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Chapter 3: Problem 49

In which quadrant or on which axis is each point located? $$ \left(-\frac{5}{2}, 0\right) $$

Short Answer

Expert verified
Negative x-axis

Step by step solution

01

- Identify the Coordinates

The given point is \(\frac{-5}{2}, 0\). The first value, \(\frac{-5}{2}\), represents the x-coordinate, and the second value, \(0\), represents the y-coordinate.
02

- Determine the Position of the Point

Since the x-coordinate is \(\frac{-5}{2}\), which is negative, and the y-coordinate is 0, this point is located on the negative x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadrants
The coordinate plane is divided into four sections called quadrants. These quadrants help us determine the location of a point based on its coordinates. Each quadrant is identified with a Roman numeral:
- Quadrant I: Both coordinates (x, y) are positive.
- Quadrant II: The x-coordinate is negative, but the y-coordinate is positive.
- Quadrant III: Both coordinates (x, y) are negative.
- Quadrant IV: The x-coordinate is positive, but the y-coordinate is negative.
When a point lies exactly on one of the axes (either the x-axis or y-axis), it is not in any of the quadrants.
X-Axis
The x-axis is the horizontal line in the coordinate plane. It divides the plane into the upper and lower halves. Points along the x-axis have their y-coordinate equal to 0. This means if a point lies on the x-axis, it is of the form (x, 0).
Conversely, depending on the sign of the x-coordinate, points can lie on either the positive x-axis (if x > 0) or the negative x-axis (if x < 0). For example, the point \(\frac{-5}{2}, 0\) is on the negative x-axis.
Negative Coordinates
Negative coordinates indicate that a point lies to the left of the y-axis (if x is negative) or below the x-axis (if y is negative).
This is crucial to understanding locations in the coordinate plane:
  • If the x-coordinate is negative, the point is to the left of the y-axis.
  • If the y-coordinate is negative, the point is below the x-axis.
  • If both coordinates are negative, the point lies in Quadrant III.
In our example, \(\frac{-5}{2}, 0\) means that the point is to the left of the origin on the x-axis since the x-coordinate is negative while the y-coordinate is zero.
Locating Points
To locate a point in the coordinate plane, follow these steps:
1. Identify the x-coordinate (the first number in the pair) and move horizontally from the origin (0,0) by this value.
2. Identify the y-coordinate (the second number in the pair) and move vertically from your new x-position by this value.
When coordinates are given, such as \(\frac{-5}{2}, 0\), you first move left from the origin because the x-coordinate \(\frac{-5}{2}\) is negative. You do not move up or down because the y-coordinate is zero. This places the point on the x-axis to the left of the origin.
Understanding how to locate points is essential for graphing equations and understanding geometric relationships.

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