Problem 26 - A Graphical Approach to College Algebra

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-1,2)$$

Step-by-Step Solution

Answer

The point \((-1, 2)\) is in Quadrant II.

Step 1: Identify the Point Coordinates

The given point is \((-1, 2)\). This means the x-coordinate is \(-1\) and the y-coordinate is \(2\).

Step 2: Locate the Point on the Coordinate System

To locate the point \((-1, 2)\), move \(-1\) unit left from the origin along the x-axis, and then move up \(2\) units along the y-axis.

Step 3: Determine the Quadrant Location

A point is in a quadrant based on the signs of its coordinates. The point \((-1, 2)\) has a negative x-coordinate and a positive y-coordinate, placing it in Quadrant II.

Key Concepts

Quadrants

A rectangular coordinate system is divided into four sections known as quadrants. These quadrants help in determining the exact position of a point based on the signs of its coordinates. Each quadrant has its own unique combination of positive and negative coordinates:

• Quadrant I: Both x and y coordinates are positive (e.g., (2, 3)).
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive (e.g., (-1, 2)).
• Quadrant III: Both x and y coordinates are negative (e.g., (-3, -4)).
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative (e.g., (4, -5)).

If a point lies on one of the axes, it is not considered to be in any quadrant. The placement of a point in a quadrant is determined by checking the signs of its coordinates. This assists in graphing and understanding spatial relationships.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, allows us to plot points on a plane based on two numerical values - the x-coordinate and the y-coordinate. This system creates a grid-like layout for easy location of any point. It consists of two number lines:

• The x-axis: A horizontal line where values increase from left to right.
• The y-axis: A vertical line where values increase from bottom to top.

These axes intersect at a point called the origin, labeled as (0,0). This origin acts as a central reference point. Each point on the coordinate system is determined by an ordered pair \((x, y)\).Using the Cartesian coordinate system helps solve geometric problems and visualize mathematical concepts in two dimensions. Movement from the origin along these axes helps locate and represent various points.

X-coordinate Y-coordinate

In the rectangular coordinate system, every point is identified by a pair of numbers, known as the x-coordinate and the y-coordinate. These coordinates are essential for pinpointing the exact location of a point:

• The **x-coordinate** indicates the horizontal position, dictating how far left or right a point is from the origin.
• The **y-coordinate** shows the vertical position, specifying how far up or down the point lies from the origin.

When you read or write coordinates, always present them in the form \((x, y)\).For example, in our exercise, the point \((-1, 2)\) has an x-coordinate of \(-1\), placing it to the left of the origin, and a y-coordinate of \(2\), indicating it's above the origin. Understanding these coordinates allows us to navigate the rectangular coordinate system effectively.