### Points in the Coordinate Plane

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**

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, and 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 (also known as 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}\\ & (\text{-}2.5, 4) && \text{An ordered pair with an} \ x\text{-value of -2.5} \ \text{and a} \ y\text{-value of four}\\ & (\text{-}107.2, \text{-}0.005) && \text{An ordered pair with an} \ x\text{-value of -107.2} \ \text{and a} \ y\text{-value of -}0.005\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!

#### Finding the Coordinates of a Point

1.

Find the coordinates of the point labeled \begin{align*}P\end{align*} in the diagram above

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).**

2.

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

In order to get to \begin{align*}Q,\end{align*} move three units to the **right** (the positive \begin{align*}x\end{align*} direction) then two units **down** (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, so the ordered pair is (3, -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, so the ordered pair is (5, -2)

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

#### Finding the Coordinates of Vertices

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} = \text{-}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} = \text{-}3\end{align*}

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

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

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

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

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

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

**Plotting 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.

Plot the following points on the coordinate plane.

\begin{align*}A(2,7) \quad B(\text{-}4, 6) \quad D(\text{-}3, \text{-}3) \quad E(0, 2)\quad F(7, \text{-}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(\text{-}4, 6)\end{align*} is 4 units left, 6 units up. It is in Quadrant II.

Point \begin{align*}D(\text{-}3, \text{-}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, \text{-}5)\end{align*} is 7 units right, 5 units down. It is in Quadrant IV.

### Examples

#### Example 1

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*}

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.

### Review

- 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)

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.1.