# 3.1: The Cartesian Plane

**Advanced**Created by: CK-12

**Practice**Points in the Coordinate Plane

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 ** y-axis** The other number line is a vertical line and it is called the

**The two number lines are referred to as the**y-axis.

**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

Every point that is plotted on a Cartesian plane has two values associated with it. The first value represents the **coordinates** of the point and are written as the ordered pair

To plot a point on the Cartesian plane:

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

Examine the points

A(-4,2) From the origin, move to the left four units (along the red line on thex -axis). Now, move vertically upward two units. Plot the pointA .B(-2,-1) From the origin, move to the left two units (along the red line on thex -axis). Now, move vertically downward one unit. Plot the pointB .C(3,-4) From the origin, move to the right three units (along the red line on thex -axis). Now, move vertically downward four units. Plot the pointC .D(6,3) From the origin, move to the right six units (along the red line on thex -axis). Now, move vertically upward three units. Plot the pointD .

The graph below shows which quadrants

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

#### 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

- 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

- (-5, 4)

The

- (7, 2)

The

- (-6, -9)

The

- (-3, 3)

The

- (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.

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axis

The axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.axis

The -axis is the vertical number line of the Cartesian plane.Abscissa

The abscissa is the coordinate of the ordered pair that represents a plotted point on a Cartesian plane. For the point (3, 7), 3 is the abscissa.Cartesian Plane

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.Coordinate Plane

The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.Ordinate

The ordinate is the -coordinate of the ordered pair that represents a plotted point on a Cartesian plane. For the point (3, 7), 7 is the ordinate.Origin

The origin is the point of intersection of the and axes on the Cartesian plane. The coordinates of the origin are (0, 0).### Image Attributions

Here you will learn about the Cartesian plane and review how to plot points on the Cartesian plane.

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