property Example

1.

2.

3.

4.

5.

6.

7.

The exponent says how many times to use the number in multiplication.

A negative exponent means divide, because the opposite of multiplying is dividing

A fractional exponent like

means take the root:

Notes #3-2: Exponential and Logistic Functions day 2 (pgs. 258&260)

Today we are going to work with transformations of exponential functions.

Identifying and

+

is the growth/decay rate

is the transformation

moves horizontal

asymptote

“Parent” Function

Horizontal asymptote @

Because (growth/decay rate)

varies in exponential functions,

there is no “true parent” function

today.

ID base

ID

ID

ID

ID

Match the following terms to or

Moves function

up/down

Vertical

stretches/shrinks

Multiply by

reciprocal

Moves function

left/right

Add/subtract

to

Flips axis

Horizontal

stretches/shrinks

Add/subtract

to

Multiply Flips axis

Example 1: Transforming Exponential Functions

Transform each function; provide a final table of values and equation of the asymptote.

a.

b.

c.

We can solve for …. Yes it is painful, but it can be done

Example 2: Solving for (example 3 from notes #3a-1)

Find for the following exponential functions

(a) (b) (c) (d) *

Bonus

We can use exponential functions to describe unrestricted population growth.

We use the form: ; where is the initial value, and is the

growth/decay rate.

Example 3: Modeling Columbus’ and Indianapolis’s population

Find the growth factor for Columbus’ population Find the growth factor for Indianapolis’s’

population

Columbus’ population equation Indianapolis’ population equation

Solve graphically ….

Use the population equation to find when Columbus’ population will pass

850,000 people.

What does represent

in the equation?

____________

What does represent

in the equation?

____________

Use the population equation to find when Indianapolis’ population will pass

850,000 people.

What does represent

in the equation?

____________

What does represent

in the equation?

____________

Columbus Population

Indianapolis Population

Notes #3-3: Exponential & Logistic Functions (day 3) pgs. 257, 259-60

Describe the transformations of the function:

1. 2. 3.

Go to page 257 and start reading …..

I. The natural base

a. The letter is the initial of the last name of Leonhard Euler

(pronounced oiler) who introduced the notation.

b. Because has special calculus properties that simplify

many calculations, is the __________________ of exponential functions for calculus

purposes and is considered the _______________________________________.

DEFINITION The natural base

Type into your calculator, what is the decimal approximation

of (round to the hundredths)

We are usually more interested in the exponential function ________________ and variations of

this function than in the __________________ number . In fact, any exponential function can

be expressed in terms of the _____________________ .

Comparing the exponential function and the Natural Exponential Function

Exponential Function THE NATURAL exponential Function

Domain: Domain: Range: Range:

Horizontal Asym @ Horizontal Asym @

Moved by

THEOREM Exponential Functions and the Base

Any exponential function ( ) can be rewritten as

For an appropriately chosen real number constant .

If _____ 0 and _____ 0 , then it is an exponential growth function

If _____ 0 and _____ 0 , the it is an exponential decay function

Sketch Sketch

This is ________________ because ___ This is ______________ because ___

Equation of asympotote:_____________ Equation of asymptote: _____________

Example 1: Identifying growth or decay

a. Sketch below Domain: Range:

Continuous: yes Inc/Dec:

Asymptote:

End Behavior:

ID : _________

Growth or decay? _________

b. Sketch below Domain: Range:

Continuous: yes Inc/Dec:

Asymptote:

End Behavior:

ID : ________

Growth or decay? _________

c. Sketch below Domain: Range:

Continuous: yes Inc/Dec:

Asymptote:

End Behavior:

ID : ________

Growth or decay? _________

We learn how to find

in section 3.3

Example 2: Transformations of THE exponential function

Provide a final table of values, and the final equation of the asymptote.

From Notes #3-2, fill in everything YOU need for transformations of exponential functions:

Start reading again on page 258, under logistic functions and their graphs (in red).

