“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
“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
The branch of mathematics which deals with location of objects in 2-D (dimensional) plane is called coordinate geometry. Need to present your work in most impressive & informative manner i.e. through Power Point Presentation call us at skype Id: kumar_sukh79 or mail us: clintech2011@gmail.com for using my service.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
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.
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.
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.
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.
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
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.
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.
2. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
Rectangular Coordinate System
3. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
4. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
5. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis.
6. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis.
7. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
8. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
9. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
x = amount to move
right (+) or left (–).
10. A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
The rectangular coordinate system for the plane consists of a
rectangular grid where each point in the plane is addressed by
an ordered pair of numbers (x, y).
Rectangular Coordinate System
The horizontal axis is called
the x-axis. The vertical axis
is called the y-axis. The point
where the axes meet
is called the origin.
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:
x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
11. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
Rectangular Coordinate System
12. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3)
Rectangular Coordinate System
13. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right,
Rectangular Coordinate System
14. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
(4, –3)
P
15. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
16. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
17. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
18. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
C
19. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
20. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P
Q
R
21. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P(4, 5),
P
Q
R
22. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P(4, 5), Q(3, -5),
P
Q
R
23. x = amount to move
right (+) or left (–).
y = amount to move
up (+) or down (–).
For example, the point P
corresponds to (4, –3) is
4 right, and 3 down from
the origin.
Rectangular Coordinate System
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
The ordered pair (x, y) corresponds to a point is called the
coordinate of the point, x is the x-coordinate and y is the
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P(4, 5), Q(3, -5), R(-6, 0)
P
Q
R
24. The coordinate of the
origin is (0, 0).
(0,0)
Rectangular Coordinate System
25. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(0,0)
Rectangular Coordinate System
26. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)
(0,0)
Rectangular Coordinate System
27. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
(0,0)
Rectangular Coordinate System
28. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).(0,0)
Rectangular Coordinate System
29. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).
(0, 6)
(0,0)
Rectangular Coordinate System
30. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
has coordinate of the
form (0, y).
(0, -4)
(0, 6)
(0,0)
Rectangular Coordinate System
31. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV
32. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV
(+,+)
33. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV
(+,+)(–,+)
34. The coordinate of the
origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
Any point on the y-axis
has coordinate of the
form (0, y).
Rectangular Coordinate System
Q1Q2
Q3 Q4
(+,+)(–,+)
(–,–) (+,–)
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
Respectively, the signs of
the coordinates of each
quadrant are shown.
35. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
(5,4)
Rectangular Coordinate System
36. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
(5,4)(–5,4)
Rectangular Coordinate System
37. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
Rectangular Coordinate System
38. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
(5, –4)
Rectangular Coordinate System
39. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
(5, –4)
(–x, –y) is the reflection of
(x, y) across the origin.
Rectangular Coordinate System
40. When the x-coordinate of the
a point (x, y) is changed to
its opposite as (–x , y), the
new point is the reflection
across the y-axis.
When the y-coordinate of
the a point (x, y) is changed
to its opposite as (x , –y),
the new point is the
reflection across the x-axis.
(5,4)(–5,4)
(5, –4)
(–x, –y) is the reflection of
(x, y) across the origin.
(–5, –4)
Rectangular Coordinate System
43. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3)
Rectangular Coordinate System
A
(2, 3)
44. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
Rectangular Coordinate System
A B
(2, 3) (6, 3)
45. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
x–coord.
increased
by 4
(2, 3) (6, 3)
46. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3)
x–coord.
increased
by 4
(2, 3) (6, 3)
47. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
48. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
49. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
Hence we conclude that changes in the x–coordinates of a point
move the point right and left.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
50. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
Hence we conclude that changes in the x–coordinates of a point
move the point right and left.
If the x–change is +, the point moves to the right.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
51. Movements and Coordinates
Let A be the point (2, 3).
Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
Rectangular Coordinate System
A B
Similarly if the x–coordinate of
(2, 3) is decreased by 4 to
(2 – 4, 3) = (–2, 3) - to the point C,
this corresponds to moving A to the
left by 4.
Hence we conclude that changes in the x–coordinates of a point
move the point right and left.
If the x–change is +, the point moves to the right.
If the x–change is – , the point moves to the left.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)
52. Again let A be the point (2, 3).
Rectangular Coordinate System
A
(2, 3)
53. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7)
Rectangular Coordinate System
A
(2, 3)
54. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
Rectangular Coordinate System
A
D
y–coord.
increased
by 4
(2, 3)
(2, 7)
55. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
y–coord.
increased
by 4
(2, 3)
(2, 7)
56. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
57. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
58. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
Hence we conclude that changes in the y–coordinates of a point
move the point right and left.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
59. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
Hence we conclude that changes in the y–coordinates of a point
move the point right and left.
If the y–change is +, the point moves up.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
60. Again let A be the point (2, 3).
Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
Rectangular Coordinate System
A
D
Similarly if the y–coordinate of
(2, 3) is decreased by 4 to
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
Hence we conclude that changes in the y–coordinates of a point
move the point right and left.
If the y–change is +, the point moves up.
If the y–change is – , the point moves down.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)
62. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
63. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4)
64. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
65. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
66. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
67. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100)
68. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
69. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D that is 50 to the right
and 30 below A?
70. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D that is 50 to the right
and 30 below A?
We need to add 50 to the x–coordinate (to the right)
71. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D that is 50 to the right
and 30 below A?
We need to add 50 to the x–coordinate (to the right)
and subtract 30 from the y–coordinate (to go down).
72. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D that is 50 to the right
and 30 below A?
We need to add 50 to the x–coordinate (to the right)
and subtract 30 from the y–coordinate (to go down).
Hence D has coordinate (–2 + 50, 4 – 30)
73. Rectangular Coordinate System
Example. C.
a. Let A be the point (–2, 4). What is the coordinate of
the point B that is 100 units directly left of A?
Moving left corresponds to decreasing the x-coordinate.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?
Moving up corresponds to increasing the y-coordinate.
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
c. What is the coordinate of the point D that is 50 to the right
and 30 below A?
We need to add 50 to the x–coordinate (to the right)
and subtract 30 from the y–coordinate (to go down).
Hence D has coordinate (–2 + 50, 4 – 30) = (48, –26).
74. Exercise. A.
a. Write down the coordinates of the following points.
Rectangular Coordinate System
AB
C
D
E
F
G
H
75. Ex. B. Plot the following points on the graph paper.
Rectangular Coordinate System
2. a. (2, 0) b. (–2, 0) c. (5, 0) d. (–8, 0) e. (–10, 0)
All these points are on which axis?
3. a. (0, 2) b. (0, –2) c. (0, 5) d. (0, –6) e. (0, 7)
All these points are on which quadrant?
4. a. (5, 2) b. (2, 5) c. (1, 7) d. (7, 1) e. (6, 6)
All these points are in which quadrant?
5. a. (–5, –2) b. (–2, –5) c. (–1, –7) d. (–7, –1) e. (–6, –6)
All these points are in which quadrant?
6. List three coordinates whose locations are in the 2nd
quadrant and plot them.
7. List three coordinates whose locations are in the 4th
quadrant and plot them.
76. C. Find the coordinates of the following points. Draw both
points for each problem.
Rectangular Coordinate System
The point that’s
8. 5 units to the right of (3, –2).
10. 4 units to the left of (–1, –5).
9. 6 units to the right of (–4, 2).
11. 6 units to the left of (2, –6).
12. 3 units to the left and 6 units down from (–2, 5).
13. 1 unit to the right and 5 units up from (–3, 1).
14. 3 units to the right and 3 units down from (–3, 4).
15. 2 units to the left and 6 units up from (4, –1).