Problem 33 - 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. $$(-2,0)$$

Step-by-Step Solution

Answer

The point (-2,0) lies on the x-axis, not in any quadrant.

Step 1: Plotting the Point

Begin by drawing a rectangular coordinate system with an x-axis and y-axis that intersect at the origin (0,0). Locate the point (-2,0) on this coordinate plane. The x-coordinate is -2, which means you need to move 2 units to the left of the origin along the x-axis.

Step 2: Verifying the y-coordinate

The y-coordinate is 0, indicating no movement up or down from the x-axis. Therefore, the point (-2,0) is on the x-axis itself.

Step 3: Identifying the Location

Since the point (-2,0) has a y-coordinate of 0, it does not lie in any of the four quadrants. Instead, it lies on the x-axis, specifically at the point where x equals -2.

Key Concepts

Plotting Points

In a rectangular coordinate system, plotting points involves using two numbers, known as coordinates, to identify a specific location on a plane. These coordinates are usually written as an ordered pair, \(x, y\). The first number in the pair is the x-coordinate, which tells you how far to move left or right from the origin. The second number is the y-coordinate, which tells you how far to move up or down. For example, with the point (-2,0), the x-coordinate is -2, indicating a movement 2 units to the left of the origin along the x-axis. The y-coordinate is 0, meaning there's no movement up or down from the x-axis. Together, they pinpoint the exact spot on the plane.

Quadrants

The rectangular coordinate system is divided into four regions called quadrants, which help in identifying the position of points. These quadrants are labeled as follows:

• Quadrant I: The region where both x and y coordinates are positive.
• Quadrant II: The region where the x-coordinate is negative and the y-coordinate is positive.
• Quadrant III: The region where both x and y coordinates are negative.
• Quadrant IV: The region where the x-coordinate is positive and the y-coordinate is negative.

If a point has a coordinate of zero, such as (-2,0), it does not lie within any of these quadrants. Instead, it is situated on one of the axes. In the case of (-2,0), it lies directly on the x-axis.

Coordinate Axes

The rectangular coordinate system consists of two perpendicular lines known as the coordinate axes: the x-axis and the y-axis. These axes intersect at a point called the origin, which is represented by (0,0). The x-axis runs horizontally, allowing movement along it to identify the x-coordinate of a point.
The y-axis runs vertically, determining the y-coordinate of a point. Together, these axes form a grid where every point is defined by its coordinates.
When a point like (-2,0) is plotted, the x-axis serves as a reference for the horizontal placement, and since the y-coordinate is 0, the point stays on the x-axis.

• Points on the x-axis will have the format \(x, 0\).
• Points on the y-axis will have the format \(0, y\).

Understanding the role of these axes is crucial for accurately plotting and identifying points within the coordinate system.