Problem 24 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(2,-6)$$

Step-by-Step Solution

Answer

Quadrant IV

Step 1: Identify the Quadrants

The x-y coordinate plane is divided into four quadrants: Quadrant I (top right), Quadrant II (top left), Quadrant III (bottom left), and Quadrant IV (bottom right).

Step 2: Determine the Sign of Coordinates

Check the signs of the coordinates of the point (2, -6). The x-coordinate (2) is positive, and the y-coordinate (-6) is negative.

Step 3: Compare with Quadrant Criteria

A point with a positive x-coordinate and a negative y-coordinate lies in Quadrant IV. Compare the signs of (2, -6) with the criteria for each quadrant to confirm.

Key Concepts

Quadrants

In coordinate geometry, we divide the coordinate plane into four sections called quadrants. These quadrants help us quickly locate points based on their coordinates. Let's look at each quadrant in detail.

• Quadrant I: This is the top-right section. Points here have both x and y coordinates positive.
• Quadrant II: This top-left section contains points with a negative x-coordinate and a positive y-coordinate.
• Quadrant III: Located at the bottom left, points in this quadrant have both coordinates negative.
• Quadrant IV: Found in the bottom right, this quadrant includes points with a positive x-coordinate and a negative y-coordinate.

Every point on the coordinate plane, except those on the axes, falls into one of these four quadrants. Knowing this division allows us to understand the position of any point easily.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is fundamental in coordinate geometry. It consists of two perpendicular lines called axes that intersect at a point called the origin.

The horizontal axis is called the x-axis, and the vertical axis is known as the y-axis. These axes allow us to define the position of any point in the plane through pairs of numbers, known as coordinates. Coordinates are written in the form (x, y) where 'x' represents the horizontal position and 'y' the vertical position.

• Origin (0,0): The point where the x-axis and y-axis intersect.
• X-axis: Horizontal line.
• Y-axis: Vertical line.

Understanding the coordinate plane helps us graph equations, locate points, and solve geometric problems.

Sign of Coordinates

The sign of coordinates is key to determining where a point lies on the coordinate plane. Coordinates are represented as (x, y), where 'x' and 'y' can be positive, negative, or zero, and their signs indicate their position relative to the origin.

Here's a quick guide on how the sign impacts the location:

• Positive x and positive y: Point is in Quadrant I.
• Negative x and positive y: Point is in Quadrant II.
• Negative x and negative y: Point is in Quadrant III.
• Positive x and negative y: Point is in Quadrant IV.
• x or y = 0: The point lies on the x-axis or y-axis respectively.

This knowledge simplifies locating points and identifying which quadrant they belong to. For example, the point (2, -6) has a positive x-coordinate and a negative y-coordinate. According to the criteria, it lies in Quadrant IV.