2.0 rectangular coordinate system

Rectangular Coordinate System
Each point in the plane
may be addressed by
two numbers (x, y) called
an ordered pair.

may be addressed by
an ordered pair.
(4, -3)
an ordered pair

may be addressed by
an ordered pair.
To locate (x, y), start
from the origin (0, 0),
(4, -3)
an ordered pair

may be addressed by
an ordered pair.
x = the amount to move
right (+) or left (–),(4, -3)
an ordered pair

may be addressed by
an ordered pair.
right (+) or left (–),
y = the amount to move
up (+) or down (–).
(4, -3)
an ordered pair

For example, the point corresponding to (4, -3) is
4 right, and 3 down from the origin.
may be addressed by
an ordered pair.
(4, -3)
4 right
3 down
an ordered pair

may be addressed by
an ordered pair.
Points on the x-axis have the form (#, 0)
(#, 0)
(4, -3)
4 right
3 down
an ordered pair

may be addressed by
an ordered pair.
Points on the x-axis have the form (#, 0) and
points on the y-axis have the form (0, #).
(#, 0)
(0, #)
(4, -3)
4 right
3 down
an ordered pair

The axes divide the
plane into four quadrants,
numbered 1, 2, 3, and 4
counter-clockwise.

The axes divide the
counter-clockwise.
Q1Q2
Q3 Q4

The axes divide the
counter-clockwise.
Respectively, the sign of the
coordinates of each quadrant
are shown.
Q1Q2
Q3 Q4

The axes divide the
counter-clockwise.
are shown.
Q1Q2
Q3 Q4
(+,+)

The axes divide the
counter-clockwise.
are shown.
Q1Q2
Q3 Q4
(+,+)(–,+)

The axes divide the
counter-clockwise.
are shown.
Q1Q2
Q3 Q4
(+,+)(–,+)
(–,–) (+,–)

Changing the coordinate (x, y)
of the point P corresponds to
moving P.

moving P.
i. Changing the sign of x or y
reflects the point P.

moving P.
The points (x, y) and (–x , y) are reflections of each
other across the y-axis.

(5,4)(–5,4) Changing the coordinate (x, y)
moving P.

The points (x, y) and (x , –y) are reflections of each
other across the x-axis.
(5,4)(–5,4) Changing the coordinate (x, y)
moving P.

(5,4)(–5,4)
(5, –4)
moving P.

The points (x, y) and (–x , –y) are reflections of each
other across the origin.
(5,4)(–5,4)
(5, –4)
moving P.

The points (x, y) and (–x , –y) are reflections of
each other across the origin.
(5,4)(–5,4)
(5, –4)(–5, –4)
moving P.

The points (x, y) and (–x , –y) are reflections of
each other across the origin.
(5,4)(–5,4)
(5, –4)(–5, –4)
moving P.
ii. Changing the value of x or y
moves P right/left/up/down.

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

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

this corresponds to
A B
Similarly if the x–coordinate 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)
this corresponds to moving A to the left by 4.

this corresponds to
A B
Hence we conclude that changes in the x–coordinates
correspond to moving the point right and left.
C
x–coord.
increased
by 4
x–coord.
decreased
by 4
(2, 3) (6, 3)(–2, 3)

this corresponds to
A B
Hence we conclude that changes in the x–coordinates
correspond to moving 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)

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

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

this corresponds to
moving A up by 4.
A
D
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)
this corresponds to moving A down by 4.

this corresponds to
moving A up by 4.
A
D
Hence we conclude that changes in the y–coordinates
correspond to moving the point up and down.
E
y–coord.
increased
by 4
y–coord.
decreased
by 4
(2, 3)
(2, 7)
(2, –1)

this corresponds to
moving A up by 4.
A
D
Hence we conclude that changes in the y–coordinates
correspond to moving the point up and down.
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)

Let (x1, y1) and (x2, y2) be two points and
D = the distance between them,
The Distance Formula

DD
(2, –4)
(–1, 3)

Example A. Find the distance
between (–1, 3) and (2, –4).
DD
(2, –4)
(–1, 3)

D = the distance between them, then D2 = Δx2 + Δy2,
where Δx = difference in the x's = x2 – x1,
Δy = difference in the y's = y2 – y1,
between (–1, 3) and (2, –4).
DD
(2, –4)
(–1, 3)

between (–1, 3) and (2, –4).
(–1, 3)
– ( 2, –4)
DD
(2, –4)
(–1, 3)

between (–1, 3) and (2, –4).
(–1, 3)
– ( 2, –4)
–3, 7
DD
(2, –4)
(–1, 3)
Δx Δy

between (–1, 3) and (2, –4).
(–1, 3)
– ( 2, –4)
–3, 7
DD
7
-3
(2, –4)
(–1, 3)
Δx Δy

between (–1, 3) and (2, –4).
(–1, 3)
– ( 2, –4)
–3, 7
D = (–3)2 + 72
DD
7
-3
(2, –4)
(–1, 3)
Δx Δy

between (–1, 3) and (2, –4).
(–1, 3)
– ( 2, –4)
–3, 7
D = (–3)2 + 72
= 58  7.62
DD
7
-3
(2, –4)
(–1, 3)
Δx Δy

