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
Excel in the concepts of coordinate geometry by knowing everything from important topics, preparation tips and formulas to practical applications. Read the complete article to know more:
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
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
Excel in the concepts of coordinate geometry by knowing everything from important topics, preparation tips and formulas to practical applications. Read the complete article to know more:
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
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
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.
➽=ALL False flag-War Machine-War profiteering-Energy (oil/Gas) Iraq, Iran,…oil and gas
USA invades other countries just to own their natural resources and to place them in the hands of American corporations. Facebook doesn’t call that terrorism. They call it democracy. BBC, CNN, FOX NEWS, FR 24, ITV/CH 4, SKY, EURO NEWS, ITV trash Sun paper,… Facebook all are protector and preserver of the propaganda classifying IR Iran as a dangerous terrorist organization. But FB, BBC, CNN, FOX NEWS, FR 24, ITV/CH 4-SKY, EURO NEWS, ITV do know well, that USA is the biggest terrorist country in the world.
‘terrorism’ the unlawful use of violence and intimidation, especially against civilians, in the pursuit of political aims.
"the fight against terrorism" is the fight against the unlawful use of violence and intimidation and carpet bombing.
Ever since the beginning of the 19th century, the West has been sucking on the jugular vein of the Moslem body politic like a veritable vampire whose thirst for Moslem blood is never sated and who refused to let go. Since 1979, Iran, which has always played the role of the intellectual leader of the Islamic world, has risen up to put a stop to this outrage against God’s law and will, and against all decency.
MY NEWS PUNCH DR F DEJAHANG 28/12/2019
PART 1 (IN TOTAL 12 PARTS)
NEWS YOU WON’T FIND ON BBC-CNN-FOX NEWS, FR 24, EURO NEWS, ITV…
ALL In My Documents: https://www.edocr.com/user/drdejahang02
Also in https://www.edocr.com/v/jqmplrpj/drdejahang02/LINKS-08-12-2019-PROJECT-ONE Click on Social Websites of mine >60
Articles for Political Science, Mathematics and Productivity the Student Room BSc, MSc & PhD Project Mangers etc
PPTs in SLIDESHARE International Studies Research Degrees (MPhilPhD) ➽➜R⇢➤=RESEARCH ➽=ALL
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NEWS YOU WON’T FIND ON BBC-CNN-FOX NEWS, FRNACE 24, EURO NEWS
Articles for Political Science, Mathematics and Productivity the Student Room BSc, MSc & PhD Project Mangers etc
PPTs in SLIDESHARE International Studies Research Degrees (MPhilPhD)
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.
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.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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.
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.
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1578 parabolas-03
1. Parabolas
Algebra II, Sections 11.1 and 11.5
State Standards: Algebra II, Standards 16.0 and 17.0
Mathematical Analysis, Standard 5.1
2. Definition of a Parabola
A parabola is a collection of all the points that are equidistant from
a fixed point known as the focus and a line known as the directrix.
3. Parts of a Parabola
The parabola
The Focus
1/4a units from the
vertex
The Directrix
1/4a units from the vertex
Latus Rectum
Passes through the focus
1/a units long
Axis of
Symmetry
4. Recognizing a Parabola: General Form
2
2
0
0
Ax Ex Dy F or
Cy Ex Dy F
+ + + =
+ + + =
Either A or C is missing
Standard Forms
2 2
2
2
or Center at (0, 0)
( )
( ) +h Center at (h, k)
y ax x ay
y a x h k or
x a y k
= =
= − +
= −
The focus and directix are located
1/(4a) units from the vertex in the
direction of the plain (unsquared)
variable..
Tip:
y = parabolas open in the y direction. Everything
happens to y.
x = parabolas open in the x direction. Everything
happens to x
Note the positions of h and
k. h goes with x and k goes
with y
5. List of Formulas
Form of Equation
Vertical Horizontal
2
( )y a x h k= − +
Vertex (h, k)
Focus 1
,
4
h k
a
+ ÷
Directrix
1
4
y k
a
= −
Length of Latus Rectum 1
a
Axis of Symmetry x = h
0
0
a
a
>
<
U
I
Direction
2
( )x a y k h= − +
(h, k)
1
,
4
h k
a
+ ÷
1
4
x h
a
= −
1
a
y = k
0
0
a
a
> ⊂
< ⊃
p = 1/(4a)
6. Graphing Parabolas
Acceptable
• Find the vertex
• Find the direction
• Make some
acknowledgement of a
and the width of the
parabola. (i.e. wide or
narrow)
Exemplary
• Find the vertex
• Find the direction
• Use a to roughly count the
“slope” of the parabola.
