This document discusses rational functions and their graphs. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It explains that the domain of a rational function excludes any values that would make the denominator equal to 0. It describes how to find vertical, horizontal, and oblique asymptotes of a rational function by comparing the degrees of the polynomials in the numerator and denominator. Vertical asymptotes occur where the denominator is 0, and horizontal or oblique asymptotes depend on whether the degree of the numerator is less than, equal to, or greater than the degree of the denominator. Examples are provided to illustrate these concepts.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about functions and its operations such as addition, subtraction, multiplication, division and composition. It is also comprised of some examples and exercises to be done for the said topic.
Basic Calculus 11 - Derivatives and Differentiation RulesJuan Miguel Palero
It is a powerpoint presentation that discusses about the lesson or topic of Derivatives and Differentiation Rules. It also encompasses some formulas, definitions and examples regarding the said topic.
The Sum of Two Functions
The Difference of Two functions
The Product of Two Functions
The Quotient of Two Functions
The Product of A constant and a Function
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about functions and its operations such as addition, subtraction, multiplication, division and composition. It is also comprised of some examples and exercises to be done for the said topic.
Basic Calculus 11 - Derivatives and Differentiation RulesJuan Miguel Palero
It is a powerpoint presentation that discusses about the lesson or topic of Derivatives and Differentiation Rules. It also encompasses some formulas, definitions and examples regarding the said topic.
This presentation tackles the controversy regarding Lyme disease by reviewing the evidence for immune evasion and persistent infection by the Lyme spiorchete, Borrelia burgdorferi. The evidence shows that physicians called upon to assist patients with this potential diagnosis should be open to the possibility of persistent infection even in patients who have already received antibiotic treatment for their condition. Lacking evidence on how best to treat a chronic infection of this kind, physicians should be allowed to rely on their experience and to exercise their best clinical judgment in managing patients with Lyme disease.
When Office 365 files are uploaded as a submission, later changes made to the...YohannesAndualem1
If your course has enabled Microsoft Office 365, you can upload a file from your Microsoft OneDrive for an assignment.
Like other file upload submissions, files uploaded from Office 365 are uploaded into your Canvas user files submissions folder.
Canvas accepts Microsoft Word, Microsoft PowerPoint, Microsoft Excel, and PDF types.
Notes:
If the Office 365 tab is not available in your submission, your institution has not enabled this feature.
Canvas will require you to authorize access to your OneDrive account.
In Office 365 assignments, you can only submit one file for your submission.
When Office 365 files are uploaded as a submission, later changes made to the file in OneDrive will not be updated in the submission.
If enabled in your account, Canvas plays a celebration animation when you submit an assignment on time. However, if you prefer, you can disable this feature setting in your user settings.
If the assignment you are accessing displays differently, your assignment may be using the Assignment Enhancements feature. Please view this guide for more information.
If your course has enabled Microsoft Office 365, you can upload a file from your Microsoft OneDrive for an assignment.
Like other file upload submissions, files uploaded from Office 365 are uploaded into your Canvas user files submissions folder.
Canvas accepts Microsoft Word, Microsoft PowerPoint, Microsoft Excel, and PDF types.
Notes:
If the Office 365 tab is not available in your submission, your institution has not enabled this feature.
Canvas will require you to authorize access to your OneDrive account.
In Office 365 assignments, you can only submit one file for your submission.
When Office 365 files are uploaded as a submission, later changes made to the file in OneDrive will not be updated in the submission.
If enabled in your account, Canvas plays a celebration animation when you submit an assignment on time. However, if you prefer, you can disable this feature setting in your user settings.
If the assignment you are accessing displays differently, your assignment may be using the Assignment Enhancements feature. Please view this guide for more information.
If your course has enabled Microsoft Office 365, you can upload a file from your Microsoft OneDrive for an assignment.
Like other file upload submissions, files uploaded from Office 365 are uploaded into your Canvas user files submissions folder.
Canvas accepts Microsoft Word, Microsoft PowerPoint, Microsoft Excel, and PDF types.
Notes:
If the Office 365 tab is not available in your submission, your institution has not enabled this feature.
Canvas will require you to authorize access to your OneDrive account.
In Office 365 assignments, you can only submit one file for your submission.
When Office 365 files are uploaded as a submission, later changes made to the file in OneDrive will not be updated in the submission.
hhhw hjjbmb bjkbjk kjhk j
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
2. ( ) ( )
( )xq
xp
xR =
What would the domain of a
rational function be?
