Problem 22 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(14,-5)$$

Step-by-Step Solution

Answer

The point (14,-5) lies in Quadrant IV.

Step 1: Quadrants Identification

In the cartesian coordinate system, we have four quadrants. Quadrant I: both x and y coordinates are positive. Quadrant II: x is negative, y is positive. Quadrant III: both x and y coordinates are negative. Quadrant IV: x is positive, y is negative.

Step 2: Identify the x and y Coordinates

Our given point is (14,-5). The first value which is 14 is the x coordinate, and the second value which is -5 is the y coordinate.

Step 3: Analyze the Coordinates

The x-coordinate, 14, is positive while the y-coordinate, -5, is negative.

Step 4: Final Identification

Since the x coordinate is positive and the y coordinate is negative, the point (14,-5) lies in Quadrant IV.

Key Concepts

Cartesian Coordinate System

The Cartesian Coordinate System is a two-dimensional space defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). With their intersection at the origin point (0,0), these axes divide the plane into four sections known as quadrants. This system is fundamental in graphing and essential for locating the position of any point in a plane by assigning it an ordered pair of numbers—typically (x, y). This ordered pair identifies the point's position in relation to the origin and axes.

The quadrants are numbered I through IV, proceeding counterclockwise:

• Quadrant I: both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

Understanding these quadrants is crucial for determining the location and behavior of points in this plane.

Positive and Negative Coordinates

Coordinates in the Cartesian plane can be either positive or negative, which signifies their position relative to the x-axis and y-axis. The first value in an ordered pair represents the x-coordinate, while the second value represents the y-coordinate. Each coordinate describes the point's distance and direction from the origin.

• Positive x-coordinates are found to the right of the y-axis.
• Negative x-coordinates are situated to the left of the y-axis.
• Positive y-coordinates are above the x-axis.
• Negative y-coordinates are below the x-axis.

These positive and negative designations directly affect which quadrant a point falls into, as the combination dictates its specific quadrant.

Point Identification in Quadrants

Identifying the correct quadrant for any point without plotting involves interpreting its coordinates. For example, consider the point (14, -5). Here, 14 is the x-coordinate, while -5 is the y-coordinate.

• The x-coordinate of 14 is positive, indicating a direction to the right of the y-axis.
• The y-coordinate of -5 is negative, placing the point below the x-axis.

Given this combination, we conclude that the point resides in Quadrant IV. This quadrant is specifically defined by a positive x-coordinate and a negative y-coordinate. Thus, understanding the signs of each coordinate is key to identifying the point's quadrant reliably, without the need for visual plotting.