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Chapter 2: Problem 18

In which quadrant or on which axis is each point located? \((-1,-4)\)

Short Answer

Expert verified
Quadrant III.

Step by step solution

01

- Understand the Coordinate System

In the Cartesian coordinate system, there are four quadrants divided by the x-axis and y-axis. Quadrant I is (+,+), Quadrant II is (-,+), Quadrant III is (-,-), and Quadrant IV is (+,-). Points can also lie on the x-axis or y-axis if one of their coordinates is zero.
02

- Identify the Coordinates of the Point

The given point is (-1, -4). Here, the x-coordinate is -1 (negative) and the y-coordinate is -4 (negative).
03

- Determine the Location Based on Signs of the Coordinates

Since both coordinates are negative, the point (-1, -4) is located in Quadrant III.

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Key Concepts

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

Cartesian coordinate system
The Cartesian coordinate system is a two-dimensional system created by the intersection of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). This system allows us to represent any point on a plane using a pair of coordinates (x, y). The coordinates specify the position relative to the origin (0,0), where both axes intersect.

Each point is described using an ordered pair:
  • The first number (x-coordinate) describes the horizontal distance from the origin.
  • The second number (y-coordinate) represents the vertical distance from the origin.
Understanding this system is crucial as it forms the foundation for identifying the location of points on the plane.
quadrant identification
Quadrant identification is about finding which of the four sections of the Cartesian plane a point is located in. The plane is divided into four quadrants:
  • Quadrant I: Here, both x and y coordinates are positive (+,+). For example, (3,4).
  • Quadrant II: This quadrant has a negative x and positive y coordinate (-,+), like (-2,5).
  • Quadrant III: Both coordinates are negative (-,-), for instance, (-1,-4).
  • Quadrant IV: It has a positive x and a negative y coordinate (+,-), such as (4,-3).
A point can also lie directly on the x-axis or y-axis if one of its coordinates is zero. For instance, (0,4) lies on the y-axis, and (5,0) lies on the x-axis. Knowing how to identify the correct quadrant helps in simplifying graphical interpretation and analysis.
negative coordinates
Negative coordinates tell us the position of a point in relation to the origin, particularly when dealing with the left and downward directions from the origin.

  • The negative x-coordinate means the point is to the left of the y-axis.
  • A negative y-coordinate indicates the point is below the x-axis.
For example, with the point (-1, -4), -1 directs us 1 unit to the left of the y-axis, and -4 directs us 4 units below the x-axis. This understanding is critical for accurate plotting and interpretation of points in various quadrants, especially Quadrant II and Quadrant III where negative coordinates are common.

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