SlideShare uma empresa Scribd logo
1 de 24
Baixar para ler offline
Exponential Functions
Section 4.1
JMerrill, 2005
Revised 2008
Definition of Exponential Functions
The exponential function f with a base
b is defined by f(x) = bx
where b is a
positive constant other than 1 (b > 0,
and b ≠ 1) and x is any real number.
So, f(x) = 2x
, looks like:
Graphing Exponential Functions
Four exponential
functions have
been graphed.
Compare the
graphs of functions
where b > 1 to
those where b < 1
2x
y =
7x
y =
1
2
x
y =
1
7
x
y =
Graphing Exponential Functions
So, when b > 1, f(x)
has a graph that
goes up to the right
and is an
increasing function.
When 0 < b < 1,
f(x) has a graph
that goes down to
the right and is a
decreasing
function.
Characteristics
The domain of f(x) = bx
consists of all real
numbers (-∞, ∞). The range of f(x) = bx
consists of all positive real numbers
(0, ∞).
The graphs of all exponential functions
pass through the point (0,1). This is
because f(o) = b0
= 1 (b≠o).
The graph of f(x) = bx
approaches but
does not cross the x-axis. The x-axis is a
horizontal asymptote.
f(x) = bx
is one-to-one and has an inverse
that is a function.
Transformations
Vertical
translation
f(x) = bx
+ c
Shifts the graph up
if c > 0
Shifts the graph
down if c < 0
2x
y =
2 3x
y = +
2 4x
y = −
Transformations
Horizontal
translation:
g(x)=bx+c
Shifts the graph to
the left if c > 0
Shifts the graph to
the right if c < 0
2x
y =
( 3)
2 x
y +
=
( 4)
2 x
y −
=
Transformations
Reflecting
g(x) = -bx
reflects
the graph about the
x-axis.
g(x) = b-x
reflects
the graph about the
y-axis.
2x
y =
2x
y = −
2 x
y −
=
Transformations
Vertical
stretching or
shrinking,
f(x)=cbx
:
Stretches the
graph if c > 1
Shrinks the graph if
0 < c < 1
2x
y =
4(2 )x
y =
1
(2 )
4
x
y =
Transformations
Horizontal
stretching or
shrinking, f(x)=bcx
:
Shinks the graph if
c > 1
Stretches the
graph if 0 < c < 1
2x
y =
4(2 )x
y =
1
(2 )
4
x
y =
You Do
Graph the function f(x)
= 2(x-3)
+2
Where is the horizontal
asymptote?
y = 2
You Do, Part Deux
Graph the function f(x)
= 4(x+5)
- 3
Where is the horizontal
asymptote?
y = - 3
The Number e
The number e is known as Euler’s number.
Leonard Euler (1700’s) discovered it’s
importance.
The number e has physical meaning. It
occurs naturally in any situation where a
quantity increases at a rate proportional to
its value, such as a bank account producing
interest, or a population increasing as its
members reproduce.
The Number e - Definition
An irrational number, symbolized by the letter
e, appears as the base in many applied
exponential functions. It models a variety of
situations in which a quantity grows or decays
continuously: money, drugs in the body,
probabilities, population studies, atmospheric
pressure, optics, and even spreading rumors!
The number e is defined as the value that
approaches as n gets larger and larger.
1
1
n
n
 
+ ÷
 
The Number e - Definition
n
1 2
2 2.25
5 2.48832
10 2.59374246
100 2.704813829
1000 2.716923932
10,000 2.718145927
100,000 2.718268237
1,000,000 2.718280469
1,000,000,000 2.718281827
1
1
n
n
 
+ ÷
 
0
1
1
n
A
n
 
+ ÷
 
1
, 1
n
As n e
n
 
→∞ + → ÷
 
The table shows
the values of
as n gets
increasingly large.
n → ∞As , the
approximate
value of e (to 9
decimal places)
is ≈
2.718281827
The Number e - Definition
For our purposes,
we will use
e ≈ 2.718.
e is 2nd
function on
the division key on
your calculator.
y = e
1
1
n
y
n
 
= + ÷
 
The Number e - Definition
Since 2 < e < 3, the
graph of y = ex
is
between the
graphs of y = 2x
and y = 3x
ex
is the 2nd
function
on the ln key on
your calculator
y =e
y = 2x
y = 3x
y = ex
Natural Base
The irrational number e, is called the
natural base.
The function f(x) = ex
is called the
natural exponential function.
Compound Interest
The formula for compound interest:
( ) 1
 
