Problem 29 - A Graphical Approach to College Algebra

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(0,5)$$

Step-by-Step Solution

Answer

Point (0,5) is on the Y-axis, not in any quadrant.

Step 1: Understanding the Coordinate Plane

The rectangular coordinate system is composed of an X-axis (horizontal) and a Y-axis (vertical), which intersect at the origin (0,0). This creates four quadrants where points can be located.

Step 2: Plotting the Point

The point (0,5) has an X-coordinate of 0 and a Y-coordinate of 5. To plot this point, start at the origin, then move 0 units along the X-axis (stay at the origin) and 5 units up along the Y-axis.

Step 3: Determining the Location

Since the X-coordinate is 0, (0,5) lies on the Y-axis, specifically 5 units above the origin. It does not belong to any of the four quadrants.

Step 4: Final Conclusion about Quadrant

Any point lying on one of the axes does not belong to a specific quadrant. The point (0,5) is on the Y-axis and thus is not in any quadrant.

Key Concepts

Quadrants of a Coordinate Plane

In a rectangular coordinate system, the plane is divided into four distinct sections known as quadrants. These quadrants help us understand and identify the location of points relative to the origin. The origin is the point where the X-axis and Y-axis intersect, noted as (0,0).

• Quadrant I: This is the section where both X and Y coordinates are positive. Points here appear in the format (x, y) where x > 0 and y > 0.
• Quadrant II: Here, the X-coordinate is negative, and the Y-coordinate is positive. Points are in the form (-x, y) where x > 0 and y > 0.
• Quadrant III: Both X and Y coordinates are negative in this quadrant. Points look like (-x, -y) where x > 0 and y > 0.
• Quadrant IV: The X-coordinate is positive, and the Y-coordinate is negative, appearing as (x, -y) where x > 0 and y > 0.

Understanding which quadrant a point belongs to gives crucial insight into its location on the plane. However, some points, like (0,5), do not belong to any quadrant as they are situated on the axes themselves, establishing a unique position.

Plotting Points

Plotting points on the coordinate plane is a basic yet essential skill in understanding how different positions relate within the system. Each point is described by an ordered pair r you first have the X-coordinate, followed by the Y-coordinate. For example, in the point (0,5), 0 is the X-coordinate, and 5 is the Y-coordinate.
To plot this point:

• Start at the origin, which is the center of the coordinate plane at (0,0).
• Move along the X-axis horizontally by the value of the X-coordinate. In this case, you stay at 0, meaning you do not move left or right.
• From there, move vertically along the Y-axis by the Y-coordinate value. Here, you move 5 units up since the Y-coordinate is 5.

This movement will help you accurately locate (0,5) in physical space, directly on the Y-axis.

Axis Identification

The coordinate plane is formed by two main lines - the X-axis and the Y-axis - which make up the framework for plotting and identifying points. Identifying these axes is critical for understanding the position of any point.

• X-axis: This is the horizontal line running from left to right. It helps measure how far a point moves side to side.
• Y-axis: The vertical line running from bottom to top. It measures how far a point moves up or down.

A point directly on any axis like (0,5) or (4,0) is easy to identify because it will have at least one coordinate as zero. Such points do not belong to any quadrant as they lie on the axis itself, establishing a direct relationship to either the X-axis or the Y-axis. Specifically: - If the X-coordinate is zero, the point is on the Y-axis. - If the Y-coordinate is zero, the point is on the X-axis.
Understanding where a point sits in relation to these axes is key in positioning and comprehending its role within the coordinate system.