Problem 25 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(-5,-2)$$

Step-by-Step Solution

Answer

The point (-5,-2) is located in Quadrant III.

Step 1: Recognize the coordinates

First, let's look at the point provided, which is (-5, -2). This point consists of an x-coordinate (the first position, -5) and a y-coordinate (the second position, -2).

Step 2: Identify the quadrant

The point has both x and y coordinates as negative. This means it falls into the third quadrant, as per our initial analysis, because Quadrant III contains all points with both negative x and y coordinates.

Key Concepts

Quadrants

In coordinate geometry, the coordinate plane is divided into four sections, called quadrants. These quadrants help us to easily locate points on the plane based on their coordinates. Each quadrant corresponds to a combination of positive and negative values of the x and y axes.

• Quadrant I: Both x and y coordinates are positive. It's located in the upper right section of the coordinate plane.
• Quadrant II: The x coordinate is negative, and the y coordinate is positive. This quadrant is in the upper left section.
• Quadrant III: Both x and y coordinates are negative. It is positioned in the lower left section, just like our point (-5, -2).
• Quadrant IV: The x coordinate is positive, while the y coordinate is negative, marking the lower right section.

Understanding quadrants helps in quickly identifying where a point is located without plotting it on the plane. Each point uniquely fits into one quadrant according to the signs of its x and y coordinates.

Coordinate System

To understand the placement of points on a plane, we use a coordinate system, particularly the Cartesian coordinate system. It consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These two axes intersect at what is called the origin, denoted by the coordinates (0,0).
The coordinate system helps us define the position of any point using two numbers, typically referred to as a pair ewline (x, y). Here:

• The first number, x, determines the horizontal position.
• The second number, y, determines the vertical position.

Points to the right of the origin have positive x-coordinates, and points above the origin have positive y-coordinates. Similarly, points to the left have negative x-coordinates, and points below have negative y-coordinates. This system provides a universal method for addressing locations in a two-dimensional space.

Negative Coordinates

Negative coordinates are an essential aspect of understanding the positioning of points within the different quadrants of the coordinate plane. A coordinate is negative when it represents a position relative to an axis that is below (for y) or to the left (for x) of the origin.
Considering the point (-5, -2):

• The x-coordinate of -5 indicates that the point is 5 units to the left of the y-axis.
• The y-coordinate of -2 shows it is 2 units below the x-axis.

Negative coordinates are fundamental in recognizing in which quadrant a point is located. For instance, a point with both coordinates negative, such as (-5, -2), will always fall within Quadrant III, as this quadrant is defined by negatively valued x and y coordinates.