The document discusses function graphs and the rules for transforming graphs of basic functions like f(x)=x^2 and f(x)=x^3. It explains how to sketch the graphs of transformed functions like f(x)+2, f(x-2), -f(x), and f(-x) based on shifting, reflecting, and stretching/squashing the original graph. Examples are provided to demonstrate applying these rules to sketch related function graphs like y=f(x-2)+1.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Pitching slots in the formal line up are limited.
Even if you are not slotted to pitch, you will learn a lot from listening to Bill's presentation and hearing his feedback to presenting startups. All are welcome to attend and learn from Bill!
O tema recuperação pós-exercício (RPE) tem sido foco de intensas reflexões, em razão da importância que representa dentro dos atuais programas de treinamento físico em diferentes níveis de desempenho, especialmente no alto nível em que os atletas treinam mais de uma vez por dia
The goalkeeper’s importance in the development of
football is evident from times gone by as well as the
more recent past. Some goalkeepers have contributed
to the development of the game through their amazing
accomplishments: the names of Yashin, Banks, Maier,
Fillol, Zoff, N’Kono, Schmeichel, Barthez, Kahn, Buffon and
Casillas are deeply etched in the memories of football fans.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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2. What is to be Learned?
• How to sketch images of given function
graph f(x) under rules such as
f(x) + 2
f(x + 2)
-f(x)
f(-x)
Will also have peek at y = 2f(x) and y = f(2x)
5. The Rules
y = f(x)
(-3 , 4)
( 2 , 1)
y = f(x) + 2
(-3 , 6)
( 2 , 3)
2
6. The Rules
y = f(x)
(-3 , 4)
( 2 , 1)
y = f(x – 2)
(-1 , 4)
( 4, 1)
2
7. The Rules
y = f(x)
(-3 , 4)
( 2 , 1)
y = -f(x)
(-3 , -4)
( 2, -1)
Reflects in
x axis
8. The Rules
y = f(x)
(-3 , 4)
( 2 , 1)
y = f(-x)
(3 , 4)
( -2, 1)
Reflects in
y axis
9. Y = 2f(x) and y = f(2x)
Remember Trig Graphs
f(x) = sinx
f(x) = 2sinx
Graph “stretches”
10. Y = 2f(x) and y = f(2x)
Remember Trig Graphs
f(x) = sinx
f(x) = sin2x
Graph Squashes
11. Function Graph Families
Given graph y = f(x) we can sketch family
graphs by following these rules
y = f(x) – 2
y = f(x + 2)
y = -f(x)
y = f(-x)
y =2f(x)
y = f(2x)
2 down
2 to left
reflects in x axis
reflects in y axis
stretches
squashes
*
*
* Think of trig graphs
12. sketch y = 3 – f(x)
y = f(x)
Ex
(3 , -1)
2
Cunningly change to y = -f(x) + 3
(3 , 1)
-2
y = -f(x)
y = -f(x) + 3
(3 , 4)
1
13. y = f(x)(-2 , 5)
( 1 , 2)
y = f(x – 2)
(0 , 5)
(3 , 2)
Key Question
Sketch y = f(x – 2) + 1 showing new key points
y = f(x – 2) + 1
(0 , 6)
(3 , 3)