Exponential growth is __________________. An exponential function increases/decreases at an

ever-increasing rate and is not bounded above. In many growth situations, however,

_______________________________________________________. A plant can only grow so tall. The

number of goldfish in an aquarium is limited by the size of the aquarium. In such situations,

growth often begins in an __________________ manner, but the growth eventually slows and

the graph levels out. The associated growth function is bounded both ______________ and

_______________by horizontal asymptotes.

Logistic Growth Functions – RESTRICTED GROWTH

DEFINITION Logistic Growth Function

Let and be positive constants, with . A logistic growth function in is a function

that can be written of the form:

__________________________________ or __________________________

where the constant is the limit to growth

If ___ 0 or these formulas yield logistic ______________ functions ….. conversely

If ___ 0 or these formulas yield logistic ______________ functions (not in book)

Example 3: Graphing Logistic Growth Functions

Find the y-intercept and horizontal asymptotes. Use your grapher to confirm your answer!

(pg. 259).

a.

b.

The Logistic Function

Domain: Range:

Horizontal Asymptotes: and

No vertical asymptotes

End Behavior:

is from

the

equation

of the

function!

Example 4: Restricted Population Growth (pg. 261)

While Columbus’ population is soaring, other major cities, such as Dallas, the population is

slowing. The once sprawling Dallas is now constrained by its neighboring cities.

A logistic function is often an appropriate model for restricted growth, such as the growth

Dallas is experience.

Based on recent census data, a logistic model for the population of Dallas, years after

1900, is modeled by the above equation.

When will the population reach 1 million? SOLVE GRAPHICALLY!

What is the maximum population Dallas can reach? SOLVE GRAPHICALLY/LOOK @

EQUATION

Notes #3-4/#3-5: Exponential and Logistic Modeling

1. Graph the function, list the transformations, provide a final table of values and the

equation of the asymptote.

Notes #3-1 - #3-3 (LOOK IT UP!)

Exponential Perfect Exponential

What is ? What is ?

When is it growth? When is it growth?

When is it decay? When is it decay?

Unrestricted population growth … is exponential

Exponential Population

Model If a population, , is changing at a constant

percentage rate each year, then

Where is the initial population

is expressed as a decimal,

And is time in years.

If , then is an exponential growth

function. The growth factor is the base of

the exponential function,

If , then is an exponential decay

function. The decay factor is the base of

the exponential function,

Special Growth - Doubling When a population doubles the growth

rate is 100% or 1

is divided by the doubling time

Special Decay – Half Life When a population is cut in half (half life)

the decay rate is -50% or –0.50 is divided by the half life

Example 1: Finding growth and decay rates

Tell if the population model is growth or decay, and find the constant percentage rate of

growth or decay.

a. Phoenix:

Growth or decay?

=

b. Dallas:

Growth or decay?

=

Example 2: Finding an exponential function

Initial value: 12, increasing at a rate of 8% per year

Example 3: Modeling Bacteria Growth

Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour.

Predict when the number of bacteria will be 350,000. (pg. 266)

Example 3b: Modeling bacteria growth

You pick up a pencil and contract 100 bacteria containing the flu virus. The bacteria double

every three hours. Predict the day you will have flu symptoms (when the bacteria reach a

population of 350,000)

Viewing Window:

[-5x100] by

[100000x400000]

35.32 hours; 1.47 days

from picking up the

pencil

a. Growth; 0.78%

b. Decay; -4.08%

Example 4: Modeling Radioactive Decay

Suppose the half-life of a certain radioactive substance is 20 days and there are 4 grams

present initially. Find the time when there will be 1 gram of the substance remaining. (pg.

266-67).

Example 4b: Modeling Radioactive Decay

Suppose the half-life of a certain radioactive substance is 16 days and there are 25 g

(grams) present initially. Find the time when there will be 2 g of the substance remaining.

Logistic Functions and Population

Logistic – population growth that is restricted/limited

What is ? When is the function representing decay?

Where are the asymptotes? When does the function representing

growth?

How do you find the y-int?

Find the -intercept and horizontal asymptotes of the following logistic functions

In 58.30 days there will

be 2 grams remaining.

Example 1: Writing Logistic Functions

a. Initial Value =

Limit to Growth =

Passing through

b.

