56 the rectangular coordinate system

Rectangular Coordinate System

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


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).




The horizontal axis is called the x-axis.



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



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.



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:



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 (–).



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 moveup (+) or down (–).

x = amount to move right (+) or left (–). y = amount to moveup (+) or down (–).


x = amount to move right (+) or left (–). y = amount to moveup (+) or down (–).For example, the point P corresponds to (4, –3)


x = amount to move right (+) or left (–). y = amount to moveup (+) or down (–).For example, the point P corresponds to (4, –3) is4 right,


x = amount to move right (+) or left (–). y = amount to moveup (+) or down (–).For example, the point P corresponds to (4, –3) is4 right, and 3 down from the origin.


(4, –3)P



Example A. Label the points A(-1, 2), B(-3, -2),C(0, -5).




A




A

B




A

B

C




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





A

B

C

Example B: Find the coordinate of P, Q, R as shown.

P

Q

R





A

B

C

Example B: Find the coordinate of P, Q, R as shown.P(4, 5),

P

Q

R





A

B

C

Example B: Find the coordinate of P, Q, R as shown.P(4, 5), Q(3, -5),

P

Q

R





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

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

(0,0)


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

(0,0)



(5, 0) (0,0)



(5, 0)(-6, 0) (0,0)



(5, 0)(-6, 0)

Any point on the y-axishas coordinate of the form (0, y). (0,0)



(5, 0)(-6, 0)

Any point on the y-axishas coordinate of the form (0, y).

(0, 6)

(0,0)



(5, 0)(-6, 0)

Any point on the y-axishas coordinate of the form (0, y).

(0, -4)

(0, 6)

(0,0)


The coordinate of the origin is (0, 0).Any point on the x-axishas coordinate of the form (x, 0).Any point on the y-axishas coordinate of the 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




QIQII

QIII QIV

(+,+)




QIQII

QIII QIV

(+,+)(–,+)



(+,+)(–,+)

(–,–) (+,–)

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.

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 reflectionacross the y-axis.

(5,4)


When the x-coordinate of the a point (x, y) is changed to its opposite as (–x , y), the new point is the reflectionacross the y-axis.

(5,4)(–5,4)


When the x-coordinate of the a point (x, y) is changed to its opposite as (–x , y), the new point is the reflectionacross 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)(–5,4)

(5, –4)



(5,4)(–5,4)

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



(5,4)(–5,4)

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

(–5, –4)


Movements and CoordinatesRectangular Coordinate System

Movements and CoordinatesLet A be the point (2, 3).


A(2, 3)

Movements and CoordinatesLet A be the point (2, 3). Suppose it’s x–coordinate is increased by 4 to (2 + 4, 3) = (6, 3)


A(2, 3)

Movements and CoordinatesLet 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,


A B(2, 3) (6, 3)

Movements and CoordinatesLet 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.


A B

x–coord. increased by 4

(2, 3) (6, 3)



A B

Similarly if the x–coordinate of (2, 3) is decreased by 4 to (2 – 4, 3) = (–2, 3)


(2, 3) (6, 3)



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. decreased by 4

(2, 3) (6, 3)(–2, 3)



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



(2, 3) (6, 3)(–2, 3)



A B


Hence we conclude that changes in the x–coordinates of a point move the point right and left.

C



(2, 3) (6, 3)(–2, 3)



A B


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



(2, 3) (6, 3)(–2, 3)



A B


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



(2, 3) (6, 3)(–2, 3)

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


A(2, 3)

Again let A be the point (2, 3). Suppose its y–coordinate is increased by 4 to (2, 3 + 4) = (2, 7)


A(2, 3)

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,


A

Dy–coord. increased by 4

(2, 3)

(2, 7)

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.


A

Dy–coord. increased by 4

(2, 3)

(2, 7)



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)



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



(2, 3)

(2, 7)

(2, –1)



A

D


Hence we conclude that changes in the y–coordinates of a point move the point right and left.

E



(2, 3)

(2, 7)

(2, –1)



A

D


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



(2, 3)

(2, 7)

(2, –1)



A

D


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



(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.


Moving left corresponds to decreasing the x-coordinate.Hence B is (–2 – 100, 4)


Moving left corresponds to decreasing the x-coordinate.Hence B is (–2 – 100, 4) = (–102, 4).


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.



Moving up corresponds to increasing the y-coordinate.Hence C is (–2, 4) = (–2, 4 +100)



Moving up corresponds to increasing the y-coordinate.Hence C is (–2, 4) = (–2, 4 +100) = (–2, 104).



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)




We need to add 50 to the x–coordinate (to the right) and subtract 30 from the y–coordinate (to go down).




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)




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).




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).

Coordinate systems are used beside the geometric context

Rectangular Coordinate SystemMeasurements often are recorded as two related numbers.

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.

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. In example C. we plotted the following recorded temperatures 35o, 27o, –40o, –25o, 16o, 21o at some location on a line.


1 35

2 27

3 –40

4 –25

5 16

6 21

0o

40o

–40o

20o

–20o

10o

30o

–10o

–30o


pm

temp.


Example. D. a. Plot the table using the time and temperature axes.

Measurements 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.


1 35

2 27

3 –40

4 –25

5 16

6 21

0o

40o

–40o

20o

–20o

10o

30o

–10o

–30o


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

EF

GH

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.

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


The point that’s8. 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 temperaturechange from 1 pm to 2 pm?17. What’s the temperaturechange 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?

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56 the rectangular coordinate system

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