LOCATING POINTS IN QUADRANTS AND ON AXES

An *ordered pair*
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$\,(x,y)\,$ is a pair of numbers,
separated by a comma, and enclosed in parentheses.

The order that the numbers are listed makes a difference:
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$\,(5,3)\,$ 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
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$\,{x}$-*value*.

The number that is listed second is called the *second coordinate* or the
$\,y$-*value*.

For example,
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$\,(5,3)\,$ 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
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$\,a\,$, $\,b\,$, $\,c\,$, and $\,d\,$:
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$$
(a,b) = (c,d) \ \ \ \ \text{if and only if}\ \ \ \
(a = c\ \ \text{and}\ \ b = d)
$$

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
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$\,xy$-*plane*) is a device to ‘picture’ ordered pairs.

Each ordered pair corresponds to a point in the coordinate plane,

and each point in the coordinate plane corresponds to an ordered pair.

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 [beautiful math coming... please be patient] $\,(1,-2)\,$: - Start at the point $\,(0,0)\,$ (look at the diagram at right).
- Move $\,1\,$ to the right.
- Move down $\,2\,$.
- Start at the point $\,(0,0)\,$.
- Move $\,2\,$ to the left.
- Move up $\,1\,$.
if the $\,x$-value is positive, move right; if the $\,x$-value is negative, move left. Notice that the [beautiful math coming... please be patient] $\,y$-value tells you how to move up/down: if the $\,y$-value is positive, move up; if the $\,y$-value is negative, move down. |

The *quadrants* (see below) divide the coordinate plane into four regions.

Quadrant I is the set of all points
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$\,(x,y)\,$ with $\,x\gt 0\,$ and $\,y\gt 0\,$.

Quadrant II is the set of all points
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$\,(x,y)\,$ with $\,x\lt 0\,$ and $\,y\gt 0\,$.

Quadrant III is the set of all points
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$\,(x,y)\,$ with $\,x\lt 0\,$ and $\,y\lt 0\,$.

Quadrant IV is the set of all points
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$\,(x,y)\,$ with $\,x\gt 0\,$ and $\,y\lt 0\,$.

Roman numerals (I, II, III, IV) are conventionally used to talk about the four quadrants.

You start numbering the quadrants in the upper right, and then proceed counter-clockwise.

The
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$\,x$-axis is the set of all points $\,(x,0)\,$, for all real numbers $\,x\,$.

The
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$\,x$-axis is the *horizontal* axis (think of the horizon).

The
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$\,x$-axis separates the upper two quadrants (I and II) from the bottom two quadrants (III and IV).

Points on the
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$\,x$-axis do not belong to *any* quadrant.

The
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$\,y$-axis is the set of all points $\,(0,y)\,$, for all real numbers $\,y\,$.

The
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$\,y$-axis is the *vertical* axis.

The
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$\,y$-axis separates the right two quadrants (I and IV) from the left two quadrants (II and III).

Points on the
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$\,y$-axis do not belong to *any* quadrant.

The *origin* is the point
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$\,(0,0)\,$.

The origin is the *only* point that lies on *both* the $\,x$-axis and the $\,y$-axis.

Points with positive
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$\,x$-values lie to the RIGHT of the $\,y$-axis.

Points with negative
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$\,x$-values lie to the LEFT of the $\,y$-axis.

Points with positive
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$\,y$-values lie ABOVE the $\,x$-axis.

Points with negative
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$\,y$-values lie BELOW the $\,x$-axis.

EXAMPLES:

Question:
In what quadrant does the point
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$\,(-1,3)\,$ lie?

Answer:
Quadrant II

Question:
Suppose that
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$\,a\gt 0\,$ and $\,b\lt 0\,$.

Then, in what quadrant does the point $\,(a,b)\,$ lie?

Then, in what quadrant does the point $\,(a,b)\,$ lie?

Answer:
Quadrant IV

Question:
Let
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$\,t\,$ be a nonzero real number.

Does the point $\,(0,t)\,$ lie on the $\,x$-axis?

Does the point $\,(0,t)\,$ lie on the $\,x$-axis?

Answer:
no

Question:
Does the point
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$\,(0,0)\,$ lie on the $\,y$-axis?

Answer:
yes

Question:
Does the point
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$\,(-3,5)\,$ lie below the $\,y$-axis?

Answer:
no

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

Answer:
yes

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Practice With Points

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Practice With Points

On this exercise, you will not key in your answer.

However, you can check to see if your answer is correct.

However, you can check to see if your answer is correct.