notes #3-1: exponential and logistic...
TRANSCRIPT
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)