What if you were given the x - and y -coordinates of a point like (-2, 3). How could you determine in which quadrant of the coordinate plane this point would lie? After completing this Concept, you'll be able to plot points like this one given their coordinates.
Watch This
CK-12 Foundation: 0401S Points in the Coordinate Plane (H264)
Try This
For more practice locating and naming points on the coordinate plane, try playing the Coordinate Plane Game at http://www.math-play.com/Coordinate%20Plane%20Game/Coordinate%20Plane%20Game.html .
Guidance
Lydia lives 2 blocks north and one block east of school; Travis lives three blocks south and two blocks west of school. What’s the shortest line connecting their houses?
The Coordinate Plane
We’ve seen how to represent numbers using number lines; now we’ll see how to represent sets of numbers using a coordinate plane. The coordinate plane can be thought of as two number lines that meet at right angles. The horizontal line is called the axis and the vertical line is the axis . Together the lines are called the axes , and the point at which they cross is called the origin . The axes split the coordinate plane into four quadrants , which are numbered sequentially (I, II, III, IV) moving counter-clockwise from the upper right.
Identify Coordinates of Points
When given a point on a coordinate plane, it’s easy to determine its coordinates . The coordinates of a point are two numbers - written together they are called an ordered pair . The numbers describe how far along the axis and axis the point is. The ordered pair is written in parentheses, with the coordinate (also called the abscissa ) first and the coordinate (or the ordinate ) second.
Identifying coordinates is just like reading points on a number line, except that now the points do not actually lie on the number line! Look at the following example.
Example A
Find the coordinates of the point labeled in the diagram above
Solution
Imagine you are standing at the origin (the point where the axis meets the axis). In order to move to a position where was directly above you, you would move 3 units to the right (we say this is in the positive direction).
The coordinate of is +3.
Now if you were standing at the 3 marker on the axis, point would be 7 units above you (above the axis means it is in the positive direction).
The coordinate of is +7.
The coordinates of point are (3, 7).
Example B
Find the coordinates of the points labeled and in the diagram to the right.
Solution
In order to get to we move three units to the right, in the positive direction, then two units down . This time we are moving in the negative direction. The coordinate of is +3, the coordinate of is −2.
The coordinates of are found in a similar way. The coordinate is +5 (five units in the positive direction) and the coordinate is again −2.
The coordinates of are (3, −2). The coordinates of are (5, −2).
Example C
Triangle is shown in the diagram to the right. Find the coordinates of the vertices and .
Point :
Point :
Point :
Solution
Plot Points in a Coordinate Plane
Plotting points is simple, once you understand how to read coordinates and read the scale on a graph. As a note on scale, in the next two examples pay close attention to the labels on the axes.
Example D
Plot the following points on the coordinate plane.
Point is 2 units right, 7 units up. It is in Quadrant I.
Point is 4 units left, 6 units up. It is in Quadrant II.
Point is 3 units left, 3 units down. It is in Quadrant III.
Point is 2 units up from the origin. It is right on the axis, between Quadrants I and II.
Point is 7 units right, 5 units down. It is in Quadrant IV.
Watch this video for help with the Examples above.
CK-12 Foundation: Points in the Coordinate Plane
Vocabulary
- The coordinate plane is a two-dimensional space defined by a horizontal number line (the axis ) and a vertical number line (the axis ). The origin is the point where these two lines meet. Four areas, or quadrants, are formed as shown in the diagram above.
- Each point on the coordinate plane has a set of coordinates, two numbers written as an ordered pair which describe how far along the axis and axis the point is. The coordinate is always written first, then the coordinate , in the form .
- Functions are a way that we can relate one quantity to another. Functions can be plotted on the coordinate plane.
Guided Practice
Plot the following points on the coordinate plane.
Solution:
Here we see the importance of choosing the right scale and range for the graph. In Example 4, our points were scattered throughout the four quadrants. In this case, all the coordinates are positive, so we don’t need to show the negative values of or . Also, there are no values bigger than about 3.14, and 1.75 is the largest value of . We can therefore show just the part of the coordinate plane where and .
Here are some other important things to notice about this graph:
- The tick marks on the axes don’t correspond to unit increments (i.e. the numbers do not go up by one each time). This is so that we can plot the points more precisely.
- The scale on the axis is different than the scale on the axis, so distances that look the same on both axes are actually greater in the direction. Stretching or shrinking the scale in one direction can be useful when the points we want to plot are farther apart in one direction than the other.
Practice
- Identify the coordinates of each point, , on the graph below.
- Draw a line on the above graph connecting point with the origin. Where does that line intersect the line connecting points and ?
Plot the following points on a graph and identify which quadrant each point lies in:
- (4, 2)
- (-3, 5.5)
- (4, -4)
- (-2, -3)
Without graphing the following points, identify which quadrant each lies in:
- (5, 3)
- (-3, -5)
- (-4, 2)
- (2, -4)