A#3-4:

#19 and #21 want an exponential function like ; look at Notes #3-1, ex #3

Corrected answers to book: #17:

Plug in Knowns Find Find *use initial value *use passing through

(

Plug in Knowns Find Find *use initial value *use passing through

is NOT THE INITIAL VALUE!!!!!

We cannot find it by looking at the graph!

Notes #3-6 Logarithmic Functions and their Graphs

If a function passes the horizontal line test, then the inverse; is also a function.

The inverse of an exponential function, is the logarithmic function with base .

When we find the inverse of the function we switch and !

Comparison of an exponential function and it’s inverse

EXPONENTIAL FUNCTION LOGARITHMIC FUNCTION

-intercept: DNE -intercept:

-intercept: -intercept: DNE

Domain: Domain:

Range: Range:

Horizontal Asymptote: ; Horizontal Asymptote: DNE

Vertical Asymptote: DNE Vertical Asymptote:

A logarithm is a form of an exponent, and uses exponent rules!

Function: Inverse:

Exponential Form

Logarithmic Form

Example 1: Changing from logarithmic form to exponential form

a.

b.

c. d.

Example 2: Changing from Exponential form to logarithmic form

a. b.

c.

d.

When working with logarithms – switch between the two forms to solve!

Base stays the same … switch and

Exponent is an

exponent

Exponent is

the answer

Example 3: Finding the exact value of a logarithm

Find the exact value of

a. b. c.

Solution (a):

The logarithm has some value. Let’s call it .

Change from logarithmic to exponential form.

3 raised to what power is 81?

Solution (b):

The logarithm has some value. Let’s call it x.

Change from logarithmic to exponential form.

169 raised to what power is 13?

Solution (c):

The logarithm has some value. Let’s call it .

Change from logarithmic to exponential form.

5 raised to what power is

?

Basic Properties of Logarithms

LOGARITHMIC FORM EXPONENTIAL

FORM

Example 5: Using logarithm laws

a. b. c. d. e.

When working with logarithms – switch between the two forms to solve!

Example 5: Solving simple logarithmic equations

a. b.

c. d.

Common and Natural Logarithms

a. 2 common bases are 10 and

i. Common logarithmic function:

1.

ii. Natural logarithmic function:

1.

Example 5: Using a calculator to evaluate common and natural logarithms

Round answers to thousandths place.

a. b.

c. d.

e. f.

Your calculator uses common

and natural log when making

calculations!

Notes #3-7: Logarithmic Functions and Their Graphs (guided)

Rewrite in terms of or as the base on each side

a. b. c.

d.

e.

f.

Evaluate the logarithmic expression without using a calculator

(notes #3-6 ex #3/5)

Sketch a graph by hand, provide a final table of values and the equation of the asymptote

CHANGE FORMS WHEN

SOLVING!

Exp Log

Log Exp

Convert to have same base on

each side … rewrite!

Remember,

will move

the

horizontal

asymptotes

Exponential Form Logarithmic Form

EXPONENTIAL FUNCTION LOGARITHMIC FUNCTION

-intercept: -intercept:

-intercept: -intercept:

Domain: Domain:

Range: Range:

Horizontal Asymptote: Horizontal Asymptote:

Equation of HA:

Vertical Asymptote: Vertical Asymptote:

Equation of VA:

Sketch Sketch

TRANSFORMATIONS FOR

LOGARITHMIC FUNCTIONS

*Horizontal shifts move VA*

TABLE OF VALUES FOR

LOGARITHMIC FUNCTIONS

EXPONENTIAL table of

values

LOGARITHMIC functions are

inverses; flip and

Example 1: Identifying and base

ID base

ID

ID

ID

ID

Example 1: Graphing Logarithmic Functions by Hand

Sketch the function. Provide a final table of values

a. b.

c. d.

Example 2: Analyzing functions

Graph the following functions (by hand), then analyze the graphs for the following

information:

a.

1. Domain 2. Range

3. -intercept

4. Increasing or decreasing behavior

5. Extrema 6. Symmetry

7. Asymptote (equation)

8. End behavior (use limit notation)

b. 1. Domain 2. Range

3. -intercept

4. Increasing or decreasing behavior

5. Extrema 6. Symmetry

7. Asymptote (equation)

8. End behavior (use limit notation)