= + ÷
 
nt
r
A t P
n
Where n is the number of times per
year interest is being compounded
and r is the annual rate.
Compound Interest - Example
Which plan yields the most interest?
Plan A: A $1.00 investment with a 7.5% annual
rate compounded monthly for 4 years
Plan B: A $1.00 investment with a 7.2% annual
rate compounded daily for 4 years
A:
B:
12(4)
0.075
1 1 1.3486
12
 
+ ≈ ÷
 
365(4)
0.072
1 1 1.3337
365
 
+ ≈ ÷
 
$1.35
$1.34
Interest Compounded Continuously
If interest is compounded “all the
time” (MUST use the word
continuously), we use the formula
where P is the initial principle (initial
amount)
( ) = rt
A t Pe
( ) = rt
A t Pe
If you invest $1.00 at a 7% annual
rate that is compounded
continuously, how much will you have
in 4 years?
You will have a whopping $1.32 in 4
years!
(.07)(4)
1* 1.3231e ≈
You Do
You decide to invest $8000 for 6
years and have a choice between 2
accounts. The first pays 7% per year,
compounded monthly. The second
pays 6.85% per year, compounded
continuously. Which is the better
investment?
You Do Answer
1st
Plan:
2nd Plan:
0.0685(6)
(6) 8000 $12,066.60P e= ≈
12(6)
0.07
(6) 8000 1 $12,160.84
12
A
 
= + ≈ ÷
 

Mais conteúdo relacionado

Mais procurados

Rational functions
Rational functionsRational functions
Rational functionszozima
 
Exponential equations
Exponential equationsExponential equations
Exponential equationsnovember12
 
8.4 logarithmic functions
8.4 logarithmic functions8.4 logarithmic functions
8.4 logarithmic functionshisema01
 
1) exponential equations
1) exponential equations1) exponential equations
1) exponential equationsestelav
 
2.8 Absolute Value Functions
2.8 Absolute Value Functions2.8 Absolute Value Functions
2.8 Absolute Value Functionshisema01
 
Exponential functions
Exponential functionsExponential functions
Exponential functionsRon Eick
 
Inverse functions
Inverse functionsInverse functions
Inverse functionsPLeach
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functionsNjabulo Nkabinde
 
Exponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptxExponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptxRoqui Gonzaga
 
Absolute value functions
Absolute value functionsAbsolute value functions
Absolute value functionsJessica Garcia
 
Piecewise functions
Piecewise functionsPiecewise functions
Piecewise functionsLori Rapp
 
General Mathematics - Rational Functions
General Mathematics - Rational FunctionsGeneral Mathematics - Rational Functions
General Mathematics - Rational FunctionsJuan Miguel Palero
 
Logarithmic Functions
Logarithmic FunctionsLogarithmic Functions
Logarithmic Functionsswartzje
 
Math presentation on domain and range
Math presentation on domain and rangeMath presentation on domain and range
Math presentation on domain and rangeTouhidul Shawan
 
Functions and Relations
Functions and RelationsFunctions and Relations
Functions and Relationssheisirenebkm
 
Composition Of Functions
Composition Of FunctionsComposition Of Functions
Composition Of Functionssjwong
 
Evaluating functions
Evaluating functionsEvaluating functions
Evaluating functionsEFREN ARCHIDE
 

Mais procurados (20)

One-to-one Functions.pptx
One-to-one Functions.pptxOne-to-one Functions.pptx
One-to-one Functions.pptx
 
Rational functions
Rational functionsRational functions
Rational functions
 
Exponential equations
Exponential equationsExponential equations
Exponential equations
 
8.4 logarithmic functions
8.4 logarithmic functions8.4 logarithmic functions
8.4 logarithmic functions
 
1) exponential equations
1) exponential equations1) exponential equations
1) exponential equations
 
Math12 lesson11
Math12 lesson11Math12 lesson11
Math12 lesson11
 
2.8 Absolute Value Functions
2.8 Absolute Value Functions2.8 Absolute Value Functions
2.8 Absolute Value Functions
 
Exponential functions
Exponential functionsExponential functions
Exponential functions
 