Hence D = Δx2 + Δy2
or
D = (x2 – x1)2+(y2 – y1)2
between (–1, 3) and (2, –4).
(–1, 3)
– ( 2, –4)
–3, 7
D = (–3)2 + 72
= 58  7.62
DD
7
-3
(2, –4)
(–1, 3)
Δx Δy

The Mid-Point Formula
The mid-point m between two numbers a and b is the
average of them, that is m = .a + b
2

2
For example, the mid-point of 2 and 4 is (2 + 4)/2 = 3.

2
In picture: a b(a+b)/2
mid-pt.

2
mid-pt.
The mid-point formula
extends to higher
dimensions.

2
mid-pt.
extends to higher
dimensions.
In 2D
(x1, y1)
(x2, y2)
x1
y1
y2
x2

2
mid-pt.
extends to higher
dimensions.
In the x&y coordinate
the mid-point of
(x1, y1) and (x2, y2) is
x1 + x2
2 ,(
y1 + y2
2
)
In 2D
(x1, y1)
(x2, y2)
x1
y1
y2
x2

2
mid-pt.
extends to higher
dimensions.
In the x&y coordinate
the mid-point of
(x1, y1) and (x2, y2) is
x1 + x2
2 ,(
y1 + y2
2
)
In 2D
(x1, y1)
(x2, y2)
x1
y1
y2
x2
(x1 + x2)/2
(y1 + y2)/2

A. Find the coordinates of the following points.
Sketch both points for each problem.
1. Point A that is 3 units to the left and 6 units down
from (–2, 5).
2. Point A that is 1 unit to the right and 5 units up
from (–3, 1).
3. a. Point B is 3 units to the left and 6 units up from
point A(–8, 4). Find the coordinate of point B.
b. Point A(–8, 4) is 3 units to the left and 6 units up from
point C, find the coordinate of point C
4. a. Point A is 37 units to the right and 63 units down from
point B(–38, 49), find the coordinate of point A.
b. Point A(–38, 49) is 37 units to the right and 63 units down
from point C, find the coordinate of point C.

1. x – y = 3 2. 2x = 6 3. –y – 7= 0
4. y = 8 – 2x 5. y = –x + 4 6. 2x – 3 = 6
7. 2 = 6 – 2y 8. 4y – 12 = 3x 9. 2x + 3y = 0
10. –6 = 3x – 2y 11.
B. Graph the following equations by doing the following steps:
i. graph the horizontal lines (x = #) and
vertical lines (y = #) by inspection.
ii. identify which tilted lines may be graphed
using the x&y intercepts by completing the table:
iii. graph the other tilted lines passing
thru the origin using the following table:
3x = 4y 12. 5x + 2y = –10
Linear Equations and Lines
13. 3(2 – x) = 3x – y 14. 3(y – x) + y = 4y + 1
15. 5(x + 2) – 2y = 10

C. Find the coordinates of the following points assuming
all points are evenly spaced.
1.
1 4
2.
–1 5
1 3 11
3. a. Find x and y.
x zy
The number z is a “weighted average” of {1, 3, 11}
whose average is 5. In this case z is the average of
{1, 3, 3,11} instead because “3” is used both for
calculating x and y.
1 3 11
b. Find z the mid-point of x and y.
x y
Find all the locations of the points in the figures.
(–4, 7)
(2, 3) (0, 0) (8, 0)
(2, 6)4. 5.

(Answers to odd problems) Exercise A.
1. B=(-5,-1) 3. B=(-11,10), C=(-5,-2)

1. x – y = 3 3. –y – 7= 0 5. y = –x + 4
Exercise B.
x y
0 -3
3 0
x y
0 4
4 0y=7

7. 2 = 6 – 2y 9. 2x + 3y = 0 11. 3x = 4y
x y
0 0
1 -2/3
x y
0 0
1 3/4
y=4

x y
0 -6
1 0
x y
0 0
1 5/2
13. 3(2 – x) = 3x – y 15. 5(x + 2) – 2y = 10

Exercise C.
1.
1 4
1 3 11
3. a.
2 4.57
1 3 11
b.
2 7
1.75 2.5 3.25
(0, 0) (8, 0)
(2, 6)5.
(4, 0)
(1, 3)
(6, 6)
(10, 6)
(9, 3)
(5, 3)
(3.5, 4.5) (7.5, 4.5)
(6.5, 1.5)(2.5, 1.5)

2.0 rectangular coordinate system

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2.0 rectangular coordinate system