Problem 34 - 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. $$(3,-3)$$

Step-by-Step Solution

Answer

The point (3, -3) is in Quadrant IV.

Step 1: Understanding the Axes

In a rectangular coordinate system, the horizontal line is called the x-axis and the vertical line is called the y-axis. Each point is represented as \(x, y\), where \(x\) is the position on the x-axis and \(y\) is the position on the y-axis.

Step 2: Plotting the Point

To locate the point (3, -3), start from the origin (0, 0). Move 3 units to the right along the x-axis because the x-coordinate is 3. Then, move 3 units down because the y-coordinate is -3, hence reaching the point (3, -3).

Step 3: Identifying the Quadrant

The rectangular coordinate system is divided into four quadrants: - Quadrant I: x > 0, y > 0 - Quadrant II: x < 0, y > 0 - Quadrant III: x < 0, y < 0 - Quadrant IV: x > 0, y < 0 The point (3, -3) has a positive x-coordinate and a negative y-coordinate, placing it in Quadrant IV.

Key Concepts

Quadrants

In the rectangular coordinate system, understanding the concept of quadrants is essential in locating points. The coordinate plane is divided into four quadrants by the x-axis and y-axis intersecting at the origin, (0, 0). Each quadrant has specific rules for the signs of coordinates:

• Quadrant I: Both coordinates are positive ( x > 0, y > 0 ).
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive ( x < 0, y > 0 ).
• Quadrant III: Both coordinates are negative ( x < 0, y < 0 ).
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative ( x > 0, y < 0 ).

This division helps us categorize points based on their location relative to the origin. Knowing the quadrant can also give insights into the direction and positioning of graphical data points.
In our example, the point (3, -3) lies in Quadrant IV, as it has a positive x-coordinate and a negative y-coordinate.

Plotting Points

Plotting points on a rectangular coordinate system is like finding a position on a grid. Each point's location is defined by an ordered pair ( x, y ), telling us exactly where to place it. To start plotting a point like (3, -3), you begin at the origin, which is the central point (0, 0).

• Move Horizontally: The first number, or the x-coordinate, tells you how far right (positive) or left (negative) to move along the x-axis from the origin. For (3, -3), we move 3 units to the right.
• Move Vertically: The second number, or the y-coordinate, tells you how far up (positive) or down (negative) to move along the y-axis from your new position. In this case, we move 3 units down.

These movements bring you to the point (3, -3), and with practice, this plotting process becomes intuitive and quick.

x-axis and y-axis

The x-axis and y-axis are the two fundamental components of the rectangular coordinate system. These two lines form the grid where all coordinates are plotted.

• x-axis: This horizontal line runs left and right through the origin, representing potential horizontal placements of points. Positive coordinates are to the right of the origin, while negative coordinates are to the left.
• y-axis: This vertical line runs up and down through the origin, designating vertical placements. Positive coordinates are above the origin, with negative ones below.

Points on this plane are defined by how far they are along these axes. As such, any point is expressed as ( x, y ), the first element being its horizontal position and the second its vertical position. Together, they create a complete picture of a point's exact location on the grid. By understanding the role of these axes, you can better visualize mathematical problems and solutions on a coordinate plane.