54 the rectangular coordinate system

Rectangular Coordinate System

Back to Algebra–Ready Review Content.

http://www.lahc.edu/math/algebra/alg-ready-placement-review.html

A coordinate system is a system of assigning addresses for

positions in the plane (2 D) or in space (3 D).



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).







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




ordered pair (x, y) where:x = amount to move

right (+) or left (–).









where the axes meet




ordered pair (x, y) where: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).


(+,+)(–,+)

(–,–) (+,–)





Respectively, the signs of

the coordinates of each

quadrant are shown.

QIQII

QIII QIV

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)

Rectangular Coordinate SystemExample. 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)





Hence B is (–2 – 100, 4) = (–102, 4).





Hence B is (–2 – 100, 4) = (–102, 4).

b. What is the coordinate of the point C that is 100 units

directly above A?





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?

Moving up corresponds to increasing the y-coordinate.





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?


Hence C is (–2, 4) = (–2, 4 +100)





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?


Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).





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?





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?


Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).


and 30 below A?

We need to add 50 to the x–coordinate (to the right)





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?


Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).


and 30 below A?


and subtract 30 from the y–coordinate (to go down).





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?


Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).


and 30 below A?



Hence D has coordinate (–2 + 50, 4 – 30)





Hence B is (–2 – 100, 4) = (–102, 4).


directly above A?


Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).


and 30 below A?



Hence D has coordinate (–2 + 50, 4 – 30) = (48, –26).

Rectangular Coordinate SystemMeasurements often are recorded as two related numbers.


In example C. we plotted the following recorded temperatures

35o, 27o, –40o, –25o, 16o, 21o at some location on a line.




time (pm) temperature

1 35

2 27

3 –40

4 –25

5 16

6 21

Suppose these numbers

were recorded at different

times shown on the table

below, then we can plot the

data using the rectangular

coordinate systems.


Example. D. a. Plot the table using

the time and temperature axes.

Measurements often are recorded as two related numbers.




1 35

2 27

3 –40

4 –25

5 16

6 21

0o

40o

–40o

20o

–20o

10o

30o

–10o

–30o






coordinate systems.

pm

temp.


Example. D. a. Plot the table using

the time and temperature axes.

Measurements often are recorded as two related numbers.




1 35

2 27

3 –40

4 –25

5 16

6 21

0o

40o

–40o

20o

–20o

10o

30o

–10o

–30o






coordinate systems.

pm

temp.

(1, 35)

(2, 27)

(3, –40)

(4, –25)

(5, 15)

(6, 21)

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).

Rectangular Coordinate SystemD. Given the following graph, answer the following questions.

16. What’s the temperature

change from 1 pm to 2 pm?

17. What’s the temperature

change from 3 pm to 4 pm?

18. Between what one hour

period the temperature had

risen the most?

19. Between what one hour

period the temperature had

fallen the most?

20. What is the largest temperature difference through out

the afternoon?

63

54 the rectangular coordinate system

Education

system of assigning

rectangular grid

ordered pair x

ordered pair of numbers

horizontal axis

vertical axis

axes meet

algebraready review