LOCATING POINTS IN QUADRANTS AND ON AXES
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An ordered pair [beautiful math coming... please be patient] $\,(x,y)\,$ is a pair of numbers, separated by a comma, and enclosed in parentheses.
The order that the numbers are listed makes a difference: [beautiful math coming... please be patient] $\,(5,3)\,$ 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 [beautiful math coming... please be patient] $\,{x}$-value.
The number that is listed second is called the second coordinate or the $\,y$-value.

For example, [beautiful math coming... please be patient] $\,(5,3)\,$ 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 [beautiful math coming... please be patient] $\,a\,$, $\,b\,$, $\,c\,$, and $\,d\,$: [beautiful math coming... please be patient] $$ (a,b) = (c,d) \ \ \ \ \text{if and only if}\ \ \ \ (a = c\ \ \text{and}\ \ b = d) $$

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 [beautiful math coming... please be patient] $\,xy$-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 $\,2\,$.
To plot the point [beautiful math coming... please be patient] $\,(-2,1)\,$:
  • Start at the point $\,(0,0)\,$.
  • Move $\,2\,$ to the left.
  • Move up $\,1\,$.
Notice that the [beautiful math coming... please be patient] $\,x$-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 [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 [beautiful math coming... please be patient] $\,(x,y)\,$ with $\,x\gt 0\,$ and $\,y\gt 0\,$.
Quadrant II is the set of all points [beautiful math coming... please be patient] $\,(x,y)\,$ with $\,x\lt 0\,$ and $\,y\gt 0\,$.
Quadrant III is the set of all points [beautiful math coming... please be patient] $\,(x,y)\,$ with $\,x\lt 0\,$ and $\,y\lt 0\,$.
Quadrant IV is the set of all points [beautiful math coming... please be patient] $\,(x,y)\,$ 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 [beautiful math coming... please be patient] $\,x$-axis is the set of all points $\,(x,0)\,$, for all real numbers $\,x\,$.
The [beautiful math coming... please be patient] $\,x$-axis is the horizontal axis (think of the horizon).
The [beautiful math coming... please be patient] $\,x$-axis separates the upper two quadrants (I and II) from the bottom two quadrants (III and IV).
Points on the [beautiful math coming... please be patient] $\,x$-axis do not belong to any quadrant.

The [beautiful math coming... please be patient] $\,y$-axis is the set of all points $\,(0,y)\,$, for all real numbers $\,y\,$.
The [beautiful math coming... please be patient] $\,y$-axis is the vertical axis.
The [beautiful math coming... please be patient] $\,y$-axis separates the right two quadrants (I and IV) from the left two quadrants (II and III).
Points on the [beautiful math coming... please be patient] $\,y$-axis do not belong to any quadrant.

The origin is the point [beautiful math coming... please be patient] $\,(0,0)\,$.
The origin is the only point that lies on both the $\,x$-axis and the $\,y$-axis.

Points with positive [beautiful math coming... please be patient] $\,x$-values lie to the RIGHT of the $\,y$-axis.
Points with negative [beautiful math coming... please be patient] $\,x$-values lie to the LEFT of the $\,y$-axis.
Points with positive [beautiful math coming... please be patient] $\,y$-values lie ABOVE the $\,x$-axis.
Points with negative [beautiful math coming... please be patient] $\,y$-values lie BELOW the $\,x$-axis.

EXAMPLES:
Question: In what quadrant does the point [beautiful math coming... please be patient] $\,(-1,3)\,$ lie?
Answer: Quadrant II
Question: Suppose that [beautiful math coming... please be patient] $\,a\gt 0\,$ and $\,b\lt 0\,$.
Then, in what quadrant does the point $\,(a,b)\,$ lie?
Answer: Quadrant IV
Question: Let [beautiful math coming... please be patient] $\,t\,$ be a nonzero real number.
Does the point $\,(0,t)\,$ lie on the $\,x$-axis?
Answer: no
Question: Does the point [beautiful math coming... please be patient] $\,(0,0)\,$ lie on the $\,y$-axis?
Answer: yes
Question: Does the point [beautiful math coming... please be patient] $\,(-3,5)\,$ lie below the $\,y$-axis?
Answer: no
Question: Does the point [beautiful math coming... please be patient] $\,(-3,5)\,$ 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

 
 
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 33; there are 33 different problem types)