Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-5,-2)$$

Step-by-Step Solution

Answer

The point (-5, -2) lies in Quadrant III.

Step 1: Understand the Coordinate System

In a rectangular coordinate system, we have a horizontal axis (x-axis) and a vertical axis (y-axis). A point on this system is represented by an ordered pair \(x, y\), where \(x\) is the horizontal position and \(y\) is the vertical position.

Step 2: Identify the Point's Coordinates

The given point is \((-5, -2)\). This means it is located at \(-5\) on the x-axis and \(-2\) on the y-axis.

Step 3: Locate the Point on the Coordinate System

Start at the origin \(0, 0\). Since \(-5\) is negative, move 5 units to the left on the x-axis. Since \(-2\) is negative, move 2 units down on the y-axis. The point is now located in the coordinate system.

Step 4: Determine the Quadrant

The coordinate system is divided into four quadrants. Quadrant I has positive x and y values, Quadrant II has negative x and positive y values, Quadrant III has negative x and y values, and Quadrant IV has positive x and negative y values. The point \((-5, -2)\) has both negative x and negative y values.

Step 5: Conclusion

Since the point has both negative x and negative y values, it is located in Quadrant III of the coordinate system.

Key Concepts

Quadrants in the Rectangular Coordinate System

In the rectangular coordinate system, the grid is divided into four distinct sections known as quadrants. These quadrants help us identify the location of points based on their x and y values. Visualize the coordinate system with two perpendicular lines crossing at the origin. The horizontal line is the x-axis, and the vertical line is the y-axis. Together, they split the plane into four sections.

Each quadrant is uniquely identified:

• Quadrant I: Both x and y coordinates are positive \((x > 0, y > 0)\). If a point falls in this quadrant, you have to move right and upward from the origin.
• Quadrant II: The x coordinate is negative, but the y coordinate is positive \((x < 0, y > 0)\). This means moving left, then upward from the origin.
• Quadrant III: Both coordinates are negative \((x < 0, y < 0)\). Here, one moves left and downward from the origin.
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative \((x > 0, y < 0)\). It requires moving right and then downward from the origin.

Remember, knowing the signs of an ordered pair can quickly help you determine which quadrant the point lies in.

Understanding Ordered Pairs

An ordered pair is a fundamental concept in the rectangular coordinate system. It defines the exact location of a point on the grid using two numbers: one for the x-axis and the other for the y-axis. You'll see this typically in the format \((x, y)\).

The first number in the ordered pair, often called the x-coordinate, indicates horizontal movement. It tells you how far to move left or right starting from the origin. The second number, known as the y-coordinate, shows vertical movement. It determines how far up or down you should go from your horizontal position. Both of these things combined let you pin down a precise location on the coordinate plane.

For instance, for the ordered pair \((-5, -2)\), the "-5" suggests moving five units left, while the "-2" suggests moving two units downward. Together, they navigate you to a specific spot. In cases where at least one number is negative, understanding which direction to move is crucial. This ultimately determines the quadrant where the point lies.

The Role of Negative Coordinates

Negative coordinates play a significant role when locating points in the rectangular coordinate system. These coordinates appear as negative values in an ordered pair, which dictate a specific movement direction.

When you encounter a negative x-coordinate, it means moving left from the origin along the x-axis, instead of right. For instance, in the point \((-5, -2)\), the negative "-5" signifies that you count five steps to the left.

A negative y-coordinate indicates downward movement along the y-axis. So, with \((-5, -2)\), the negative "-2" guides you to move two units downward from your current horizontal position.

Both negative coordinates cause the point \((-5, -2)\) to situate itself in the third quadrant, where both axis values are negative. Recognizing these movements can greatly simplify the process of accurately locating points on the grid, making your understanding clearer and navigation easier. Remember that the combination of negative coordinates always has a specific interpretation, crucial for identifying the quadrant the point resides in.