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Chapter 1: Problem 30

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-2,-4)$$

Short Answer

Expert verified
The point (-2, -4) lies in Quadrant III.

Step by step solution

01

Understanding the Coordinate System

The rectangular coordinate system is composed of an x-axis (horizontal) and a y-axis (vertical). The origin, where the axes intersect, is at the point (0,0). Points are plotted as (x, y), where x is the horizontal distance from the origin, and y is the vertical distance.
02

Identifying the Coordinates

The given point is (-2, -4). This means the point is 2 units to the left of the y-axis and 4 units below the x-axis. Here, x = -2 and y = -4.
03

Plotting the Point

To plot the point (-2, -4) on the coordinate system, start at the origin (0,0). Move 2 units to the left along the x-axis (because x = -2) and then move 4 units down parallel to the y-axis (because y = -4). Mark the point where you reach.
04

Determining the Quadrant

The coordinate system is divided into four quadrants: 1. Quadrant I: (+,+) 2. Quadrant II: (-,+) 3. Quadrant III: (-,-) 4. Quadrant IV: (+,-) Since the point (-2, -4) has both coordinates negative, it lies in Quadrant III.

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Key Concepts

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

Understanding Quadrants in a Coordinate System
A coordinate system is divided into four quadrants. These are formed by the intersection of two axes鈥攖he x-axis and the y-axis鈥攃rossing at the origin point (0,0). Each quadrant is a unique section of the plane, defined by the signs of points' coordinates.
  • Quadrant I is located in the top right corner, where both x and y coordinates are positive: \((+,+)\).
  • Quadrant II is in the top left, where x is negative and y is positive: \((-,+)\).
  • Quadrant III is found in the bottom left, where both x and y are negative: \((-,-)\). This is where you'd locate the point \((-2,-4)\).
  • Quadrant IV occupies the bottom right, where x is positive and y is negative: \((+,-)\).
By understanding these quadrants, you can find out where any given point will lie on the plane just by looking at the signs of its coordinates.
Using Rectangular Coordinates
In the rectangular coordinate system, every point can be identified with an ordered pair of the form \((x, y)\). The x-coordinate tells us how far left or right a point is from the origin, while the y-coordinate indicates how far up or down it is.Here鈥檚 how it works:
  • The x-value represents the horizontal position of the point, with positive values to the right and negative values to the left.
  • The y-value stands for the vertical position of the point, with positive numbers going upwards and negatives going downwards from the origin.
Let's examine the point \((-2, -4)\):
  • Its x-coordinate is \(-2\), meaning it is two units to the left of the y-axis.
  • Its y-coordinate is \(-4\), indicating it is four units below the x-axis.
Understanding these coordinates makes it easy to navigate and plot points on the graph definitively.
Plotting Points on a Coordinate Plane
Plotting points on a coordinate plane is like connecting dots through instructions. Start at the origin, which is the center point where the x-axis and y-axis meet.To plot a point like \((-2, -4)\):
  • Begin at the origin, represented by \((0,0)\).
  • Move horizontally along the x-axis. Since the x-coordinate is \(-2\), you move 2 units to the left.
  • Next, move vertically along the y-axis. With the y-coordinate being \(-4\), shift 4 units downward.
  • Mark your spot. This plotted point represents the coordinates \((-2, -4)\).By following these simple steps, you can confidently place any point accurately in its respective spot on the coordinate plane.

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