# 3.3: The Many Points of the Cartesian Plane

Difficulty Level: At Grade Created by: CK-12

## The make-up of the Cartesian Plane

Introduction

In this lesson you will learn about the Cartesian plane. You will learn what is meant by horizontal and vertical number lines on a grid, the origin, the four quadrants and the coordinates of a point.

In this lesson you will learn how to plot points and how to graph coordinates on a Cartesian plane. You will also learn how to determine the coordinates of points that have been plotted on the plane.

Objectives

The lesson objectives for The Many Points of the Cartesian Plane are:

• Understanding the Cartesian Plane
• Plotting points on the Cartesian Plane
• Determining the coordinates of points plotted on the Cartesian Plane.

Introduction

In this concept you will begin your study of graphing points by being introduced to the Cartesian plane. You will learn the various terms associated with the Cartesian plane and the role that each plays in the world of graphing. Once you understand the makeup of the Cartesian plane, you will learn how to plot points on the grid. You will learn that the process of plotting a point involves two steps – locate and move. The final step in this lesson will be for you to learn to identify the coordinates of points that have been plotted on the plane.

Watch This

Guidance

Kaitlyn walked into Math class and saw the following image displayed from the overhead projector. Her teacher asked everyone in the class to duplicate the picture on the blank sheet of paper that she had placed on each student’s desk.

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.

Example A

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*}x-\end{align*}axis. The other number line is a vertical line and it is called the \begin{align*}y-\end{align*}axis. 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.

Using the above information, fill in the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in each quadrant. In other words, state whether \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are positive or negative values in each quadrant.

Example B

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-\end{align*}coordinate is positive, move vertically upward from the \begin{align*}x-\end{align*}coordinate, 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, the number of units displayed by the \begin{align*}y-\end{align*}coordinate. The point is can now be plotted. Follow the steps presented above to 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 (-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 (-2, -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, -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*}.

On the following Cartesian plane, draw an \begin{align*}x-y\end{align*} axis that is 6 round and plot the following points.

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

To draw a Cartesian that is six round means to have six numbers on all sides of the origin.

Example C

The coordinates of points that are plotted on a Cartesian plane can be determined in the same way that they were plotted. The first step is to determine the \begin{align*}x-\end{align*}coordinate of the point. To do this, locate the point and move vertically up or down to the \begin{align*}x-\end{align*}axis. The \begin{align*}x-\end{align*}value is the \begin{align*}x-\end{align*}coordinate of the point. The \begin{align*}y-\end{align*}coordinate can be determined by counting horizontally from the point to the \begin{align*}y-\end{align*}axis.

Determine the coordinates of each of the plotted points on the following graph.

Example D

Now, let us return to the beginning of the lesson to find out the solution that the teacher had for the students.

Now that the students can see the picture on a Cartesian plane, the reproduction process should be much easier.

Vocabulary

Abscissa
The abscissa is the \begin{align*}x-\end{align*}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
A Cartesian plane is a system of four areas or quadrants produced by the perpendicular intersection of two number lines. A Cartesian plane is the grid on which points are plotted.
Coordinates
The coordinates are the ordered pair \begin{align*}(x, y)\end{align*} that represent a point on the Cartesian plane.
Coordinate Plane
The coordinate plane is another name for the Cartesian plane.
Ordinate
The ordinate is the \begin{align*}y\end{align*} 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 \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes on the Cartesian plane. The coordinates of the origin are (0, 0).
\begin{align*}x-\end{align*}axis
The \begin{align*}x-\end{align*}axis is the horizontal number line of the Cartesian plane.
\begin{align*}y-\end{align*}axis
The \begin{align*}y-\end{align*}axis is the vertical number line of the Cartesian plane.

Guided Practice

1. 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

2. In which quadrant would the following points be located?

i) (3, -8)

ii) (-5, 4)

iii) (7, 2)

iv) (-6, -9)

v) (-3, 3)

vi) (9, -7)

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

1. 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.

2. i) (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.

ii) (-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.

iii) (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.

iv) (-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.

v) (-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.

vi) (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.

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

Summary

The Cartesian plane is the grid that is used in mathematics for plotting points. The grid consists of four quadrants that are numbered counterclockwise. You have learned how to plot points on this grid by using the \begin{align*}(x, y)\end{align*} coordinates of a point. In addition to plotting points, you have also learned to determine the coordinates of points that have been plotted on a Cartesian plane. To enhance this lesson, you have also plotted given points to create a picture on the grid. As you continue this chapter, you will learn more results of plotting points.

Problem Set

1. With a partner, create a picture on a Cartesian plane that is numbered ten round. Using the coordinates, list the points for at least five lines necessary for a classmate to complete this same picture. (Go back to the directions for the pumpkin)
2. On each of the following graphs, select three points and state the coordinates of these points.
3. Answer the following questions with respect to the Cartesian plane:
1. What name is given to the horizontal number line on the Cartesian plane?
2. What name is given to the four areas of the Cartesian plane?
3. What are the coordinates of the origin?
4. What name is given to the vertical number line on the Cartesian plane?
5. What other name is often used to refer to the \begin{align*}x-\end{align*}coordinate of a point on the Cartesian plane?

1. With a partner... Answers to this question will vary. It is a good activity for students to strengthen their ability to plot points on the Cartesian plane. It will also help them to identify coordinates.
2. On each of the following graphs... Answers to this question will vary. Here are some examples of the responses that students may give:
3. Answer the following questions... (a) The \begin{align*}x-\end{align*}axis is the horizontal number line. (c) The coordinates of the origin are (0, 0). (e) The abscissa is the other name used for the \begin{align*}x-\end{align*}coordinate of a point on the Cartesian plane.

## Summary

In this lesson you have learned the composition of the Cartesian plane. This grid is used in mathematics for plotting points. The points that are plotted have coordinates that correspond to the \begin{align*}x-\end{align*}axis and the \begin{align*}y-\end{align*}axis of the Cartesian plane. To plot the coordinates of a point, the \begin{align*}x-\end{align*}value is located on the horizontal number line and from here, you move up or down the value of the \begin{align*}y-\end{align*}coordinate. At this location, the point is plotted.

You have also learned to identify the coordinates of points that have been plotted on the Cartesian plane. To enhance the whole process, these skills were combined to create pictures or to duplicate pictures that were plotted on a Cartesian plane. In later lessons, these skills will be applied to graphing various lines and curves.

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