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4.1: Points in the Coordinate Plane

Difficulty Level: At Grade Created by: CK-12
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Practice Points in the Coordinate Plane
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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.

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 $x-$ axis and the vertical line is the $y-$ 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 $x-$ axis and $y-$ axis the point is. The ordered pair is written in parentheses, with the $x-$ coordinate (also called the abscissa ) first and the $y-$ coordinate (or the ordinate ) second.

$& (1,7) && \text{An ordered pair with an} \ x-\text{value of one and a} \ y-\text{value of seven}\\& (0, 5) && \text{An ordered pair with an} \ x-\text{value of zero and a} \ y-\text{value of five}\\& (-2.5, 4) && \text{An ordered pair with an} \ x-\text{value of -2.5} \ \text{and a} \ y-\text{value of four}\\& (-107.2, -.005) && \text{An ordered pair with an} \ x-\text{value of -107.2} \ \text{and a} \ y-\text{value of} \ -.005$

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 $P$ in the diagram above

Solution

Imagine you are standing at the origin (the point where the $x-$ axis meets the $y-$ axis). In order to move to a position where $P$ was directly above you, you would move 3 units to the right (we say this is in the positive $x-$ direction).

The $x-$ coordinate of $P$ is +3.

Now if you were standing at the 3 marker on the $x-$ axis, point $P$ would be 7 units above you (above the axis means it is in the positive $y$ direction).

The $y-$ coordinate of $P$ is +7.

The coordinates of point $P$ are (3, 7).

Example B

Find the coordinates of the points labeled $Q$ and $R$ in the diagram to the right.

Solution

In order to get to $Q$ we move three units to the right, in the positive $x-$ direction, then two units down . This time we are moving in the negative $y-$ direction. The $x-$ coordinate of $Q$ is +3, the $y-$ coordinate of $Q$ is −2.

The coordinates of $R$ are found in a similar way. The $x-$ coordinate is +5 (five units in the positive $x-$ direction) and the $y-$ coordinate is again −2.

The coordinates of $Q$ are (3, −2). The coordinates of $R$ are (5, −2).

Example C

Triangle $ABC$ is shown in the diagram to the right. Find the coordinates of the vertices $A, B$ and $C$ .

Point $A$ :

$x-\text{coordinate} = -2$

$y-\text{coordinate} = +5$

Point $B$ :

$x-\text{coordinate} = +3$

$y-\text{coordinate} = -3$

Point $C$ :

$x-\text{coordinate} = -4$

$y-\text{coordinate} = -1$

Solution

$A(-2, 5)$

$B(3, -3)$

$C(-4, -1)$

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.

$A(2,7) \quad B(-4, 6) \quad D(-3, -3) \quad E(0, 2)\quad F(7, -5)$

Point $A(2,7)$ is 2 units right, 7 units up. It is in Quadrant I.

Point $B(-4, 6)$ is 4 units left, 6 units up. It is in Quadrant II.

Point $D(-3, -3)$ is 3 units left, 3 units down. It is in Quadrant III.

Point $E(0, 2)$ is 2 units up from the origin. It is right on the $y-$ axis, between Quadrants I and II.

Point $F(7, -5)$ is 7 units right, 5 units down. It is in Quadrant IV.

Watch this video for help with the Examples above.

Vocabulary

• The coordinate plane is a two-dimensional space defined by a horizontal number line (the $x-$ axis ) and a vertical number line (the $y-$ 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 $x-$ axis and $y-$ axis the point is. The $x-$ coordinate is always written first, then the $y-$ coordinate , in the form $(x, y)$ .
• 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.

$A(2.5, 0.5) \quad B(\pi, 1.2) \quad C(2, 1.75) \quad D(0.1, 1.2) \quad E(0, 0)$

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 $x$ or $y$ . Also, there are no $x-$ values bigger than about 3.14, and 1.75 is the largest value of $y$ . We can therefore show just the part of the coordinate plane where $0 \le x \le 3.5$ and $0 \le y \le 2$ .

• 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 $x-$ axis is different than the scale on the $y-$ axis, so distances that look the same on both axes are actually greater in the $x-$ 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

1. Identify the coordinates of each point, $A-F$ , on the graph below.
2. Draw a line on the above graph connecting point $B$ with the origin. Where does that line intersect the line connecting points $C$ and $D$ ?

Plot the following points on a graph and identify which quadrant each point lies in:

1. (4, 2)
2. (-3, 5.5)
3. (4, -4)
4. (-2, -3)

Without graphing the following points, identify which quadrant each lies in:

1. (5, 3)
2. (-3, -5)
3. (-4, 2)
4. (2, -4)

Aug 13, 2012

Aug 21, 2014