Problem 20 - Algebra 1: Concepts and Skills

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$ (-2,7) $$

Step-by-Step Solution

Answer

The point (-2, 7) is located in Quadrant II.

Step 1: Identify the given coordinates

The point given is (-2, 7). Here, -2 is the x-coordinate and 7 is the y-coordinate.

Step 2: Use quadrant determination rules

Based on the rules for quadrant determination, compare the sign of the x-coordinate and the y-coordinate. Here, the x-coordinate is negative (x<0) and the y-coordinate is positive (y>0).

Step 3: Determine the Quadrant

Since x<0 and y>0, this point falls in Quadrant II based on the quadrant determination rules.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional plane formed by the intersection of a vertical line called the y-axis and a horizontal line called the x-axis. These axes divide the plane into four sections known as quadrants. The origin, where the x-axis and y-axis intersect, is the center of the coordinate plane and is designated as the point (0, 0).

The coordinate plane is used for plotting points, lines, and curves. It is critical to understanding the layout of the coordinate plane in order to determine in which quadrant a point lies. Remember, the plane is divided into four quadrants with each offering a unique combination of positive and negative values for x and y coordinates.

Cartesian Coordinates

Cartesian coordinates are used to identify the location of points on the coordinate plane. Each point is represented by a pair of numbers, referred to as the x-coordinate and the y-coordinate. For example, in the point (-2, 7), -2 is the x-coordinate and 7 is the y-coordinate.

The x-coordinate indicates the position along the horizontal axis, while the y-coordinate indicates the position along the vertical axis. This system, developed by René Descartes, allows us to precisely map the relationships between points in two-dimensional space, a foundation for algebra and geometry.

Quadrants in Mathematics

In mathematics, particularly in the study of coordinate geometry, quadrants are the four regions of a coordinate plane. They are numbered counterclockwise, starting from the upper right-hand side. Here are the quadrants:

• Quadrant I (QI): Both x and y coordinates are positive.
• Quadrant II (QII): x is negative, y is positive.
• Quadrant III (QIII): Both x and y coordinates are negative.
• Quadrant IV (QIV): x is positive, y is negative.

Determining which quadrant a point belongs to helps in understanding its relationship to the other points and the axes. Points lying on the axes are not considered to be in any quadrant.

Negative and Positive Coordinates

Coordinates can be either negative or positive, and this sign gives us valuable information about their location in a quadrant. In Cartesian coordinates, a negative x-coordinate suggests that the point is to the left of the y-axis, while a negative y-coordinate means that the point is below the x-axis. Conversely, a positive x-coordinate indicates a position to the right of the y-axis, and a positive y-coordinate signifies a position above the x-axis.

Understanding the significance of positive and negative coordinates is crucial for quadrant determination. For instance, a point with a negative x-coordinate and a positive y-coordinate, such as (-2, 7), is located in Quadrant II. This knowledge helps to quickly visualize the position of a point without plotting it and is also essential for solving problems related to distance, angle, and area in coordinate geometry.