Problem 26 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(3,0)$$

Step-by-Step Solution

Answer

The point (3,0) lies on the x-axis.

Step 1: Understand Coordinate Axes

The coordinate plane is divided into four quadrants by the x-axis and y-axis. The quadrants are labeled as Quadrant I (top-right), Quadrant II (top-left), Quadrant III (bottom-left), and Quadrant IV (bottom-right). Points on the axes do not belong to any quadrant.

Step 2: Identify Point Location

The given point is \(3, 0\). Observe the coordinates: the x-coordinate is 3 and the y-coordinate is 0.

Step 3: Determine the Position

Since the y-coordinate is 0, the point \(3,0\) lies on the x-axis.

Key Concepts

Quadrants

In a coordinate plane, we divide the plane into four areas known as quadrants. These quadrants help us locate points in a systematic way. Let’s explore the basic structure:

• Quadrant I - Top-right section; here, both the x-coordinate and y-coordinate values are positive.
• Quadrant II - Top-left section; here, the x-coordinate is negative and the y-coordinate is positive.
• Quadrant III - Bottom-left section; in this quadrant, both x and y coordinates are negative.
• Quadrant IV - Bottom-right section; here, the x-coordinate is positive and the y-coordinate is negative.

These quadrants are important in graphing and solving problems in coordinate geometry as they give us a way to categorize where points are located based on their signs.

X-axis

The x-axis is one of the two axes in the coordinate plane. It acts as a horizontal guideline, splitting the plane into the top and bottom halves. Here are some key points to know about the x-axis:

• The x-axis is a horizontal number line.
• Values on the right of the y-axis are positive.
• Values on the left of the y-axis are negative.

At any point where the y-coordinate is zero, the point lies on the x-axis. For example, in our exercise, the point (3,0) lies on the x-axis because its y-coordinate is 0. The x-axis is crucial for determining if a point is located above, below, or directly on the horizontal reference line.

Y-axis

The y-axis complements the x-axis in creating a coordinate plane. It runs vertically and divides the plane into left and right halves. Key features of the y-axis include:

• The y-axis is a vertical number line.
• Values above the x-axis are positive.
• Values below the x-axis are negative.

Points that have an x-coordinate of zero will lie on the y-axis. For example, a point such as (0,4) would lie on the y-axis since its x-coordinate is zero. Knowing how to identify points relative to the y-axis is crucial for graphing and solving various geometry problems.