Problem 1 - College Algebra with Corequisite Support

Question

Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.

Step-by-Step Solution

Answer

Yes, points on the axes or at the origin do not lie in any quadrant.

Step 1: Understanding the Quadrants

The Cartesian coordinate system is divided into four quadrants by the x-axis and y-axis. Quadrant I is where both x and y coordinates are positive, Quadrant II is where x is negative and y is positive, Quadrant III is where both x and y are negative, and Quadrant IV is where x is positive and y is negative.

Step 2: Exploring Points Outside the Quadrants

A point that lies directly on one of the axes does not fall into any of the four quadrants. This includes points where either the x-coordinate is zero (lying on the y-axis) or the y-coordinate is zero (lying on the x-axis).

Step 3: Considering Points at the Origin

The origin of the Cartesian coordinate system, which is the point \(0, 0\), does not belong to any of the four quadrants. It is a special point where both the x and y coordinates are zero, and it is considered to be at the intersection of the axes.

Key Concepts

Quadrants

In the Cartesian coordinate system, the plane is divided into four sections called quadrants. Each quadrant is defined by the x-axis and y-axis. These axes are like dividing lines that intersect at the origin, splitting the plane into four parts. Here’s how each quadrant works:

• Quadrant I: Both the x and y coordinates are positive. A point in this quadrant can be something like (2, 3).
• Quadrant II: Here, the x coordinate is negative, and the y coordinate is positive. For instance, a point could be (-2, 3).
• Quadrant III: In this quadrant, both x and y coordinates are negative, such as (-2, -3).
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative, for example (2, -3).

Quadrants help in easily identifying the location of a point based on the sign of its coordinates.

The x-axis

The x-axis is a crucial component of the Cartesian coordinate system. It is the horizontal line that runs left to right across the plane. Any point that lies directly on the x-axis has a y-coordinate of zero. For instance, the point (5, 0) is on the x-axis. Here are some key points about the x-axis:

• Separates Quadrants I and IV from II and III.
• Any movement along the x-axis only affects the x-coordinate, while the y-coordinate remains zero.
• The positive direction of the x-axis extends to the right, and the negative direction extends to the left.

The x-axis serves as a reference line for understanding the horizontal placement of points on the plane.

The y-axis

The y-axis is another fundamental part of the Cartesian coordinate system, representing the vertical direction. It runs up and down across the plane and any point positioned directly on the y-axis has an x-coordinate of zero, like (0, 4). You should know the following about the y-axis:

• The y-axis separates Quadrants I and II from III and IV.
• Movement along the y-axis affects only the y-coordinate, not the x-coordinate, keeping the latter at zero.
• The positive y direction points upwards, while the negative y direction points downwards.

The y-axis is crucial for determining how high or low a point is placed in the coordinate plane.

Origin

The origin is the central point of the Cartesian coordinate system, where the x-axis and y-axis intersect. Its coordinates are specifically (0, 0). The origin acts as a starting point for measuring the location of any point on the plane. Here are several noteworthy aspects of the origin:

• It is a unique point because it does not belong to any quadrant.
• All distances on the graph originate from this point, making it crucial for plotting any object or figure.
• It serves as a reference point indicating zero distance both horizontally and vertically.

The origin plays a pivotal role in simplifying mathematical tasks and understanding geometric shapes on the coordinate plane.