Problem 2 - Algebra and Trigonometry

Question

If x is negative and y is positive, then the point (x, y) is in Quadrant ________.

Step-by-Step Solution

Answer

Quadrant II.

Step 1: Understand the Coordinate System

The Cartesian coordinate system consists of four quadrants. Quadrant I is where both x and y are positive. Quadrant II, III, and IV are where the signs of x and y are different or both negative.

Step 2: Identify the Significance of Signs

Each quadrant is defined by the sign of x and y coordinates. Quadrant II has x negative and y positive, Quadrant III has both x and y negative, and Quadrant IV has x positive and y negative.

Step 3: Locate the Quadrant for Given Conditions

Given that x is negative and y is positive, examine the quadrant where these conditions apply. Check which quadrant has these sign characteristics based on our earlier step.

Step 4: Conclude the Correct Quadrant

From the previous analysis, Quadrant II is where x is negative and y is positive. Thus, the point (x, y) falls in Quadrant II.

Key Concepts

Understanding the Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane made to locate and represent points using two numbers: the x-coordinate and the y-coordinate. These coordinates are drawn on two perpendicular lines, known as axes, which intersect at a point called the origin. The x-axis runs horizontally, while the y-axis runs vertically. This system helps us describe the position of points precisely in a plane. Imagine it as a map where each point has a unique address defined by its coordinates. The plane is divided into four sections, called quadrants, which help in identifying the position of points based on the signs of their coordinates. They are labeled in a counterclockwise manner:

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

Understanding this setup is crucial for plotting points and analyzing their positions relative to both the axes and the origin.

The Role of Signs in Coordinates

In the Cartesian coordinate system, the signs of the coordinates (x and y) are vital to understanding and locating points accurately. The sign of each coordinate indicates which direction from the origin it lies. The x-coordinate:

• If positive, the point is to the right of the y-axis.
• If negative, the point is to the left of the y-axis.

The y-coordinate:

• If positive, the point is above the x-axis.
• If negative, the point is below the x-axis.

By knowing whether the coordinates are positive or negative, you can determine the point's exact location on the coordinate plane. This understanding also aids in identifying which quadrant the point resides in. For example, if x is negative and y is positive, the point falls in Quadrant II.

Locating Points in Quadrants

To efficiently locate points within the Cartesian coordinate system, you must understand the quadrant in which they lie. This is determined by the signs of its coordinates, as previously discussed. Let's explore how these signs help locate points:

• For Quadrant I, both coordinates are positive, so any point like (3, 2) is here.
• For Quadrant II, x is negative, and y is positive, placing points such as (-3, 2) in this quadrant.
• For Quadrant III, both x and y are negative, so (-3, -2) would lie here.
• In Quadrant IV, x is positive, y is negative, so a point like (3, -2) belongs here.

By examining the combination of signs for x and y, you can identify where points are situated within the coordinate plane. This ability is essential for graphing, solving geometry problems, and understanding various mathematical concepts tied to the coordinate system.