1 rectangular coordinate system x

A coordinate system is a system of assigning addresses for
positions in the plane (2 D) or in space (3 D).
Rectangular Coordinate System

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).

The horizontal axis is called
the x-axis.

the x-axis. The vertical axis
is called the y-axis.

is called the y-axis. The point
where the axes meet
is called the origin.

where the axes meet
Starting from the origin, each
point is addressed by its
ordered pair (x, y) where:

where the axes meet
x = amount to move
right (+) or left (–).

where the axes meet
x = amount to move
y = amount to move
up (+) or down (–).

x = amount to move
y = amount to move

x = amount to move
y = amount to move
For example, the point P
corresponds to (4, –3)

x = amount to move
y = amount to move
corresponds to (4, –3) is
4 right,

x = amount to move
y = amount to move
4 right, and 3 down from
the origin.
(4, –3)
P

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
A
B
C

x = amount to move
y = amount to move
the origin.
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

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
y-coordinate.
A
B
C
Example B: Find the coordinate of P, Q, R as shown.
P
Q
R

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
y-coordinate.
A
B
C
P(4, 5),
P
Q
R

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
y-coordinate.
A
B
C
P(4, 5), Q(3, -5),
P
Q
R

x = amount to move
y = amount to move
the origin.
Example A.
Label the points
A(-1, 2), B(-3, -2),C(0, -5).
y-coordinate.
A
B
C
P(4, 5), Q(3, -5), R(-6, 0)
P
Q
R

The coordinate of the
origin is (0, 0).
(0,0)

origin is (0, 0).
Any point on the x-axis
has coordinate of the
form (x, 0).
(0,0)

origin is (0, 0).
form (x, 0).
(5, 0)
(0,0)

origin is (0, 0).
form (x, 0).
(5, 0)(-6, 0)
(0,0)

origin is (0, 0).
form (x, 0).
(5, 0)(-6, 0)
Any point on the y-axis
form (0, y).(0,0)

origin is (0, 0).
form (x, 0).
(5, 0)(-6, 0)
form (0, y).
(0, 6)
(0,0)

origin is (0, 0).
form (x, 0).
(5, 0)(-6, 0)
form (0, y).
(0, -4)
(0, 6)
(0,0)

origin is (0, 0).
form (x, 0).
form (0, y).
The axes divide the plane
into four parts. Counter
clockwise, they are denoted
as quadrants I, II, III, and IV.
QIQII
QIII QIV

origin is (0, 0).
form (x, 0).
form (0, y).
QIQII
QIII QIV
(+,+)

origin is (0, 0).
form (x, 0).
form (0, y).
QIQII
QIII QIV
(+,+)(–,+)

origin is (0, 0).
form (x, 0).
form (0, y).
Q1Q2
Q3 Q4
(+,+)(–,+)
(–,–) (+,–)
Respectively, the signs of
the coordinates of each
quadrant are shown.

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)

across the y-axis.
(5,4)(–5,4)

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)

across the y-axis.
(5,4)(–5,4)
(5, –4)

across the y-axis.
(5,4)(–5,4)
(5, –4) (–x, –y) is the reflection of
(x, y) across the origin.

across the y-axis.
(5,4)(–5,4)
(5, –4) (–x, –y) is the reflection of
(x, y) across the origin.
(–5, –4)

Movements and Coordinates

Let A be the point (2, 3).
A
(2, 3)

Suppose it’s x–coordinate is
increased by 4 to
(2 + 4, 3) = (6, 3)
A
(2, 3)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
A B
(2, 3) (6, 3)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
this corresponds to moving A to the
right by 4.
A B
x–coord.
increased
by 4
(2, 3) (6, 3)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
right by 4.
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)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
right by 4.
A B
(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)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
right by 4.
A B
(2 – 4, 3) = (–2, 3) - to the point C,
left by 4.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
right by 4.
A B
(2 – 4, 3) = (–2, 3) - to the point C,
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)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
right by 4.
A B
(2 – 4, 3) = (–2, 3) - to the point C,
left by 4.
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)

increased by 4 to
(2 + 4, 3) = (6, 3) - to the point B,
right by 4.
A B
(2 – 4, 3) = (–2, 3) - to the point C,
left by 4.
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)

Again let A be the point (2, 3).
A
(2, 3)

Suppose its y–coordinate is
increased by 4 to
(2, 3 + 4) = (2, 7)
A
(2, 3)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
A
D
y–coord.
increased
by 4
(2, 3)
(2, 7)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
this corresponds to moving A up
by 4.
A
D
y–coord.
increased
by 4
(2, 3)
(2, 7)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
by 4.
A
D
Similarly if the y–coordinate of
(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)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
by 4.
A
D
(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)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
by 4.
A
D
(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
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
by 4.
A
D
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
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)

increased by 4 to
(2, 3 + 4) = (2, 7) - to the point D,
by 4.
A
D
(2, 3 – 4) = (2, –1) - to the point E,
this corresponds to
moving A down by 4.
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)

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?

Example. C.
Moving left corresponds to decreasing the x-coordinate.

Example. C.
Hence B is (–2 – 100, 4)

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
b. What is the coordinate of the point C that is 100 units
directly above A?

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Moving up corresponds to increasing the y-coordinate.

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100)

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
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?

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
Here is the vertical format for the calculation:
(–2, 4)
point A

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
adding 50 to the x–coordinate to move right,
and –30 to the y–coordinate to move down.
(–2, 4)
point A

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
(–2, 4)
+ (50, –30)
+ the “moves”
point A

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
(–2, 4)
+ (50, –30)
(48, –26)
point A
+ the “moves”

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).
Hence D has the coordinate (–2 + 50, 4 – 30) = (48, –26).
(–2, 4)
+ (50, –30)
(48, –26)
point A
+ the “moves”

d. The point A(–2, 4) is 50 to the right and 30 below the point E
What’s the coordinate of the point E?

Let the coordinate of E be (a, b). (a , b) point E

Let the coordinate of E be (a, b).
In the vertical format we have:
(a , b)
+ (50, –30)
(–2, 4)
the “moves”
point A
point E

In the vertical format we have
Hence a + 50 = –2 so a = –52
and that b + (–30) = 4 so b = 34.
(a , b)
+ (50, –30)
(–2, 4)
the “moves”
point A
point E

In the vertical format we have
Hence a + 50 = –2 so a = –52
and that b + (–30) = 4 so b = 34.
Hence E is (–52 , 34).
(a , b)
+ (50, –30)
(–2, 4)
the “moves”
point A
point E

Example. C.
Hence B is (–2 – 100, 4) = (–102, 4).
directly above A?
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).

Exercise. A.
a. Write down the coordinates of the following points.
AB
C
D
E
F
G
H

Ex. B. Plot the following points on the graph paper.
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.

C. Find the coordinates of the following points. Draw both
points for each problem.
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).

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