Inverse functions
Inverse functionsInverse functions
Inverse functions
 
7 functions
7   functions7   functions
7 functions
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functions
 
Exponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptxExponential Equation & Inequalities.pptx
Exponential Equation & Inequalities.pptx
 
Absolute value functions
Absolute value functionsAbsolute value functions
Absolute value functions
 
Piecewise functions
Piecewise functionsPiecewise functions
Piecewise functions
 
General Mathematics - Rational Functions
General Mathematics - Rational FunctionsGeneral Mathematics - Rational Functions
General Mathematics - Rational Functions
 
Logarithmic Functions
Logarithmic FunctionsLogarithmic Functions
Logarithmic Functions
 
Math presentation on domain and range
Math presentation on domain and rangeMath presentation on domain and range
Math presentation on domain and range
 
Functions and Relations
Functions and RelationsFunctions and Relations
Functions and Relations
 
Composition Of Functions
Composition Of FunctionsComposition Of Functions
Composition Of Functions
 
Evaluating functions
Evaluating functionsEvaluating functions
Evaluating functions
 

Destaque

5 4 the number e
5 4 the number e5 4 the number e
5 4 the number ehisema01
 
Cal2 ba dinh_hai_slides_ch1
Cal2 ba dinh_hai_slides_ch1Cal2 ba dinh_hai_slides_ch1
Cal2 ba dinh_hai_slides_ch1Khoa Đăng
 
Tutorials--Exponential Functions in Tabular and Graph Form
Tutorials--Exponential Functions in Tabular and Graph FormTutorials--Exponential Functions in Tabular and Graph Form
Tutorials--Exponential Functions in Tabular and Graph FormMedia4math
 
Exponential functions
Exponential functionsExponential functions
Exponential functionskvillave
 
2.2 exponential function and compound interest
2.2 exponential function and compound interest2.2 exponential function and compound interest
2.2 exponential function and compound interestmath123c
 
Lesson 3.1
Lesson 3.1Lesson 3.1
Lesson 3.1kvillave
 
Simple interest
Simple interestSimple interest
Simple interestAnju Soman
 
61 exponential functions
61 exponential functions61 exponential functions
61 exponential functionsmath126
 
The Exponential and natural log functions
The Exponential and natural log functionsThe Exponential and natural log functions
The Exponential and natural log functionsJJkedst
 
Lesson 14: Exponential Functions
Lesson 14: Exponential FunctionsLesson 14: Exponential Functions
Lesson 14: Exponential FunctionsMatthew Leingang
 
Compound Interest
Compound InterestCompound Interest
Compound Interestitutor
 
Properties of logarithms
Properties of logarithmsProperties of logarithms
Properties of logarithmsJessica Garcia
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functionsawesomepossum7676
 
Simple and compound interest student
Simple and compound interest studentSimple and compound interest student
Simple and compound interest studentleblanjo
 

Destaque (20)

5 4 the number e
5 4 the number e5 4 the number e
5 4 the number e
 
Cal2 ba dinh_hai_slides_ch1
Cal2 ba dinh_hai_slides_ch1Cal2 ba dinh_hai_slides_ch1
Cal2 ba dinh_hai_slides_ch1
 
Tutorials--Exponential Functions in Tabular and Graph Form
Tutorials--Exponential Functions in Tabular and Graph FormTutorials--Exponential Functions in Tabular and Graph Form
Tutorials--Exponential Functions in Tabular and Graph Form
 
Exponential functions
Exponential functionsExponential functions
Exponential functions
 
2.2 exponential function and compound interest
2.2 exponential function and compound interest2.2 exponential function and compound interest
2.2 exponential function and compound interest
 
Hprec5 4
Hprec5 4Hprec5 4
Hprec5 4
 
Lesson 3.1
Lesson 3.1Lesson 3.1
Lesson 3.1
 
Simple interest
Simple interestSimple interest
Simple interest
 
61 exponential functions
61 exponential functions61 exponential functions
61 exponential functions
 
Indices & logarithm
Indices & logarithmIndices & logarithm
Indices & logarithm
 
Hprec5 5
Hprec5 5Hprec5 5
Hprec5 5
 
The Exponential and natural log functions
The Exponential and natural log functionsThe Exponential and natural log functions
The Exponential and natural log functions
 
Lesson 14: Exponential Functions
Lesson 14: Exponential FunctionsLesson 14: Exponential Functions
Lesson 14: Exponential Functions
 
