Problem 32 - 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. $$(3,0)$$

Step-by-Step Solution

Answer

The point (3,0) is on the x-axis, not in any quadrant.

Step 1: Understanding the Coordinate System

A rectangular coordinate system consists of two axes: the x-axis (horizontal) and the y-axis (vertical). The point (3,0) consists of two numbers, where 3 is the x-coordinate and 0 is the y-coordinate.

Step 2: Plotting the Point

To plot the point (3,0), start from the origin (0,0). Move 3 units to the right along the x-axis and do not move up or down, since the y-coordinate is 0.

Step 3: Identifying the Point's Position

The point (3,0) lies on the x-axis since the y-coordinate is 0. Points on the x-axis are not in any quadrant; they are between Quadrant I and Quadrant IV.

Key Concepts

Quadrants

The rectangular coordinate system is divided into four distinct quadrants. These quadrants help us to easily identify the position of points based on their coordinates. The x-axis (horizontal) and y-axis (vertical) intersect at the origin, dividing the plane into these four areas:

• Quadrant I: Contains points where both x and y coordinates are positive (x > 0, y > 0).
• Quadrant II: Contains points where the x coordinate is negative and the y coordinate is positive (x < 0, y > 0).
• Quadrant III: Contains points where both x and y coordinates are negative (x < 0, y < 0).
• Quadrant IV: Contains points where the x coordinate is positive and the y coordinate is negative (x > 0, y < 0).

Points that lie directly on the axes do not belong to any quadrant. For instance, the point (3,0) is directly on the x-axis and is therefore between Quadrant I and Quadrant IV.

Plotting Points

Plotting points on a rectangular coordinate system is a basic yet crucial skill in understanding coordinate geometry. Each point is given in the form of an ordered pair (x, y), where 'x' is the position on the x-axis and 'y' is the position on the y-axis. To plot a point like (3,0):

• Start at the origin (0,0), which is the intersecting point of the x and y axes.
• Move 3 units to the right along the x-axis because the x-coordinate is positive 3.
• Since the y-coordinate is 0, do not move up or down, as it remains on the x-axis.

By following these steps, you can accurately locate any point in the coordinate plane.

Axes

The axes of the rectangular coordinate system serve as reference lines for determining the positions of points. There are two main axes:

• x-axis: This is the horizontal line that spans from left to right through the origin. It indicates the horizontal component of a point's position.
• y-axis: This is the vertical line that runs from top to bottom through the origin. It indicates the vertical component of a point's position.

The point where the x-axis and y-axis intersect is called the origin, denoted as (0,0). The axes are fundamental for identifying coordinates, as each point on the plane is a combination of a location on the x-axis and a location on the y-axis. For example, in the point (3,0), '3' indicates its position on the x-axis, while '0' signifies that it sits directly on the x-axis, with no deviation along the y-axis.