We’d need to make sure the
denominator ≠ 0
( )
x
x
xR
+
=
3
5 2
Find the domain.{ }3: −≠ℜ∈ xx
( )
( )( )22
3
−+
−
=
xx
x
xH { }2,2: ≠−≠ℜ∈ xxx
( )
45
1
2
++
−
=
xx
x
xF
If you can’t see it in your
head, set the denominator = 0
and factor to find “illegal”
values.
( )( ) 014 =++ xx { }1,4: −≠−≠ℜ∈ xxx
3. The graph of looks like this:( ) 2
1
x
xf =
Since x ≠ 0, the graph approaches 0 but never crosses or
touches 0. A vertical line drawn at x = 0 is called a
vertical asymptote. It is a sketching aid to figure out the
graph of a rational function. There will be a vertical
asymptote at x values that make the denominator = 0
If you choose x values close to 0, the graph gets
close to the asymptote, but never touches it.
4. Let’s consider the graph ( )
x
xf
1
=
We recognize this function as the reciprocal
function from our “library” of functions.
Can you see the vertical asymptote?
Let’s see why the graph looks
like it does near 0 by putting in
some numbers close to 0.
10
10
1
1
10
1
==
f
100
100
1
1
100
1
==
f
10
10
1
1
10
1
−=
−
=
−f
100
100
1
1
100
1
−=
−
=
−f
The closer to 0 you get
for x (from positive
direction), the larger the
function value will be Try some negatives
5. Does the function have an x intercept?( )
x
xf
1
=
There is NOT a value that you can plug in for x that
would make the function = 0. The graph approaches
but never crosses the horizontal line y = 0. This is
called a horizontal asymptote.
A graph will NEVER cross a
vertical asymptote because the
x value is “illegal” (would make
the denominator 0)
x
1
0 ≠
A graph may cross a horizontal
asymptote near the middle of
the graph but will approach it
when you move to the far right
or left
6. Graph ( )
x
xQ
1
3+=
This is just the reciprocal function transformed. We can
trade the terms places to make it easier to see this.
3
1
+=
x
vertical translation,
moved up 3
( )
x
xf
1
=
( )
x
xQ
1
3+=
The vertical asymptote
remains the same because in
either function, x ≠ 0
The horizontal asymptote
will move up 3 like the graph
does.
7. Finding Asymptotes
VERTICALASYMPTOTES
There will be a vertical asymptote at any
“illegal” x value, so anywhere that would make
the denominator = 0
( )
43
52
2
2
−−
++
=
xx
xx
xR
Let’s set the bottom = 0
and factor and solve to
find where the vertical
asymptote(s) should be.
( )( ) 014 =+− xx
So there are vertical
asymptotes at x = 4
and x = -1.
8. If the degree of the numerator is
less than the degree of the
denominator, (remember degree
is the highest power on any x
term) the x axis is a horizontal
asymptote.
If the degree of the numerator is
less than the degree of the
denominator, the x axis is a
horizontal asymptote. This is
along the line y = 0.
We compare the degrees of the polynomial in the
numerator and the polynomial in the denominator to tell
us about horizontal asymptotes.
( )
43
52
2
+−
+
=
xx
x
xR
degree of bottom = 2
HORIZONTAL ASYMPTOTES
degree of top = 1
1
1 < 2
9. If the degree of the numerator is
equal to the degree of the
denominator, then there is a
horizontal asymptote at:
y = leading coefficient of top
leading coefficient of bottom
degree of bottom = 2
HORIZONTAL ASYMPTOTES
degree of top = 2
The leading coefficient
is the number in front of
the highest powered x
term.
horizontal asymptote at:
1
2=
( )
43
542
2
2
+−
++
=
xx
xx
xR
1
2
=y
10. ( )
43
532
2
23
+−
+−+
=
xx
xxx
xR
If the degree of the numerator is
greater than the degree of the
denominator, then there is not a
horizontal asymptote, but an
oblique one. The equation is
found by doing long division and
the quotient is the equation of
the oblique asymptote ignoring
the remainder.
degree of bottom = 2
OBLIQUE ASYMPTOTES
degree of top = 3
532 23
+−+ xxx432
−− xx
remaindera5 ++x
Oblique asymptote
at y = x + 5
11. SUMMARY OF HOW TO FIND ASYMPTOTES
Vertical Asymptotes are the values that are NOT in the
domain. To find them, set the denominator = 0 and solve.
To determine horizontal or oblique asymptotes, compare
the degrees of the numerator and denominator.
1. If the degree of the top < the bottom, horizontal
asymptote along the x axis (y = 0)
2. If the degree of the top = bottom, horizontal asymptote
at y = leading coefficient of top over leading
coefficient of bottom
3. If the degree of the top > the bottom, oblique
asymptote found by long division.
12. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au