Compound Interest
Compound InterestCompound Interest
Compound Interest
 
Simple Interest
Simple InterestSimple Interest
Simple Interest
 
Properties of logarithms
Properties of logarithmsProperties of logarithms
Properties of logarithms
 
Compound Interest
Compound InterestCompound Interest
Compound Interest
 
Simple interest
Simple interestSimple interest
Simple interest
 
Exponential and logarithmic functions
Exponential and logarithmic functionsExponential and logarithmic functions
Exponential and logarithmic functions
 
Simple and compound interest student
Simple and compound interest studentSimple and compound interest student
Simple and compound interest student
 

Semelhante a 4.1 exponential functions 2

exponential functions (copied)
exponential functions (copied)exponential functions (copied)
exponential functions (copied)hossameldeen ahmed
 
2nd-year-Math-full-Book-PB.pdf
2nd-year-Math-full-Book-PB.pdf2nd-year-Math-full-Book-PB.pdf
2nd-year-Math-full-Book-PB.pdfproacademyhub
 
8.2 Exploring exponential models
8.2 Exploring exponential models8.2 Exploring exponential models
8.2 Exploring exponential modelsswartzje
 
Exponential_Functions.ppt
Exponential_Functions.pptExponential_Functions.ppt
Exponential_Functions.pptLynSumonod1
 
grph_of_polynomial_fnctn.ppt
grph_of_polynomial_fnctn.pptgrph_of_polynomial_fnctn.ppt
grph_of_polynomial_fnctn.pptLunaLedezma3
 
Exponential_Functions.ppt
Exponential_Functions.pptExponential_Functions.ppt
Exponential_Functions.pptHasanAhtesham2
 
logarithmic, exponential, trigonometric functions and their graphs.ppt
logarithmic, exponential, trigonometric functions and their graphs.pptlogarithmic, exponential, trigonometric functions and their graphs.ppt
logarithmic, exponential, trigonometric functions and their graphs.pptYohannesAndualem1
 
7.2 Alg 2 exploring exponential models
7.2 Alg 2 exploring exponential models7.2 Alg 2 exploring exponential models
7.2 Alg 2 exploring exponential modelsswartzje
 
237654933 mathematics-t-form-6
237654933 mathematics-t-form-6237654933 mathematics-t-form-6
237654933 mathematics-t-form-6homeworkping3
 
Exponential functions
Exponential functionsExponential functions
Exponential functionskvillave
 
phuong trinh vi phan d geometry part 2
phuong trinh vi phan d geometry part 2phuong trinh vi phan d geometry part 2
phuong trinh vi phan d geometry part 2Bui Loi
 
Module1 exponential functions
Module1  exponential functionsModule1  exponential functions
Module1 exponential functionsdionesioable
 
Exponential functions
Exponential functionsExponential functions
Exponential functionsNagendrasahu6
 
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docxMATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docxTatianaMajor22
 

Semelhante a 4.1 exponential functions 2 (20)

exponential functions (copied)
exponential functions (copied)exponential functions (copied)
exponential functions (copied)
 
3.1 2
3.1 23.1 2
3.1 2
 
Maths 12
Maths 12Maths 12
Maths 12
 
2nd-year-Math-full-Book-PB.pdf
2nd-year-Math-full-Book-PB.pdf2nd-year-Math-full-Book-PB.pdf
2nd-year-Math-full-Book-PB.pdf
 
2018-G12-Math-E.pdf
2018-G12-Math-E.pdf2018-G12-Math-E.pdf
2018-G12-Math-E.pdf
 
8.2 Exploring exponential models
8.2 Exploring exponential models8.2 Exploring exponential models
8.2 Exploring exponential models
 
Exponential_Functions.ppt
Exponential_Functions.pptExponential_Functions.ppt
Exponential_Functions.ppt
 
grph_of_polynomial_fnctn.ppt
grph_of_polynomial_fnctn.pptgrph_of_polynomial_fnctn.ppt
grph_of_polynomial_fnctn.ppt
 
Exponential_Functions.ppt
Exponential_Functions.pptExponential_Functions.ppt
Exponential_Functions.ppt
 
logarithmic, exponential, trigonometric functions and their graphs.ppt
logarithmic, exponential, trigonometric functions and their graphs.pptlogarithmic, exponential, trigonometric functions and their graphs.ppt
logarithmic, exponential, trigonometric functions and their graphs.ppt
 
