Kaitlyn walked into Math class and saw the following image displayed on the board. Her teacher asked everyone in the class to duplicate the picture on a blank sheet of paper.

When the teacher felt that the students had completed the drawing, she asked them to share their results with the class. Most of the students had difficulty reproducing the picture. Kaitlyn told the class that she could not make the picture the same size as the one shown. She also said that she had a problem locating the leaves in the same places on the stem. Her teacher said that she could offer a solution to these problems. What was the solution?

### The Cartesian Plane

The **Cartesian plane** is a system of four areas or quadrants produced by the perpendicular intersection of two number lines. The two number lines intersect at right angles. The point of intersection is known as the **origin**. One number line is a horizontal line and this is called the **\begin{align*}y\text{-axis}\end{align*}** The other number line is a vertical line and it is called the **\begin{align*}y\text{-axis}.\end{align*}** The two number lines are referred to as the **axes** of the Cartesian plane. The Cartesian plane, also known as the **coordinate plane**, has four quadrants that are labeled counterclockwise.

The value of the origin on the \begin{align*}x\end{align*}-axis is zero. If you think of the \begin{align*}x\end{align*}-axis as a number line, the numbers to the right of zero are positive values, and those to the left of zero are negative values. The same can be applied to the \begin{align*}y\end{align*}-axis. The value of the origin on the \begin{align*}y\end{align*}-axis is zero. The numbers above zero are positive values and those below zero are negative values.

Every point that is plotted on a Cartesian plane has two values associated with it. The first value represents the \begin{align*}x\end{align*}-value and the second value represents the \begin{align*}y\end{align*}-value. These two values are called the **coordinates** of the point and are written as the ordered pair \begin{align*}(x, y)\end{align*}.

To plot a point on the Cartesian plane:

- Start at zero (the origin) and locate the \begin{align*}x\end{align*}-coordinate on the \begin{align*}x\end{align*}-axis.
- If the \begin{align*}x\end{align*}-coordinate is positive, move to the right of the origin the number of units displayed by the number. If the \begin{align*}x\end{align*}-coordinate is negative, move to the left of the origin the number of units displayed by the number.
- Once the \begin{align*}x\end{align*}-coordinate (also called the
**abscissa**) has been located, move vertically the number of units displayed by the \begin{align*}y\end{align*}-coordinate (also called the**ordinate**). If the \begin{align*}y\text{-coordinate}\end{align*} is positive, move vertically upward from the \begin{align*}x\end{align*}-coordinate by the number of units displayed by the \begin{align*}y\end{align*}-coordinate. If the \begin{align*}y\end{align*}-coordinate is negative, move vertically downward from the \begin{align*}x\end{align*}-coordinate by the number of units displayed by the \begin{align*}y\text{-coordinate}.\end{align*} - The point can now be plotted.

Examine the points \begin{align*}A, B, C\end{align*} and \begin{align*}D\end{align*} that have been plotted on the graph below.

- \begin{align*}A (\text{-}4, 2)\end{align*} From the origin, move to the left four units (along the red line on the \begin{align*}x\end{align*}-axis). Now, move vertically upward two units. Plot the point \begin{align*}A\end{align*}.
- \begin{align*}B (\text{-}2, \text{-}1)\end{align*} From the origin, move to the left two units (along the red line on the \begin{align*}x\end{align*}-axis). Now, move vertically downward one unit. Plot the point \begin{align*}B\end{align*}.
- \begin{align*}C (3, \text{-}4)\end{align*} From the origin, move to the right three units (along the red line on the \begin{align*}x\end{align*}-axis). Now, move vertically downward four units. Plot the point \begin{align*}C\end{align*}.
- \begin{align*}D (6, 3)\end{align*} From the origin, move to the right six units (along the red line on the \begin{align*}x\end{align*}-axis). Now, move vertically upward three units. Plot the point \begin{align*}D\end{align*}.

The graph below shows which quadrants \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values are positive and negative. Remembering this will help you easily plot and name points.

