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LOCATING POINTS IN QUADRANTS AND ON AXES

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An ordered pair  ( x , y )  is a pair of numbers, separated by commas, and enclosed in parentheses.
The order that the numbers are listed makes a difference: (5,3) is different from (3,5).
Thus, the name ordered pair is appropriate.

The number that is listed first is called the first coordinate or the x-value.
The number that is listed second is called the second coordinate or the y-value.

For example,  (5,3)  is an ordered pair; the first coordinate is  5  and the second coordinate is  3 .
Alternately, the x-value is  5  and the y-value is  3 .

EQUALITY OF ORDERED PAIRS

For all real numbers  a ,  b ,  c  and  d ,

(a,b) = (c,d)   if and only if   (a = c and b = d).


Partial translation:
For two ordered pairs to be equal, the first coordinates must be equal, and the second coordinates must be equal.

The coordinate plane (also called the xy-plane) is a device to "picture" ordered pairs.
Each ordered pair corresponds to a point in the coordinate plane.
(For this reason, ordered pairs are often called points.)

The process of showing where a point "lives" in a coordinate plane is called plotting the point.

To plot the point (1,-2):
  • Start at the point (0,0) (look at the diagram at right).
  • Move  1  to the right.
  • Move down  2 .
To plot the point (-2,1):
  • Start at the point (0,0).
  • Move  2  to the left.
  • Move up  1 .
Notice that the x-value tells you how to move left/right:
if the x-value is positive, move right;
if the x-value is negative, move left.

Notice that the y-value tells you how to move up/down:
if the y-value is positive, move up;
if the y-value is negative, move down.

Quadrant I is the set of all points  ( x , y )  with  x > 0  and  y > 0 .
Quadrant II is the set of all points  ( x , y )  with  x < 0  and  y > 0 .
Quadrant III is the set of all points  ( x , y )  with  x < 0  and  y < 0 .
Quadrant IV is the set of all points  ( x , y )  with  x > 0  and  y < 0 .

Roman numerals (I, II, III, IV) are always used to talk about the four quadrants.
You start numbering the quadrants in the upper right, and then proceed counter-clockwise.

The  x-axis  is the set of all points  ( x , 0 ) .
The  x-axis  is the horizontal axis (think of the horizon).

The  y-axis  is the set of all points  ( 0 , y ) .
The  y-axis  is the vertical axis.

The  origin  is the point  (0,0) .

Points with positive x-values lie to the RIGHT of the y-axis.
Points with negative x-values lie to the LEFT of the y-axis.
Points with positive y-values lie ABOVE the x-axis.
Points with negative y-values lie BELOW the x-axis.


EXAMPLES:

Question: In what quadrant does the point  (-1,3)  lie?
Answer: Quadrant II

Question: Suppose that  a > 0  and  b > 0.
Then, in what quadrant does the point  (a,b)  lie?
Answer: Quadrant I

Question: Suppose that  a > 0  and  b > 0.
Then, in what quadrant does the point  (a,-b)  lie?
Answer: Quadrant IV

Question: Let  t  be a nonzero real number.
Does the point  (0,t)  lie on the x-axis?
Answer: no

Question: Does the point  (0,0)  lie on the y-axis?
Answer: yes

Question: Does the point  (-3,5)  lie below the x-axis?
Answer: no

Question: Does the point  (-3,5)  lie to the left of the y-axis?
Answer: yes

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.