Graph a function
Graph a functionGraph a function
Graph a function
 
7.2 Alg 2 exploring exponential models
7.2 Alg 2 exploring exponential models7.2 Alg 2 exploring exponential models
7.2 Alg 2 exploring exponential models
 
237654933 mathematics-t-form-6
237654933 mathematics-t-form-6237654933 mathematics-t-form-6
237654933 mathematics-t-form-6
 
Exponential functions
Exponential functionsExponential functions
Exponential functions
 
phuong trinh vi phan d geometry part 2
phuong trinh vi phan d geometry part 2phuong trinh vi phan d geometry part 2
phuong trinh vi phan d geometry part 2
 
Unit 3.1
Unit 3.1Unit 3.1
Unit 3.1
 
Module1 exponential functions
Module1  exponential functionsModule1  exponential functions
Module1 exponential functions
 
Exponential functions
Exponential functionsExponential functions
Exponential functions
 
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docxMATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docx
 
Unit 2.6
Unit 2.6Unit 2.6
Unit 2.6
 

Último

UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6DianaGray10
 
OpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability AdventureOpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability AdventureEric D. Schabell
 
Cybersecurity Workshop #1.pptx
Cybersecurity Workshop #1.pptxCybersecurity Workshop #1.pptx
Cybersecurity Workshop #1.pptxGDSC PJATK
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024SkyPlanner
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
 
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfJamie (Taka) Wang
 
Designing A Time bound resource download URL
Designing A Time bound resource download URLDesigning A Time bound resource download URL
Designing A Time bound resource download URLRuncy Oommen
 
Videogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdfVideogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdfinfogdgmi
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1DianaGray10
 
Comparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and IstioComparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and IstioChristian Posta
 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-pyJamie (Taka) Wang
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxUdaiappa Ramachandran
 
UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7DianaGray10
 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8DianaGray10
 
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Will Schroeder
 
Machine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdfMachine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdfAijun Zhang
 
VoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXVoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXTarek Kalaji
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesMd Hossain Ali
 
Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™Adtran
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationIES VE
 

Último (20)

UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6
 
OpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability AdventureOpenShift Commons Paris - Choose Your Own Observability Adventure
OpenShift Commons Paris - Choose Your Own Observability Adventure
 
Cybersecurity Workshop #1.pptx
Cybersecurity Workshop #1.pptxCybersecurity Workshop #1.pptx
Cybersecurity Workshop #1.pptx
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
 
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
 
Designing A Time bound resource download URL
Designing A Time bound resource download URLDesigning A Time bound resource download URL
Designing A Time bound resource download URL
 
Videogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdfVideogame localization & technology_ how to enhance the power of translation.pdf
Videogame localization & technology_ how to enhance the power of translation.pdf
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1
 
Comparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and IstioComparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and Istio
 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-py
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptx
 
UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7UiPath Studio Web workshop series - Day 7
UiPath Studio Web workshop series - Day 7
 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8
 
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
 
Machine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdfMachine Learning Model Validation (Aijun Zhang 2024).pdf
Machine Learning Model Validation (Aijun Zhang 2024).pdf
 
VoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXVoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBX
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
 
Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™Meet the new FSP 3000 M-Flex800™
Meet the new FSP 3000 M-Flex800™
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
 

