# Pre-Algebra : Graphing Points

## Example Questions

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### Example Question #1 : Graphing Points

How many of the following points are in Quadrant II?

Explanation:

The key is solving this problem is remembering the division of quadrants and how the sign of the x- and y-coordinates determines the quadrant.  The picture diagram below helps greatly.

Using the diagram as a guide, we remember that a point is only in Quadrant II if the x-coordinate is negative while the y-coordinate is positive.  We therefore need only to look at all of the points in our list to determine how many satisfy these conditions.  Two of the points,  and , both have x-coordinates that are negative.  However, neither of these have positive y-coordinates.  In , the y-coordinate is negative, which means the point is in Quadrant III.  in , the y-coordinate is , meaning that the point lies on the x-axis rather than in any quadrant.  Therefore, none of the listed points are in Quadrant II, meaning our answer is .

### Example Question #2 : Graphing Points

Determine the quadrant in which the point (-5,-3) lies.

The point lies on an axis.

Explanation:

The point given in the problem is expressed as an ordered pair in the form . The number in the place of the  is called the x-coordinate and tells how far from the origin to move horizontally, while the number in place of the  is called the y-coordinate and tells how far to move from the origin vertically.  The coorinate plane (the set of axes we graph points on) is divided into four regions (or quadrants) by the x-axis and y-axis.  The top right region is Quadrant I, the top left region is Quadrant II, the bottom left region is Quadrant III, and the bottom right region is Quadrant IV.

Going back to our x-coordinate and y-coordinate, a postive x-coordinate means we move right from the origin the same number of units as the value.  A negative x-coordinate means we move left from the origin.  With y-coordinates, a positive value means we move up, while a negative value means we move down the indicated number of units.  Since the ordered pair given in the problem is   , our x-coordinate is  and our y-coordinate is .  That means we should move left from the origin five units and then down three units. The resulting plotted point is shown.

Having plotted the point we can clearly see that we are in Quadrant III, giving us the correct answer.  In general, we can actually determine the quadrant of a point without plotting if we keep some general rules in mind.  If the x- and y-coordinates are both positive, we are in Quadrant I.  If the x-coordinate is negative while the y-coordinate remains positive, we are in Quadrant II.  If both coordinates are negative, we are in Quadrant III.  Finally, if the x-coordinate is positive while the y-coordinate is negative, we are in Quadrant IV.  If either of the coordinates happens to be zero, our point will lie on one of the axes.

### Example Question #1 : Graphing Points

The point  is on the graph of the line . Evaluate .

Explanation:

Substitute  for  and 8 for , and solve for :

### Example Question #4 : Graphing Points

Which of the following is not a point on the graph of the equation  ?

Explanation:

For each of the five choices, substitute the coordinates for  and  in that order, and test the validity of the resulting equation. Below is the proof that  is on the graph:

:

This is true, so  is on the graph.

The points , and  can all be proved by the same procedure to be part of the graph.

However, we can prove that  is not on the graph by the same procedure:

This is false, so  is proved to be not on the graph. This is the correct choice.

### Example Question #2 : Graphing Points

The point  is on the graph of the line . Evaluate .

Explanation:

Substitute 10 for  and  for , and then solve for :

### Example Question #1 : How To Find The Points On A Coordinate Plane

Which of the following points would be located in Quadrant III?

Explanation:

By definition, a point on the coordinate plane that is in Quadrant III must have both a negative  coordinate and a negative  coordinate. The only answer choice that satisfies both of these conditions is .

### Example Question #3 : Graphing Points

In which quadrant or on what axis will you find the point ?

The -axis

Explanation:

The point  has a negative  coordinate and a positive  coordinate. By definition, any point on a coordinate plane with these characteristics is located in Quadrant II.

### Example Question #2 : How To Find The Points On A Coordinate Plane

In which quadrant or on which axis will you find the point ?

The -axis

The -axis

The -axis

Explanation:

The point  does not move in any direction on the -axis, but does move  units up on the -axis. Therefore, the point is located only on the -axis.

### Example Question #2 : How To Find The Points On A Coordinate Plane

In which quadrant or on which axis will you find the point ?

The -axis

Explanation:

By definition, a point with a negative -coordinate and a negative -coordinate lies in Quadrant III on the coordinate plane.

### Example Question #4 : Graphing Points

Which of the following points is in Quadrant II on the coordinate plane?