The document discusses the geometric and algebraic definitions of a parabola, noting that a parabola is the set of all points equidistant from a fixed point (the focus) and a line (the directrix). It also provides steps for writing the standard form equation of a parabola given its vertex and focus, as well as for graphing a parabola by plotting its vertex, focus, directrix, axis of symmetry, and sketching the curve through these points.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
More Related Content
Similar to Conic Sections Parabolas FCIT compat.ppt
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
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.
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.
2. Review: The geometric definition relies on a
cone and a plane intersecting it.
Algebraic definition: All points that are
equidistant from a given line (the directrix)and
a fixed point not on the directrix (the focus)
3. Focus
Directrix
x
y
Any point on
the parabola is
equidistant to
the focus and
the directrix.
Example:
Point A: d1=d2
Point B: d1=d2
A
d1
d2
d1
d2
B
5. One variable is squared and one is not. (How
does this differ from linear equations?)
There are many ways the equation of a
parabola can be written. We will get the
quadratic part (variable that is squared) on the
left of the equal sign and the linear part
(variable is to the first power) on the right of
the equal sign.
Equation:
(x - h)2 = c(y – k) OR (y - k)2 = c(x – h)
6. To graph:
1. Put in standard form (above) – squared term
on left
2. Decide which way the parabola opens.
Look at the right side. If y: + c → opens up
If y: - c → opens down
If x: + c → opens right
If x: - c → opens left
7. To graph:
3. Plot the vertex (h,k) Note what happens to the
signs.
4. Plot the focus: move │¼ c │ from the vertex in
the direction that the parabola opens. Mark
with an f.
5. Draw the directrix: │¼ c │ from the vertex in
the opposite direction of the focus (Remember
that the directrix is a line.)
8. To graph:
6. Plot the endpoints of the latus rectum/focal
chord (width at the focus). The width is the
│c│ at the focus.
7. Sketch the parabola by going through the
vertex and the endpoints of the latus rectum.
(Be sure to extend the curve and put arrows.)
8. Identify the axis of symmetry. (The line that
goes through the vertex dividing the parabola
in half.)
9. To graph:
1. Put in standard form– squared term on left
Done
2. Decide which way the parabola opens.
Look at the right side. If y: + c → opens up
If y: - c → opens down
If x: + c → opens right
If x: - c → opens left
Up because y is on the right and 12 is positive
10. To graph:
3. Plot the vertex (h,k) Note what happens to the
signs.
(5,6)
4. Plot the focus: move │¼ c │ from the vertex in the
direction that the parabola opens. Mark with an f.
(5,9): found by moving up 3 from the vertex
5. Draw the directrix: │¼ c │ from the vertex in the
opposite direction of the focus (Remember that the
directrix is a line.)
y = 3: found by moving down 3 from the vertex
11. To graph:
6. Plot the endpoints of the latus rectum/focal chord
(width at the focus). The width is the │c│ at the
focus.
L.R. = 12 with endpoints at (-1,9) & (11,9)
7. Sketch the parabola by going through the vertex
and the endpoints of the latus rectum. (Be sure to
extend the curve and put arrows.)
8. Identify the axis of symmetry. (The line that goes
through the vertex dividing the parabola in half.)
x = 5
13. To graph:
1. Put in standard form
Done
2. Decide which way the parabola opens.
Left because x is on the right and 4 is negative
3. Plot the vertex (h,k)
(2,-3)
4. Plot the focus:
(1,-3): found by moving left 1 from the vertex
5. Draw the directrix:
x = 3: found by moving right 1 from the vertex
14. To graph:
6. Plot the endpoints of the latus rectum
L.R. = 4 with endpoints at (1,-1) & (1,-5)
7. Sketch the parabola
8. Identify the axis of symmetry.
y = -3
15.
x
y
Vertex: (2,-3)
Focus: (1,-3)
Directrix: x = 3
L.R.: 4
Axis: y = -3
f
16. What’s the first step?
Put in standard form.
y2 - 4y + 1 = x
y2 - 4y + 4 = x - 1 + 4 Complete the square.
(y – 2)2 = x + 3
(y – 2)2 = 1(x + 3)
Now you try graphing the parabola and labeling
all the parts.
17. Given the following information, write the
equation of the parabola.
Vertex is (0,0) and Focus is at (0,2)
18. 1) How can you tell the graph of an equation will
be a parabola?
2) What’s the standard form of a parabola?
3) What are the steps for graphing a parabola?
4) What are common errors people make when
graphing parabolas?