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 **\begin{align*}x-\end{align*}axis** and the vertical line is the **\begin{align*}y-\end{align*}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 \begin{align*}x-\end{align*}axis and \begin{align*}y-\end{align*}axis the point is. The ordered pair is written in parentheses, with the \begin{align*}x-\end{align*}**coordinate** (also called the **abscissa**) first and the \begin{align*}y-\end{align*}**coordinate** (or the **ordinate**) second.

\begin{align*}& (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\end{align*}

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 \begin{align*}P\end{align*} in the diagram above*

**Solution**

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

The \begin{align*}x-\end{align*}coordinate of \begin{align*}P\end{align*} is +3.

Now if you were standing at the 3 marker on the \begin{align*}x-\end{align*}axis, point \begin{align*}P\end{align*} would be 7 units **above** you (above the axis means it is in the **positive** \begin{align*}y\end{align*} direction).

The \begin{align*}y-\end{align*}coordinate of \begin{align*}P\end{align*} is +7.

**The coordinates of point \begin{align*}P\end{align*} are (3, 7).**

#### Example B

*Find the coordinates of the points labeled \begin{align*}Q\end{align*} and \begin{align*}R\end{align*} in the diagram to the right.*

**Solution**

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

The coordinates of \begin{align*}R\end{align*} are found in a similar way. The \begin{align*}x-\end{align*}coordinate is +5 (five units in the positive \begin{align*}x-\end{align*}direction) and the \begin{align*}y-\end{align*}coordinate is again −2.

**The coordinates of \begin{align*}Q\end{align*} are (3, −2). The coordinates of \begin{align*}R\end{align*} are (5, −2).**

#### Example C

*Triangle \begin{align*}ABC\end{align*} is shown in the diagram to the right. Find the coordinates of the vertices \begin{align*}A, B\end{align*} and \begin{align*}C\end{align*}.*

Point \begin{align*}A\end{align*}:

\begin{align*}x-\text{coordinate} = -2\end{align*}

\begin{align*}y-\text{coordinate} = +5\end{align*}

Point \begin{align*}B\end{align*}:

\begin{align*}x-\text{coordinate} = +3\end{align*}

\begin{align*}y-\text{coordinate} = -3\end{align*}

Point \begin{align*}C\end{align*}:

\begin{align*}x-\text{coordinate} = -4\end{align*}

\begin{align*}y-\text{coordinate} = -1\end{align*}

**Solution**

\begin{align*}A(-2, 5)\end{align*}

\begin{align*}B(3, -3)\end{align*}

\begin{align*}C(-4, -1)\end{align*}

**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.*

\begin{align*}A(2,7) \quad B(-4, 6) \quad D(-3, -3) \quad E(0, 2)\quad F(7, -5)\end{align*}

Point \begin{align*}A(2,7)\end{align*} is 2 units right, 7 units up. It is in Quadrant I.

Point \begin{align*}B(-4, 6)\end{align*} is 4 units left, 6 units up. It is in Quadrant II.

Point \begin{align*}D(-3, -3)\end{align*} is 3 units left, 3 units down. It is in Quadrant III.

Point \begin{align*}E(0, 2)\end{align*} is 2 units up from the origin. It is right on the \begin{align*}y-\end{align*}axis, between Quadrants I and II.

Point \begin{align*}F(7, -5)\end{align*} 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**\begin{align*}x-\end{align*}axis**) and a vertical number line (the**\begin{align*}y-\end{align*}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 \begin{align*}x-\end{align*}axis and \begin{align*}y-\end{align*}axis the point is. The \begin{align*}x-\end{align*}**coordinate**is always written first, then the \begin{align*}y-\end{align*}**coordinate**, in the form \begin{align*}(x, y)\end{align*}. -
**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.*

\begin{align*}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)\end{align*}

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

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 \begin{align*}x-\end{align*}axis is different than the scale on the \begin{align*}y-\end{align*}axis, so distances that look the same on both axes are actually greater in the \begin{align*}x-\end{align*}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.

### Explore More

- Identify the coordinates of each point, \begin{align*}A-F\end{align*}, on the graph below.
- Draw a line on the above graph connecting point \begin{align*}B\end{align*} with the origin. Where does that line intersect the line connecting points \begin{align*}C\end{align*} and \begin{align*}D\end{align*}?

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)