Problem 29 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(0,0)$$

Step-by-Step Solution

Answer

The point \(0,0\) is located at the origin of the coordinate system.

Step 1: Understand the Coordinate System

A coordinate system is divided into four quadrants by the x-axis and y-axis. Quadrant I contains points where both coordinates are positive. Quadrant II has points where the x-coordinate is negative and the y-coordinate is positive. Quadrant III contains points with both coordinates negative. Quadrant IV has points with a positive x-coordinate and negative y-coordinate. Points on the x-axis have a y-coordinate of 0, and points on the y-axis have an x-coordinate of 0.

Step 2: Identify the Given Point

The given point is \(0,0\). This point represents the origin of the coordinate system.

Step 3: Determine Position of the Origin

Since the origin \(0,0\) is the point where the x-axis and y-axis intersect, it does not belong to any of the four quadrants. The origin is a special point.

Step 4: Indicate the Location of the Origin

Thus, the point \(0,0\) is located at the origin of the coordinate system and is not in any quadrant.

Key Concepts

Quadrants

In a coordinate system, the plane is divided into four quadrants. Each quadrant represents a unique combination of positive and negative values for the coordinates.

• **Quadrant I**: Both the x-coordinate and y-coordinate are positive. (e.g., (3, 4)).
• **Quadrant II**: The x-coordinate is negative, and the y-coordinate is positive. (e.g., (-2, 5)).
• **Quadrant III**: Both coordinates are negative. (e.g., (-3, -6)).
• **Quadrant IV**: The x-coordinate is positive, and the y-coordinate is negative. (e.g., (4, -3)).

Understanding which quadrant a point belongs to helps in pinpointing its location on the coordinate plane. Points that lie on the axes are not part of any quadrant.

Origin

The origin is the point \((0,0)\) where the x-axis and y-axis intersect. It is the central point that serves as a reference for the entire coordinate system. Unlike other points, the origin does not belong to any specific quadrant. Here are some unique features of the origin:

• It is the starting point for measuring distances on the coordinate plane.
• Coordinates of the origin are always \(0,0\).
• It is a neutral point, meaning neither the x-coordinate nor the y-coordinate is positive or negative.

The origin is crucial for understanding the position and movement of points within the coordinate system.

Axes

The coordinate system is structured around two main lines called axes—the x-axis and the y-axis.
**The X-Axis**
The x-axis is the horizontal line that runs from left to right. Points lying on the x-axis have a zero y-coordinate. For example, the point \((5, 0)\) lies on the x-axis.
**The Y-Axis**
The y-axis is the vertical line running from top to bottom. Points on the y-axis have a zero x-coordinate. For example, \((0, -3)\) lies on the y-axis.
The x-axis and y-axis divide the coordinate plane into four quadrants, and their intersection at \(0,0\) is known as the origin. Understanding these axes is fundamental to mastering the coordinate system.