Graphing Points - Pre-Algebra

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.

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 .

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.

Substitute 10 for and for , and then solve for :

Substitute for and 8 for , and solve for :

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.

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 .

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.

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

By definition, any point in the coordinate plane that has a non-zero value -coordinate and a zero-value -coordinate is on the -axis.

All points in Quadrant II have negative -coordinates and positive -coordinates. The only answer that fulfills these criteria is .

An ordered is always written in the following format:

where x is the position on the x-axis, and y is the position on the y-axis.

If we look at point a, we find where it falls on the x-axis. We can see that it's position on the x-axis is 1. Then, we see where it falls on the y-axis. It's position on the y-axis is 4. So, we can write the ordered pair for point a as

We will do the same for b. The position of point b on the x-axis is -5. The position of point b on the y-axis is 2. So, we can write the ordered pair for point b as

And finally for c. The position of point c on the x-axis is zero. The position of point c on the y-axis is -5. So, we can write the ordered pair of point c as

When looking at a coordinate plane, draw the letter C in there making sure you connect all four quarters of the grid. The C should start in the top right corner, move to the top left, then bottom left, then finish in the bottom right corner.

The quadrants start with and finish with .

Therefore the coordinates means to the right and down which would place the coordinates in quadrant 4.

The following coordinate plane displays the different quandrants.

You can also see there is a point that has been graphed.

That point is (-3, 4).

Therefore, (-3, 4) can be found in the second quandrant of a coordinate plane.

The midpoint formula is

where and are the points given. So, using the points

(-5, 6) and (1, 2) we can substitute into the midpoint formula and get

Therefore, the midpoint is (-2, 4).