On a Cartesian plane, the following points would be located as labeled on the graph below:

\begin{align*}A \ (5,3) \quad B \ (\text{-}3,\text{-}2) \quad C \ (4,\text{-}5) \quad D \ (\text{-}4,1)\end{align*}

#### Let's determine the coordinates of each of the plotted points on the following graph:

### Examples

#### Example 1

Earlier, you were told about some students who were asked to duplicate the image below:

Many of the students had difficulty with being precise. The teacher mentioned that she had a solution for the students that struggled.

If the teacher lets the students see the picture on a Cartesian plane, the reproduction process should be much easier.

#### Example 2

Draw a Cartesian plane that displays only positive values. Number the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes to twelve. Plot the following coordinates and connect them in order. Use a straight edge to connect the points. When the word “STOP” appears, begin the next line. Plot the points in the order they appear in each Line row.

- LINE 1 (6, 0) (8, 0) (9, 1) (10, 3) (10, 6) (9, 8) (7, 9) (5, 9)
**STOP** - LINE 2 (6, 0) (4, 0) (3, 1) (2, 3) (2, 6) (3, 8) (5, 9)
**STOP** - LINE 3 (7, 9) (6, 12) (4, 11) (5, 9)
**STOP** - LINE 4 (4, 8) (3, 6) (5, 6) (4, 8)
**STOP** - LINE 5 (8, 8) (7, 6) (9, 6) (8, 8)
**STOP** - LINE 6 (5, 5) (7, 5) (6, 3) (5, 5)
**STOP** - LINE 7 (3, 2) (4, 1) (5, 2) (6, 1) (7, 2) (8, 1) (9, 2)
**STOP** - LINE 8 (4, 1) (6, 1) (8, 1)
**STOP**

The following picture is the result of plotting the coordinates and joining them in the order in which they were plotted. Your pumpkin can be any color you like.

#### Example 3

In which quadrant would the following points be located?

- (3, -8)

The \begin{align*}x\end{align*}-coordinate is positive and the \begin{align*}y\end{align*}-coordinate is negative. This point will be located in Quadrant IV.

- (-5, 4)

The \begin{align*}x\end{align*}-coordinate is negative and the \begin{align*}y\end{align*}-coordinate is positive. This point will be located in Quadrant II.

- (7, 2)

The \begin{align*}x\end{align*}-coordinate is positive and the \begin{align*}y\end{align*}-coordinate is positive. This point will be located in Quadrant I.

- (-6, -9)

The \begin{align*}x\end{align*}-coordinate is negative and the \begin{align*}y\end{align*}-coordinate is negative. This point will be located in Quadrant III.

- (-3, 3)

The \begin{align*}x\end{align*}-coordinate is negative and the \begin{align*}y\end{align*}-coordinate is positive. This point will be located in Quadrant II.

- (9, -7)

The \begin{align*}x\end{align*}-coordinate is positive and the \begin{align*}y\end{align*}-coordinate is negative. This point will be located in Quadrant IV.

#### Example 4

State the coordinates of the points plotted on the following Cartesian plane.

\begin{align*}A(4,4) \quad B(\text{-}10,8) \quad C(8,\text{-}1) \quad D(\text{-}6,-6) \quad E(0,5) \quad F(\text{-}3,0) \quad G(2,\text{-}5) \quad H(0,0)\end{align*}

### Review

Answer the following questions with respect to the Cartesian plane:

- What name is given to the horizontal number line on the Cartesian plane?
- What name is given to the four areas of the Cartesian plane?
- What are the coordinates of the origin?
- What name is given to the vertical number line on the Cartesian plane?
- What other name is often used to refer to the \begin{align*}x\end{align*}-coordinate of a point on the Cartesian plane?

On a Cartesian plane, plot each of the following points. For each point, name the quadrant it is in or axis it is on.

- \begin{align*}(2, 0)\end{align*}
- \begin{align*}(\text{-}3, 1)\end{align*}
- \begin{align*}(0, 4)\end{align*}
- \begin{align*}(1, \text{-}2)\end{align*}
- \begin{align*}(5, 5)\end{align*}

Use the graph below for #s 11 - 13.

- The coordinates of point A are ______.
- The coordinates of point B are ______.
- The coordinates of point C are ______.

For each of the following graphs, select three points on the graph and state the coordinates of these points.

- .

- .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.