Problem 21 - 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 I

Step 1: Determine the Signs of the Coordinates

The given point is \(2, 6\). Determine the signs of the x-coordinate and the y-coordinate. Here, both the x-coordinate (2) and the y-coordinate (6) are positive.

Step 2: Identify the Corresponding Quadrant

Recall the coordinate system quadrants:1. Quadrant I: x > 0, y > 02. Quadrant II: x < 0, y > 03. Quadrant III: x < 0, y < 04. Quadrant IV: x > 0, y < 0Since both coordinates of the point \(2, 6\) are positive, it lies in Quadrant I.

Key Concepts

Coordinate System

Coordinate Geometry is a field of mathematics that uses a coordinate system to define and describe geometric shapes. A 'Coordinate System' allows us to locate points in a plane using pairs of numbers, called coordinates.
The most common coordinate system is the Cartesian coordinate system. It consists of:

• An x-axis (horizontal line)
• A y-axis (vertical line)
• An origin (the point where the x-axis and y-axis intersect)

By using these axes, we can define the position of any point in the plane based on how far it is from the origin along the x and y directions.
For example, in the given problem, the point (2, 6) means the point is 2 units to the right of the origin and 6 units up. Each pair of points (x, y) uniquely identifies a location in this system.

Quadrants

In the Cartesian coordinate system, the plane is divided into four sections called quadrants. Each quadrant is defined by the signs of the coordinates.
The quadrants are numbered counterclockwise starting from the top right:

• Quadrant I (top right): Both x and y coordinates are positive (x > 0, y > 0)
• Quadrant II (top left): x is negative, y is positive (x < 0, y > 0)
• Quadrant III (bottom left): Both x and y are negative (x < 0, y < 0)
• Quadrant IV (bottom right): x is positive, y is negative (x > 0, y < 0)

In the exercise, the point (2, 6) lies in Quadrant I because both coordinates are positive.
Knowing which quadrant a point lies in helps us understand its relative position on the coordinate plane.

Positive and Negative Coordinates

Coordinates can be either positive or negative, based on their position relative to the origin and the axes. Positive and negative signs tell you in which direction to move from the origin to reach the point.

• Positive x-coordinates (x > 0) indicate a move right from the origin
• Negative x-coordinates (x < 0) indicate a move left from the origin
• Positive y-coordinates (y > 0) indicate a move up from the origin
• Negative y-coordinates (y < 0) indicate a move down from the origin

By understanding the signs of the coordinates, you can accurately pinpoint any location on the Cartesian plane.
For instance, in the point (2, 6), '2' indicates a movement 2 units to the right, and '6' indicates a movement 6 units up. Hence, the coordinates tell us exactly where the point is placed.