Question

Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane. Your paragraphs should include the terms "coordinate axes," "ordered pair," "coordinates," "plot," " \(x\) -coordinate," and " \(y\) -coordinate."

Short answer

A Cartesian plane is defined by the x and y coordinate axes. To plot points, use the x-coordinate and y-coordinate to determine the position.

Step-by-step solution

Describe the Cartesian Plane

A Cartesian plane is a two-dimensional surface defined by two perpendicular lines called coordinate axes. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. These axes intersect at a point called the origin, which has coordinates (0,0). Each point on the Cartesian plane is represented by an ordered pair \((x, y)\), where \(x\) and \(y\) are the coordinates showing the point's distance from the y-axis and x-axis, respectively.

Explain How to Plot Points

To plot points on the Cartesian plane, start at the origin. The first number in the ordered pair, known as the x-coordinate, indicates how many units to move horizontally from the origin. If the x-coordinate is positive, move right; if negative, move left. The second number, the y-coordinate, shows how many units to move vertically from the new position. If the y-coordinate is positive, move up; if negative, move down. Mark the point where the movements end.

Key concepts

coordinate axes

The Cartesian plane functions on two main lines known as coordinate axes. These axes are important as they provide a reference system for the plane. They are both perpendicular to each other, which means they intersect at a right angle precisely. The horizontal line is called the x-axis, while the vertical line is known as the y-axis. The point where these two axes intersect is called the origin, and it has coordinates (0,0). The coordinate axes help us determine the location of any point on the plane by utilizing these two reference lines.

ordered pair

On the Cartesian plane, every point is identified by an ordered pair \( (x, y) \). This pair is called an ordered pair because the order of the numbers is very important. The first number in the pair, \( x \), tells us how far to move horizontally from the origin. The second number, \(y \), tells us how far to move vertically. For instance, the ordered pair \( (3, -2) \) means you go 3 units to the right along the x-axis and 2 units down along the y-axis.

x-coordinate

The x-coordinate is the first number in any ordered pair. It shows the horizontal position of a point relative to the origin. It determines how many units to move left or right from the origin. If the x-coordinate is positive, you move to the right. If it's negative, you move to the left. For example, in the ordered pair \( (4, 1) \), the x-coordinate is 4, indicating you move 4 units to the right from the origin.

y-coordinate

The y-coordinate is the second number in an ordered pair. It indicates the vertical position of a point. To plot the y-coordinate, you move up or down from the current x-coordinate position. A positive y-coordinate means moving upwards, whereas a negative y-coordinate means moving downwards. For instance, in the ordered pair \( (3, -2) \), the y-coordinate is -2, directing you to move 2 units down from the x-coordinate point.

plot points

To plot points on the Cartesian plane, we first locate the x-coordinate and then the y-coordinate. Here's a step-by-step guide:

• Start at the origin, which is the point \( (0,0) \).
• Move horizontally to the x-coordinate. If the x-coordinate is positive, move right. If it's negative, move left.
• From this new position, move vertically to the y-coordinate. If the y-coordinate is positive, move upward. If it's negative, move downward.
• Mark the final position where you end up; this is the precise location of your point.

For example, to plot \( (2, 3) \), you go 2 units right and 3 units up from the origin, and then you mark that point on the plane.