4.1 exponential functions 2

  • 2. Definition of Exponential Functions The exponential function f with a base b is defined by f(x) = bx where b is a positive constant other than 1 (b > 0, and b ≠ 1) and x is any real number. So, f(x) = 2x , looks like:
  • 3. Graphing Exponential Functions Four exponential functions have been graphed. Compare the graphs of functions where b > 1 to those where b < 1 2x y = 7x y = 1 2 x y = 1 7 x y =
  • 4. Graphing Exponential Functions So, when b > 1, f(x) has a graph that goes up to the right and is an increasing function. When 0 < b < 1, f(x) has a graph that goes down to the right and is a decreasing function.
  • 5. Characteristics The domain of f(x) = bx consists of all real numbers (-∞, ∞). The range of f(x) = bx consists of all positive real numbers (0, ∞). The graphs of all exponential functions pass through the point (0,1). This is because f(o) = b0 = 1 (b≠o). The graph of f(x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f(x) = bx is one-to-one and has an inverse that is a function.
  • 6. Transformations Vertical translation f(x) = bx + c Shifts the graph up if c > 0 Shifts the graph down if c < 0 2x y = 2 3x y = + 2 4x y = −
  • 7. Transformations Horizontal translation: g(x)=bx+c Shifts the graph to the left if c > 0 Shifts the graph to the right if c < 0 2x y = ( 3) 2 x y + = ( 4) 2 x y − =
  • 8. Transformations Reflecting g(x) = -bx reflects the graph about the x-axis. g(x) = b-x reflects the graph about the y-axis. 2x y = 2x y = − 2 x y − =
  • 9. Transformations Vertical stretching or shrinking, f(x)=cbx : Stretches the graph if c > 1 Shrinks the graph if 0 < c < 1 2x y = 4(2 )x y = 1 (2 ) 4 x y =
  • 10. Transformations Horizontal stretching or shrinking, f(x)=bcx : Shinks the graph if c > 1 Stretches the graph if 0 < c < 1 2x y = 4(2 )x y = 1 (2 ) 4 x y =
  • 11. You Do Graph the function f(x) = 2(x-3) +2 Where is the horizontal asymptote? y = 2
  • 12. You Do, Part Deux Graph the function f(x) = 4(x+5) - 3 Where is the horizontal asymptote? y = - 3
  • 13. The Number e The number e is known as Euler’s number. Leonard Euler (1700’s) discovered it’s importance. The number e has physical meaning. It occurs naturally in any situation where a quantity increases at a rate proportional to its value, such as a bank account producing interest, or a population increasing as its members reproduce.
  • 14. The Number e - Definition An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. It models a variety of situations in which a quantity grows or decays continuously: money, drugs in the body, probabilities, population studies, atmospheric pressure, optics, and even spreading rumors! The number e is defined as the value that approaches as n gets larger and larger. 1 1 n n   + ÷  
  • 15. The Number e - Definition n 1 2 2 2.25 5 2.48832 10 2.59374246 100 2.704813829 1000 2.716923932 10,000 2.718145927 100,000 2.718268237 1,000,000 2.718280469 1,000,000,000 2.718281827 1 1 n n   + ÷   0 1 1 n A n   + ÷   1 , 1 n As n e n   →∞ + → ÷   The table shows the values of as n gets increasingly large. n → ∞As , the approximate value of e (to 9 decimal places) is ≈ 2.718281827
  • 16. The Number e - Definition For our purposes, we will use e ≈ 2.718. e is 2nd function on the division key on your calculator. y = e 1 1 n y n   = + ÷  
  • 17. The Number e - Definition Since 2 < e < 3, the graph of y = ex is between the graphs of y = 2x and y = 3x ex is the 2nd function on the ln key on your calculator y =e y = 2x y = 3x y = ex
  • 18. Natural Base The irrational number e, is called the natural base. The function f(x) = ex is called the natural exponential function.
  • 19. Compound Interest The formula for compound interest: ( ) 1   = + ÷   nt r A t P n Where n is the number of times per year interest is being compounded and r is the annual rate.
  • 20. Compound Interest - Example Which plan yields the most interest? Plan A: A $1.00 investment with a 7.5% annual rate compounded monthly for 4 years Plan B: A $1.00 investment with a 7.2% annual rate compounded daily for 4 years A: B: 12(4) 0.075 1 1 1.3486 12   + ≈ ÷   365(4) 0.072 1 1 1.3337 365   + ≈ ÷   $1.35 $1.34
  • 21. Interest Compounded Continuously If interest is compounded “all the time” (MUST use the word continuously), we use the formula where P is the initial principle (initial amount) ( ) = rt A t Pe
  • 22. ( ) = rt A t Pe If you invest $1.00 at a 7% annual rate that is compounded continuously, how much will you have in 4 years? You will have a whopping $1.32 in 4 years! (.07)(4) 1* 1.3231e ≈
  • 23. You Do You decide to invest $8000 for 6 years and have a choice between 2 accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?
  • 24. You Do Answer 1st Plan: 2nd Plan: 0.0685(6) (6) 8000 $12,066.60P e= ≈ 12(6) 0.07 (6) 8000 1 $12,160.84 12 A   = + ≈ ÷  