• Able to find (when asked)
– Focus
– Directrix
– Axis of symmetry
– Size of latus rectum
• Finds intercepts when
practicable.
7. Graphing Parabolas
Example 1
2
2y x=
1. Find the vertex.
• If there is nothing in the h or k position,
the coordinate is 0.
• Vertex (0, 0)
2. Find the direction
• Since y is plain, it is a vertical parabola.
• Since a > 0, it opens up.
• Since a > 1, it will be somewhat skinny.
• The red points were found by counting
“slope.”
3. The focus is 1/(4(2))
units from the vertex (0,
1/8)
4. The directrix is
y = -1/(4(2)) =
-1/8
5. Axis of symmetry is
the y-axis
6. Latus rectum = ½
7. Intercept at (0, 0)
8. Graphing Parabolas
Example 2
2
( 3) 2x y= − − +
1. Find the vertex.
• The variable in the parentheses lets you
know which coordinate you are finding.
• Only numbers in ( ) change signs
• Vertex (2, 3)
2. Find the direction
• Since x is plain, it is a horizontal parabola.
• Since a < 0, it opens left.
• Since a = 1, it will be standard width.
• The red points were found by counting
“slope.”
3. The focus is 1/(4(1)) units
from the vertex (1 ¾, 3)
4. The directrix is
x = 2 +1/(4(1)) = 2 ¼
5. Axis of symmetry is
the y = 3
6. Latus rectum = 1/4
7. y-intercepts irrational,
x-intercept = -7
9. Graphing Parabolas
Example 3
21
( 2) 1
4
y x= − − +
1. Find the vertex.
• The variable in the parentheses lets you
know which coordinate you are finding.
• Only numbers in ( ) change signs
• Vertex (2, 1)
2. Find the direction
• Since y is plain, it is a vertical parabola.
• Since a < 0, it opens downward.
• Since a < 1, it will be fat/wide.
• The red points were found by counting
“slope.”
3. The focus is 1/(4( ¼)) = 1
unit from the vertex (2, 0)
4. The directrix is
y = 1 +1/(4( ¼ )) = 2
5. Axis of symmetry is
the x=2
6. Latus rectum = 1/ ¼ =4
7. Intercepts are (0, 0)
and (4, 0)
10. Completing the Square on a Parabola
1. Move the “plain”variable to one side and everything else to the
other. Divide by the coefficient of this variable if needed.
2. Group the variables.
3. Factor out the “squared” coefficient.
4. Complete the square
a) Write down the x (or y) coefficient
b) Divide it by 2 (or multiply by ½)
c) Square and add the result inside the parentheses. Subtract what you really added
on the end
d) Factor as a perfect square trinomial (it’s the middle number)
5. Read your information.
We want to turn this
2
2
0
0
Ax Ex Dy F or
Cy Ex Dy F
+ + + =
+ + + =
into
2
2
( )
( ) +h
y a x h k or
x a y k
= − +
= −
11. Example 1
2
3 18 5 0x x y− − + =
1. Move the plain variable (y) to one
side and everything else to the other
2
3 18 5y x x= − +
2. Group the variables ( )2
3 18 5y x x= − +
3. Factor out the squared coefficient ( )2
3 6 5y x x= − +
4. Complete the square
Write down the coefficient -6
Divide it by 2 -3
Square and add inside the ( ) 9
Subtract what you truly added
on the back 3(9)
Factor as a perfect square trinomial
(middle number)
( )2
3 6 5 3(9)9y x x + −= − +
( )
2
33 22y x −= −
12. 5. Read your info and graph
( )
2
33 22y x −= −
Vertex (3, -22)
Opens upward
Skinny
13. Example 2
2
8 2 48 10 0y x y− − + − =
1. Move the plain variable (x) to one
side and everything else to the other.
Divide by the x coefficient.
2
2
2 8 48 10
4 24 5
x y y
x y y
= − + −
= − + −
2. Group the variables
( )2
4 24 5x y y= − + −
3. Factor out the squared coefficient ( )2
4 6 5x y y= − − −
4. Complete the square
Write down the coefficient -6
Divide it by 2 -3
Square and add inside the ( ) 9
Subtract what you truly added
on the back -4(9)
Factor as a perfect square trinomial
(middle number)
( )2
4(4 6 959 )x y y + += − − −
( )
2
4 313x y −= − +
14. ( )
2
4 313x y −= − +5. Read your info and graph
Vertex (31, 3)
Opens left
Skinny
Not visible on the graph, but y =
0 gives x =- 5. Intercept (-5, 0)